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author:
- ZEUS Collaboration
date: July 2008
title: ' Angular correlations in three-jet events in [$ep$]{} collisions at HERA '
---
[ The ZEUS Collaboration ]{}
S. Chekanov, M. Derrick, S. Magill, B. Musgrave, D. Nicholass$^{ 1}$, , R. Yoshida\
[*Argonne National Laboratory, Argonne, Illinois 60439-4815, USA*]{} $^{n}$
M.C.K. Mattingly\
[*Andrews University, Berrien Springs, Michigan 49104-0380, USA*]{}
P. Antonioli, G. Bari, L. Bellagamba, D. Boscherini, A. Bruni, G. Bruni, F. Cindolo, M. Corradi, , A. Margotti, R. Nania, A. Polini\
[*INFN Bologna, Bologna, Italy*]{} $^{e}$
S. Antonelli, M. Basile, M. Bindi, L. Cifarelli, A. Contin, S. De Pasquale$^{ 2}$, G. Sartorelli, A. Zichichi\
[*University and INFN Bologna, Bologna, Italy*]{} $^{e}$
D. Bartsch, I. Brock, H. Hartmann, E. Hilger, H.-P. Jakob, M. Jüngst, , E. Paul, U. Samson, V. Schönberg, R. Shehzadi, M. Wlasenko\
[*Physikalisches Institut der Universität Bonn, Bonn, Germany*]{} $^{b}$
N.H. Brook, G.P. Heath, J.D. Morris\
[*H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom*]{} $^{m}$
M. Capua, S. Fazio, A. Mastroberardino, M. Schioppa, G. Susinno, E. Tassi\
[*Calabria University, Physics Department and INFN, Cosenza, Italy*]{} $^{e}$
J.Y. Kim\
[*Chonnam National University, Kwangju, South Korea*]{}
Z.A. Ibrahim, B. Kamaluddin, W.A.T. Wan Abdullah\
[*Jabatan Fizik, Universiti Malaya, 50603 Kuala Lumpur, Malaysia*]{} $^{r}$
Y. Ning, Z. Ren, F. Sciulli\
[*Nevis Laboratories, Columbia University, Irvington on Hudson, New York 10027*]{} $^{o}$
J. Chwastowski, A. Eskreys, J. Figiel, A. Galas, K. Olkiewicz, P. Stopa,\
[*The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland*]{} $^{i}$
L. Adamczyk, T. Bołd, I. Grabowska-Bołd, D. Kisielewska, J. Łukasik, , L. Suszycki\
[*Faculty of Physics and Applied Computer Science, AGH-University of Science and , Cracow, Poland*]{} $^{p}$
A. Kotański$^{ 3}$, W. S[ł]{}omiński$^{ 4}$\
[*Department of Physics, Jagellonian University, Cracow, Poland*]{}
O. Behnke, U. Behrens, C. Blohm, A. Bonato, K. Borras, R. Ciesielski, N. Coppola, S. Fang, J. Fourletova$^{ 5}$, A. Geiser, P. Göttlicher$^{ 6}$, J. Grebenyuk, I. Gregor, T. Haas, W. Hain, A. Hüttmann, F. Januschek, B. Kahle, I.I. Katkov, U. Klein$^{ 7}$, U. Kötz, H. Kowalski, , B. Löhr, R. Mankel, I.-A. Melzer-Pellmann, , A. Montanari, T. Namsoo, D. Notz$^{ 8}$, A. Parenti, L. Rinaldi$^{ 9}$, P. Roloff, I. Rubinsky, R. Santamarta$^{ 10}$, , A. Spiridonov$^{ 11}$, D. Szuba$^{ 12}$, J. Szuba$^{ 13}$, T. Theedt, G. Wolf, K. Wrona, , C. Youngman, $^{ 8}$\
[*Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany*]{}
V. Drugakov, W. Lohmann,\
[*Deutsches Elektronen-Synchrotron DESY, Zeuthen, Germany*]{}
G. Barbagli, E. Gallo\
[*INFN Florence, Florence, Italy*]{} $^{e}$
P. G. Pelfer\
[*University and INFN Florence, Florence, Italy*]{} $^{e}$
A. Bamberger, D. Dobur, F. Karstens, N.N. Vlasov$^{ 14}$\
[*Fakultät für Physik der Universität Freiburg i.Br., Freiburg i.Br., Germany*]{} $^{b}$
P.J. Bussey$^{ 15}$, A.T. Doyle, W. Dunne, M. Forrest, M. Rosin, D.H. Saxon, I.O. Skillicorn\
[*Department of Physics and Astronomy, University of Glasgow, Glasgow, United* ]{} $^{m}$
I. Gialas$^{ 16}$, K. Papageorgiu\
[*Department of Engineering in Management and Finance, Univ. of Aegean, Greece*]{}
U. Holm, R. Klanner, E. Lohrmann, P. Schleper, , J. Sztuk, H. Stadie, M. Turcato\
[*Hamburg University, Institute of Exp. Physics, Hamburg, Germany*]{} $^{b}$
C. Foudas, C. Fry, K.R. Long, A.D. Tapper\
[*Imperial College London, High Energy Nuclear Physics Group, London, United* ]{} $^{m}$
T. Matsumoto, K. Nagano, K. Tokushuku$^{ 17}$, S. Yamada, Y. Yamazaki$^{ 18}$\
[*Institute of Particle and Nuclear Studies, KEK, Tsukuba, Japan*]{} $^{f}$
A.N. Barakbaev, E.G. Boos, N.S. Pokrovskiy, B.O. Zhautykov\
[*Institute of Physics and Technology of Ministry of Education and Science of Kazakhstan, Almaty,* ]{}
V. Aushev$^{ 19}$, O. Bachynska, M. Borodin, I. Kadenko, A. Kozulia, V. Libov, M. Lisovyi, D. Lontkovskyi, I. Makarenko, Iu. Sorokin, A. Verbytskyi, O. Volynets\
[*Institute for Nuclear Research, National Academy of Sciences, Kiev and Kiev National University, Kiev, Ukraine*]{}
D. Son\
[*Kyungpook National University, Center for High Energy Physics, Daegu, South Korea*]{} $^{g}$
J. de Favereau, K. Piotrzkowski\
[*Institut de Physique Nucléaire, Université Catholique de Louvain, Louvain-la-Neuve,* ]{} $^{q}$
F. Barreiro, C. Glasman, M. Jimenez, L. Labarga, J. del Peso, E. Ron, M. Soares, J. Terrón,\
[*Departamento de Física Teórica, Universidad Autónoma de Madrid, Madrid, Spain*]{} $^{l}$
F. Corriveau, C. Liu, J. Schwartz, R. Walsh, C. Zhou\
[*Department of Physics, McGill University, Montréal, Québec, Canada H3A 2T8*]{} $^{a}$
T. Tsurugai\
[*Meiji Gakuin University, Faculty of General Education, Yokohama, Japan*]{} $^{f}$
A. Antonov, B.A. Dolgoshein, D. Gladkov, V. Sosnovtsev, A. Stifutkin, S. Suchkov\
[*Moscow Engineering Physics Institute, Moscow, Russia*]{} $^{j}$
R.K. Dementiev, P.F. Ermolov $^{\dagger}$, L.K. Gladilin, Yu.A. Golubkov, L.A. Khein, , V.A. Kuzmin, B.B. Levchenko$^{ 20}$, O.Yu. Lukina, A.S. Proskuryakov, L.M. Shcheglova, D.S. Zotkin\
[*Moscow State University, Institute of Nuclear Physics, Moscow, Russia*]{} $^{k}$
I. Abt, A. Caldwell, D. Kollar, B. Reisert, W.B. Schmidke\
[*Max-Planck-Institut für Physik, München, Germany*]{}
G. Grigorescu, A. Keramidas, E. Koffeman, P. Kooijman, A. Pellegrino, H. Tiecke, M. Vázquez$^{ 8}$,\
[*NIKHEF and University of Amsterdam, Amsterdam, Netherlands*]{} $^{h}$
N. Brümmer, B. Bylsma, L.S. Durkin, A. Lee, T.Y. Ling\
[*Physics Department, Ohio State University, Columbus, Ohio 43210*]{} $^{n}$
P.D. Allfrey, M.A. Bell, A.M. Cooper-Sarkar, R.C.E. Devenish, J. Ferrando, , K. Korcsak-Gorzo, K. Oliver, A. Robertson, C. Uribe-Estrada, R. Walczak\
[*Department of Physics, University of Oxford, Oxford United Kingdom*]{} $^{m}$
A. Bertolin, F. Dal Corso, S. Dusini, A. Longhin, L. Stanco\
[*INFN Padova, Padova, Italy*]{} $^{e}$
P. Bellan, R. Brugnera, R. Carlin, A. Garfagnini, S. Limentani\
[*Dipartimento di Fisica dell’ Università and INFN, Padova, Italy*]{} $^{e}$
B.Y. Oh, A. Raval, J. Ukleja$^{ 21}$, J.J. Whitmore$^{ 22}$\
[*Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802*]{} $^{o}$
Y. Iga\
[*Polytechnic University, Sagamihara, Japan*]{} $^{f}$
G. D’Agostini, G. Marini, A. Nigro\
[*Dipartimento di Fisica, Università ’La Sapienza’ and INFN, Rome, Italy*]{} $^{e}~$
J.E. Cole$^{ 23}$, J.C. Hart\
[*Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, United Kingdom*]{} $^{m}$
H. Abramowicz$^{ 24}$, R. Ingbir, S. Kananov, A. Levy, A. Stern\
[*Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics, Tel Aviv University, Tel Aviv, Israel*]{} $^{d}$
M. Kuze, J. Maeda\
[*Department of Physics, Tokyo Institute of Technology, Tokyo, Japan*]{} $^{f}$
R. Hori, S. Kagawa$^{ 25}$, N. Okazaki, S. Shimizu, T. Tawara\
[*Department of Physics, University of Tokyo, Tokyo, Japan*]{} $^{f}$
R. Hamatsu, H. Kaji$^{ 26}$, S. Kitamura$^{ 27}$, O. Ota$^{ 28}$, Y.D. Ri\
[*Tokyo Metropolitan University, Department of Physics, Tokyo, Japan*]{} $^{f}$
M. Costa, M.I. Ferrero, V. Monaco, R. Sacchi, A. Solano\
[*Università di Torino and INFN, Torino, Italy*]{} $^{e}$
M. Arneodo, M. Ruspa\
[*Università del Piemonte Orientale, Novara, and INFN, Torino, Italy*]{} $^{e}$
S. Fourletov$^{ 5}$, J.F. Martin, T.P. Stewart\
[*Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7*]{} $^{a}$
S.K. Boutle$^{ 16}$, J.M. Butterworth, C. Gwenlan$^{ 29}$, T.W. Jones, J.H. Loizides, M. Wing$^{ 30}$\
[*Physics and Astronomy Department, University College London, London, United* ]{} $^{m}$
B. Brzozowska, J. Ciborowski$^{ 31}$, G. Grzelak, P. Kulinski, P. [Ł]{}użniak$^{ 32}$, J. Malka$^{ 32}$, R.J. Nowak, J.M. Pawlak, A.F. Żarnecki\
[*Warsaw University, Institute of Experimental Physics, Warsaw, Poland*]{}
M. Adamus, P. Plucinski$^{ 33}$, A. Ukleja\
[*Institute for Nuclear Studies, Warsaw, Poland*]{}
Y. Eisenberg, D. Hochman, U. Karshon\
[*Department of Particle Physics, Weizmann Institute, Rehovot, Israel*]{} $^{c}$
E. Brownson, T. Danielson, A. Everett, D. Kçira, D.D. Reeder, P. Ryan, A.A. Savin, W.H. Smith, H. Wolfe\
[*Department of Physics, University of Wisconsin, Madison, Wisconsin 53706*]{}, USA $^{n}$
S. Bhadra, C.D. Catterall, Y. Cui, G. Hartner, S. Menary, U. Noor, J. Standage, J. Whyte\
[*Department of Physics, York University, Ontario, Canada M3J 1P3*]{} $^{a}$
$^{\ 1}$ also affiliated with University College London, United Kingdom\
$^{\ 2}$ now at University of Salerno, Italy\
$^{\ 3}$ supported by the research grant no. 1 P03B 04529 (2005-2008)\
$^{\ 4}$ This work was supported in part by the Marie Curie Actions Transfer of Knowledge project COCOS (contract MTKD-CT-2004-517186)\
$^{\ 5}$ now at University of Bonn, Germany\
$^{\ 6}$ now at DESY group FEB, Hamburg, Germany\
$^{\ 7}$ now at University of Liverpool, UK\
$^{\ 8}$ now at CERN, Geneva, Switzerland\
$^{\ 9}$ now at Bologna University, Bologna, Italy\
$^{ 10}$ now at BayesForecast, Madrid, Spain\
$^{ 11}$ also at Institut of Theoretical and Experimental Physics, Moscow, Russia\
$^{ 12}$ also at INP, Cracow, Poland\
$^{ 13}$ also at FPACS, AGH-UST, Cracow, Poland\
$^{ 14}$ partly supported by Moscow State University, Russia\
$^{ 15}$ Royal Society of Edinburgh, Scottish Executive Support Research Fellow\
$^{ 16}$ also affiliated with DESY, Germany\
$^{ 17}$ also at University of Tokyo, Japan\
$^{ 18}$ now at Kobe University, Japan\
$^{ 19}$ supported by DESY, Germany\
$^{ 20}$ partly supported by Russian Foundation for Basic Research grant no. 05-02-39028-NSFC-a\
$^{ 21}$ partially supported by Warsaw University, Poland\
$^{ 22}$ This material was based on work supported by the National Science Foundation, while working at the Foundation.\
$^{ 23}$ now at University of Kansas, Lawrence, USA\
$^{ 24}$ also at Max Planck Institute, Munich, Germany, Alexander von Humboldt Research Award\
$^{ 25}$ now at KEK, Tsukuba, Japan\
$^{ 26}$ now at Nagoya University, Japan\
$^{ 27}$ member of Department of Radiological Science, Tokyo Metropolitan University, Japan\
$^{ 28}$ now at SunMelx Co. Ltd., Tokyo, Japan\
$^{ 29}$ PPARC Advanced fellow\
$^{ 30}$ also at Hamburg University, Inst. of Exp. Physics, Alexander von Humboldt Research Award and partially supported by DESY, Hamburg, Germany\
$^{ 31}$ also at Łódź University, Poland\
$^{ 32}$ member of Łódź University, Poland\
$^{ 33}$ now at Lund Universtiy, Lund, Sweden\
$^{\dagger}$ deceased\
-------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$^{a}$ supported by the Natural Sciences and Engineering Research Council of Canada (NSERC)
$^{b}$ supported by the German Federal Ministry for Education and Research (BMBF), under contract numbers 05 HZ6PDA, 05 HZ6GUA, 05 HZ6VFA and 05 HZ4KHA
$^{c}$ supported in part by the MINERVA Gesellschaft für Forschung GmbH, the Israel Science Foundation (grant no. 293/02-11.2) and the U.S.-Israel Binational Science Foundation
$^{d}$ supported by the Israel Science Foundation
$^{e}$ supported by the Italian National Institute for Nuclear Physics (INFN)
$^{f}$ supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and its grants for Scientific Research
$^{g}$ supported by the Korean Ministry of Education and Korea Science and Engineering Foundation
$^{h}$ supported by the Netherlands Foundation for Research on Matter (FOM)
$^{i}$ supported by the Polish State Committee for Scientific Research, project no. DESY/256/2006 - 154/DES/2006/03
$^{j}$ partially supported by the German Federal Ministry for Education and Research (BMBF)
$^{k}$ supported by RF Presidential grant N 8122.2006.2 for the leading scientific schools and by the Russian Ministry of Education and Science through its grant for Scientific Research on High Energy Physics
$^{l}$ supported by the Spanish Ministry of Education and Science through funds provided by CICYT
$^{m}$ supported by the Science and Technology Facilities Council, UK
$^{n}$ supported by the US Department of Energy
$^{o}$ supported by the US National Science Foundation. Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
$^{p}$ supported by the Polish Ministry of Science and Higher Education as a scientific project (2006-2008)
$^{q}$ supported by FNRS and its associated funds (IISN and FRIA) and by an Inter-University Attraction Poles Programme subsidised by the Belgian Federal Science Policy Office
$^{r}$ supported by the Malaysian Ministry of Science, Technology and Innovation/Akademi Sains Malaysia grant SAGA 66-02-03-0048
-------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Introduction
============
Quantum chromodynamics (QCD) is based on the non-Abelian group SU(3) which induces the self-coupling of the gluons. Investigations of the triple-gluon vertex (TGV) were carried out at LEP [@pl:b284:151; @*zfp:c76:1; @*epj:c27:1; @*zfp:c59:357; @*pl:b414:401; @*pl:b449:383; @*pl:b248:227; @*zfp:c49:49; @*zfp:c65:367; @*epj:c20:601; @pl:b255:466] using angular correlations in four-jet events from $\z0$ hadronic decays. At HERA, the effects of the different colour configurations arising from the underlying gauge structure can be studied in a clean way in three-jet production in neutral current (NC) deep inelastic scattering (DIS) and photoproduction ($\gp$).
Neutral current DIS at high $\q2$ ($\q2\gg\Lambda_{\rm QCD}^2$, where $\q2$ is the virtuality of the exchanged photon) up to leading order (LO) in the strong coupling constant, $\as$, proceeds as in the quark-parton model ($Vq\rightarrow q$, where $V=\gamma^*$ or $\z0$) or via the boson-gluon fusion ($Vg\rightarrow\qq$) and QCD-Compton ($Vq\rightarrow qg$) processes. Photoproduction is studied at HERA by means of $ep$ scattering at low four-momentum transfers ($\q2\approx 0$). In $\gp$ reactions, two types of QCD processes contribute to jet production at LO [@pl:b79:83; @*np:b166:413; @*pr:d21:54; @*zfp:c6:241; @proc:hera:1987:331; @*prl:61:275; @*prl:61:682; @*pr:d39:169; @*zfp:c42:657; @*pr:d40:2844]: either the photon interacts directly with a parton in the proton (the direct process) or the photon acts as a source of partons which scatter off those in the proton (the resolved process).
A subset of resolved subprocesses with two jets in the final state are described by diagrams with a TGV; however, such events are difficult to distinguish from two-jet events without such a contribution. Three-jet final states in direct $\gp$ processes also contain contributions from TGVs and are easier to identify. Since three-jet production in NC DIS proceeds via the same diagrams as in direct $\gp$, such processes can also be used to investigate the underlying gauge symmetry. Examples of diagrams contributing to four colour configurations are shown in Fig. \[fig1\]: (A) double-gluon bremsstrahlung from a quark line, (B) the splitting of a virtual gluon into a pair of final-state gluons, (C) the production of a $\qq$ pair through the exchange of a virtual gluon emitted by an incoming quark, and (D) the production of a $\qq$ pair through the exchange of a virtual gluon arising from the splitting of an incoming gluon.
Other possible diagrams and interferences correspond to one of the four configurations. The production rate of all contributions is proportional to the so-called colour factors, $C_F$, $C_A$ and $T_F$, which are a physical manifestation of the underlying group structure. For QCD, these factors represent the relative strengths of the processes $q\rightarrow qg$, $g\rightarrow gg$ and $g\rightarrow \qq$. The contributions of the diagrams of Fig. \[fig1\] are proportional to $C_F^2$, $C_FC_A$, $C_FT_F$ and $T_FC_A$, respectively, independently of the underlying gauge symmetry.
Three-jet cross sections were previously measured in $\gp$ [@pl:b443:394; @*np:b792:1] and in NC DIS [@epj:c44:183; @pl:b515:17]. The shape of the measured cross sections was well reproduced by perturbative QCD (pQCD) calculations and a value of $\as$ was extracted [@epj:c44:183]. In this paper, measurements of angular correlations in three-jet events in $\gp$ and NC DIS are presented. The comparison between the measurements and fixed-order $\oass$ and $\oasss$ perturbative calculations based on different colour configurations provides a stringent test of pQCD predictions directly beyond LO and gives insight into the underlying group symmetry. Phase-space regions where the angular correlations show potential sensitivity to the presence of the TGV were identified.
Theoretical framework
=====================
The dynamics of a gauge theory such as QCD are completely defined by the commutation relations between its group generators $T^i$,
$$[T^i,T^j]=i \sum_k f^{ijk} \cdot T^k,$$
where $f^{ijk}$ are the structure constants. The generators $T^i$ can be represented as matrices. In perturbative calculations, the average (sum) over all possible colour configurations in the initial (final) states leads to the appearance of combinatoric factors $C_F$, $C_A$ and $T_F$, which are defined by the relations
$$\sum_{k,\eta} T^k_{\alpha\eta} T^k_{\eta\beta}=\delta_{\alpha\beta} C_F,\
\sum_{j,k} f^{jkm} f^{jkn}=\delta^{mn} C_A,$$
$$\sum_{\alpha,\beta} T^m_{\alpha\beta} T^n_{\beta\alpha}=\delta^{mn} T_F.$$
Measurements of the ratios between the colour factors allow the experimental determination of the underlying gauge symmetry of the strong interactions. For SU($N$), the predicted values of the colour factors are:
$$C_A=N,\ \ C_F=\frac{N^2-1}{2N}\ \ {\rm and}\ \ T_F=1/2,$$ where $N$ is the number of colour charges. In particular, SU(3) predicts $C_A/C_F=9/4$ and $T_F/C_F=3/8$. In contrast, an Abelian gluon theory based on U(1)$^3$ would predict $C_A/C_F=0$ and $T_F/C_F=3$. A non-Abelian theory based on SO(3) predicts $C_A/C_F=1$ and $T_F/C_F=1$.
The $\oass$ calculations of three-jet cross sections for direct $\gp$ and NC DIS processes can be expressed in terms of $C_A$, $C_F$ and $T_F$ as [@np:b286:553]:
$$\sigma_{ep \rightarrow 3{\rm jets}} = C_F^2 \cdot \sigma_A +
C_F C_A \cdot \sigma_B +
C_F T_F \cdot \sigma_C + T_F C_A \cdot \sigma_D,\label{one}$$
where $\sigma_A$, ..., $\sigma_D$ are the partonic cross sections for the different contributions (see Fig. \[fig1\]).
Definition of the angular correlations
======================================
Angular-correlation observables were devised to distinguish the contributions from the different colour configurations. They are defined in terms of the three jets with highest transverse energy in an event and the beam direction as:
- [$\th$]{}, the angle between the plane determined by the highest-transverse-energy jet and the beam and the plane determined by the two jets with lowest transverse energy [@pr:d52:3894];
- [$\a34$]{}, the angle between the two lowest-transverse-energy jets. This variable is based on the angle $\alpha_{34}^{\ele}$ for $\ele\rightarrow {4\ \rm jets}$ [@pl:b255:466];
- [$\pksw$]{}, the angle defined via the equation\
$\cos(\pksw)=\cos \left [ \frac{1}{2} \left ( \angle[(\vec p_1 \times \vec p_3),(\vec p_2 \times \vec p_B)] + \angle[(\vec p_1 \times \vec p_B),(\vec p_2 \times \vec p_3)] \right ) \right ]$,
where $\vec p_i,\ i=1,...,3$ is the momentum of jet $i$ and $\vec p_B$ is a unit vector in the direction of the beam; the jets are ordered according to decreasing transverse energy. This variable is based on the Körner-Schierholz-Willrodt angle $\Phi_{\rm KSW}^{\ele}$ for $\ele\rightarrow {4\ \rm jets}$ [@np:b185:365];
- [$\etajmax$]{}, the maximum pseudorapidity of the three jets.
For three-jet events in $ep$ collisions, the variable $\th$ was designed [@pr:d52:3894] to be sensitive to the TGV in quark-induced processes (see Fig. \[fig1\]B). In $\ele$ annihilation into four-jet events, the distribution of $\Phi_{\rm KSW}^{\ele}$ is sensitive to the differences between $\qq gg$ and $\qq\qq$ final states whereas that of $\alpha_{34}^{\ele}$ distinguishes between contributions from double-bremsstrahlung diagrams and diagrams involving the TGV.
Experimental set-up
===================
The data samples used in this analysis were collected with the ZEUS detector at HERA and correspond to an integrated luminosity of $44.9\pm 0.8\ (65.1\pm 1.5)$ 1 for $e^+p$ collisions taken during 1995–97 (1999–2000) and $16.7\pm 0.3$ 1 for $e^-p$ collisions taken during 1998–99. During 1995–97 (1998–2000), HERA operated with protons of energy $E_p=820$ ($920$) GeV and positrons or electrons[^1] of energy $E_e=27.5$ GeV, yielding a centre-of-mass energy of $\sqrt s=300$ ($318$) GeV.
A detailed description of the ZEUS detector can be found elsewhere [@pl:b293:465; @zeus:1993:bluebook]. A brief outline of the components that are most relevant for this analysis is given below. Charged particles were tracked in the central tracking detector (CTD) [@nim:a279:290; @*npps:b32:181; @*nim:a338:254], which operated in a magnetic field of $1.43\Tesla$ provided by a thin superconducting solenoid. The CTD consisted of 72 cylindrical drift-chamber layers, organised in nine superlayers covering the polar-angle[^2] region . The transverse-momentum resolution for full-length tracks was parameterised as $\sigma(p_T)/p_T=0.0058p_T\oplus0.0065\oplus0.0014/p_T$, with $p_T$ in $\Gev$. The tracking system was used to measure the interaction vertex with a typical resolution along (transverse to) the beam direction of 0.4 (0.1) cm and to cross-check the energy scale of the calorimeter.
The high-resolution uranium–scintillator calorimeter (CAL) [@nim:a309:77; @*nim:a309:101; @*nim:a321:356; @*nim:a336:23] covered $99.7\%$ of the total solid angle and consisted of three parts: the forward (FCAL), the barrel (BCAL) and the rear (RCAL) calorimeters. Each part was subdivided transversely into towers and longitudinally into one electromagnetic section (EMC) and either one (in RCAL) or two (in BCAL and FCAL) hadronic sections (HAC). The smallest subdivision of the calorimeter was called a cell. Under test-beam conditions, the CAL single-particle relative energy resolutions were $\sigma(E)/E=0.18/\sqrt E$ for electrons and $\sigma(E)/E=0.35/\sqrt E$ for hadrons, with $E$ in GeV.
The luminosity was measured from the rate of the bremsstrahlung process $ep\rightarrow e\gamma p$. The resulting small-angle energetic photons were measured by the luminosity monitor [@desy-92-066; @*zfp:c63:391; @*acpp:b32:2025], a lead–scintillator calorimeter placed in the HERA tunnel at $Z=-107$ m.
Data selection and jet search
=============================
A three-level trigger system was used to select events online [@zeus:1993:bluebook; @proc:chep:1992:222]. At the third level, jets were reconstructed using the energies and positions of the CAL cells. Events with at least one (two) jet(s) with transverse energy in excess of $10\ (6)$ GeV and pseudorapidity below $2.5$ were accepted. For trigger-efficiency studies, no jet algorithm was applied and events with a total transverse energy, excluding the energy in the eight CAL towers immediately surrounding the forward beampipe, of at least $25$ GeV were selected in the $\gp$ sample; for the NC DIS sample, events were selected in which the scattered-electron candidate was identified using localised energy depositions in the CAL.
In the offline selection, a reconstructed event vertex consistent with the nominal interaction position was required and cuts based on tracking information were applied to reduce the contamination from beam-induced and cosmic-ray background events. The selection criteria of the $\gp$ and NC DIS samples were analogous to previous publications [@pl:b560:7; @pl:b649:12].
The selected $\gp$ sample consisted of events from $ep$ interactions with $\q2<1$ 2 and a median $\q2\approx
10^{-3}$ 2. The event sample was restricted to the kinematic range $0.2<y<0.85$, where $y$ is the inelasticity.
Events from NC DIS interactions were selected from the 1998–2000 data. Two samples were studied: $\q2>125$ 2 and $500<\q2<5000$ 2. For both samples, $|\cgh|$ was restricted to be below $0.65$, where $\gamma_h$, which corresponds to the angle of the scattered quark in the quark-parton model, is defined as
$$\cgh= \frac{(1-y)x E_p - y E_e}{(1-y)x E_p + y E_e}$$ and $x$ is the Bjorken variable.
The $\kt$ cluster algorithm [@np:b406:187] was used in the longitudinally invariant inclusive mode [@pr:d48:3160] to reconstruct jets in the measured hadronic final state from the energy deposits in the CAL cells (calorimetric jets). The axis of the jet was defined according to the Snowmass convention [@proc:snowmass:1990:134].
For $\gp$ events, the jet search was performed in the $\etaphi$ plane of the laboratory frame. Corrections [@pl:b560:7] to the jet transverse energy, $\etjet$, were applied to the calorimetric jets as a function of the jet pseudorapidity, $\etajet$, and $\etjet$ and averaged over the jet azimuthal angle. Events with at least three jets of $\etjet>14$ GeV and $\etar$ were retained. Direct $\gp$ events were further selected by requiring $\xo>0.8$, where $\xo$, the fraction of the photon momentum participating in the production of the three jets with highest $\etjet$, is defined as
$$\xo=\frac{1}{2 y E_e}\left ( \etJ e^{-\etaJ}+\etjj e^{-\etajj}+\etjjj
e^{-\etajjj}\right ) .$$ The final $\gp$ data sample contained 1888 events.
For NC DIS events, the $\kt$ jet algorithm was applied after excluding those cells associated with the scattered-electron candidate and the search was conducted in the Breit frame. Jet transverse-energy corrections were computed using the method developed in a previous analysis [@pl:b649:12]. Events were required to have at least three jets satisfying $\etjbj>8$ GeV, $E^{\rm jet2,3}_{T,{\rm B}}>5$ GeV and $-2<\etajb<1.5$, where $\etjb$ and $\etajb$ are the jet transverse energy and pseudorapidity in the Breit frame, respectively. The final NC DIS data sample with $\q2>125$ ($500<\q2<5000$) 2 contained 1095 (492) events.
Monte Carlo simulation {#mc}
======================
Samples of Monte Carlo (MC) events were generated to determine the response of the detector to jets of hadrons and the correction factors necessary to obtain the hadron-level jet cross sections. The hadron level is defined by those hadrons with lifetime $\tau\geq 10$ ps. For the NC DIS sample, the MC events were also used to correct the measured cross sections for QED radiative effects and the running of $\alpha_{\rm em}$.
The generated events were passed through the [ Geant]{} 3.13-based [@tech:cern-dd-ee-84-1] ZEUS detector- and trigger-simulation programs [@zeus:1993:bluebook]. They were reconstructed and analysed by the same program chain as the data. The $\kt$ jet algorithm was applied to the MC simulated events using the CAL cells in the same way as for the data. The jet algorithm was also applied to the final-state particles (hadron level) and the partons available after the parton shower (parton level).
The programs [Pythia]{} 6.1 [@cpc:82:74; @*cpc:135:238] and [Herwig]{} 6.1 [@cpc:67:465; @*jhep:0101:010] were used to generate $\gp$ events for resolved and direct processes. Events were generated using GRV-HO [@pr:d45:3986; @*pr:d46:1973] for the photon and CTEQ4M [@pr:d55:1280] for the proton parton distribution functions (PDFs). In both generators, the partonic processes are simulated using LO matrix elements, with the inclusion of initial- and final-state parton showers. Fragmentation into hadrons is performed using the Lund string model [@prep:97:31] as implemented in [Jetset]{} [@cpc:82:74; @*cpc:135:238; @cpc:39:347; @*cpc:43:367] in the case of [Pythia]{}, and a cluster model [@np:b238:492] in the case of [Herwig]{}.
Neutral current DIS events including radiative effects were simulated using the [Heracles]{} 4.6.1 [@cpc:69:155; @*spi:www:heracles] program with the [ Djangoh]{} 1.1 [@cpc:81:381; @*spi:www:djangoh11] interface to the hadronisation programs. [Heracles]{} includes corrections for initial- and final-state radiation, vertex and propagator terms, and two-boson exchange. The QCD cascade is simulated using the colour-dipole model (CDM) [@pl:b165:147; @*pl:b175:453; @*np:b306:746; @*zfp:c43:625] including the LO QCD diagrams as implemented in [Ariadne]{} 4.08 [@cpc:71:15; @*zfp:c65:285]; additional samples were generated with the MEPS model of [Lepto]{} 6.5 [@cpc:101:108]. Both MC programs use the Lund string model for the hadronisation. The CTEQ5D [@epj:c12:375] proton PDFs were used for these simulations.
Fixed-order calculations {#nlo}
========================
The calculations of direct $\gp$ processes used in this analysis are based on the program by Klasen, Kleinwort and Kramer (KKK) [@epjdirectcbg:c1:1]. The number of flavours was set to five; the renormalisation, $\mu_R$, and factorisation scales, $\mu_F$, were set to $\mu_R=\mu_F=E_T^{\rm max}$, where $E_T^{\rm max}$ is the highest $\etjet$ in an event. The calculations were performed using the ZEUS-S [@pr:d67:012007] parameterisations of the proton PDFs; $\as$ was calculated at two loops using $\Lambda^{(5)}_{\overline{\rm
MS}}=226$ MeV, which corresponds to $\asz=0.118$. These calculations are $\oass$ and represent the lowest-order contribution to three-jet $\gp$. Full $\oasss$ corrections are not yet available for three-jet cross sections in $\gp$.
The calculations of NC DIS processes used in this analysis are based on the program [Nlojet++]{} [@prl:87:082001], which provides $\oass$ and $\oasss$ predictions for three-jet cross sections. The scales were chosen to be $\mu_R=\mu_F=Q$. Other parameters were set as for the $\gp$ program.
In general, the programs mentioned above are very flexible and provide observable-independent computations that allow a complete analytical cancellation of the soft and collinear singularities encountered in the calculations of jet cross sections. However, these programs were written assuming the SU(3) gauge group and the different ingredients necessary to perform a calculation according to Eq. (1) were not readily available. The programs were rewritten in order to disentagle the colour components to make separate predictions for $\sigma_A$, ..., $\sigma_D$.
The $\kt$ jet algorithm was applied to the partons in the events generated by KKK and [Nlojet++]{} in order to compute the jet cross-section predictions. Thus, these predictions refer to jets of partons. Since the measurements refer to jets of hadrons, the calculations were corrected to the hadron level. The multiplicative correction factors, defined as the ratios between the cross section for jets of hadrons and that for jets of partons, were estimated using the MC samples described in Section \[mc\]. The normalised cross-section calculations changed typically by less than $\pm 5\ (10)\%$ for the predictions in $\gp$ (NC DIS) upon application of the parton-to-hadron corrections. Therefore, the effect of the parton-to-hadron corrections on the angular distributions is small. In NC DIS processes, other effects not accounted for in the calculations, namely $\z0$ exchange, were also corrected for using the MC samples.
The predictions for jet cross sections are expressed as the convolution of the PDFs and the matrix elements, which depend on $\as$. Both the PDFs and $\as$ evolve with the energy scale. In the calculations performed for this analysis, QCD evolution via the DGLAP and the renormalisation group equations, respectively, were used. These evolution equations also depend on the colour factors. This procedure introduces an additional dependence on the colour factors with respect to that shown in Eq. (1); this dependence is suppressed by considering normalised cross sections (see Section \[cs\] for the definition of the cross sections). The remaining dependence was estimated by comparing to calculations with fixed $\mu_F$ or $\mu_R$. The values chosen for $\mu_F$ and $\mu_R$ were the mean values of the data distributions, $\langle E_T^{\rm max}\rangle_{\rm data}=27.8$ GeV for $\gp$ and $\sqrt{\langle\q2\rangle_{\rm data}}=31.3\ (36.6)$ GeV for NC DIS with $\q2>125$ ($500<\q2<5000$) 2.
Figure \[fig7\] shows the relative difference of the $\oass$ $\gp$ calculations with $\mu_F$ ($\mu_R$) fixed[^3] to those in which $\mu_F=E_T^{\rm max}$ ($\mu_R=E_T^{\rm max}$) as a function of the angular variables studied. Figures \[fig8\](a) to \[fig8\](d) show the same relative difference for the $\oass$ [ Nlojet++]{} calculations for $\q2>125$ 2.
Very small differences are observed for the $\mu_F$ variation. Sizeable differences for the $\mu_R$ variation are seen in some regions; in particular, a trend is observed for the relative difference as a function of $\etajmax$: this trend is due to the fact that the mean values of $\q2$ in each bin of $\etajmax$ increase as $\etajmax$ decreases.
These studies demonstrate that the normalised cross sections have little sensitivity to the evolution of the PDFs. However, there is still some sensitivity to the running of $\as$. Figures \[fig8\](e) to \[fig8\](h) show the relative difference for $500<\q2<5000$ 2. The restriction of the phase space further reduces the dependence on the running of $\as$; thus, this region is more suitable to extract the colour factors in NC DIS at $\oass$. At $\oasss$ (see Fig. \[fig9\]), the effect due to the running of $\as$ is already very small for $\q2>125$ 2. Therefore, the wider phase-space region can be kept in an extraction of the colour factors at $\oasss$.
The following theoretical uncertainties were considered (as an example of the size of the uncertainties, an average value of the effect of each uncertainty on the normalised cross section as a function of $\th$ is shown in parentheses for $\gp$, NC DIS with $\q2>125$ 2and NC DIS with $500<\q2<5000$ 2):
- the uncertainty in the modelling of the parton shower was estimated by using different models (see Section \[mc\]) to calculate the parton-to-hadron correction factors ($\pm 2.8\%$, $\pm 2.9\%$ and $\pm 5.8\%$);
- the uncertainty on the calculations due to higher-order terms was estimated by varying $\mu_R$ by a factor of two up and down ($_{-0.8}^{+0.6}\%$, $\pm 1.6\%$ and $\pm 2.2\%$);
- the uncertainty on the calculations due to those on the proton PDFs was estimated by repeating the calculations using 22 additional sets from the ZEUS analysis [@pr:d67:012007]; this analysis takes into account the statistical and correlated systematic experimental uncertainties of each data set used in the determination of the proton PDFs ($\pm 0.7\%$, $\pm 0.2\%$ and $\pm 0.1\%$);
- the uncertainty on the calculations due to that on $\asz$ was estimated by repeating the calculations using two additional sets of proton PDFs, for which different values of $\asz$ were assumed in the fits. The difference between the calculations using these various sets was scaled to reflect the uncertainty on the current world average of $\as$ [@jp:g26:r27] (negligible in all cases);
- the uncertainty of the calculations due to the choice of $\mu_F$ was estimated by varying $\mu_F$ by a factor of two up and down (negligible in all cases).
The total theoretical uncertainty was obtained by adding in quadrature the individual uncertainties listed above. The dominant source of theoretical uncertainty is that on the modelling of the parton shower.
Definition of the cross sections {#cs}
================================
Normalised differential three-jet cross sections were measured as functions of $\th$, $\a34$ and $\pksw$ using the selected data samples in $\gp$ and NC DIS. For NC DIS, the normalised differential three-jet cross section as a function of $\etajmax$ was also measured. The normalised differential three-jet cross section in bin $i$ for an observable $A$ was obtained using
$$\frac{1}{\sigma}\frac{d\sigma_i}{dA}=\frac{1}{\sigma}\frac{N_{{\rm data},i}}{{\cal L}\cdot \Delta A_i}\cdot\frac{N^{\rm had}_{{\rm MC},i}}{N^{\rm det}_{{\rm MC},i}},$$ where $N_{{\rm data},i}$ is the number of data events in bin $i$, $N^{\rm had}_{{\rm MC},i}\ (N^{\rm det}_{{\rm MC},i})$ is the number of MC events at hadron (detector) level, ${\cal L}$ is the integrated luminosity and $\Delta A_i$ is the bin width. The integrated three-jet cross section, $\sigma$, was computed using the formula:
$$\sigma=\sum_i\frac{N_{{\rm data},i}}{{\cal L}}\cdot\frac{N^{\rm had}_{{\rm MC},i}}{N^{\rm det}_{{\rm MC},i}},$$ where the sum runs over all bins.
For the $\gp$ sample, due to the different centre-of-mass energies of the two data sets used in the analysis, the measured normalised differential three-jet cross sections were combined using
$$\sigma^{\rm
comb}=\frac{\sigma_{300}\cdot{\cal{L}}_{300}+\sigma_{318}\cdot{\cal{L}}_{318}}{{\cal{L}}_{300}+{\cal{L}}_{318}},$$ where ${\cal{L}}_{\sqrt{s}}$ is the luminosity and $\sigma_{\sqrt{s}}$ is the measured cross section corresponding to $\sqrt s=300$ or $318$ GeV. This formula was applied for combining the differential and total cross sections. The same formula was used for computing the $\oass$ predictions in $\gp$.
Acceptance corrections and experimental uncertainties {#expunc}
=====================================================
The [Pythia]{} (MEPS) MC samples were used to compute the acceptance corrections to the angular distributions of the $\gp$ (NC DIS) data. These correction factors took into account the efficiency of the trigger, the selection criteria and the purity and efficiency of the jet reconstruction. The samples of [Herwig]{} and CDM were used to compute the systematic uncertainties coming from the fragmentation and parton-shower models in $\gp$ and NC DIS, respectively.
The data $\etjet$, $\etajet$ and $\xo$ distributions of the $\gp$ sample, before the $\xo>0.8$ requirement, are shown in Fig. \[fig2\] together with the MC simulations of [Pythia]{} and [ Herwig]{}. Considering that three-jet events in the MC arise only from the parton-shower approximation, the description of the data is reasonable. Figure \[fig2\](d) shows the resolved and direct contributions for the [Pythia]{} MC separately. It is observed that the region of $\xo>0.8$ is dominated by direct $\gp$ events. The remaining contribution in this region from resolved-photon events was estimated using [Pythia]{} ([Herwig]{}) simulated events to be $\approx 25\ (31)\%$.
Figure \[fig3\] shows the data distributions as functions of $\th$, $\a34$ and $\pksw$ together with the simulations of [Pythia]{} and [Herwig]{} for $\xo>0.8$. The [Pythia]{} MC predictions describe the data distributions well, whereas the description given by [ Herwig]{} is somewhat poorer. It was checked that the angular distributions of the events from resolved processes with $\xo>0.8$ were similar to those from direct processes (see Fig. \[fig4\]) and, therefore, no subtraction of the resolved processes was performed when comparing to the fixed-order calculations described in Section \[nlo\].
The data $\etjbj$, $E^{\rm jet2,3}_{T,{\rm B}}$, $\etajb$ and $\q2$ distributions of the NC DIS samples are shown in Fig. \[fig23\] (\[fig24\]) for $\q2>125$ ($500<\q2<5000$) 2 together with the MC simulations from the MEPS and CDM models. Both models give a reasonably good description of the data in both kinematic regions. The data distributions of $\th$, $\a34$, $\pksw$ and $\etajmax$ are shown in Fig. \[fig5\] (\[fig6\]) for $\q2>125$ ($500<\q2<5000$) 2. The MEPS MC predictions describe the data distributions well, whereas the description given by CDM is somewhat poorer.
A detailed study of the sources contributing to the experimental uncertainties was performed [@marcos]. The following experimental uncertainties were considered for $\gp$ (as an example of the size of the uncertainties, an average value of the effect of each uncertainty on the cross section as a function of $\th$ is shown in parentheses):
- the effect of the modelling of the parton shower and hadronisation was estimated by using [Herwig]{} instead of [ Pythia]{} to evaluate the correction factors ($\pm 6.1\%$);
- the effect of the uncertainty on the absolute energy scale of the calorimetric jets was estimated by varying $\etjet$ in simulated events by its uncertainty of $\pm 1\%$. The method used was the same as in earlier publications [@proc:calor:2002:767; @pl:b560:7; @pl:b649:12] ($\pm 1.6\%$);
- the effect of the uncertainty on the reconstruction of $y$ was estimated by varying its value in simulated events by the estimated uncertainty of $\pm 1\%$ ($\pm 1.0\%$);
- the effect of the uncertainty on the parameterisations of the proton and photon PDFs was estimated by using alternative sets of PDFs in the MC simulation to calculate the correction factors ($\pm 0.4\%$ and $\pm 2.0\%$, respectively);
- the uncertainty in the cross sections due to that in the simulation of the trigger ($\pm 0.4\%$).
For NC DIS events, the following experimental uncertainties were considered (as an example of the size of the uncertainties, an average value of the effect of each uncertainty on the cross section as a function of $\th$ is shown in parentheses for the $\q2>125$ 2 and $500<\q2<5000$ 2 kinematic regions):
- the effect of the modelling of the parton shower was estimated by using CDM instead of MEPS to evaluate the correction factors ($\pm 5.6\%$ and $\pm 9.1\%$);
- the effect of the uncertainty on the absolute energy scale of the calorimetric jets was estimated by varying $\etjet$ in simulated events by its uncertainty of $\pm 1\%$ for $\etjet>10$ GeV and $\pm 3\%$ for lower $\etjet$ values ($\pm 2.3\%$ and $\pm 1.7\%$);
- the uncertainties due to the selection cuts was estimated by varying the values of the cuts within the resolution of each variable (less than $\pm 1.6\%$ and less than $\pm 4.2\%$ in all cases);
- the uncertainty on the reconstruction of the boost to the Breit frame was estimated by using the direction of the track associated with the scattered electron instead of that derived from the impact position as determined from the energy depositions in the CAL ($\pm 1.6\%$ and $\pm 1.6\%$);
- the uncertainty in the absolute energy scale of the electron candidate was estimated to be $\pm 1\%$ [@epj:c21:443] ($\pm 0.2\%$ and $\pm 0.3\%$);
- the uncertainty in the cross sections due to that in the simulation of the trigger ($\pm 0.5\%$ and $\pm 0.5\%$).
The effect of these uncertainties on the normalised differential three-jet cross sections is small compared to the statistical uncertainties for the measurements presented in Section \[results\]. The systematic uncertainties were added in quadrature to the statistical uncertainties.
Results
=======
Normalised differential three-jet cross sections were measured in $\gp$ in the kinematic region $\q2<1$ 2, $0.2<y<0.85$ and $\xo>0.8$. The cross sections were determined for jets of hadrons with $\etjet>14$ GeV and $-1<\etajet<2.5$. In NC DIS, the cross sections were measured in two kinematic regimes: $\q2>125$ 2 and $500<\q2<5000$ 2. In both cases, it was required that $|\cgh|<0.65$. The cross sections correspond to jets of hadrons with $\etjbj>8$ GeV, $E^{\rm jet2,3}_{T,{\rm B}}>5$ GeV and $-2<\etajb<1.5$.
Colour components and the triple-gluon vertex
---------------------------------------------
Normalised differential three-jet cross sections at $\oass$ of the individual colour components from Eq. (\[one\]), $\sigma_A$, ..., $\sigma_D$, were calculated using the programs described in Section \[nlo\] and are shown separately in Fig. \[fig14\] for $\gp$ and in Fig. \[fig15\] (\[fig16\]) for NC DIS with $\q2>125$ ($500<\q2<5000$) 2 as functions of the angular variables. In these and subsequent figures, the predictions were obtained by integrating over the same bins as for the data. The curves shown join the points and are a result of a cubic spline interpolation, except in the case of $\etajmax$, for which adjacent points are connected by straight lines.
The component which contains the contribution from the TGV in quark-induced processes, $\sigma_B$, has a very different shape than the other components for all the angular variables considered. The other components have distributions in $\pksw$ and $\th$ that are similar and are best separated by the distribution of $\a34$ in $\gp$. In NC DIS with $500<\q2<5000$ 2, the different colour components as functions of $\th$ and $\pksw$ also display different shapes. In particular, the $\sigma_D$ component, which also contains a TGV, shows a distinct shape for these distributions. This demonstrates that the three-jet angular correlations studied show sensitivity to the different colour components.
In $\gp$ (NC DIS: $\q2>125$ 2, $500<\q2<5000$ 2), the SU(3)-based predictions for the relative contribution of each colour component are: (A): $0.13\ (0.23,\ 0.30)$, (B): $0.10\ (0.13,\ 0.14)$, (C): $0.45\ (0.39,\ 0.35)$ and (D): $0.32\ (0.25,\ 0.21)$. Therefore, the overall contribution from the diagrams that involve a TGV, B and D, amounts to $42\ (38,\ 35)\%$ in SU(3).
Three-jet cross sections in [$\gp$]{}
-------------------------------------
The integrated three-jet cross section in $\gp$ in the kinematic range considered was measured to be:
$$\sigma_{ep \rightarrow 3{\rm jets}}=14.59\pm 0.34\ ({\rm
stat.})\ _{-1.31}^{+1.25}\ ({\rm syst.})\ {\rm pb}.$$ The predicted $\oass$ integrated cross section, which is the lowest order for this process and contains only direct processes, is $8.90\ _{-2.92}^{+2.01}$ pb.
The measured normalised differential three-jet cross sections are presented in Fig. \[fig17\] and Tables \[tabone\] to \[tabthree\] as functions of $\th$, $\cos(\a34)$ and $\cos(\pksw)$. The measured cross section shows a peak at $\th\approx
60^{\circ}$, increases as $\cos(\a34)$ increases and shows a broad peak in the range of $\cos(\pksw)$ between $-0.5$ to $0.1$.
Three-jet cross sections in NC DIS
----------------------------------
The integrated three-jet cross sections in NC DIS for $\q2>125$ 2and $500<\q2<5000$ 2 were measured to be:
$$\sigma_{ep \rightarrow 3{\rm jets}}=11.48\pm 0.35\ ({\rm
stat.})\ \pm 1.98\ ({\rm syst.})\ {\rm pb}$$ and $$\sigma_{ep \rightarrow 3{\rm jets}}=5.73\pm 0.26\ ({\rm
stat.})\ \pm 0.60\ ({\rm syst.})\ {\rm pb}.$$ The predicted $\oasss$ integrated cross sections are $14.14\pm 3.40$ pb and $6.86\pm 1.77$ pb for the two kinematic regions, respectively.
The measured normalised differential three-jet cross sections in NC DIS for $\q2>125$ 2 and $500<\q2<5000$ 2 are presented in Figs. \[fig18\] and \[fig19\], respectively, as functions of $\th$, $\cos(\a34)$, $\cos(\pksw)$ and $\etajmax$ (see Tables \[tabfour\] to \[tabseven\]). The measured cross sections have similar shapes in the two kinematic regions considered, except for the distribution as a function of $\cos(\pksw)$: the cross section decreases as $\cos(\pksw)$ increases for $500<\q2<5000$ 2 whereas for $\q2>125$ 2 it shows an approximately constant behaviour for $-1<\cos(\pksw)<0.25$. The measured cross section as a function of $\cos(\a34)$ peaks around $0.5$ and increases as $\th$ and $\etajmax$ increase.
Comparison to fixed-order calculations
--------------------------------------
Calculations at $\oass$ in which each colour contribution in Eq. (1) was weighted according to the colour factors predicted by SU(3) ($C_F=4/3$, $C_A=3$ and $T_F=1/2$) are compared to the measurements in Figs. \[fig17\] to \[fig21\]. The theoretical uncertainties are shown in Figs. \[fig17\], \[fig20\] and \[fig21\] as hatched bands. Since the calculations are normalised to unity, the uncertainties are correlated among the points; this correlation is partially responsible for the pulsating pattern exhibited by the theoretical uncertainties. The predictions based on SU(3) give a reasonable description of the data for all angular correlations. For $\gp$, the predictions do not include resolved processes (see Section \[nlo\]), as calculations separated according to the different colour factors are not available. Monte Carlo simulations of such processes show that their contribution is most likely to be different from that of direct processes in the fifth and last bin of $(1/\sigma)(d\sigma/d\cos(\a34))$ (see Figs. \[fig4\]b and \[fig17\]b).
To illustrate the sensitivity of the measurements to the colour factors, calculations based on different symmetry groups are also compared to the data in Figs. \[fig17\] to \[fig19\]. In these calculations, the colour components were combined in such a way as to reproduce the colour structure of a theory based on the non-Abelian group SU($N$) in the limit of large $N$ ($C_F=1$, $C_A=2$ and $T_F=0$), the Abelian group U(1)$^3$ ($C_F=1$, $C_A=0$ and $T_F=3$), the non-Abelian group SO(3) ($C_F=1/3$, $C_A=3$ and $T_F=1/3$) and, as an extreme choice, a calculation with $C_F=0$. The shapes of the distributions predicted by U(1)$^3$ in $\gp$ are very similar to those by SU(3) due to the smallness of the component $\sigma_B$ and the difficulty to distinguish the component $\sigma_D$. In NC DIS, the predictions of U(1)$^3$ show differences of around $10\%$ with respect to those of SU(3), which are of the same order as the statistical uncertainties. In both regimes, the data clearly disfavour a theory based on SU($N$) in the limit of large $N$ or on $C_F=0$.
Figures \[fig20\] and \[fig21\] show the measurements in NC DIS compared to the predictions of QCD at $\oass$ and $\oasss$. This comparison provides a very stringent test of pQCD. The $\oasss$ calculations give a very good description of the data. In particular, a significant improvement in the description of the data can be observed for the first bin of the $\a34$ distribution (Figs. \[fig20\]b and \[fig21\]b).
Summary and conclusions
=======================
Measurements of angular correlations in three-jet $\gp$ and NC DIS were performed in $ep$ collisions at HERA using $127$ 1 of data collected with the ZEUS detector. The cross sections refer to jets identified with the $\kt$ cluster algorithm in the longitudinally invariant inclusive mode and selected with $\etjet>14$ GeV and $\etar$ ($\gp$) and $\etjbj>8$ GeV, $E^{\rm jet2,3}_{T,{\rm B}}>5$ GeV and $-2<\etajb<1.5$ (NC DIS). The measurements were made in the kinematic regions defined by $\q2<1$ 2, $0.2<y<0.85$ and $\xo>0.8$ ($\gp$) and $\q2>125$ 2 or $500<\q2<5000$ 2 and $|\cgh|<0.65$ (NC DIS). Normalised differential three-jet cross sections were measured as functions of $\th$, $\a34$, $\pksw$ and $\etajmax$.
The colour configuration of the strong interaction was studied for the first time in $ep$ collisions using the angular correlations in three-jet events. While the extraction of the colour factors will require the full analysis of all HERA data and complete $\oasss$ calculations, the studies presented in this paper demonstrate the potential of the method.
Fixed-order calculations separated according to the colour configurations were used to study the sensitivity of the angular correlations to the underlying gauge structure. The predicted distributions of $\th$, $\a34$ and $\pksw$ clearly isolate the contribution from the triple-gluon coupling in quark-induced processes while $\etajmax$ isolates the contribution from gluon-induced processes. The variable $\a34$ provides additional separation for the other contributions. Furthermore, the studies performed demonstrate that normalised cross sections in three-jet $ep$ collisions have reduced sensitivity to the assumed evolution of the PDFs and the running of $\as$.
The data clearly disfavour theories based on SU($N$) in the limit of large $N$ or $C_F=0$. Differences between SU(3) and U(1)$^3$ are smaller than the current statistical uncertainties. The measurements are found to be consistent with the admixture of colour configurations as predicted by SU(3). The $\oasss$ calculations give a very good description of the NC DIS data.
[**Acknowledgements**]{}
We thank the DESY Directorate for their strong support and encouragement. We appreciate the contributions to the construction and maintenance of the ZEUS detector of many people who are not listed as authors. The HERA machine group and the DESY computing staff are especially acknowledged for their success in providing excellent operation of the collider and the data-analysis environment. We would like to thank M. Fontannaz, M. Klasen and Z. Nagy for useful discussions.
‘=11 bibitem\#1[@bstsupport @iftail[\#1]{} [;]{} [@first.@bibitem[\#1]{}]{} @firstfalse @iftail[\#1]{} [@bibitem[\#1]{}]{}]{}‘=12
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$\th$ bin (deg) $(1/\sigma)\ d\sigma/d\th$ $\delta_{\rm stat}$ $\delta_{\rm syst}$ $C_{\rm had}$
----------------- ---------------------------- --------------------- -------------------------------- ---------------
0, 9 0.00264 0.00038 $\pm 0.00052$ 0.93
9, 18 0.00393 0.00044 $\pm 0.00021$ 0.94
18, 27 0.00507 0.00051 ${}_{- 0.00039}^{+ 0.00040}$ 1.00
27, 36 0.00838 0.00064 ${}_{- 0.00104}^{+ 0.00105}$ 0.93
36, 45 0.01071 0.00075 $\pm 0.00023$ 0.96
45, 54 0.01486 0.00087 ${}_{- 0.00016}^{+ 0.00021}$ 0.94
54, 63 0.01795 0.00098 ${}_{- 0.00035}^{+ 0.00036}$ 0.95
63, 72 0.01765 0.00095 $\pm 0.00062$ 0.94
72, 81 0.01517 0.00088 ${}_{- 0.00084}^{+ 0.00081}$ 0.94
81, 90 0.01473 0.00086 ${}_{- 0.00077}^{+ 0.00075}$ 0.96
: Normalised differential $ep$ cross section for three-jet photoproduction integrated over $\etjet>14$ GeV and $\etar$ in the kinematic region defined by $\q2<1$ 2, $0.2<y<0.85$ and $\xo>0.8$ as a function of $\th$. The statistical and systematic uncertainties are shown separately. The multiplicative corrections for hadronisation effects to be applied to the parton-level QCD differential cross section, $C_{\rm had}$, are shown in the last column.[]{data-label="tabone"}
$\cos(\a34)$ bin $(1/\sigma)\ d\sigma/d\cos(\a34)$ $\delta_{\rm stat}$ $\delta_{\rm syst}$ $C_{\rm had}$
------------------ ----------------------------------- --------------------- ---------------------------- ---------------
-1, -0.8 0.0138 0.0046 $\pm 0.00042$ 1.04
-0.8, -0.6 0.078 0.012 ${}_{- 0.003}^{+ 0.004}$ 0.96
-0.6, -0.4 0.198 0.022 ${}_{- 0.027}^{+ 0.026}$ 0.95
-0.4, -0.2 0.343 0.029 ${}_{- 0.040}^{+ 0.041}$ 0.93
-0.2, 0 0.360 0.029 $\pm 0.010$ 0.97
0, 0.2 0.512 0.034 ${}_{- 0.013}^{+ 0.014}$ 0.98
0.2, 0.4 0.618 0.037 ${}_{- 0.016}^{+ 0.015}$ 1.00
0.4, 0.6 0.847 0.044 $\pm 0.013$ 0.99
0.6, 0.8 0.937 0.045 ${}_{- 0.042}^{+ 0.043}$ 0.99
0.8, 1 1.092 0.049 ${}_{- 0.018}^{+ 0.019}$ 1.02
: Normalised differential $ep$ cross section for three-jet photoproduction integrated over $\etjet>14$ GeV and $\etar$ in the kinematic region defined by $\q2<1$ 2, $0.2<y<0.85$ and $\xo>0.8$ as a function of $\cos(\a34)$. Other details as in the caption to Table \[tabone\].[]{data-label="tabtwo"}
$\cos(\pksw)$ bin $(1/\sigma)\ d\sigma/d\cos(\pksw)$ $\delta_{\rm stat}$ $\delta_{\rm syst}$ $C_{\rm had}$
------------------- ------------------------------------ --------------------- ---------------------------- ---------------
-1, -0.8 0.552 0.035 $\pm 0.044$ 0.97
-0.8, -0.6 0.651 0.039 $\pm 0.026$ 0.99
-0.6, -0.4 0.745 0.042 ${}_{- 0.031}^{+ 0.032}$ 0.97
-0.4, -0.2 0.741 0.042 $\pm 0.039$ 0.93
-0.2, 0 0.784 0.042 ${}_{- 0.016}^{+ 0.014}$ 0.96
0, 0.2 0.768 0.042 $\pm 0.046$ 0.95
0.2, 0.4 0.500 0.034 $\pm 0.005$ 0.94
0.4, 0.6 0.200 0.022 $\pm 0.021$ 0.95
0.6, 0.8 0.056 0.010 ${}_{- 0.009}^{+ 0.010}$ 0.85
0.8, 1 0.0029 0.0015 $\pm 0.0037$ 0.74
: Normalised differential $ep$ cross section for three-jet photoproduction integrated over $\etjet>14$ GeV and $\etar$ in the kinematic region defined by $\q2<1$ 2, $0.2<y<0.85$ and $\xo>0.8$ as a function of $\cos(\pksw)$. Other details as in the caption to Table \[tabone\].[]{data-label="tabthree"}
[||c|ccc||c||c||]{} $\th$ bin (deg) & $(1/\sigma)\ d\sigma/d\th$ & $\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $C_{\rm QED}$ & $C_{\rm had}$\
\
0, 18 & 0.00372 & 0.00046 & $\pm 0.00031$ & 0.92 & 0.89\
18, 36 & 0.00770 & 0.00056 & $\pm 0.00095$ & 0.88 & 0.90\
36, 54 & 0.01291 & 0.00072 & $\pm 0.00045$ & 0.96 & 0.84\
54, 72 & 0.01438 & 0.00074 & $\pm 0.00042$ & 1.00 & 0.84\
72, 90 & 0.01686 & 0.00077 & $\pm 0.00160$ & 0.99 & 0.84\
\
0, 18 & 0.00481 & 0.00076 & $\pm 0.00048$ & 0.88 & 0.92\
18, 36 & 0.00993 & 0.00094 & $\pm 0.00231$ & 0.95 & 0.96\
36, 54 & 0.0141 & 0.0011 & $\pm 0.0004$ & 0.92 & 0.97\
54, 72 & 0.0134 & 0.0011 & $\pm 0.0008$ & 1.03 & 0.89\
72, 90 & 0.0133 & 0.0011 & $\pm 0.0023$ & 0.96 & 0.94\
[||c|ccc||c||c||]{} $\cos(\a34)$ bin & $(1/\sigma)\ d\sigma/d\cos(\a34)$ & $\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $C_{\rm QED}$ & $C_{\rm had}$\
\
-1, -0.6 & 0.117 & 0.015 & $\pm 0.025$ & 0.96 & 0.90\
-0.6, -0.2 & 0.338 & 0.028 & $\pm 0.035$ & 1.01 & 0.70\
-0.2, 0.2 & 0.568 & 0.032 & $\pm 0.018$ & 0.90 & 0.78\
0.2, 0.6 & 0.993 & 0.037 & $\pm 0.021$ & 0.95 & 0.88\
0.6, 1 & 0.484 & 0.030 & $\pm 0.020$ & 1.02 & 1.01\
\
-1, -0.6 & 0.199 & 0.030 & $\pm 0.018$ & 1.04 & 0.83\
-0.6, -0.2 & 0.381 & 0.043 & $\pm 0.041$ & 0.97 & 0.75\
-0.2, 0.2 & 0.589 & 0.047 & $\pm 0.074$ & 0.92 & 0.83\
0.2, 0.6 & 1.018 & 0.055 & $\pm 0.061$ & 0.95 & 1.07\
0.6, 1 & 0.313 & 0.036 & $\pm 0.022$ & 0.97 & 1.16\
[||c|ccc||c||c||]{} $\cos(\pksw)$ bin & $(1/\sigma)\ d\sigma/d\cos(\pksw)$ & $\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $C_{\rm QED}$ & $C_{\rm had}$\
\
-1, -0.6 & 0.585 & 0.031 & $\pm 0.057$ & 0.92 & 0.95\
-0.6, -0.2 & 0.691 & 0.034 & $\pm 0.094$ & 0.99 & 0.88\
-0.2, 0.2 & 0.721 & 0.035 & $\pm 0.020$ & 1.01 & 0.85\
0.2, 0.6 & 0.332 & 0.026 & $\pm 0.025$ & 0.92 & 0.74\
0.6, 1 & 0.171 & 0.020 & $\pm 0.022$ & 0.93 & 0.71\
\
-1, -0.6 & 0.770 & 0.052 & $\pm 0.076$ & 0.94 & 1.04\
-0.6, -0.2 & 0.536 & 0.045 & $\pm 0.112$ & 0.93 & 0.97\
-0.2, 0.2 & 0.497 & 0.045 & $\pm 0.037$ & 1.01 & 0.94\
0.2, 0.6 & 0.430 & 0.044 & $\pm 0.058$ & 1.01 & 0.84\
0.6, 1 & 0.267 & 0.036 & $\pm 0.061$ & 0.89 & 0.78\
[||c|ccc||c||c||]{} $\etajmax$ bin & $(1/\sigma)\ d\sigma/d\etajmax$ & $\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $C_{\rm QED}$ & $C_{\rm had}$\
\
-2, -0.1 & 0.0042 & 0.0013 & $\pm 0.0006$ & 1.07 & 0.61\
-0.1, 0.3 & 0.092 & 0.016 & $\pm 0.012$ & 1.17 & 0.77\
0.3, 0.7 & 0.267 & 0.024 & $\pm 0.054$ & 0.96 & 0.81\
0.7, 1.1 & 0.751 & 0.034 & $\pm 0.016$ & 0.93 & 0.83\
1.1, 1.5 & 1.370 & 0.038 & $\pm 0.048$ & 0.96 & 0.88\
\
-2, -0.1 & 0.0059 & 0.0021 & $\pm 0.0022$ & 1.14 & 0.62\
-0.1, 0.3 & 0.110 & 0.022 & $\pm 0.011$ & 0.96 & 0.77\
0.3, 0.7 & 0.378 & 0.040 & $\pm 0.084$ & 0.96 & 0.86\
0.7, 1.1 & 0.918 & 0.054 & $\pm 0.052$ & 0.93 & 0.93\
1.1, 1.5 & 1.066 & 0.056 & $\pm 0.035$ & 0.98 & 1.00\
(18.0,15.0) (2.8,9.0) (10.5,8.0) (3.0,0.0) (10.5,0.5) (4.5,15.0)[**(A)**]{} (12.0,15.0)[**(B)**]{} (4.5,6.7)[**(C)**]{} (12.0,6.7)[**(D)**]{}
(18.0,7.0) (0.0,1.5) (7.0,1.5) (0.0,-4.5) (7.0,10.0)[**(a)**]{} (14.0,10.0)[**(b)**]{} (7.0,4.0)[**(c)**]{}
(18.0,17.0) (0.0,10.5) (7.0,10.5) (0.0,5.5) (7.0,5.5) (0.0,0.5) (7.0,0.5) (0.0,-4.5) (7.0,-4.5) (7.0,19.0)[**(a)**]{} (14.0,19.0)[**(b)**]{} (7.0,14.0)[**(c)**]{} (14.0,14.0)[**(d)**]{} (7.0,9.0)[**(e)**]{} (14.0,9.0)[**(f)**]{} (7.0,4.0)[**(g)**]{} (14.0,4.0)[**(h)**]{}
(18.0,17.0) (0.0,10.5) (7.0,10.5) (0.0,5.5) (7.0,5.5) (0.0,0.5) (7.0,0.5) (0.0,-4.5) (7.0,-4.5) (7.0,19.0)[**(a)**]{} (14.0,19.0)[**(b)**]{} (7.0,14.0)[**(c)**]{} (14.0,14.0)[**(d)**]{} (7.0,9.0)[**(e)**]{} (14.0,9.0)[**(f)**]{} (7.0,4.0)[**(g)**]{} (14.0,4.0)[**(h)**]{}
(18.0,17.0) (-0.3,8.0)
(-1.0,7.5) (6.5,7.5) (-1.0,-0.5) (6.5,-0.5) (6.3,13.0)[**(a)**]{} (8.8,15.0)[**(b)**]{} (6.3,7.0)[**(c)**]{} (13.8,7.0)[**(d)**]{}
(18.0,17.0) (-0.3,8.0)
(-1.0,7.5) (6.5,7.5) (-1.0,-0.5) (6.3,15.0)[**(a)**]{} (8.8,15.0)[**(b)**]{} (6.3,7.0)[**(c)**]{}
(18.0,17.0) (-0.3,8.0)
(-1.0,7.5) (6.5,7.5) (-1.0,-0.5) (6.3,15.0)[**(a)**]{} (8.8,15.0)[**(b)**]{} (6.3,7.0)[**(c)**]{}
(18.0,17.0) (-0.3,8.0)
(-1.0,7.5) (-1.0,7.5) (6.5,7.5) (-1.0,-0.5) (6.5,-0.5) (6.3,15.0)[**(a)**]{} (13.8,15.0)[**(b)**]{} (6.3,7.0)[**(c)**]{} (13.8,7.0)[**(d)**]{}
(18.0,17.0) (-0.3,8.0)
(-1.0,7.5) (-1.0,7.5) (6.5,7.5) (-1.0,-0.5) (6.3,15.0)[**(a)**]{} (13.8,15.0)[**(b)**]{} (6.3,7.0)[**(c)**]{}
(18.0,17.0) (-0.3,8.0)
(-1.0,7.5) (6.5,7.5) (-1.0,-0.5) (6.5,-0.5) (6.3,15.0)[**(a)**]{} (13.8,15.0)[**(b)**]{} (6.3,7.0)[**(c)**]{} (13.8,7.0)[**(d)**]{}
(18.0,17.0) (-0.3,8.0)
(-1.0,7.5) (6.5,7.5) (-1.0,-0.5) (6.5,-0.5) (6.3,15.0)[**(a)**]{} (13.8,15.0)[**(b)**]{} (6.3,7.0)[**(c)**]{} (13.8,7.0)[**(d)**]{}
(18.0,17.0) (-0.3,8.0)
(-1.0,7.5) (6.5,7.5) (-1.0,-0.5) (5.3,15.0)[**(a)**]{} (8.8,15.0)[**(b)**]{} (6.3,7.0)[**(c)**]{}
(18.0,17.0) (-0.3,8.0)
(-1.0,7.5) (6.5,7.5) (-1.0,-0.5) (6.5,-0.5) (6.3,15.0)[**(a)**]{} (13.8,15.0)[**(b)**]{} (6.3,7.0)[**(c)**]{} (13.8,7.0)[**(d)**]{}
(18.0,17.0) (-0.3,8.0)
(-1.0,7.5) (6.5,7.5) (-1.0,-0.5) (6.5,-0.5) (6.3,15.0)[**(a)**]{} (13.8,15.0)[**(b)**]{} (6.3,7.0)[**(c)**]{} (13.8,7.0)[**(d)**]{}
(18.0,17.0) (-0.3,11.0)
(-1.0,9.5) (6.5,9.5) (-1.0,-0.5) (6.3,17.5)[**(a)**]{} (13.8,18.0)[**(b)**]{} (6.3,8.0)[**(c)**]{}
(18.0,17.0) (-0.3,11.0)
(-1.0,9.5) (6.5,9.5) (-1.0,-0.5) (6.5,-0.5) (6.3,17.5)[**(a)**]{} (13.8,18.0)[**(b)**]{} (6.3,8.0)[**(c)**]{} (13.8,8.0)[**(d)**]{}
(18.0,17.0) (-0.3,11.0)
(-1.0,9.5) (6.5,9.5) (-1.0,-0.5) (6.5,-0.5) (6.3,17.5)[**(a)**]{} (13.8,18.0)[**(b)**]{} (6.3,8.0)[**(c)**]{} (13.8,8.0)[**(d)**]{}
(18.0,17.0) (-0.3,11.0)
(-1.0,9.5) (6.5,9.5) (-1.0,-0.5) (6.5,-0.5) (6.3,17.5)[**(a)**]{} (13.8,18.0)[**(b)**]{} (6.3,8.0)[**(c)**]{} (13.8,8.0)[**(d)**]{}
(18.0,17.0) (-0.3,11.0)
(-1.0,9.5) (6.5,9.5) (-1.0,-0.5) (6.5,-0.5) (6.3,17.5)[**(a)**]{} (13.8,18.0)[**(b)**]{} (6.3,8.0)[**(c)**]{} (13.8,8.0)[**(d)**]{}
[^1]: Here and in the following, the term “electron” denotes generically both the electron ($e^-$) and the positron ($e^+$).
[^2]: The ZEUS coordinate system is a right-handed Cartesian system, with the $Z$ axis pointing in the proton beam direction, referred to as the “forward direction”, and the $X$ axis pointing left towards the centre of HERA. The coordinate origin is at the nominal interaction point.
[^3]: When $\mu_F$ was fixed, $\mu_R$ was allowed to vary with the scale, and vice-versa.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Estimation of the Saupe tensor is central to the determination of molecular structures from residual dipolar couplings (RDC) or chemical shift anisotropies. Assuming a given template structure, the singular value decomposition (SVD) method proposed in [@losonczi1999order] has been used traditionally to estimate the Saupe tensor. Despite its simplicity, whenever the template structure has large structural noise, the eigenvalues of the estimated tensor have a magnitude systematically smaller than their actual values. This leads to systematic error when calculating the eigenvalue dependent parameters, magnitude and rhombicity. We propose here a Monte Carlo simulation method to remove such bias. We further demonstrate the effectiveness of our method in the setting when the eigenvalue estimates from multiple template protein fragments are available and their average is used as an improved eigenvalue estimator. For both synthetic and experimental RDC datasets of ubiquitin, when using template fragments corrupted by large noise, the magnitude of our proposed bias-reduced estimator generally reaches at least 90% of the actual value, whereas the magnitude of SVD estimator can be shrunk below 80% of the true value.'
author:
- 'Y. Khoo[^1]'
- 'A. Singer[^2]'
- 'D. Cowburn[^3]'
bibliography:
- 'bibref.bib'
title: Bias Correction in Saupe Tensor Estimation
---
Introduction
============
Background
----------
The residual dipolar couplings (RDC) of molecules can be measured when the molecule ensemble in solution exhibits partial alignment with the magnetic field in a nuclear magnetic resonance (NMR) experiment. Due to the $r^{-3}$ dependence, such effects can be measured with high precision and provides accurate alignment information of a specific single or multiple bond vector to the magnetic field. This is in contrast to the Nuclear Overhauser Effect (NOE) which scales as $r^{-6}$. Hence the importance of RDC in obtaining high quality protein structures and studying molecular dynamics has increased considerably over the last decade. For a detailed survey of RDC and its applications we refer readers to [@lipsitz2004rdc; @blackledge2005survey; @tolman2006rdc; @salmon20154539]. We first give a brief summary of RDC. Let $v_{nm}$ be the unit vector denoting the direction of the vector between nuclei $n$ and $m$. Let $b$ be the unit vector denoting the direction of the magnetic field. The RDC $D_{nm}$ due to the interaction between nuclei $n$ and $m$ is $$D_{nm} = D_{nm}^{max} \left \langle \frac{3 (b^T v_{nm})^2 -1}{2} \right \rangle_{t,e}.$$ $D_{nm}^{max}$ is a constant depending on the gyromagnetic ratios $\gamma_n,\gamma_m$ of the two nuclei, bond length $r_{nm}$, and Planck’s constant $h$ as $$D_{nm}^{max} = -\frac{\gamma_n \gamma_m h}{2 \pi^2 r_{nm}^3},$$ and $\langle\ \cdot \rangle_{t,e}$ denotes the ensemble and time averaging operator. As presented, RDC depends on the relative angle between the magnetic field and the bond. Extracting such angular information from RDC complements NOE and other measurements for determining the molecular structure. It is conventional to interpret the RDC measurement in the molecular frame. More precisely here, for this analysis of bias, we treat the molecule as being static in some coordinate system, and the magnetic field direction being a time and sample varying vector. In this case the RDC becomes $$\label{RDC}
D_{nm} = D_{nm}^{max} v_{nm}^T S v_{nm},$$ where the Saupe tensor $S$ is defined as where the Saupe tensor $S$ [@Saupe1964] is defined as $$\label{Saupe tensor}
S = \frac{1}{2}(3 B - I_3),\qquad B = \left \langle bb^T \right \rangle_{t,e}.$$ $B$ is known as the field tensor and $I_3$ denotes the $3\times 3$ identity matrix. We note that $S$ is symmetric and ${\mathrm{Tr}}(S) = 0$. In order to use RDC for structural refinement of a protein, $S$ is usually either first determined from a proposed structure (known $v_{nm}$) that is similar to the protein under study, or estimated from a distribution of RDCs [@clore19989654491] . To satisfy the assumption that the molecule is static in the molecular frame, a rigid fragment of the known structure has to be selected. $S$ can be determined if the fragment contains sufficient RDC measurements.
Notation
--------
We summarize here the notation that is used throughout the paper. For a $3\times 3$ matrix $A$, we use $A_{ij}$, $i,j = x,y,z$ to denote the nine entries of the matrix. When $A$ is symmetric, we denote the eigen-decomposition of $A$ by $$A = U(A) \Lambda(A) U(A)^T,$$ where $U(A)$ is an orthogonal matrix (i.e. $U(A)^T U(A) = U(A) U(A)^T = I_3$) and $\Lambda(A)$ is a diagonal matrix $$\begin{bmatrix} \lambda_x(A) & 0 & 0 \\ 0 & \lambda_y(A) & 0 \\ 0 & 0 & \lambda_z(A) \end{bmatrix}$$ that contains the eigenvalues of $A$ on the diagonal in ascending order. For a matrix $A\in \mathbb{R}^{n\times n}$ we use $$\|A\|_F = \sqrt{\sum_{i,j=1}^n A_{ij}^2}$$ to denote the Frobenius norm of the matrix. For a vector $v$, $v_i$ denotes its $i$-th entry, and $i=1,\ldots,n$ if $v\in \mathbb{R}^n$. In the special case of $v\in \mathbb{R}^3$, we use $v_x,v_y,v_z$ to denote components of the vector $v$. For a matrix $A$, we use $A_i$ to denote its $i$-th column.
Previous approach
-----------------
We review the singular value decomposition (SVD) approach [@losonczi1999order] for estimating the Saupe tensor. $S$ is symmetric and ${\mathrm{Tr}}(S) = 0$, so eqn. (\[RDC\]) can be rewritten as $$\begin{gathered}
\label{RDC for bond nm}
D_{nm}/D_{nm}^{max} = ({v_{nm}}_y^2-{v_{nm}}_x^2) S_{yy} + ({v_{nm}}_z^2-{v_{nm}}_x^2) S_{zz} \cr
+ 2 {v_{nm}}_x {v_{nm}}_y S_{xy} + 2 {v_{nm}}_x {v_{nm}}_z S_{xz} + 2 {v_{nm}}_y {v_{nm}}_z S_{yz}\end{gathered}$$ where ${v_{nm}}_i$, $i=x,y,z$ are the different components of $v_{nm}$ in the molecular frame. We let $d_{nm} = D_{nm}/D_{nm}^{max}$, which are the RDC measurements. When there are $M$ RDC measurements, eq. (\[RDC for bond nm\]) results in $M$ linear equations in five unknowns ($S_{yy},S_{zz},S_{xy},S_{xz}$ and $S_{yz}$), that can be written in matrix form as $$\label{ls prestegard}
A s = d,\qquad s = \begin{bmatrix} S_{yy}\\S_{zz}\\S_{xy}\\S_{xz}\\S_{yz} \end{bmatrix} \in \mathbb{R}^5, \qquad d =\begin{bmatrix} d_{n_1m_1}\\ \vdots \\d_{n_M m_M} \end{bmatrix} \in \mathbb{R}^M,$$ and $A\in \mathbb{R}^{M\times 5}$.
Let the SVD of matrix $A$ be $$A = U\Sigma V^T,$$ where $U \in \mathbb{R}^{M\times 5}$ is a column orthogonal matrix (i.e. $U^T U = I_5$), $V\in \mathbb{R}^{5\times 5}$ is an orthogonal matrix, and $\Sigma \in \mathbb{R}^{5\times 5}$ is a positive diagonal matrix. We assume that $M\geq5$ and that $A$ has full rank for otherwise there is no unique solution to the linear system (\[ls prestegard\]). The estimator of the Saupe tensor entries $s$ proposed in [@losonczi1999order] is $$\label{SVD method}
\hat s = V \Sigma^{-1} U^T d.$$ This is equivalent to the ordinary least squares (OLS) solution to the linear system (\[ls prestegard\]), given by $$\label{OLS estimator}
\hat s = (A^T A)^{-1} A^T d.$$ For this reason, we will refer to this SVD method for Saupe tensor estimation as the OLS method. The computational aspects of employing the expressions in (\[SVD method\]) and (\[OLS estimator\]) are discussed in [@wirz2015fitting]. Notice that the Saupe tensor estimator given by (\[SVD method\]) and (\[OLS estimator\]), denoted $\hat S$, is the solution to the optimization problem $$\underset{S}{\min} \sum_{i=1}^M \vert d_{n_im_i} - v_{n_im_i}^T S v_{n_im_i} \vert^2 \quad \text{such that} \ S\ \text{is symmetric},\ {\mathrm{Tr}}(S) = 0.$$ As such, the OLS estimator is also the maximum likelihood estimator when the error on $d_{nm}$ is assumed to be white Gaussian noise.
Although the SVD procedure ensures ${\mathrm{Tr}}(S) = 0$ and $S$ symmetric, it does not ensure that $S$ is obligatorily derived from a specific linear combination of the field tensor and the identity matrix as expressed in eqn. (\[Saupe tensor\]). In contrast, we can use a positive semidefinite matrix description of the field tensor $B$ to exactly characterize $S$. A matrix is positive semidefinite (PSD) if it is symmetric and has nonnegative eigenvalues. The field tensor $B$ can be characterized by the following observation:
\[observtion:sdp\] $B = \langle b b^T \rangle_{t,e}$ where $b$ is a unit vector $\Leftrightarrow$ $B$ is positive semidefinite and ${\mathrm{Tr}}(B) = 1$.
This follows from the convexity of both the set of PSD matrices and the set of unit trace matrices. A set is convex if and only if any weighted average of the elements in the set belongs to the set. Since for any unit vector $b$, $bb^T$ is PSD and ${\mathrm{Tr}}(bb^T) = {\mathrm{Tr}}(b^T b) = 1$, the time and ensemble average of such matrices is PSD with unit trace. Using this observation, the set of physical Saupe tensors can be characterized as $$\mathcal{S} = \left\{S = \frac{1}{2}(3 B - I_3)\,\vert\, B\ \text{is PSD}, {\mathrm{Tr}}(B) = 1\right\}.$$ However, since $B= (1/3)(2 S + I_3)$, when Saupe tensor $S$ has small entries it is dominated by $I_3$ and $B$ is always positive semidefinite. This is often the case in practice, hence the SVD method suffices for Saupe tensor estimation at this approximation. We note that should we ever need to estimate $S$ with large entries of magnitude around $10^{-1}$, we can solve $$\begin{aligned}
\label{Saupe SDP}
&\underset{S\in\mathcal{S}}{\min}& \sum_{i=1}^M \vert d_{n_im_i} - v_{n_im_i}^T S v_{n_im_i} \vert^2,\end{aligned}$$ using semidefinite programming toolboxes available, e.g., in `CVX` [@cvx] so that the derived Saupe tensor remains physically reasonable.
An Alternative Approach
-----------------------
Here, we illustrate how the noise on the bond vectors $v_{nm}$ leads to bias in the OLS estimation of Saupe tensor parameters. We call such type of noise structural noise. In particular, we consider the situation when using noisy template structures differing from the true structure of the molecule for Saupe tensor fitting. This situation may arise when using homologous structure in Saupe tensor fitting, or when the protein of interest has small conformation changes due to the dynamic nature of protein in solution. Our simulation consisted of adding modest noise to the backbone torsion angles of the protein backbone. This results in the magnitude of the estimated Saupe tensor eigenvalues being typically smaller than their true value, as demonstrated in Fig. \[fig:RMS\_vs\_sigma\]. Our observation corroborates with the simulation results reported in [@zweckstetter2002evaluation], in which independently and identically distributed (i.i.d.) random noise [@wasserman2013all] is added to each bond vector instead. In linear regression, such decrease in magnitude of the estimator in the presence of noise on the regressor is commonly known as *attenuation* [@carroll2006measurement]. While the focus of [@zweckstetter2002evaluation] is mainly to use Monte Carlo simulation to evaluate the uncertainty of estimated alignment magnitude and rhombicity, we here focus on using it to correct the attenuation effect in the OLS Saupe tensor eigenvalues estimator. The method we propose bears similarity with the statistical method *simulation extrapolation* (SIMEX) [@cook1994simex; @stefanski1995simex] that is frequently used to correct for the attenuation effect. Typically this type of methods are parametric and require noise variance as input. We show that an estimator of the noise magnitude can be obtained from the root mean square (RMS) of the residual of OLS estimator. We further demonstrate the usefulness of removing such bias when estimating the Saupe tensor eigenvalue from homology fragments of ubiquitin, using RDCs measured in two different alignment medias. We note that there are other approaches to improve the estimation of Saupe tensor in the presence of structural noise by studying local bond orientations using multiple alignment media [@meiler2001mfa; @tolman2002didc; @meirovitch2012standard; @sabo2014orium]. However here, we seek to remove the bias in the Saupe tensor eigenvalues in a single alignment media, when multiple Saupe tensor estimates is available from a collection of predetermined molecular fragments.
Method {#section:method}
======
We now introduce a Monte Carlo method for correcting the bias in the eigenvalues of the OLS estimator arising from structural noise of backbone torsion angles. For a protein with $N+1$ peptide planes, we assume the $\{\phi_i,\psi_i\}_{i=1}^N$ torsion angles fully determine the backbone conformation, i.e. variations of $\{\omega_i\}_{i=1}^N$ are minimal. The template structure’s torsion angles $\phi_i^t, \psi_i^t$’s are related to the true structure via $$\label{torsion angle noise}
\phi_i^t = \phi_i + \sigma \alpha_i,\qquad \psi_i^t = \psi_i + \sigma \beta_i,\quad i=1,\ldots,N$$ where $\alpha_i, \beta_i$’s are i.i.d. random normal variables with mean 0 and variance 1, and $\sigma$ is the level of noise on the torsion angles. Henceforth for a variable $\theta$, we make explicit the dependence on the torsion angles and noise by writing $\theta$ as $\theta(\phi_i^t,\psi_i^t)$. We also assume that the normalized dipolar coupling $d$ is noiseless, i.e. $d = A(\phi_i,\psi_i) s$, where $s$ corresponds to the entries of the “ground truth” Saupe tensor $S$. The validity of this assumption is discussed below in section \[section:add noise\]. Our method consists of the following steps:
\(1) Compute $$\hat s(\phi_i^t,\psi_i^t) = (A(\phi_i^t,\psi_i^t)^T A(\phi_i^t,\psi_i^t))^{-1} A(\phi_i^t,\psi_i^t)^T d$$.
\(2) Generate $n_1$ copies of $A_{\text{sim}} = A(\phi_i^t + \sigma \alpha_i, \psi_i^t + \sigma \beta_i)$ by adding i.i.d. Gaussian noise with variance $\sigma^2$ to the torsion angles of the template structure.
\(3) Find $$\hat s_{\text{sim}} = \hat s(\phi_i^t + \sigma \alpha_i, \psi_i^t + \sigma \beta_i) = (A^T_{\text{sim}} A_{\text{sim}})^{-1} A_{\text{sim}}^T d.$$
\(4) Let $\hat S$ and $\hat S_{\text{sim}}$ be the Saupe tensor estimators corresponding to $\hat s$ and $\hat s_{\text{sim}}$. Let $$\widehat{\text{Bias}} = \langle \Lambda(\hat S_{\text{sim}}) \rangle_{\text{sim}} - \Lambda(\hat S)$$ denote the bias estimate for the eigenvalues of the OLS estimator $\hat S$, where $\langle \cdot \rangle_{\text{sim}}$ denotes the averaging over $n_1$ simulated template structures. We propose deriving $\Lambda$ (Eqn 5) from $$\tilde \Lambda = \Lambda(\hat S) - \widehat{\text{Bias}} = 2 \Lambda(\hat S) - \langle \Lambda(\hat S_{\text{sim}}) \rangle_{\text{sim}}$$ as an estimator with less bias.
The rationale relies on the notion that upon adding noise of similar magnitude to the linear system (\[ls prestegard\]), the eigenvalues of the OLS estimator for the simulated samples should be biased away from $\Lambda(\hat S)$ by an amount similar to the difference between $\Lambda(\hat S)$ and the true $\Lambda(S)$ This is also the intuition behind twicing [@RN5009; @RN5006], and related bootstrapped [@efron1994bootstrap] biased reduced estimators. Alternatively, one can understand this procedure from the viewpoint of the SIMEX technique [@cook1994simex] for correcting bias resulting from regressor noise. Under the SIMEX estimation framework one would simulate $A_{\text{sim}} = A(\phi_i^t + k\sigma \alpha_i, \psi_i^t + k\sigma \beta_i)$ with noise magnitudes of $k \sigma$ for various positive $k$ to find out the dependency of $\Lambda(\hat S_{\text{sim}})$ on $k$. The $k=0$ point corresponds to the case when no additional simulated noise is added, i.e. when the eigenvalue estimator is $\Lambda(\hat S)$. From the extrapolation of the relation between $\Lambda(\hat S_{\text{sim}})$ and $k$ one can obtain a debiased estimator at $k=-1$. Our method corresponds to the special case of SIMEX where we only add simulated noise with magnitude $k \sigma$ where $k=1$. Our numerical results shows that this suffices for the application of Saupe tensor eigenvalue estimation.
Estimating $\sigma$ {#section:sigma estimate}
-------------------
We note that there is a general caveat when using any parametric Monte Carlo method, in that it requires knowledge of the noise magnitude $\sigma$. Let the residual of the OLS estimator be defined as $$r \equiv d - A \hat s.$$ In the simple case when additive noise with variance $\sigma_{\text{add}}^2$ is added to the normalized dipolar couplings $d$, and $A$ has no structural noise, i.e. $A = A(\phi_i,\psi_i)$, the dependence between the RMS of the residual, denoted $\text{RMS}(r)$ and the noise magnitude can be readily calculated. In particular, an unbiased estimator of $\sigma_\text{add}^2$ is given by [@gross2003linear] $$\widehat{ \sigma_{\text{add}}^2} = \frac{M}{M-5} \text{RMS}(r)^2.$$ where $M$ is the number of linear equations. Now in the case when there is noise on the design matrix $A = A(\phi_i^t,\psi_i^t)$ due to noise on the torsion angles (\[torsion angle noise\]), we show that there exists a linear dependence of $\text{RMS}(r)$ on $\sigma$. We define $A_0 = A(\phi_i,\psi_i)$, and $A(\phi_i^t,\psi_i^t) = A_0 + E$. In this notation, normalized RDC $d = A_0 s$. Then $$\begin{aligned}
\|r\|^2_2 &=& \|d - A \hat s \|_2^2\cr
&=&\|A_0 s - A (A^T A)^{-1} A^T (A_0 s)\|_2^2\cr
&=&s^T A_0^T (I_M - A (A^T A)^{-1} A^T) A_0 s.\end{aligned}$$ The second equality follows from the fact that $I_M - A (A^T A)^{-1} A^T$ is a projection matrix. From $$\begin{aligned}
&\ &A_0^T (I_M - A (A^T A)^{-1} A^T) A_0\cr
&=& A_0^T A_0 - (A-E)^T A (A^T A)^{-1} A^T (A-E) \cr
&=&A_0^T A_0 - A^T A + E^T A + A^T E - E^T A (A^T A)^{-1} A^T E\cr
&=&A_0^T A_0 - (A-E)^T (A-E) +E^T E - E^T A (A^T A)^{-1} A^T E\cr
&=&E^T (I_M-A (A^T A)^{-1} A^T) E,\end{aligned}$$ we get $$\begin{aligned}
\label{RMS residual}
\|r\|^2_2 &=& s^T E^T (I_M-A (A^T A)^{-1} A^T) E s \cr
&\approx& s^T E^T (I_M-A_0 (A_0^T A_0)^{-1} A_0^T) E s \cr
&=& s^T E^T P E s\end{aligned}$$ where $P = I_M-A_0 (A_0^T A_0)^{-1} A_0^T$ is a projection operator projecting vectors in $\mathbb{R}^M$ to $\mathbb{R}^{M-5}$. We drop the terms involving entries of $E$ raised to the power greater than 2 to obtain the approximation in (\[RMS residual\]). Using Taylor expansion, $$\begin{aligned}
E_{ij} &=& A_{ij} - {A_0}_{ij}\cr
&\approx& \sum_{k=1}^N \frac{\partial A_{ij}(\phi_k^t,\psi_k^t)}{\partial \phi_k^t}\bigg\vert_{\phi_k,\psi_k} \sigma \alpha_k + \frac{\partial A_{ij}(\phi_k^t,\psi_k^t)}{\partial \psi_k^t}\bigg\vert_{\phi_k,\psi_k} \sigma \beta_k\cr
&=& F_{ij} \sigma.\end{aligned}$$ Plugging this into (\[RMS residual\]), it is clear that $\|r\|^2_2$ depends linearly on $\sigma^2$ and $$\label{Average RMS}
\langle \text{RMS}(r)^2 \rangle_{\alpha_i,\beta_i}\approx \frac{1}{M}\langle s^T F^T P F s \rangle_{\alpha_i,\beta_i} \sigma^2$$ in the small noise regime. We therefore use $$\hat \sigma = \sqrt{\frac{M}{ s^T F^T P F s}} \text{RMS}(r)$$ as the approximate noise magnitude when using the Monte Carlo method for bias reduction. Although we do not have the parameters $s, F$ and $P$ derived from the ground truth Saupe tensor and conformations, we can use $\hat s$ as surrogate of $s$, and use the noisy structure to derive an approximation of $F$ and $P$.
![Plot of the eigenvalues of the OLS estimator $\hat S$ normalized by the eigenvalues of $S$ v.s. $\sigma$. Increasing the noise level biases the eigenvalues towards zero. A fragment of ubiquitin composed of 7 peptide planes (residue 1-8) and a specific Saupe tensor $S$ is used for the simulation and each point in the plot is computed from 200 different realizations of $\alpha_i,\beta_i$’s. \] []{data-label="fig:RMS_vs_sigma"}](figures/Bias.eps){width="90.00000%"}
Numerical results
=================
We first demonstrate that $\hat \sigma$ obtained through the method described in section \[section:sigma estimate\] is a good estimate of $\sigma$. For simulation purposes, we use a segment of ubiquitin with seven peptide planes (residue 1-8) containing 21 $N-H$, $C-CA$ and $C-N$ bonds. We note that in all the simulations, we do not consider the RDC of the nonexistent $N-H$ bond for proline. In Fig. \[fig:sigma estimate\](left), we plot $\hat \sigma$ v.s. $\sigma$. The simulation shows a close agreement between $\hat \sigma$ and $\sigma$, especially when the angular noise is less than 12 degrees.
We next show that the SIMEX-like method proposed in section \[section:method\] is able to reduce the bias in eigenvalue estimation, where the bias of an estimator $\hat \theta$ of parameter $\theta$ is defined to be $$\text{Bias}(\hat \theta) = \langle \hat \theta \rangle - \theta.$$ $\langle \cdot \rangle$ denotes averaging over the distribution of data. For this simulation, we use a specific ground truth Saupe tensor and the aforementioned ubiquitin fragment to generate precise RDC measurements. From the fragment, 200 realizations of noisy conformation are generated with $\sigma=20^\circ$. To obtain $\tilde \Lambda$, we set $n_1 = 8000$ when simulating $A_b$ in step (2) of the Monte Carlo procedure. In Fig. \[fig:sigma estimate\](right), we see that the values of $\langle \tilde \Lambda \rangle_{\alpha_i,\beta_i}$ (Red dotted line) obtained from averaging over 200 samples are almost the same as the eigenvalues of $S$ (Black line), while there is a clear bias in the estimator $\Lambda(\hat S)$ (Blue dotted line).
![[**Above:**]{} Plot of $\hat \sigma$ v.s. $\sigma$. For a given noise level $\sigma$, $\hat \sigma$ is averaged over 200 different realizations of $\alpha_i,\beta_i$’s. [**Below:**]{} Histograms of the diagonal entries of $\Lambda(\hat S)$ and $\tilde \Lambda$ obtained from 200 fragment conformations with $20^\circ$ noise on the torsion angles. The values of $\Lambda(S), \langle \Lambda(\hat S)\rangle_{\alpha_i,\beta_i}$ and $\langle \tilde \Lambda \rangle_{\alpha_i,\beta_i}$ are denoted by black, blue and red line respectively.[]{data-label="fig:sigma estimate"}](figures/sigma_estimate.eps "fig:"){width="95.00000%"} ![[**Above:**]{} Plot of $\hat \sigma$ v.s. $\sigma$. For a given noise level $\sigma$, $\hat \sigma$ is averaged over 200 different realizations of $\alpha_i,\beta_i$’s. [**Below:**]{} Histograms of the diagonal entries of $\Lambda(\hat S)$ and $\tilde \Lambda$ obtained from 200 fragment conformations with $20^\circ$ noise on the torsion angles. The values of $\Lambda(S), \langle \Lambda(\hat S)\rangle_{\alpha_i,\beta_i}$ and $\langle \tilde \Lambda \rangle_{\alpha_i,\beta_i}$ are denoted by black, blue and red line respectively.[]{data-label="fig:sigma estimate"}](figures/bias_hist.eps "fig:"){width="95.00000%"}
Estimation of Saupe tensor eigenvalues from multiple molecular fragments
------------------------------------------------------------------------
While the proposed eigenvalue estimator $\tilde \Lambda$ has less bias, this does not obligate that $\tilde \Lambda$ has a lower mean squared error (MSE).This can be understood from the *bias-variance decomposition*, which is a classical way in statistics to decompose the MSE of an estimator $\hat \theta$ [@wasserman2013all]. The MSE of an estimator $\hat \theta$ admits the following decomposition $$\begin{aligned}
\text{MSE}(\hat \theta) &=& \langle (\theta - \hat \theta)^2 \rangle \cr
&=& \langle (\theta - \langle \hat \theta \rangle + \langle \hat \theta \rangle -\hat \theta)^2 \rangle\cr
&=& \text{Bias}(\hat \theta)^2 + \text{Var}(\hat \theta) + 2(\theta - \langle \hat \theta \rangle )\langle \langle \hat \theta \rangle -\hat \theta \rangle\cr
&=& \text{Bias}(\hat \theta)^2 + \text{Var}(\hat \theta)\end{aligned}$$ $\text{Var}(\hat \theta)$ denotes the variance of $\hat \theta$. Although we achieve less bias with the estimator $\tilde \Lambda$, we pay the price of having larger variance due to bias estimation involved in obtaining $\tilde \Lambda$. This increase in variance can lead to $\tilde \Lambda$ having higher MSE than $\Lambda(\hat S)$. From this point of view, when estimating the Saupe tensor eigenvalue using a single template fragment, the Monte Carlo method for debiasing may seem unnecessary or even disadvantageous. However, when multiple template fragments are available, the average of $\tilde \Lambda$ over these fragments, denoted $\tilde \Lambda_{\text{ave}}$, enjoys variance reduction proportional to the number of fragments. Therefore in the case when there are many fragments, it is worth paying the price of increased variance because the systematic bias error cannot be reduced via averaging. In the rest of the section, we use $\Lambda_{\text{ave}}(\hat S)$ to denote the average of $\Lambda(\hat S)$ over multiple fragments.
We now demonstrate the usefulness of our method under the setting of Molecular Fragment Replacement (MFR) approach [@delaglio2000mfr; @kontaxis2005molecular]. When RDCs are measured in two different alignment medias for a protein of unknown structure, the MFR method can construct its structure by combining short homologous fragments obtained from chemical shift and dipolar homology database mining. Typically for every protein fragment of seven residues, 10 homologous structures are searched based on the similarity of chemical shifts and the goodness of Saupe tensor fit to the observed RDC. When OLS is used to fit the Saupe tensor with design matrix $A$ constructed from homologous structures, one can average all OLS eigenvalue estimated to obtain improved estimators of the parameters such as alignment magnitude and rhombicity that depend on the eigenvalues [@kontaxis2005molecular]. These parameters can in turn be used in a simulated annealing procedure such as XPLOR-NIH [@schwieters2003xplor] to refine the structure.
We first use synthetic data to demonstrate our method. We generate 12 random Saupe tensors, by sampling two eigenvalues from the uniform distribution on $[-10^{-3},0]$ and $[0,10^{-3}]$ respectively, and extract the third eigenvalue by requiring $\Lambda(S)_{xx}+\Lambda(S)_{yy}+\Lambda(S)_{zz}=0$. The orthogonal matrix $U(S)$ is sampled uniformly from the group of $3\times 3$ orthogonal matrices, by computing the orthogonal factor in the polar decomposition of a $3 \times 3$ Gaussian random matrix [@blower2009random]. After obtaining the RDC $d_{nm}$’s from the clean structure and the ground truth Saupe tensor, under each simulated alignment condition we add structural noise of magnitude $\sigma$ to every fragment of seven peptide planes of the ubiquitin structure obtained from X-ray crystallography (PDB ID 1UBQ). We only consider the first 71 residues of the 76 residues of ubiquitin, as there are few RDC reported for the last five residues of ubiquitin. This gives a total of 64 fragments. We evaluate the estimators of the Saupe tensor eigenvalues $\Lambda_{\text{ave}}(\hat S)$ and $\tilde \Lambda_{\text{ave}}$ computed from the average of $\Lambda(\hat S)$ and $\tilde \Lambda$ of all fragments, by comparing their fractional errors averaged over the 12 different Saupe tensors and torsion angle noise realizations in Fig. \[fig:MFR synthetic\]. The fractional error is defined as $$\frac{\|\Lambda_{\text{ave}}(\hat S) - \Lambda(S)\|_F}{\|\Lambda(S)\|_F}\quad \text{and}\quad \frac{\|\tilde \Lambda_{\text{ave}} - \Lambda(S)\|_F}{\|\Lambda(S)\|_F}.$$ In this simulation, the fractional error of $\Lambda_{\text{ave}}(\hat S)$ is at least three times larger than $\tilde \Lambda_{\text{ave}}$.
![Plot of the fractional error of $\Lambda_{\text{ave}}(\hat S)$ and $\tilde \Lambda_{\text{ave}}$ v.s. $\sigma$. Each data point is averaged over 12 different Saupe tensor and noise realizations for 1UBQ. The plot shows a clear advantage of the bias reduced estimator over the OLS estimator.[]{data-label="fig:MFR synthetic"}](figures/synthetic_1UBQ.eps){width="90.00000%"}
We finally apply this method to estimate the Saupe tensor of ubiquitin in two different alignment medias using the experimental RDC data in [@cornilescu19981D3Z]. From 600 homologous structures returned by MFR homology search, each containing seven residues, we obtain 600 Saupe tensor estimates using OLS. Since we expect our method to have a significant effect for fragments severely corrupted by structural noise, we average the fragments with residual RMS *above* a certain threshold and plot $\Lambda_{\text{ave}}(\hat S)$ and $\tilde \Lambda_{\text{ave}}$ normalized by $\Lambda(S)$ v.s. RMS thresholds. To get an estimated (and approximate) ground truth Saupe tensor $S$, we use the high resolution ubiquitin structure 1UBQ obtained from X-ray crystallography [@vijay19871UBQ] to fit the RDC data. We demonstrate the results in Fig. \[fig:MFR real\]. Other than the estimators for $\Lambda(S)_{yy}$ of the second alignment media which has a large percent error due to the relatively small magnitude of $\Lambda(S)_{yy}$, $\tilde \Lambda_{\text{ave}}$ typically achieves 0.9 of the ground truth value, whereas $\Lambda_{\text{ave}}(\hat S)$ can shrink to 0.8 of the value of $\Lambda(S)$ when only the fragments of high RMS are used in averaging. We therefore recommend the use of our proposed bias removing method when estimating eigenvalues from multiple noisy fragments.
![Plot of the eigenvalue estimators normalized by $\Lambda(S)$ v.s. residual RMS thresholds. Estimators are obtained from experimental RDC measurements in two different alignment medias. While the magnitude of $\tilde \Lambda_{\text{ave}}$ (Red curves) and $\Lambda_{\text{ave}}(\hat S)$ (Blue dotted curves) both decrease as low quality (high RMS) fragments are solely used in averaging, $\tilde \Lambda_{\text{ave}}$ in general is within 90% of the ground truth value but $\Lambda(S)$ drops to 80% of $\Lambda_{\text{ave}}(\hat S)$. The value of $\Lambda(S)$ for both alignment medias are indicated in the plot title.[]{data-label="fig:MFR real"}](figures/mfr.eps){width="95.00000%"}
Effect of additive noise on Saupe tensor estimation {#section:add noise}
===================================================
So far we have been neglecting the presence of additive noise on $d_{nm}$, which is considered by [@losonczi1999order]. We define the noisy RDC measurements corrupted by additive noise as $$\label{additive noise}
d_\text{add} = d + \sigma_\text{add} \varepsilon$$ where entries of the column vector $\varepsilon$ are i.i.d random variables with mean zero. In this section, we show using perturbation theory that this type of additive noise biases the eigenvalue magnitude positively, therefore it cannot explain the magnitude shrinkage we see when fitting the Saupe tensor to real RDC data (Fig. \[fig:MFR real\]). Moreover, the order of magnitude of this positive bias is not sufficient to explain the error between $\hat S$ and $S$. This has been noted by the authors of [@losonczi1999order] that in order to account for the size of the OLS misfit, an uncertainly of 2-3 Hz for the RDC measurements is required although the experimental uncertainty is only about 0.2-0.5 Hz. This is the reason why in this paper we focus on removing the bias that arises from structural noise.
Let $$S = U(S) \Lambda(S) U(S)^T$$ be the eigendecomposition of $S$. Assuming the eigenvalues of $S$ are nondegenerate, the second order perturbation theory [@landau2013quantum] states $$\begin{aligned}
\lambda_j(\hat S) &\approx& \lambda_j({S}) + U({S})_j^T (\hat S - S) U({S})_j \cr
&\ & + \sum_{\substack{k=x,y,z,\\ k\neq j}} \frac{(U({S})_k^T (\hat S - S) U({S})_j)^2}{\lambda_{j}({S})-\lambda_{k}({S})}.\end{aligned}$$ Averaging the perturbation expansion over the distribution of $\varepsilon$, we get $$\begin{aligned}
\label{bias PT}
\lambda_j(\hat S) \approx \lambda_j({S}) + \sum_{\substack{k=x,y,z,\\ k\neq j}} \frac{\langle(U({S})_k^T (\hat S - S) U({S})_j)^2\rangle_\varepsilon}{\lambda_{j}({S})-\lambda_{k}({S})}.\end{aligned}$$ Here we use the fact that $\langle \hat S - S \rangle_\varepsilon = 0$ since $$\langle \hat s - s \rangle_\varepsilon= \langle(A^TA)^{-1} A^T (A s + \varepsilon) - s \rangle_\varepsilon = \langle (A^TA)^{-1} A^T\varepsilon\rangle_\varepsilon = 0$$ The expression in (\[bias PT\]) reveals that in the presence of noise, the largest eigenvalue of $\hat S$ is always greater than the largest eigenvalue of ${S}$, while the smallest eigenvalue behaves in the exact opposite manner. Such effect of bias of pushing the extreme eigenvalues outwards is also commonly seen in the context of estimating the extreme eigenvalues of covariance matrices [@schafer2005shrinkage].
For this type of bias we now give an estimate of its order of magnitude. First we bound the numerator in the second order correction term in (\[bias PT\]): $$\begin{aligned}
\label{numerator bound}
(U({S})_k^T (\hat S - S) U({S})_j)^2 &=& {\mathrm{Tr}}((\hat S - S)U({S})_j U({S})_k^T)^2\cr
&\leq& \|\hat S - S\|_F^2 \|U({S})_j U({S})_k^T\|_F^2\cr
&\leq& 3\|\hat s - s\|_2^2.\end{aligned}$$ The first inequality results from Cauchy-Schwarz inequality, and the second inequality relies on the fact that $\|U({S})_j U({S})_k^T\|_F = 1$ and $\|\hat S - S\|_F^2 \leq 3\|\hat s - s\|_2^2$, which can be verified easily. It is a classical result [@gross2003linear] that the OLS estimator has covariance matrix $$\langle (\hat s - s)(\hat s - s)^T \rangle_\varepsilon= \sigma_\text{add}^2 (A^T A)^{-1},$$ therefore $$\label{s variance}
\langle \|\hat s - s\|_2^2 \rangle_\varepsilon= \sigma_\text{add}^2 {\mathrm{Tr}}((A^T A)^{-1}).$$
Using (\[s variance\]), (\[numerator bound\]) we obtain an upperbound for the bias in (\[bias PT\]). Taking $\lambda_z(\hat S)$ for example: $$\label{actual estimate}
\lambda_z(\hat S) - \lambda_z({S}) \lesssim \frac{3{\mathrm{Tr}}((A^T A)^{-1})}{\lambda_z(S)-\lambda_y(S)} \sigma_\text{add}^2$$ We now give an estimate of the order of magnitude of the bias. Since the magnitude of the extreme eigenvalues of the Saupe tensor is around $10^{-3}$, for example for the two RDC datasets acquired for ubiquitin, we simply assume $\lambda_z(S)-\lambda_y(S) \sim 10^{-4}$. The typical experimental uncertainty for RDC measurements is about 0.2 Hz - 0.5 Hz, and the dipolar coupling constant $D_{nm}^\text{max}$ for e.g. $N-H$ bonds, is about 23 kHz, therefore the noise magnitude $\sigma_\text{add}$ of the additive noise on the normalized dipolar coupling is about $0.5/(23 \times 10^3) \approx two \times 10^{-5}$. For ubiquitin, the average value of ${\mathrm{Tr}}((A^T A)^{-1})$ for fragments containing seven peptide planes is about 1.35. Using these numbers in (\[actual estimate\]), we get $$\lambda_z(\hat S) - \lambda_z({S}) \lesssim 1.6\times10^{-5},$$ which amounts to 1-2% error when $\lambda_z(S)\sim10^{-3}$. This cannot explain the 10% or larger error in fitting Saupe tensor to real RDC datasets using homology fragments in the previous section.
We present a simulation to illustrate the bias in OLS eigenvalues estimation in the presence of additive noise. We use the Saupe tensor eigenvalues for ubiquitin in the first alignment media presented in Fig. \[fig:MFR real\], and a ubiquitin fragment consisting of seven peptide planes (residue 1-8) for this simulation. We generate noisy datasets using the noise model $$d_\text{add} = As + \sigma_\text{add} \varepsilon.$$ For every noise level, we average $\Lambda(\hat S)$ normalized by $\Lambda(S)$ over 500 different realizations of $s$ and $\varepsilon$ where entries of $\varepsilon$ are i.i.d. random normal variables. The different realization of $s$ are generated from $S = U(S) \Lambda(S) U(S)^T$ where $\Lambda(S)$ is fixed but $U(S)$ is sampled uniformly from the orthogonal group in $\mathbb{R}^3$. We vary the orientation of the Saupe tensor since it is clear from (\[bias PT\]) that the bias depends on $U(S)$. We change $\sigma_\text{add}$ from 0 to 10% of $\lambda_z(S)$ and present the results in Fig. \[fig:Bias add\]. We note again from previous calculations, $\sigma_\text{add}\sim 2 \times 10^{-5}$, which amounts to 2-3% of the $\lambda_z(S) = 0.85\times 10^{-3}$ considered. As shown in the simulation and our crude estimate, such magnitude of noise gives rise to bias error of about 1%. Even in the case of having very noisy RDC (having noise magnitude 10% of $\lambda_z(S)$), the bias error caused by additive noise is around 3%. Whereas in a typical MFR search with torsion angle tolerance being set to $\pm 20^\circ-30^\circ$ [@wu2005mfr], the simulation in Fig. \[fig:RMS\_vs\_sigma\] suggests structural noise can cause bias error sometimes much greater than 10%. Therefore in this paper we focus on removing the bias that arises from structural noise. In the case when accurate template structure is available and the additive noise is a concern, we refer readers to the appendix for the removal of such bias using an analytic expression derived from perturbation theory.
![Plot of the three eigenvalues of the OLS estimator $\hat S$ normalized by the eigenvalues of $S$ v.s. $\sigma_\text{add}$ under noise model (\[additive noise\]). Each point is averaged over 2000 noise and Saupe tensor realizations. Increasing the noise level biases the eigenvalues positively, unlike the case for structural noise. At 10% noise level, the bias is about 3%.[]{data-label="fig:Bias add"}](figures/bias_add.eps){width="95.00000%"}
Conclusion
==========
We observe a negative bias when estimating the Saupe tensor eigenvalues through the classical SVD method, in the presence of structural noise on the template structure due to torsion angle noise. We present a Monte Carlo method that simulates noise on the template structure by perturbing the torsion angles and use the simulated structure to estimate the bias in the eigenvalues. We demonstrate the effectiveness of our method in reducing the error arising from bias when estimating Saupe tensor eigenvalues from multiple protein fragments, which is a natural setting to consider when building protein structure from homologous substructures.
Acknowledgement
===============
The research of AS was partially supported by award R01GM090200 from the NIGMS, by awards FA9550-12-1-0317 and FA9550-13-1-0076 from AFOSR, by award LTR DTD 06-05-2012 from the Simons Foundation, and the Moore Foundation Data Driven Discovery Investigator award.
Appendix
========
Removing bias from additive noise
---------------------------------
Define a linear operator $L: \mathbb{R}^{5} \rightarrow \mathbb{R}^{3\times 3}$ that forms a Saupe tensor $S$ from the vector $s$ as $$\label{linear operator L}
L(s) = \begin{bmatrix} -s(1)-s(2) & s(3) & s(4) \\ s(3) & s(1) & s(5) \\ s(4) & s(5) & s(2) \end{bmatrix},\quad s \in \mathbb{R}^5.$$ For the additive noise model (\[additive noise\]) we have $$\label{S OLS}
\hat S = L(\hat s) = L ((A^T A)^{-1} A^T d_\text{add}) = S + L ((A^T A)^{-1} A^T \varepsilon).$$ We also define the adjoint operator of $L$, $L^*:\mathbb{R}^{3\times3} \rightarrow \mathbb{R}^5$ through the relation $$\label{adjoint L}
{\mathrm{Tr}}(X^T L(y)) = L^*(X)^T y,$$ for every $y\in \mathbb{R}^5$ and $X \in \mathbb{R}^{3\times 3}$. To obtain the form of $L^*$, we let $y\in \{e_1,\ldots,e_5\}$, where $e_i(i)=1$ and $e_i(j) = 0$ if $j\neq i$. Plugging such $y$ into Eq. (\[adjoint L\]), we get $$\begin{aligned}
{L^*(X)}_1 &=& -X_{xx}+X_{yy} \cr
{L^*(X)}_2 &=& -X_{xx}+X_{zz} \cr
{L^*(X)}_3 &=& X_{xy} + X_{yx} \cr
{L^*(X)}_4 &=& X_{xz} + X_{zx} \cr
{L^*(X)}_5 &=& X_{yz} + X_{zy}\end{aligned}$$ Using such notion of the adjoint operator, the perturbation series in (\[bias PT\]) can be written as $$\begin{gathered}
\label{PT Taylor}
\langle \lambda_j(\hat S) \rangle_\varepsilon \approx \lambda_j({S}) + \sum_{\substack{k=x,y,z,\\ k\neq j}} \frac{ \langle ((\hat s - {s})^T L^*(U({S})_k U({S})_j^T))^2 \rangle_\varepsilon}{\lambda_{j}({S})-\lambda_{k}({S})}\\
= \lambda_j({S}) + {\mathrm{Tr}}\bigg[ \bigg(\sum_{\substack{k=x,y,z,\\ k\neq j}} \frac{L^*(U({S})_k U({S})_j^T)L^*(U({S})_k U({S})_j^T)^T}{\lambda_{j}({S})-\lambda_{k}({S})}\bigg) \text{Var}(\hat s)\bigg]\end{gathered}$$ where $$\text{Var}(\hat s) \equiv \langle (\hat s - {s})(\hat s - {s})^T\rangle_\varepsilon = (A^TA)^{-1} \sigma_\text{add}^2.$$ Therefore we can subtract the second order term in (\[PT Taylor\]) to correct for the bias in the eigenvalues. Although we do not know the eigenvectors and eigenvalues of ${S}$, we can replace them with the eigenvectors and eigenvalues $\hat S$. This change will only affect on the higher order terms in the perturbation series.
[^1]: Department of Physics, Princeton University, Princeton, NJ 08544,USA ([[email protected]]{}).
[^2]: Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544, USA ([[email protected]]{}).
[^3]: Department of Biochemistry, Albert Einstein College of Medicine, Bronx, NY 10461, USA ([[email protected]]{}).
| {
"pile_set_name": "ArXiv"
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abstract: |
On corrige deux erreurs dans le papier [@ro]: l’une dans l’étude d’une involution sur les représentations irréductibles non ramifiées d’un groupe semi-simple, l’autre dans la description de représentations du groupe ${\rm GSp}(4)$.
**Abstract**
We correct two errors in the paper [@ro]: the first in the study of an involution on the irreducible unramified representations of a semi-simple group, the second in the description of representations of the group ${\rm
GSp}(4)$.
author:
- |
François Rodier\
Institut de Mathématiques de Luminy – C.N.R.S.\
Marseille – France
title: |
Errata à “Sur les représentations non ramifiées des groupes réductifs $p$-adiques;\
l’exemple de ${\rm GSp}(4)$”
---
Deux erreurs m’ont été signalées dans cet article, la première par Amritanshu Prasad, qui avait utilisé l’énoncé de la proposition 13 dans son papier [@ap1] et qui a dû écrire par la suite un erratum, et la seconde par Laurent Clozel. Je les remercie tous deux de m’avoir signalé ces erreurs.
L’erreur dans la Proposition 13
===============================
La première erreur concerne la proposition 13. Elle a des conséquences sur la description des composants irréductibles des représentations de ${\rm GSp}(4)$ dans le chapitre 6 et dans la remarque finale du chapitre 7, mais ni sur le nombre de ces représentations, ni sur leur multiplicité.
Elle est due à une confusion entre deux notations. La notation $\sgn$ définie dans la section 2.1 comme dénotant un caractère de $k^\times$, donc une application du groupe multiplicatif du corps local non archimédien $k$ dans $\C$, est à ne pas confondre avec la notation $\sgn q_x$ définie en 5.3, où $q_x$ représente un volume, donc un nombre réel.
Dans la démonstration de la proposition 13, l’assertion $q_t=\rho_P(t)^{-2}$ était utilisée pour prouver que $\sgn q_t=1$, alors que $\rho(t)$ n’est pas forcément entier puissance de $q$.
L’assertion de la ligne suivante, obtenue à l’aide du lemme 4, doit s’écrire par conséquent: $$((\hat\pi)_U(t)\circ A)x_U=\sgn q_t\
A(\pi((T_{w_0(t^{-1})})^{-1})x)_U.$$
Elle implique $$(\hat\pi)_U(t)\circ A=\sgn q_t\ \rho_P(t)^{2}
A\circ\pi_U(w_0(t))$$
Ou encore, en remarquant que $\sgn q_t=sgn\ \rho_P^2(t)=\sgn\nu(t)$: $$(\hat\pi)_U(t)\circ A=\sgn\circ\nu(t) \rho_P(t)^{2}
A\circ\pi_U(w_0(t))$$
D’ou l’énoncé corrigé de la proposition 13:
. — La représentation $R(\hat \pi)$ est équivalente à $(\sgn\circ\nu) R(\pi)\circ \Int w_0$ où $\Int w_0$ est l’automorphisme de $T$ défini par $w_0$.
Par la suite cette erreur affecte dans les sections postérieures la description des composants irréductibles du groupe ${\rm GSp}(4)$.
Dans la section 6.2, il faut lire que la représentation $(\sgn\circ\nu)
R(\hat\pi_\chi)$ est composée de $\chi$, avec la multiplicité 2, et de $w_\alpha \chi$ avec la multiplicité 1. Et par conséquent $I(\chi)$ est composée de 4 représentations irréductibles: $\pi_\chi$, $\pi'_\chi$, $(\sgn\circ\nu) \hat\pi_\chi$, $(\sgn\circ\nu) \hat\pi'_\chi$.
Dans la section 6.3: $I(\chi)$ est composée de 2 représentations irréductibles: $\pi_\chi$, $(\sgn\circ\nu) \hat\pi_\chi$.
C’est aussi le même cas dans les sections 6.4 et 6.5.
A la fin du paragraphe 7.2, il faut modifier la remarque finale:
[*Remarque.*]{} — Si $\chi$ est le caractère $t\mapsto|t^{2\alpha+\beta}|^{1/2}$ ou le caractère $t\mapsto\sgn\circ\nu(t)|t^{2\alpha+\beta}|^{1/2}$, alors $I(\chi)$ admet $\sgn\circ\nu\ \hat\pi_\chi$ et $\pi'_\chi$ comme composants tempérés.
Erreurs dans l’énoncé du théorème 2
===================================
Les conditions sur le représentant $x$ dans $X(T)\otimes \R$ du caractère unitaire $\chi_U$ imposées dans la section 7.2 sont traduites maladroitement dans le théorème 2. Voici les corrections.
. — Les représentations irréductibles non ramifiées de $G$ sont les suivantes.
: sans changement;
: $I(\chi)$ pour $\chi(t)=\exp(-2\pi i\mu\val t^{\alpha+\beta})
|t^{\alpha}|^\lambda$ avec $\lambda,\mu\in\R$, $0<\mu<1/2$ et $0<\lambda<1/2$;
: sans changement;
: $I(\chi)$ pour $\chi(t)=\exp(-2\pi i\mu\val t^{2\alpha+\beta})
|t^{\beta}|^\lambda$ avec $\lambda,\mu\in\R$, $0<\mu<1/2$ et $0<\lambda<1/2$;
: sans changement;
: $I(\chi)$ pour $\chi(t)=\sgn t^{\alpha+\beta}\exp(-2\pi i\mu\val t^{\alpha})
|t^{\alpha+\beta}|^\lambda$ avec $\lambda,\mu\in\R$, et $0<\lambda<1/2$;
: sans changement;
: sans changement;
: Les composants de $I(\chi)$ pour $\chi=|t^\alpha|^\lambda |t^\beta|^{1/2}$ avec $1/2\le \lambda\le 1$;
: Les composants de $I(\chi)$ pour $\chi=\sgn\circ\nu|t^\alpha|^\lambda |t^\beta|^{1/2}$ avec $1/2\le
\lambda\le 1$.
[XXXX]{}
, [*Almost unramified automorphic representations for split groups over $\f q(t)$*]{}. J. Algebra, 262(1): 253-261, 2003.
, [*Erratum to “Almost unramified automorphic representations for split groups over $\f q(t)$”*]{}, preprint, disponible sur [http://www.imsc.res.in/\~amri/erratum.pdf]{} .
, [*Sur les représentations non ramifiées des groupes réductifs $p$-adiques; l’exemple de ${\rm GSp}(4)$*]{}, Bull. Soc. Math. Fr. 116 (1988), 15–42.
| {
"pile_set_name": "ArXiv"
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---
abstract: 'An algorithm to simulate full QCD with 3 colours at nonzero chemical potential on the lattice is proposed. The algorithm works for small values of the chemical potential and can be used to extract expectation values of [CPT]{} invariant operators.'
---
[**A proposal for simulating QCD**]{}
[**at finite chemical potential on the lattice**]{}
B. Allés, E. M. Moroni
*Dipartimento di Fisica, Università di Milano-Bicocca*
*and INFN Sezione di Milano, Milano, Italy*
Introduction
============
Understanding the properties of matter at nonzero density and temperature is an essential ingredient to describe the physics of the early universe and the collapse of very massive stars [@rajagopalwnardulli]. Moreover recent heavy ion collision experiments offer a unique laboratory test for the predictions of the theory [@satzgavai].
Effective models [@rajagopalwnardulli; @barroisbailincasalbuoni] predict a rich structure in the temperature–chemical potential $T$–$\mu$ diagram of QCD with several phases. Numerical simulations on the lattice is an adequate technique to study the corresponding phase transitions. The lattice action of QCD at finite $\mu$ has been given in [@hasenfratzk; @kogut]. By using Wilson fermions within a $SU(N)$ invariant gauge theory, this action is [@wilson] $$\begin{aligned}
S_{\rm Wilson}&\equiv& \beta \sum_{P}
\left( 1 - {1\over N} \hbox{Re Tr}\, {P} \right) \nonumber \\
& & +\sum_{\rm flavours} \sum_x a^3 \Big[ \left( am + 4 \lambda\right)
\overline\psi_x \psi_x \nonumber \\
& & \qquad\qquad\qquad - {1\over 2} \sum_\nu\Big(
\left(\lambda - \gamma_\nu\right)
\overline\psi_{x} U_\nu(x) \xi_\nu(\mu) \psi_{x+\hat\nu}
\nonumber \\
& & \qquad\qquad\qquad + \left(\lambda + \gamma_\nu\right)
\overline\psi_{x+\hat\nu} U^\dagger_\nu(x) \xi_\nu(\mu)^{-1} \psi_x
\Big)\Big]\; , \nonumber \\
& & \nonumber \\
\xi_\nu(\mu) &\equiv& 1 + \delta_{4\nu} \left(\exp\left(f(a\mu)\right) - 1\right)\; ,
\label{action}\end{aligned}$$ where the first term is the pure gauge action, $a$ is the lattice spacing, $\beta$ is the inverse bare lattice coupling ($\beta=2N/g_0^2$), ${P}$ stands for the elementary plaquette, $m$ is the fermion mass, $\lambda$ is the Wilson parameter and $\mu$ is the chemical potential. $f(a\mu)$ is an odd function of the chemical potential that satisfies $f(x)= x + O(x^3)$. The simplest choice is $f(x)=x$ although others are possible (see for instance [@bilic] where $f(x)=\hbox{arg tanh} x$). Notice that $\xi_\nu(-\mu)=\xi_\nu(\mu)^{-1}$. Analogous expressions are valid for staggered and naïve fermions.
The expectation value of an operator $O$ is defined by the Feynman path integral $$\begin{aligned}
\langle O \rangle &=& {1\over Z} \int {\cal D}U_\nu {\cal D}\overline\psi_x
{\cal D}\psi_x \; O\,\exp\left(-S_{\rm Wilson}\right)\; , \nonumber \\
Z &\equiv& \int {\cal D}U_\nu {\cal D}\overline\psi_x
{\cal D}\psi_x \; \exp\left(-S_{\rm Wilson}\right)\; ,
\label{vev}\end{aligned}$$ where terms regarding gauge fixing have been skipped as they are inessential for our analysis. After integrating out fermions (we assume that $O$ can be expressed in terms of gluon fields only) we have $$\langle O \rangle = {1\over Z} \int {\cal D}U_\nu\; O\, \det D \;
\exp\left(-S_{\rm g}\right) \; ,
\label{vevO}$$ where $S_{\rm g}$ is the pure gauge action and $D$ is the fermion matrix $$\begin{aligned}
D_{xy}(\mu)&=&(ma + \lambda) \delta_{x,y}\nonumber \\
- {1\over 2}
\negthinspace\negthinspace\negthinspace\negthinspace
&& \negthinspace\negthinspace\negthinspace\negthinspace
\sum_\nu \left(\delta_{x,y+\hat\nu} U^\dagger_\nu(y) \left(\lambda +
\gamma_\nu\right) \xi_\nu(\mu)^{-1} +
\delta_{y,x+\hat\nu} U_\nu(x) \left(\lambda -
\gamma_\nu\right) \xi_\nu(\mu) \right)\; .
\label{D}\end{aligned}$$ Colour, flavour and spinor indices are placed where required. The theory with $N=3$ colours and fermions in the fundamental representation in general yields a complex value for $\det D$ if $\mu\not= 0$. This is a problem because importance sampling in the Monte Carlo integration of Eq.(\[vevO\]) cannot be applied with a complex weight. Several solutions have been proposed that allow to perform the integration in an approximate way or in particular regions of the $T$–$\mu$ diagram: reweighting methods [@fodork], calculation of the canonical partition function from simulations at imaginary chemical potential [@alford], analytical continuations of Taylor expansions in powers of imaginary chemical potential [@delial], responses of several observables to a nonzero small chemical potential [@hands] and fugacity expansions [@barbour]. Here we present another idea that should work for small values of the chemical potential. We will prove that if we restrict our attention to the calculation of expectation values of [CPT]{} invariant operators then the correct Boltzmann weight under the path integral is $\exp\left(-S_{\rm g}\right)\hbox{Re} \det D$.
The Boltzmann weight
====================
The Boltzmann weight in Eq.(\[vevO\]), $\exp\left(-S_{\rm g}\right) \det D$, is a complex number. However the expectation value of any observable represented by the operator $O$ in Eq.(\[vevO\]) must be a real quantity. Imposing that $\langle O\rangle$ be real for any observable $O$ is a strong constraint on the integration measure. Clearly all possible configurations can be gathered in sets such that the contribution to the imaginary part of $\langle O\rangle$ from all configurations in one single set cancels out. We shall assume that all sets are formed by only two configurations and that these two are related by some transformation [S]{}. Firstly we want to find this transformation. Let ${\cal C}$ denote a thermalized arbitrary configuration on the lattice and ${\cal C^{\sf S}}$ the corresponding transformed configuration by the action of [S]{}. We require that [*(i)*]{} the value of the operator $O$ calculated on ${\cal C}$ and ${\cal C^{\sf S}}$ be the same; [*(ii)*]{} the weight under the path integral for ${\cal C^{\sf S}}$ be the complex conjugate of the weight for ${\cal C}$ and [*(iii)*]{} ${\cal C}$ and ${\cal C^{\sf S}}$ have the same Haar measure, $$\begin{aligned}
O[{\cal C}] & \rightarrow & O[{\cal C^{\sf S}}] = O[{\cal C}]\;, \nonumber \\
\det D[{\cal C}] \hbox{e}^{-S_{\rm g}[{\cal C}]}
& \rightarrow & \det D[{\cal C^{\sf S}}] \hbox{e}^{-S_{\rm g}[{\cal C^{\sf S}}]}
= \left(\det D[{\cal C}]\right)^* \hbox{e}^{-S_{\rm g}[{\cal C}]} \; ,
\nonumber \\
{\cal D}U_\nu[{\cal C}] & \rightarrow & {\cal D}U_\nu[{\cal C^{\sf S}}] =
{\cal D}U_\nu[{\cal C}] \; ,
\label{conditions}\end{aligned}$$ where we have explicitely written the dependence on the configuration. We will write this dependence wherever necessary. These conditions are sufficient to guarantee that $\langle O\rangle$ is real.
A simple way to enforce the last constraint in (\[conditions\]) is by imposing that [S]{} is a discrete transformation. ${\cal D}U_\nu$ means $\Pi_x \Pi_\nu \hbox{d}U_\nu(x)$ where each single factor $\hbox{d}U_\nu(x)$ is the Haar measure over the gauge group. By a discrete transformation we mean that [S]{} does not transform each single Haar measure. In fact this would entail a jacobian and the search of the transformed configuration would become more difficult.
The matrix $D_{xy}(\mu)$ has the property $\left(\det D(\mu)\right)^* = \det D(-\mu)$. This stems from the relation $D(\mu)^\dagger=\gamma_5 D(-\mu)\gamma_5$[^1]. Notice that this implies Re det$D$ (Im det$D$) is an even (odd) function of $\mu$. Moreover it is physically clear that applying a charge conjugation operator [C]{} changes the sign of $\mu$. Then we expect that the condition $\det D[{\cal C^{\sf S}}] = \left(\det D[{\cal C}]\right)^*$ can be verified if [S]{} contains the transformation [C]{}. However the above condition is only part of the requirements in (\[conditions\]). In particular we see from the second condition in (\[conditions\]) that the implementation of [C]{} must be such that the pure gluon action $S_{\rm g}$ remains unaltered.
In order to leave $S_{\rm g}$ invariant, the correct transformation should involve some space–time rearrangement besides charge conjugation. A reasonable guess is the [CPT]{} transformation that we define in the following way: if our lattice was $d$ dimensional and the lateral sizes were finite and equal to $L_1$, $L_2$, ..., $L_d$ (in units of lattice spacing) then the [CPT]{} transformation would act in the following way $$\begin{aligned}
U_\nu(x)^{{\sf CPT}} = U_\nu^\dagger(
&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_1 - x_1 + 1,L_1) + 1, \nonumber \\
&& \negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_2 - x_2 + 1,L_2) + 1, \nonumber \\
&& \negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\cdots, \nonumber \\
&& \negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_\nu - x_\nu,L_\nu) + 1, \nonumber \\
&& \negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\cdots, \nonumber \\
&& \negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_d - x_d + 1,L_d) + 1 ) \; ,
\label{Ld}\end{aligned}$$ where mod($x,L$) gives the remainder on integer division of $x$ by $L$. This is the finite volume version of $U_\nu(x)^{\sf CPT}= U^\dagger_\nu(-x)$. This transformation fulfils all conditions in Eq.(\[conditions\]) as a straightforward computation shows.
As an example let us check that $S_{\rm g}$ is invariant under [CPT]{}. It is enough to show that under the action of [CPT]{} every plaquette of the original configuration ${\cal C}$ maps onto one and only one plaquette of ${\cal C}$. Let us denote ${P} (x;\nu,\rho)$ the plaquette starting at the site $x$ and going around through direction $\nu$ and then $\rho$. This is ${P} (x;\nu,\rho)\equiv U_\nu(x) U_\rho(x+\hat\nu) U^\dagger_\nu(x+\hat\rho)
U^\dagger_\rho(x)$. Applying transformation (\[Ld\]) on each factor we obtain $$\begin{aligned}
{P} (x;\nu,\rho) \rightarrow U^\dagger_\nu&&
\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
(\hbox{mod}(L_1 - x_1+1, L_1)+1, \cdots \nonumber \\
&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_\nu - x_\nu, L_\nu)+1, \cdots \nonumber \\
&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_\rho - x_\rho+1, L_\rho)+1, \cdots ) \times \nonumber \\
U^\dagger_\rho&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
(\hbox{mod}(L_1 - x_1+1, L_1)+1, \cdots \nonumber \\
&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_\nu - x_\nu, L_\nu)+1, \cdots \nonumber \\
&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_\rho - x_\rho, L_\rho)+1, \cdots ) \times \nonumber \\
U_\nu&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
(\hbox{mod}(L_1 - x_1+1, L_1)+1, \cdots \nonumber \\
&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_\nu - x_\nu, L_\nu)+1, \cdots \nonumber \\
&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_\rho - x_\rho, L_\rho)+1, \cdots ) \times \nonumber \\
U_\rho&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
(\hbox{mod}(L_1 - x_1+1, L_1)+1, \cdots \nonumber \\
&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_\nu - x_\nu +1, L_\nu)+1, \cdots \nonumber \\
&&\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace
\negthinspace\negthinspace\negthinspace\negthinspace
\hbox{mod}(L_\rho - x_\rho, L_\rho)+1, \cdots ) \; , \nonumber\\\end{aligned}$$ which, by the cyclic property of the trace, is equivalent to the plaquette $$\begin{aligned}
&& {P} (\hbox{mod}(L_1-x_1+1, L_1)+1,\cdots , \hbox{mod}(L_\nu-x_\nu, L_\nu)+1, \cdots ,
\nonumber \\
&&\quad\, \hbox{mod}(L_\rho-x_\rho, L_\rho)+1, \cdots , \hbox{mod}(L_d-x_d+1, L_d)+1 ;
\nu ,\rho)\end{aligned}$$ in the original configuration ${\cal C}$. This is clearly a one–to–one mapping from plaquettes to plaquettes in ${\cal C}$. This completes the proof.
Then a configuration ${\cal C}$ with weight $\det D\exp(-S_{\rm g})$ transforms under [CPT]{} into another configuration ${\cal C}^{{\sf CPT}}$ with weight $\left(\det D\right)^* \exp(-S_{\rm g})$ and the same Haar measure. This means that both configurations have the same odds to be selected by an updating algorithm. If moreover we restrict our interest only to [CPT]{} invariant observables $O$ then we can say that the numerical value $O[{\cal C}]$ appears with a probability proportional to $\exp(-S_{\rm g}[{\cal C}])\left(\det D[{\cal C}] + \left(\det D[{\cal C}]\right)^*\right)$. Barring a factor 2 and dispensing with the [CPT]{} transformed configuration one can just count ${\cal C}$ with the weight $\exp(-S_{\rm g})\hbox{Re}\det D$. This is the main result of our paper.
When we say [CPT]{} invariant operators we mean that the operator $O$ is [CPT]{} invariant configuration by configuration. Most of the operators are [CPT]{} invariant after averaging over configurations (i.e. the corresponding observable is [CPT]{} invariant), but our constraint is stronger than this.
Focusing our attention only on [CPT]{} invariant operators still allows to study several interesting problems. Indeed the average plaquette, chiral condensate, Polyakov loop, etc. can be calculated by evaluating expectation values of [CPT]{} invariant operators. On the other hand space–time valued operators $O(x)$ are in general not permitted because our method requires that the equality $O(x)=O(-x)^\dagger$ holds configuration by configuration which in general is not true. In particular all correlation functions are not admissible.
Are there other discrete transformations such that the imaginary part of the determinant is eliminated within groups of two configurations? We have performed the following test. We discretized a 2–dimensional gauge theory[^2] on a $2^2$ lattice and loaded it with an arbitrary configuration (an arbitrary $SU(3)$ matrix on each of the 8 links). We calculated $\det D$ obtaining a complex number. Then we rearranged the 8 $SU(3)$ matrices in all possible ways (8!=40320) and for each permutation we placed them again on the 8 links and recalculated the determinant. Only the transformation described in (\[Ld\]) yielded the result that satisfies conditions (\[conditions\]) (permutations of the 8 links that can be viewed as spatial or temporal translations led to the same result but they are clearly uninteresting). We conclude that there are no other simple discrete transformations providing the complex conjugate of the original configuration weight.
A note on Monte Carlo simulation
================================
For the importance sampling to work it is necessary that the weight be positive (it must behave as a probability). However Re $\det D$ can take negative values and this fact limits the applicability of the method. By continuity we expect that Re $\det D$ is positive in the majority of configurations when $\mu$ is small enough. This suspicion is confirmed by numerical simulation. In [@toussaint] results from a simulation with 4 flavours of staggered fermions are presented. A $4^4$ lattice is used at $\beta=4.8$ and $am=0.025$. In Fig. 1 we show the fraction of configurations with a positive (negative) weight $N_+/N$ ($N_-/N$) as a function of $a\mu$ ($N\equiv N_++N_-$ is the total number of configurations). It indicates that our method can be used at moderate values of $\mu$. This may include the region in the phase diagram $T$–$\mu$ where present and future heavy ion experiments (RHIC, LHC) are going to be run (large $T$ and small $\mu$).
![[]{data-label="fig:4histo"}](gmachgraphs.eps){width="80.00000%"}
Monte Carlo simulations with the weight Re $\det D$ would be greatly facilitated if we were able to find a new matrix $\Delta$ such that Re $\det D=\det \Delta$ because then fast and well–known simulation methods for fermions [@gottlieb] could be used. We have not found a general and efficient algorithm to construct the matrix $\Delta$ starting from $D$. Consequently we have to resort to algorithms which explicitely calculate the determinant of $D$.
We shall not insist in these aspects of the problem as they will be analysed in a future publication containing several numerical studies [@prep].
Conclusions
===========
We have proved that the correct Boltzmann weight for updating full QCD in lattice simulations at finite chemical potential is $$\exp\left(-S_{\rm g}\right)\hbox{Re }\det D
\label{we}$$ where $S_{\rm g}$ is the pure gluon action and $D$ is the fermion matrix. We have shown that this is true when we calculate expectation values of operators that are [CPT]{} invariant. For other operators our proof does not work. Possibly in this case the assumption that the imaginary part of the Boltzmann weight is eliminated by combining couples of configurations should be relaxed. Nonetheless the class of [CPT]{} invariant operators include many observables usually studied in the context of finite density systems.
Our algorithm has two problems which deserves further improvement: on one hand the weight (\[we\]) is not positive in general. We have shown that it is mostly positive for moderate values of the chemical potential. On the other hand the present method requires the explicit calculation of the determinant of the fermion matrix $D$ which is very time consuming. In a future publication [@prep] we will give numerical results obtained by using Eq.(\[we\]).
Acknowledgements
================
It is a pleasure to thank G. Marchesini for useful discussions.
[99]{} See for instance K. Rajagopal, F. Wilczek, hep–ph/0011333; G. Nardulli, hep–ph/0206065. H. Satz, Nucl. Phys. B (Proc. Suppl.) 94 (2001) 204; R. V. Gavai, hep–ph/0010048. B. Barrois, Nucl. Phys. B129 (1977) 390; D. Bailin, A. Love, Phys. Rep. 107 (1984) 325; R. Casalbuoni, R. Gatto, G. Nardulli, Phys. Lett. B498 (2001) 179. P. Hasenfratz, F. Karsch, Phys. Lett. B125 (1983) 308. J. Kogut, H. Matsuoka, M. Stone, H. W. Wyld, S. Shenker, J. Shigemitsu, D. K. Sinclair, Nucl. Phys. B225 \[FS9\] (1983) 93. K. G. Wilson, Phys. Rev. D10 (1974) 2445; in “New Phenomena in Subnuclear Physics”, ed. A. Zichichi (Plenum Press, New York) (1975) pg. 69. N. Bilic, R. V. Gavai, Z. Phys. C23 (1984) 77. Z. Fodor, S. D. Katz, hep–lat/0204029. M. Alford, A. Kapustin, F. Wilczek, Phys. Rev. D59 (1999) 054502. Ph. de Forcrand, O. Philipsen, hep–lat/0205016; M. D’Elia, M.–P. Lombardo, hep–lat/0205022. S. Choe, Ph. de Forcrand, M. García–Pérez, S. Hioki, Y. Liu, H. Matsufuru, O. Miyamura, I.-O. Stamatescu, T. Takaishi, T. Umeda, Nucl. Phys. B (Proc. Suppl.) 106 (2002) 462; C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, Ch. Schmidt, L. Scorzato, hep–lat/0204010. I. M. Barbour, C. T. H. Davies, Z. Sabeur, Phys. Lett. B215 (1988) 567. D. Toussaint, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 248. S. Gottlieb, W. Liu, D. Toussaint, R. L. Renken, R. L. Sugar, Phys. Rev. D35 (1987) 2531. B. Allés, E. M. Moroni, in preparation.
[^1]: This is true for Wilson and naïve fermions; for staggered fermions a matrix other than $\gamma_5$ must be used but the result is the same. We use the euclidean definition of the gamma matrices such that $\gamma_\nu=\gamma_\nu^\dagger$ holds.
[^2]: In 2 dimensions the action that introduces the chemical potential on the lattice through the term $\mu\psi^\dagger\psi$ does not present the problems discussed in [@hasenfratzk]. The density of energy obtained with this action on the lattice is $\mu^2/2\pi$, the same result than in the continuum. However this is irrelevant for the purpose of the present test and we used the action in Eq.(\[action\]).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possesses an index system. Because of their topological robustness, they can be used to obtain rigorous results from computer approximations of a dynamical system.'
address:
- |
Dickinson College\
Carlisle, PA 17013
- |
Agnes Scott College\
Decatur, GA 30030
author:
- David Richeson
- Jim Wiseman
bibliography:
- 'indexsystems.bib'
title: Symbolic dynamics for nonhyperbolic systems
---
Introduction
============
Hyperbolicity is one of the most important ideas in dynamical systems. Every hyperbolic diffeomorphism admits a Markov partition—a finite collection of rectangles that stretch or shrink in different directions, and map nicely onto each other. The interaction of rectangles under a single application of the map is sufficient to generate symbolic dynamics, which in turn gives global information about the dynamical system.
In this paper we introduce index systems, a topological generalization of hyperbolicity and Markov partitions. Index systems are composed of finitely many index pairs, a fundamental object from Conley index theory. The mapping of the index pairs mimics the expansion and contraction of the Markov rectangles, and they may be used to generate symbolic dynamics.
The benefit of this topological approach is that it applies in much more general situations than do Markov partitions. Hyperbolicity is a strong condition (requiring at least a manifold and a differentiable map) that can be difficult to verify. However, index systems can be constructed on metric spaces in which the map is not differentiable. In particular, every discrete dynamical system that satisfies a weak form of expansiveness (one of the properties of every hyperbolic system) is guaranteed to have a nontrivial index system. Moreover, unlike Markov partitions, index systems are robust under slight perturbations of the map. This will enable us (in future work) to obtain rigorous results about dynamical systems from computer approximations. As we will see, however, the expense of this generality is that whereas a Markov partition produces a subshift of finite type, an index system generates a cocyclic subshift.
The paper is organized as follows. In Section \[sec:background\] we recall necessary information about expansiveness and the Conley index. We discuss index systems and their properties in Section \[sec:systems\]. We show how to use them to detect orbits of the dynamical system in Section \[sec:detecting\] and how they can be used to generate symbolic dynamics in Section \[sec:symb\]. In Section \[sec:existence\] we show that all maps with a weak form of expansiveness have index systems. In Section \[sec:examples\] we give examples.
Background {#sec:background}
==========
Unless otherwise specified, throughout this paper we let every space $X$ be a compact metric space and every dynamical system $f:X\to X$ be a continuous map. An *orbit* of $f$ is a bi-infinite sequence $(x_{i}\in X:i\in{\mathbb Z})$ with the property that $f(x_{i})=x_{i+1}$ for all $i\in{\mathbb Z}$.
Expansiveness {#ssect:expansiveness}
-------------
A homeomorphism $f:X\to X$ is *expansive* if there exists $\rho>0$ such that for any distinct points $x,y\in X$, there is an integer $n$ with $d(f^n(x),f^n(y))>\rho$. In other words, if the orbits of two points stay close together for all time, then they must be the same point. Expansiveness is a strong form of sensitive dependence on initial conditions since any two distinct points must eventually move apart in either forward or backward time. In practical terms, this means that any small initial measurement error will lead to large errors in predicting behavior. It is not difficult to show that if ${S}$ is a hyperbolic invariant set for a diffeomorphism $f$, then $f$ restricted to ${S}$ is expansive ([@KH; @Ro]).
Although expansiveness is defined as a metric property, on compact spaces it is independent of the metric (of those compatible with the topology), and there is a very simple and useful topological characterization. Moreover, the topological definition can be extended trivially to continuous maps. To state it we need some preliminary definitions.
We begin with the definition of an [isolated invariant set]{}, a notion that is also central to Conley index theory and will be discussed in more detail in Section \[sec:conley\]. A set $S\subset X$ is an *isolated invariant set* for a continuous map $f:X\to X$ provided there is a compact set $I$ with $S=\operatorname{Inv}I\subset\operatorname{Int}I$ (where $\operatorname{Inv}I$ denotes the maximal invariant subset of $I$). The set $I$ is called an *isolating neighborhood* for $S$. Examples of isolated invariant sets include hyperbolic periodic orbits, attractors, and the invariant Cantor set inside the Smale horseshoe.
For any dynamical system $f:X\to X$, let $f\times f:X\times X\to X\times X$ be the map $(f\times f)(x_{1},x_{2})=(f(x_{1}),f(x_{2}))$. Let $1_{X}=\{(x,x)\in X\times X\}$ denote the diagonal of $X\times X$.
A continuous map $f:X\to X$ is [*expansive*]{} if the diagonal $1_{X}\subset X\times X$ is an isolated invariant set with respect to $f\times f$.
Note that when $f$ is a homeomorphism this definition of expansive is equivalent to the metric definition ([@A Def. 11.4]).
We have defined expansiveness for a map $f:X\to X$. If ${S}$ is an isolated invariant subset of $X$ we say that [*$f$ is expansive on ${S}$*]{} if $f$ restricted to ${S}$ is expansive. Note that this is equivalent to the condition that the set $1_{S}=\{(x,x) : x\in {S}\}\subset X\times X$ is an isolated invariant set for $f\times f:X\times X\to X\times X$ ([@A Ex. 11.5]).
Conley index {#sec:conley}
------------
The discrete Conley index is a powerful topological tool for studying isolated invariant sets. Roughly speaking, the Conley index assigns to each isolated invariant set for $f$ a pointed topological space and a base-point preserving map on it, $f_P$, which is unique up to an equivalence relation. By studying the simpler map $f_P$ we can draw conclusions about the original map $f$. Our discussion of the discrete Conley index is based on that in [@FR], where one can find more details and proofs of the theorems below.
Ideally one would like to place an isolated invariant set, $S$, between levels of a filtration. In other words, we would like sets $N_{0}\subset N_{1}$, both of which map into their interiors, such that $S=\operatorname{Inv}(N_{1}{\backslash}N_{0})$. In practice this may not be possible. Instead we must settle for a topological pair that behaves locally like a filtration.
\[filpair\] Let $S$ be an isolated invariant set and suppose $L \subset N$ are compact sets. The pair $(N,L)$ is an [*index pair*]{} for $S$ provided $N$ and $L$ are each the closures of their interiors and
1. $\operatorname{cl}(N{\backslash}L)$ is an isolating neighborhood for $S$,
2. $L$ is a neighborhood of the [*exit set*]{}, $N^{-}=\{x\in N: f(x)
\notin \operatorname{Int}N\}$, in $N$, and
3. $f(L) \cap \operatorname{cl}(N{\backslash}L) = \emptyset$.
The definition of index pair given above was introduced in [@FR] (where they were called filtration pairs). This definition is similar to those in [@BF; @E; @Mr1; @Sz; @RS]. We see examples of index pairs in Figure \[fig:conley\].
![Index pairs: $(N,L)$ for a fixed point saddle, $(N^{\prime},L^{\prime})$ for the horseshoe Cantor set, and their images under $f$.[]{data-label="fig:conley"}](conley.eps)
Given a neighborhood $U$ of an isolated invariant set $S$ there exists an index pair $P=(N,L)$ with $N{\backslash}L\subset U$. Given such an index pair we form the pointed space $N_{L}$ by collapsing $L$ to a point, $[L]$ (see Figure \[fig:conley2\]). Thus $f$ induces a continuous map $f_{P}:N_{L}\to
N_{L}$. The base point $[L]$ is an attracting fixed point of $f_{P}$. The choice of index pairs for $S$ is not unique, and different choices can lead to topologically different maps $f_P$. However, the choice is unique up to *shift equivalence* (we will use $\cong$ to denote shift equivalence). The resulting equivalence class is called the [*homotopy Conley index*]{}. For a definition of shift equivalence and a proof of the facts in this paragraph see [@FR].
![The pointed spaces $N_{L}$ and $N^{\prime}_{L^{\prime}}$ obtained from the filtration pairs for the saddle and the horseshoe.[]{data-label="fig:conley2"}](conley2.eps)
In practice, one may wish to compute the relative homology (or cohomology) of the pointed space and the induced map $(f_{P})_{*}:H_{*}(N_{L},[L])\to H_{*}(N_{L},[L])$. Again, this map is unique up to shift equivalence. We call this equivalence class the *homology Conley index*, ${\operatorname{Con}}_{*}(S)$. (Note: we may apply the Leray functor to $(f_{P})_{*}$ to obtain an automorphism of a graded group, which is an invariant for $S$ (see [@Mr1]).)
For instance, the fixed point saddle in Figure \[fig:conley\] has ${\operatorname{Con}}_{*}(S)\cong ({\mathbb Z},{\operatorname{Id}})$ where the ${\mathbb Z}$ is in dimension 1. The invariant Cantor set in the horseshoe has ${\operatorname{Con}}_{*}(S)\cong 0$ (from a homological perspective the two arms of the horseshoe cancel one another). Note that that if ${\operatorname{Con}}_{*}(S)\not\cong 0$, then $S\ne\emptyset$ (this is called Ważewski’s theorem), but the horseshoe shows that the converse is not true.
An important feature of the Conley index is that if $I$ is an isolating neighborhood for $f$, then $I$ is also an isolating neighborhood for $\operatorname{Inv}(I,g)$ for any $g$ that is $C^0$-close to $f$, and in this case $\operatorname{Inv}(I,f)$ and $\operatorname{Inv}(I,g)$ have the same Conley index ([@FR]). This robustness allows one to compute the Conley index from a suitable numerical approximation of the map ([@Ka; @Mi; @Mr2; @Mr3]).
Index systems {#sec:systems}
=============
For differentiable maps it is frequently useful to look at the behavior of orbits on the manifold and also the behavior of the derivative on the tangent space following these orbits. Indeed, this is precisely what one does with hyperbolicity. As we see in this section, for an expansive map $f:X\to X$ we may use $f\times f:X\times X\to X\times X$ to obtain a nice analogue of the differentiable situation. We will see that slices of a neighborhood $N$ of the diagonal (that is, intersections of $\{x\}\times X$ with $N$) are the analogues of the tangent spaces and it is from these slices that we build the index system.
Suppose ${\mathcal I}=(I_i:i\in{\mathbb Z})$ is a sequence of compact sets in $X$ (in practice $\{I_{i}:i\in {\mathbb Z}\}$ will be a finite set of sets). We say that an orbit $(x_{i})$ *follows* ${\mathcal I}$ if $x_{i}\in I_{i}$ for all $i\in {\mathbb Z}$. We call such a sequence, ${\mathcal I}$, an *isolating neighborhood chain* if for any orbit $(x_{i})$ that follows ${\mathcal I}$, $x_{i}\in\operatorname{Int}(I_i)$ for all $i\in{\mathbb Z}$. For ${\mathcal I}$ an isolating neighborhood chain, let $\operatorname{Inv}_0({\mathcal I}) \subset \operatorname{Int}(I_0)$ denote the set of points whose orbits follow ${\mathcal I}$; that is, $$\operatorname{Inv}_{0}({\mathcal I})= \{x\in I_0 : \text{there exists an orbit $(x_i)$, with $x_0=x$, that follows $\mathcal I$}\}.$$ Note that ${\operatorname{Inv}_0}({\mathcal I})$ is not, in general, $f$-invariant.
An index system is a collection of compact pairs. Each is similar to an index pair, but instead of necessarily mapping to itself under $f$, it maps to one or more of the pairs in the index system (see Figure \[fig:fsystems\]). More precisely, we have the following definition.
![$(N_{a_i}, L_{a_i})$ precedes $(N_{a_{i+1}}, L_{a_{i+1}})$ which precedes $(N_{a_{i+2}}, L_{a_{i+2}})$.[]{data-label="fig:fsystems"}](fsystems)
An [*index system*]{} is a finite collection of compact pairs, ${\mathcal{P}}=\{P_a=(N_a,L_a):a\in{\mathcal A}\}$, such that
1. for each $a\in\mathcal A$, there exists at least one $b\in\mathcal A$ such that $P_a$ *precedes* $P_b$, that is, such that
1. $L_a$ is a neighborhood of the exit set, $N_{ab}^- = \{x\in N_a: f(x) \not\in \operatorname{Int}N_b\}$, in $N$, and
2. $f(L_a)\cap \operatorname{cl}(N_b{\backslash}L_b)=\emptyset$, and
2. any sequence $(I_i=\operatorname{cl}(N_{a_i}{\backslash}L_{a_i}): P_{a_i}\text{ precedes }P_{a_{i+1}}\text{ for all }i\in{\mathbb Z})$ is an isolating neighborhood chain.
For each $a$, we form the pointed space ${N_a}_{L_a}$ by collapsing $L_a$ to a point, $[L_a]$. If $P_a$ precedes $P_b$, then $f$ induces a continuous map $f_{a,b}:{N_a}_{L_a} \to {N_b}_{L_b}$.
We can think of an index system as a directed graph. Each pointed space ${N_a}_{L_a}$ is a vertex, with an edge from ${N_a}_{L_a}$ to ${N_b}_{L_b}$ if $P_a$ precedes $P_b$. (The induced maps $f_{a,b}$ are analogues of the induced maps on the tangent bundle for differentiable systems.) We say that a finite or infinite sequence $(a_i)$ is [*allowable*]{} if it corresponds to a path in the graph, that is, if $P_{a_i}$ precedes $P_{a_{i+1}}$ for all $i$.
Given an index system ${\mathcal{P}}$, we define its [*invariant set*]{} by $\operatorname{Inv}({\mathcal{P}})=\bigcup \operatorname{Inv}_0((I_i=\operatorname{cl}(N_{a_i}{\backslash}L_{a_i})))$, where the union is over all allowable sequences $(a_i:i\in{\mathbb Z})$. Thus $\operatorname{Inv}({\mathcal{P}})$ is the set of points on orbits following an allowable sequence of pairs. Let ${\mathcal{P}}$ be an index system. Define $\operatorname{Inv}^m({\mathcal{P}})$ to be the set of $x$ such that there exists an orbit segment $(x_i)_{i=-m}^m$ with $x_0=x$ and $x_i \in \operatorname{cl}(N_{a_{i}}{\backslash}L_{a_{i}})$ for some finite allowable sequence $(a_i)_{i=-m}^m$.
\[lem:inv\] Let ${\mathcal{P}}$ be an index system. Then $\operatorname{Inv}({\mathcal{P}}) = \bigcap_{m=0}^\infty\operatorname{Inv}^m({\mathcal{P}})$.
The proof is based on that of [@FR Prop. 2.2], which is the same result for index pairs instead of index systems. It is obvious that $\operatorname{Inv}({\mathcal{P}}) \subset \bigcap_{m=0}^\infty\operatorname{Inv}^m({\mathcal{P}})$. To prove the opposite inclusion, let $x$ be an element of $\bigcap_{m=0}^\infty\operatorname{Inv}^m({\mathcal{P}})$; we must show that $x \in \operatorname{Inv}({\mathcal{P}})$.
For an allowable sequence $(a_{-1}, a_0)$ with $x \in \operatorname{cl}(N_{a_0}{\backslash}L_{a_0})$, define $X_1^{(a_{-1}, a_0)}=f^{-1}(x) \cap \operatorname{cl}(N_{a_{-1}}{\backslash}L_{a_{-1}})$. Then, for $(a_{-k},\dots, a_0)$ an allowable sequence with $x \in \operatorname{cl}(N_{a_0}{\backslash}L_{a_0})$, inductively define $X_k^{(a_{-k}, \dots,a_0)}=f^{-1}(X_{k-1}^{(a_{-k+1}, \dots,a_0)}) \cap \operatorname{cl}(N_{a_{-k}}{\backslash}L_{a_{-k}})$. Each $X_k^{(a_{-k}, \dots,a_0)}$ is compact and $f(X_k^{(a_{-k}, \dots,a_0)}) \subset X_{k-1}^{(a_{-k+1}, \dots,a_0)}$. Set $X_k = \bigcup X_k^{(a_{-k}, \dots,a_0)}$, where the union is over all allowable sequences $(a_{-k},\dots, a_0)$ with $x \in \operatorname{cl}(N_{a_0}{\backslash}L_{a_0})$; then $X_k$ is a non-empty, compact set with $f(X_k) \subset X_{k-1}$.
Now define $Y_k = \bigcap_{n\ge1}f^{n+1}(X_{n+k})$. As the intersection of a nested sequence of non-empty compact subsets of $X_k$, $Y_k$ is a compact, non-empty subset of $X_k$, so $f(Y_1)=\{x\}$ and for $k>1$, $$f(Y_k)=\bigcap_{n\ge1}f^{n+1}(X_{n+k}) = \bigcap_{n\ge2}f^{n}(X_{n+k-1}) = Y_{k-1}.$$ Finally, define $x_{-1}$ to be any point of $Y_1$, and inductively define $x_{-k}$ to be any point in $Y_k$ with $f(x_{-k})=x_{-k+1}$. If we set $x_k=f^k(x)$ for $k\ge0$, then $(x_i)_{i=-\infty}^\infty$ is an orbit following an allowable sequence, so $x\in \operatorname{Inv}({\mathcal{P}})$.
The following result follows immediately from Lemma \[lem:inv\].
If ${\mathcal{P}}$ is an index system, then $\operatorname{Inv}({\mathcal{P}})$ is a compact invariant set.
Let ${S}\subset X$ be an isolated invariant set. An index system ${\mathcal{P}}$ is an [*index system for ${S}$*]{} if $\operatorname{Inv}({\mathcal{P}}) = {S}$.
In Section \[sec:existence\] we present situations in which we can guarantee the existence of index systems, and we give a procedure for constructing them that could be implemented on a computer.
Detecting Orbits {#sec:detecting}
================
In this section we show how to prove the existence of an orbit following a given allowable sequence of sets $(\operatorname{cl}(N_{a_i}{\backslash}L_{a_i}))$. If $f$ had the shadowing property and the sets $\operatorname{cl}(N_{a_i}{\backslash}L_{a_i})$ were small enough, then we could guarantee the existence of an orbit that follows this sequence ([@Ro §9.3]). However, the shadowing property is difficult to verify in practice. Instead, we use the Conley index to verify that there is an $f$-orbit that “shadows” the sequence in the sense that each iterate is in the appropriate set $\operatorname{cl}(N_{a_i}{\backslash}L_{a_i})$.
An allowable sequence $(a_i)$ yields a directed system $$\begin{CD} \cdots \longrightarrow H_{*}(P_{a_i}) @>f_{a_{i},a_{i+1}*}>>H_{*}(P_{a_{i+1}}) @>f_{a_{i+1},a_{i+2}*}>>H_{*}(P_{a_{i+2}}) \longrightarrow \cdots
\end{CD}.$$ The sequence $(P_{a_i})$ has a *nonzero orbital Conley index* if any finite composition $f_{a_{n-1},a_{n}*}\circ \dots \circ f_{a_{m+1},a_{m+2}*} \circ f_{a_{m},a_{m+1}*}$ is nonzero (note that if we take coefficients in a field and $H_{*}(P_a)$ is finitely generated for each $a$, then the maps are simply matrices).
We have the following theorem, which is the orbital analogue of Ważewski’s theorem.
\[thm:waz\] Let ${\mathcal{P}}$ be an index system for the isolated invariant set ${S}$, and let $(a_i)$ be an allowable sequence. If $(P_{a_i})$ has a nonzero orbital Conley index, then there is an orbit in ${S}$ following $(\operatorname{cl}(N_{a_i}{\backslash}L_{a_i}))$. If $f$ is expansive and the diameters of the sets $\operatorname{cl}(N_{a}{\backslash}L_{a})$ are sufficiently small for all $a$, then this orbit is unique.
If the set $\bigcap_{i=-m}^m f^{-i} (\operatorname{cl}(N_{a_i}{\backslash}L_{a_i}))$ were empty, then the induced map would send every point to the basepoint, and the corresponding map on homology would be zero. Since $(P_{a_i})$ has a nonzero orbital Conley index, the set $\bigcap_{i=-m}^m f^{-i}(\operatorname{cl}(N_{a_i}{\backslash}L_{a_i}))$ is nonempty for all $m$. The same argument used to prove Lemma \[lem:inv\] then shows that there is an orbit in ${S}$ following $(\operatorname{cl}(N_{a_i}{\backslash}L_{a_i}))$.
Suppose $f$ is expansive on ${S}$ and $(x_{i})$ and $(y_{i})$ are two orbits following $(\operatorname{cl}(N_{a_i}{\backslash}L_{a_i}))$. Because $f$ is expansive, $1_{{S}}$ is an isolated invariant set for $f\times f$ with some isolating neighborhood $I$. Assuming the diameters of $\operatorname{cl}(N_{a_i}{\backslash}L_{a_i})$ are small enough, that means that $(f\times f)$-orbit $((x_{i},y_{i}))$ remains in $I$ for all $i$. But this implies that for all $i$, $(x_{i},y_{i})\in 1_{{S}}$, or equivalently $x_{i}=y_{i}$.
Symbolic dynamics {#sec:symb}
=================
Many authors have used the Conley index or related techniques (for example, Easton’s windows ([@Eas81])) to detect symbolic dynamics; an incomplete list of references is [@DFT; @Gi; @Mr2; @Wis1; @WZ; @Z; @CKM; @WisSquare; @Sz96; @MM; @Eas81; @GR; @ZG; @vVW; @Gia; @Eas75; @Gid99]. Most treatments resemble index systems, except that all of the index pairs are pairwise disjoint. This is an important difference—disjoint index pairs make the conclusions stronger, but they are significantly less useful because they cover less of the space. In [@Gid99], Gidea developed a notion similar to index systems, of an orbital Conley index for compact sets traveling within a prescribed sequence of neighborhoods, which he applied to detect periodic orbits and symbolic dynamics. An important difference is that index systems are used to study invariant sets, while Gidea’s orbital Conley index applies to non-invariant sets that travel along a bi-infinite sequence of boxes. In addition, there is no result similar to Theorem \[thm:systemsexist\] for constructing the neighborhood sequence for Gidea’s orbital Conley index, and the method for detecting symbolic dynamics requires that the neighborhoods be pairwise disjoint. As we see in this section, we can relax this requirement with index systems.
The mapping of rectangles in a Markov partition generates a directed graph, and paths through this graph become the elements of a subshift of finite type. The elements of this subshift can be paired uniquely with points in the dynamical system. The situation for index systems is similar, but more complicated. Intuitively, we would like allowable sequences from our directed graph of pointed spaces to generate symbolic dynamics. However, we must use the Conley index (as in Section \[sec:detecting\]) to determine when there is a true orbit corresponding to this allowable sequence, and then we must show that each point corresponds to only one symbol sequence.
In order to put this in proper context we must introduce the notion of a cocyclic subshift, a generalization of sofic shifts and subshifts of finite type. See [@Kwa] for more information about cocylic subshifts and [@CKM; @Wis1; @WisSquare] for examples of cocyclic subshifts in other dynamical and Conley index contexts.
Let $G$ be a directed graph with vertices $\{1,\ldots,n\}$ and at most one edge from any vertex to any other. Assign to each vertex $i$ a vector space $V_i$ and to the edge from $i$ to $j$ a linear transformation $T_{i,j}:V_i\to V_j$. The [*cocyclic subshift*]{} associated to $G$ consists of all words $(\ldots, \omega_{i-1}, \omega_i, \omega_{i+1},\ldots)$ such that
1. there is an edge from vertex $\omega_{i}$ to $\omega_{i+1}$ for all $i$, and
2. any finite composition $T_{i+m-1,i+m}\circ \cdots \circ T_{i,i+1}$ is nonzero,
together with the usual shift map.
An index system generates the cocyclic subshift corresponding to the graph with vertex set $\mathcal A$, vector spaces $H_*(P_a)$, edges from $a$ to $b$ if $P_a$ precedes $P_b$, and linear transformations $f_{a,b*}$. In other words, we take the homology of our directed graph of pointed spaces. At each vertex we place the homology of the given pair, $H_*(P_{a}) = H_*({N_a}_{L_a},[L_a])$, and we label each directed edge from $H_*(P_a)$ to $H_*(P_b)$ with the corresponding induced map on homology $f_{a,b*}$. Then elements of the cocyclic subshift correspond to allowable sequences with nonzero orbital Conley index.
A further complication is that in general we do not get a conjugacy (or semiconjugacy) from $f$ to the cocyclic subshift because the sets $\{\operatorname{cl}(N_a{\backslash}L_a)\}$ are usually not pairwise disjoint. For example, say that the period-two word $(1,2,1,2,\dots)$ is in the cocyclic subshift, which implies that there is a point $x\in X$ such that $x\in
\operatorname{cl}(N_{1}{\backslash}L_{1})$, $f(x)\in \operatorname{cl}(N_{2}{\backslash}L_{2})$, $f^2(x)\in \operatorname{cl}(N_{1}{\backslash}L_{1})$, and so on. But if the two sets are not disjoint, then $x$ could be a fixed point. The ${\mathcal{P}}$-itinerary of a point is not necessarily unique.
One way around this issue is simply to remove overlapping index pairs. That is, take a maximal subgraph of the cocyclic subshift graph consisting of vertices for which all of the corresponding sets $
\operatorname{cl}(N_a{\backslash}L_a)$ are pairwise disjoint. Then, as desired, we will get a semi-conjugacy from an isolated invariant set in $X$ to the cocyclic subshift induced by the subgraph. The disadvantage of this approach is that we are losing information about $f$ when we throw away vertices.
Another approach is to look at powers $f^n$. In this case, only some, not all, of the index pairs must be disjoint in order to detect positive-entropy symbolic dynamics. For example, let $\omega=(\omega_0,\dots,\omega_n)$ be a word in the cocyclic subshift, and define ${S}_\omega\subset X$ to be the subset $\operatorname{Inv}(\{x\in X : f^i(x) \in \operatorname{cl}(N_{\omega_i}), 0\le i\le n
\} , f^{n+1})$. If, say, the sets $\operatorname{cl}(N_{1}{\backslash}L_{1})$ and $\operatorname{cl}(N_{3}{\backslash}L_{3})$ are disjoint, then the sets ${S}_{(1,1,1)}$ and ${S}_{(1,2,3)}$ are disjoint (even if $\operatorname{cl}(N_{1}{\backslash}L_{1})$, $\operatorname{cl}(N_{2}{\backslash}L_{2})$, and $\operatorname{cl}(N_{3}{\backslash}L_{3})$ are not pairwise disjoint), so we can define symbolic dynamics and get a semiconjugacy for $f^3$.
Using either method for a generic index system, we get only a semiconjugacy from the invariant set to the cocyclic subshift. Thus, while we can get a lower bound for the topological entropy of $f$, we cannot conclude that a periodic word for the subshift actually corresponds to a periodic point for $f
$. However, if $f$ is expansive and the index pairs are sufficiently small, then by the uniqueness of orbits guaranteed by Theorem \[thm:waz\], we do get a conjugacy. Even in some cases when we do not have expansiveness (in particular, if the homology is nontrivial on exactly one level), we can use a version of the Lefschetz fixed point theorem ([@Sz96]) instead of Theorem \[thm:waz\] to detect periodic points for $f$. For the sake of brevity, we omit the details.
There are further details on techniques for generating symbolic dynamics from index systems in [@RWindexsymb].
Existence of index systems {#sec:existence}
==========================
As we have said, hyperbolic diffeomorphisms admit Markov partitions. In fact, homeomorphisms that are expansive and have the shadowing property also admit Markov partitions ([@AH]), but expansivity alone is not sufficient. Furthermore, even when they exist, Markov partitions can be difficult to construct.
One of the benefits of index systems is that under certain general circumstances their existence is guaranteed. Moreover, the proof is constructive—it requires only the ability to find a single index pair. There are computer algorithms that will do that (see [@DJM2], for example).
It is immediate that every isolated invariant set ${S}$ has at least a trivial index system: the system consisting of the single pair $(N,L)$, where $(N,L)$ is an index pair for ${S}$. In the case that $f$ is expansive, we can do better.
\[thm:systemsexist\] Let $f$ be expansive on an isolated invariant set ${S}$. Then there exists an index system ${\mathcal{P}}=\{(N_a,L_a)\}$ for ${S}$ of arbitrarily small diameter (that is, the diameter of $N_a$ is arbitrarily small for all $a$).
The idea of the proof is simple. Recall from Section \[ssect:expansiveness\] that since ${S}$ is expansive, the set $1_{S}=\{(x,x) : x\in {S}\} \subset X\times X$ is an isolated invariant set for $f\times f$. Thus any neighborhood of $1_{S}$ contains an index pair $(N,L)$ for $1_{S}$ under $f\times f$. By taking cross-sections, we get pairs $(N_x,L_x)=(\pi_2((\{x\}\times X) \cap N), \pi_2((\{x\}\times X) \cap L))$ of arbitrarily small diameter (where $\pi_2:X\times X\to X$ is the projection onto the second coordinate). We would like these pairs to make up the index system. However, an index system must be finite, and in general this construction may give us infinitely many pairs, one for each $x\in{S}$.
By essentially discretizing the space and using the robustness of the Conley index we can modify the original index pair $(N,L)$ to yield only finitely many cross-sections. Let $D$ be the metric on $X\times X$ given by $D((x,y),(x',y'))=\max(d(x,x'),d(y,y'))$. By the definition of index pair, there exists an ${\varepsilon}>0$ such that $D(1_{{S}},L)>{\varepsilon}$, $D((f\times f)(L), \operatorname{cl}(N{\backslash}L))>{\varepsilon}$, and $D(N^-,\operatorname{cl}(N{\backslash}L))>{\varepsilon}$. Pick $\delta<{\varepsilon}/3$ such that for any points $(x,y)$ and $(x',y')$ within ${\varepsilon}$ of $N$, if $D((x,y),(x',y'))<\delta$, then $D((f\times f)(x,y),(f\times f)(x',y'))<{\varepsilon}/3$.
Let $\{V_i\}$ be a finite collection of compact subsets of $X$ of diameter less than $\delta$ such that $\pi_1(N)\cup\pi_2(N)\subset\bigcup_i\operatorname{Int}(V_i)$. This gives a product cover of $N$, $\{V_{ij}=V_i\times V_j\}$. We now construct a new index pair for $1_{S}$, $(\tilde N,\tilde L)$, by setting $\tilde N = \bigcup_{\{ij:V_{ij}\cap N\ne \emptyset\}} V_{ij}$ and $\tilde L = \bigcup_{\{ij:V_{ij}\cap L\ne \emptyset\}} V_{ij}$. (See Figure \[fig:discretized\].) The set of cross-sections of $(\tilde N, \tilde L)$ is finite, and it is straightforward to check that it forms an index system for ${S}$.
![The given index pair for $1_{{S}}$, a discretized index pair nearby, and one of the finite number of slices. []{data-label="fig:discretized"}](discretized)
Theorem \[thm:systemsexist\] says that if $f$ is expansive on an isolated invariant set ${S}$, then it has an index system. Expansiveness is a strong condition that may be difficult to verify in practice. It turns out that a condition weaker than expansiveness guarantees the existence of index systems. In Theorem \[thm:systemsexist\] we assumed that $1_{{S}}\subset X\times X$ was an isolated invariant set (i.e., that $f$ was expansive on ${S}$), but the technique for constructing the index system requires only that $1_{{S}}$ be contained in some isolated invariant set $\Lambda$. Then the slices of an index pair $(N,L)$ for $\Lambda$ give an index system for ${S}$. At one extreme we have $\Lambda=1_{S}$ ($f$ is expansive), in which case we can find an index system with index pairs arbitrarily small. At the other extreme, for *any* isolated invariant set ${S}$ we could take $\Lambda={S}\times{S}$; this would produce a trivial index system consisting of one index pair. There may be times when $1_{{S}}\subsetneq \Lambda\subsetneq {S}\times{S}$, and the resulting index system is useful.
If $1_{{S}}$ is not an isolated invariant set, then we will not be able to make the index pairs arbitrarily small, but that is not necessarily a big disadvantage. On the one hand, the smaller the sets $\operatorname{cl}(N_a{\backslash}L_a)$, the stronger the conclusion of Theorem \[thm:waz\], which is one reason the small diameters guaranteed by Theorem \[thm:systemsexist\] are important. Furthermore, the smaller the sets, the more disjoint pairs we have, which, as we have seen, make it easier to detect symbolic dynamics. On the other hand, small diameters can lead to a system with many small, individually unimportant pieces. This would produce a large graph, and thus a very complicated cocyclic subshift. So, in practice there is a trade-off involved in the size of the sets of the index system.
Examples {#sec:examples}
========
Let $f:S^1\to S^1$ be the doubling map on the circle, considered as ${\mathbb R}/{\mathbb Z}$. The collection ${\mathcal{P}}=\{(N_i,L_i)\}_{i=0}^9$ is an index system, where $N_i = [\frac{i-3-3{\varepsilon}}{10},\frac{i+3+3{\varepsilon}}{10}]$ and $L_i= [\frac{i-3-3{\varepsilon}}{10},\frac{i-1-{\varepsilon}}{10}] \cup [\frac{i+1+{\varepsilon}}{10},\frac{i+3+3{\varepsilon}}{10}]$. $P_i$ precedes $P_j$ for $j=2i-1$, $2i$, or $2i+1\mod 10$ (see Figure \[fig:circle\]).
![An example showing that $P_{0}$ precedes $P_{9}$, $P_{0}$, and $P_{1}$, and the directed graph for this index system.[]{data-label="fig:circle"}](circle)
Each pointed space ${N_i}_{L_i}$ is homeomorphic to a circle. So, for every $i$, the only nonzero homology is $H_{1}(P_{i})={\mathbb Z}$, and the induced maps are $f_{i,j*}=1$ if $P_i$ precedes $P_j$, and $f_{i,j*}=0$ otherwise. Thus any concatenation of the maps $f_{(1,1,1)*}= f_{1,1*} \circ f_{1,1*} \circ f_{1,1*}$ and $f_{(1,3,5)*}= f_{5,1*} \circ f_{3,5*} \circ f_{1,3*}$ is nonzero. Since the sets $\operatorname{cl}((N_1{\backslash}L_1))$ and $\operatorname{cl}((N_5{\backslash}L_5))$ are disjoint, so are the sets ${S}_{(1,1,1)}$ and ${S}_{(1,3,5)}$, and thus we see that the map $f^3:{S}_{(1,1,1)} \cup {S}_{(1,3,5)} \to {S}_{(1,1,1)} \cup {S}_{1,3,5}$ factors onto the full shift on two symbols.
\[ex:tent\] Let $f:{\mathbb R}\to{\mathbb R}$ be the tent map given by $$f(x)=\begin{cases} 3x \text{, if } x\le 1/2,\\
3-3x \text{, if } x\ge 1/2.\end{cases}$$
The map $f\times f$ has an index pair shown in Figure \[fig:product\].
![An index pair for $f\times f$, where $f$ is the tent map.[]{data-label="fig:product"}](product)
Taking slices we obtain the index system ${\mathcal{P}}=\{(N_i,L_i)\}_{i=1}^4$ for $f$ (as shown in Figure \[fig:tent\]), where $$\begin{array}{l}
N_1 = N_2 = [0-4{\varepsilon}, \frac13+4{\varepsilon}]\\
L_1 = [0-4{\varepsilon},0-{\varepsilon}] \cup [\frac19 + {\varepsilon}, \frac13 + 4{\varepsilon}]\\
L_2 = [0-4{\varepsilon},\frac29-{\varepsilon}] \cup [\frac13 + {\varepsilon}, \frac13 + 4{\varepsilon}]\\
N_3 = N_4 = [\frac23 - 4{\varepsilon}, 1 + 4{\varepsilon}]\\
L_3 = [\frac23 - 4{\varepsilon}, \frac23 -{\varepsilon}] \cup [\frac79 + {\varepsilon}, 1+4{\varepsilon}]\\
L_4 = [\frac23 - 4{\varepsilon}, \frac89 -{\varepsilon}] \cup [1 + {\varepsilon}, 1+4{\varepsilon}].
\end{array}$$
Again, each pointed space ${N_i}_{L_i}$ is homeomorphic to a circle. $P_1$ and $P_4$ precede $P_1$ and $P_2$, while $P_2$ and $P_3$ precede $P_3$ and $P_4$. In homology, in dimension one, the induced maps are $f_{1,1*}= f_{1,2*} = f_{2,3*} = f_{2,4*} = 1$ and $f_{3,3*} = f_{3,4*} = f_{4,1*} = f_{4,2*} = -1$. Thus the tent map restricted to $\operatorname{Inv}[0-{\varepsilon},1+{\varepsilon}]$ factors onto the shift given by the graph in Figure \[fig:tent\], which is conjugate to the full shift on two symbols ([@LM §2.4]).
![An index system for the tent map, and the associated directed graph.[]{data-label="fig:tent"}](tent)
Let $f$ be the tent map from Example \[ex:tent\]. The pair $(N_0,L_0)=([-4{\varepsilon},1+4{\varepsilon}],[-4{\varepsilon},-{\varepsilon}]\cup[\frac{1}{3}+{\varepsilon},\frac{2}{3}-{\varepsilon}]\cup[1+{\varepsilon},1+4{\varepsilon}])$ is an index pair, and thus also a (trivial) index system. The only nonzero homology is $H_1(N_0,L_0)={\mathbb Z}^2$, and the induced map is given by $f_{00*}=\begin{pmatrix}1&-1\\1&-1\end{pmatrix}$. Since $(f_{00*})^2$ is the zero matrix, the cocyclic subshift is empty. Thus this index system fails to detect any invariant set.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'S.Borgniet'
- 'N.Meunier'
- 'A.-M. Lagrange'
bibliography:
- 'Simon\_Biblio.bib'
date: 'Received ... ; accepted ...'
subtitle: 'V. Parameterizing the impact of solar activity components on radial velocities.'
title: 'Using the Sun to estimate Earth-like planets detection capabilities.'
---
[Stellar activity induced by active structures such as stellar spots and faculae is known to strongly impact the radial velocity (RV) time series. It is thereby a strong limitation to the detection of small planetary RV signals, such as that of an Earth-mass planet in the habitable zone of a solar-like star. In a serie of previous papers, we studied the detectability of such planets around the Sun observed as a star in an edge-on configuration. For that purpose, we computed the RV, photometric and astrometric variations induced by solar magnetic activity, using all active structures observed over one entire cycle.]{} [Our goal is to perform similar studies on stars with different physical and geometrical properties. As a first step, we focus on Sun-like stars seen with various inclinations, and on estimating detection capabilities with forthcoming instruments.]{} [To do so, we first parameterize the solar active structures with the most realistic pattern so as to obtain results consistent with the observed ones. We simulate the growth, evolution and decay of solar magnetic features (spots, faculae and network), using parameters and empiric laws derived from solar observations and literature. We generate the corresponding structures lists over a full solar cycle. We then build the resulting spectra and deduce the RV and photometric variations, first in the case of a “Sun” seen edge-on and then with various inclinations. The produced RV signal takes into account the photometric contribution of spots and faculae as well as the attenuation of the convective blueshift in faculae. We then use these patterns to study solar-like stars with various inclinations.]{} [The comparison between our simulated activity pattern and the observed one validates our model. We show that the inclination of the stellar rotation axis has a significant impact on the photometric and RV time series. RV long-term amplitudes as well as short-term jitters are significantly reduced when going from edge-on to pole-on configurations. Assuming spin-orbit alignment, the optimal configuration for planet detection is an inclined star ($i \simeq 45\degr$).]{}
Introduction
============
Thanks to the radial velocity (RV) technique, more than 500 exoplanets have been discovered in the two last decades (http://exoplanets.eu). First limited to Jupiter-like planets on short period orbits, the RV technique allows today to detect Neptune-like planets (10-40 [M$_{\rm Earth}$]{}) and Super Earths (1.2-10 [M$_{\rm Earth}$]{}) on longer period orbits. This progress was made possible thanks to the important improvements on the sensitivity and stability of instruments, as well as observational strategies to average out known sources of stellar signals [@dumusque11b]. For example, the High-Accuracy Radial velocity Planet Searcher ([H[ARPS]{}]{}) spectrograph located on the ESO 3.6 m telescope in La Silla [@pepe02] gives a precision of 1 [ms$^{\rm -1}$]{} for a Solar-type star in average conditions, compared to 5-10 [ms$^{\rm -1}$]{} with previous instruments. Furthermore, future instruments, such as [E[SPRESSO]{}]{} on the Very Large Telescope (VLT) or [G[-CLEF]{}]{} on the Giant Magellan Telescope (GMT) are expected to reach a precision down to 0.1 [ms$^{\rm -1}$]{} [@megevand10; @frez14]. Such a level of precision will theorically give access to lower planetary masses far from their host stars such as Earth-like planets in the habitable zone (hereafter HZ).
However, low-amplitude RV planet signals (such as RV signatures of Earth-mass planets in the HZ) are much more sensitive to stellar perturbations than giant planet RV signatures. In the case of a low-mass planet RV signature with an amplitude in the 0.1-1 [ms$^{\rm -1}$]{} range, the stellar noise or “jitter” is high enough to either mask or mimic the planet-induced RV variations, even in the case of a chromospherically nearly inactive star [@isaacson10; @lagrange10; @meunier10a].
The so-called stellar jitter mainly comes from three different sources: stellar oscillations or pulsations, granulation, and stellar magnetic activity.
Stellar pulsations or oscillations dominate the RV jitter on the shorter timescales. For FGK dwarfs, they are mainly driven by accoustic or pressure waves ([*p*]{}-modes). Pressure waves are commonly believed to originate from turbulent convective motions occuring in the stellar outer layers and propagate through the star. They induce RV shifts with an amplitude of a few [cms$^{\rm -1}$]{} to one [ms$^{\rm -1}$]{}. The oscillation periods range from a few minutes ([[*e.g.*]{}]{}, five minutes in the case of the Sun) to a few tens of minutes. The amplitude and the period of the oscillations increase with stellar mass. Long exposure times and observational strategies have shown to be efficient in averaging the RV noise induced by pulsations in the case of late-type stars [@santos04; @dumusque11a].
The photospheric granulation phenomenon accounts for the convective plasma motions occuring in the outer envelop of solar-mass stars. The quick rise and fall of the plasma results in a pattern of bright granules and darker lanes at the stellar surface, and in rapidly evolving RV shifts, with amplitudes up to a few [kms$^{\rm -1}$]{} locally. When integrated over the entire stellar disc, upflows and downflows average out, leaving a residual RV jitter at the level of the [ms$^{\rm -1}$]{}. Up to now, there have been few tentatives to estimate the impact of granulation on the RV. [@dumusque11a] made a first estimation for different spectral type stars based on the study of asteroseismology measurements. [@cegla13] developped a four-component model of granulation, building absorption line profiles from three-dimensional magnetohydrodynamic solar simulations. This study was continued by [@cegla14], who derived the corresponding RV time series as well as other time series of observables such as the bisector amplitude. They concluded first that granulation has a strong impact on RV at the [cms$^{\rm -1}$]{} to [ms$^{\rm -1}$]{} levels and would be a potentially significant limitation to low-mass planets detection. Second, they found correlations between the RV time series and other observables ([[*e.g.*]{}]{} bisector curvature or bisector inverse slope) that already allow to partially correct the granulation signal and reduce the corresponding jitter by one third. Meunier et al ([*submitted*]{}) simulated a collection of both granules and supergranules on the solar visible hemisphere over a whole solar cycle and derived the induced RV and photometric time series. These authors concluded that the granulation noise is hard to average out even over an entire night and will therefore have a significant impact on detection limits.
From a few days to several years timescales, the stellar jitter is dominated by the so-called stellar magnetic activity. Active regions such as dark spots an bright faculae (colder and hotter than the quiet photosphere, respectively) grow and evolve on the stellar surface, inducing changes in the stellar irradiance and RV variations (as the flux loss or excess in the active regions distorts the CCF of the stellar spectra). The stellar rotation makes one see these active regions moving across the stellar disc, thus inducing an apparent Doppler shift. The first studies of the impact of starspots on RV and line bisectors were made by [@saar97] and [@hatzes02]. [@saar97] built a semi-empirical law that directly bound the spot-induced RV semi-amplitude, the spot filling factor and the stellar rotational velocity for cool F-G stars. [@saar03; @saar09] made similar studies for bright faculae. The first spot model was developed by [@desort07]. The authors computed synthetic stellar spectra and applied a black-body law to take into account the contribution of a colder spot. They quantified accurately the RV amplitude, RV bisector and photometric variations induced by a starspot for a range of different stellar and spot properties.
Such simple spot models have proved to be useful and efficient to simulate the RV signature of one or a few active regions during a timescale of the order of the stellar rotational period. They are well adapted to stars that host a single main spot or a few ones, such as active young dwarf stars [[[*e.g.*]{}]{} , see @dumusque14b and ref. therein]. They are however not sufficient to reproduce the total activity-induced RV signal for Sun-like stars, due to the presence of multiple evolving active regions on the stellar disc, and due to the strong contribution of faculae because of slow rotation [@meunier10a; @dumusque14a]. A complete model of the stellar activity pattern for longer timescales (of the order of the activity cycles) is needed. For most stars though, we have no detailed information on the activity properties as active structures are not directly observable. Doppler and Zeeman-Doppler imaging technics allow to recover the largest active structures (or active structure clusters) only for fast-rotating stars ([[*i.e.*]{}]{}, young, active ones) but not for old, slow-rotating solar-type stars [@rice02]. The structure properties, as well as the convection ones, are mostly unknown and we rely only on indirect and global estimators such as the Calcium (Ca) index [see [[*e.g.*]{}]{} @baliunas83; @noyes84; @hall04].
The Sun is an exceptional star, for which we have a lot more of information. Indeed, both solar dark spots and bright features are well observed and the various solar activity properties are thus well described. The Sun, if seen as a moderately active star, therefore represents an ideal prototype to study the impact of activity on low-mass planets detectability. In a series of papers, we modelled with a great accuracy the solar activity pattern using detailed observations ([[*i.e.*]{}]{}, spots catalogs and magnetograms) extending over the full solar Cycle 23. We then rebuilt the induced photometric variations and the corresponding spectra, and deduced from the latter the RV time series. We first considered only cold spots in [@lagrange10] (hereafter Paper I), studying their impact on the detectability of an Earth-mass planet in the HZ of a Solar-like star. In [@meunier10a] (hereafter Paper II), we extended this study by considering in addition the contribution of bright faculae and, for the RV, the attenuation of the convective blueshift in magnetically active regions. Combining the effect of the three activity components, we found out that the attenuation of convective blueshift in faculae (hereafter the convective component) dominated the induced RV variations (this happens as the Sun is a slow rotator: for larger [$v\sin{i}$]{}, the photospheric effect of spots becomes dominant). We showed that unless with correction tools (still to be identified and tested) it would be impossible to detect an Earth-like planet in the HZ. Using the same simulation as in Paper II, [@lagrange11] (hereafter Paper III) estimated the astrometric effect of stellar magnetic activity. Finally, in [@meunier13] (Paper IV), we used the strong correlation between the convective component and the Ca index (as both are directly related to the chromospheric plage or photospheric facula filling factor) as a tool to correct RV time series from the convection-induced jitter. We showed that an Earth-like planet in the HZ would become detectable given that: [*i*]{}) there is an excellent signal-to-noise ratio (hereafter S/N) on the Ca index data; and [*ii*]{}) the temporal coverage of the RV observations over one activity cycle is very dense (typically one night out of four during the cycle).
The objective of the present paper is to extend the previous study to a Sun-like star seen under any inclination. To do so, we fully parameterize the activity pattern of a Sun-like star. A few activity parametrizations have already been developed. [@barnes10] undertook the first tentative to parameterize dark spot distributions at low and high activity levels in the case of M-type stars. A similar simple model was used by [@jeffers14], this time for young and active G and K dwarfs. A parametrization of an activity pattern was made by [@dumusque11b], who simulated dark spot distributions over the stellar surface for different levels of activity of a solar-like star. The authors derived detection limits for different activity levels and observational strategies. This study was however limited to dark spots and did not include the flux effect of bright features (faculae and network), nor the inhibition of the convective blueshift in active structures.
Using the Sun as a template presents several important advantages: [*i*]{}) we have much more information on solar activity than for any other star; [*ii*]{}) the validity of such a model can be easily asserted by a comparison with the previous papers; and [*iii*]{}) such an activity model can be adapted to other spectral types and other activity levels than the Sun, if the active region configuration is supposed similar. In addition, it allows studying the impact of stellar inclination on the results, contrary to the solar observations for which only one hemisphere is observed at any given time.
In Sect. \[simus\], we build the spot and facula distributions on the Sun surface over a full solar cycle. We describe in details the laws and parameters we used to make these simulations. To test our approach, we then compare the obtained structure distributions with the observed solar distributions used in paper II. In Sect. \[results\], we compute the induced RV variations, taking into account both the photometric contribution of spots and faculae and the attenuation of the convective blueshift in active structures. Then we compare our results with those previously obtained using observed solar magnetic features. In Sect. \[inclin\], we study the influence of the stellar inclination on the resulting RV and photometry. The astrometric time series will be treated in a separate paper. We also compute the corresponding detection limits. We finally discuss our results and the use of our model for other stars in Sect. \[conclu\].
Building the activity pattern {#simus}
=============================
Approach
--------
We consider the solar activity pattern during Cycle 23. We use the Sun as a template and as a mean of comparison with the observed activity pattern used in Papers I and II. We parameterize an extended range of activity levels and timescales. We start from the general distribution of the activity level over the cycle, then we parametrize the spatial and temporal distributions of active structures and their dynamics, and we finally modelize the individual behaviour of the structures, including the dark spots, the bright faculae around these spots and the network smaller features. The two outputs of our simulations are the lists of the dark spots and of the bright features, respectively. Each file gives, for each time step of the simulation, the structure sizes (in millionth of hemisphere, hereafter $\mu$Hem), latitudes and longitudes. For a given set of parameters, these files will then represent the inputs of our simulation tool to produce the spectra and the corresponding RV at each time step, as it was done in Papers I and II with observed solar structure lists.
Building the spot and facula catalogs: input parameters {#listparam}
-------------------------------------------------------
All input parameters are summarized in Table. \[parameters\]. We detail them hereafter.
1. [*Global activity level.*]{} The sunspot-induced solar activity follows a cycle of $12 \pm 1$ years. To mimic as accurately as possible the global activity level and its distribution over such a cycle, we used the relative sunspot number, or Wolf number (noted $R$), as a proxy. For this study, we based our simulations on solar Cycle 23. This allowed us both to build a realistic activity pattern and to compare our results with those previously obtained in Paper II. The daily Wolf number $R$ for Cycle 23 was recovered at the Solar Influences Analysis Data Center (SIDC) and is displayed in Fig. \[wolfnbr\] (black dots, upper panel). We first smoothed the daily $R$ data to obtain a long-term reference $R_{\rm smth}$ (Fig. \[wolfnbr\], solid line, two top panels) for the cycle shape.
![[*Top*]{}: daily ([*dots*]{}) and smoothed ([*bold line*]{}) Wolf number over solar cycle 23. [*Middle*]{}: same smoothed Wolf number ([*black line*]{}), and when randomly adding a second order polynomial dispersion ([*red dots*]{}). [*Bottom*]{}: Power spectra of the observed daily ([*black*]{}) and simulated ([*red*]{}) Wolf numbers (respectively $R$ and $R^{'}$). The solar rotation period ([*blue solid line*]{}) and its first harmonic ([*blue dashed line*]{}) are also displayed for comparison.[]{data-label="wolfnbr"}](wolf.ps){width="1\hsize"}
The observed daily dispersion of the Wolf number comes from two contributions: the first comes from the rotational modulation and will naturally be present in the model (hence the smoothing); the second is due to the fact that appearing spots do not follow a smooth curve and present some dispersion originating in the dynamo process. To take this contribution into account, we calculated a second order polynomial dispersion $d$ to the smoothed curve $R_{\rm smth}$, expressed as: $$d = d_{0} + d_{1}R_{\rm smth} + d_{2}R_{\rm smth}^{2}$$ where $d_{0}$, $d_{1}$, $d_{2}$ are derived empirically from observations. We then randomly added this dispersion $d$ to the smoothed data $R_{\rm smth}$ to obtain a daily input Wolf number $R^{'}$ (Fig. \[wolfnbr\], red dots, middle panel). The Wolf number ($R$, $R^{'}$) is a combination of the number of individual spots and of the number of spot groups. As the input of our simulation is the total number of spots $N$ (which accounts both for the individual ones and for those in groups), we have derived a calibration to deduce this total number of spots from an activity level expressed in Wolf number units: $$\label{wolfeq}
N = n_{0} +n_{1}R^{'}$$ where $n_{0}$, $n_{1}$ are derived empirically from observations. Using Eq. \[wolfeq\], we obtained the theoretical total number of sunspots expected on the solar surface at each time step. The conversion $n_{0,1}$ and dispersion $d_{0,1,2}$ coefficients used in our simulations are given in Table. \[parameters\]. By subtracting the number of sunspots already existing at this step, one can finally determine the number of new sunspots to be generated at this step in the simulation.\
We finally display in Fig. \[wolfnbr\] (bottom panel) the power spectra of the observed $R$ and simulated input $R^{'}$ Wolf number for comparison purpose. The periodograms are quite different, especially at the rotational period and its first harmonic. This is expected as we do not introduce the rotational modulation (which is naturally present in the observed Wolf number) in our input Wolf number $R^{'}$ but rather at the next step of the simulations (see above). Concerning the high-amplitude (and short-period) variations of the observed Wolf number in the high activity solar period, we consider that these variations are not due to a lack of spots at some moments in our model compared to the observations, but rather to the presence at these times of a group of large spots or an active cluster in the solar observations (see below for the spot size and spot filling factor distributions). Our model would require a more complex spot distribution function to reproduce such time-to-time appearances of large spot groups and we decided not to complicate it at this stage.\
We assumed an activity cycle length of 12.5 years (length of solar Cycle 23) and a time step of one day, leading to a total number of 4566 iterations or days. Furthermore, we added a random scattering of the time steps, with an amplitude of 4 hours around the regular daily period, to mimic real observations.\
2. [*Spatial and temporal distributions of the structures*]{}. At the beginning of the activity cycle, sunspots appear at medium latitudes, with some dispersion. As the activity level increases, the mean appearance latitude of sunspot slowly decreases in absolute value. This migration of active regions towards the equator continues during the whole cycle leading to a low mean appearance latitude at the end of the cycle. In our simulations, the mean latitudes of sunspot appearance at the beginning and at the end of the cycle were fixed at $\pm $22 and $\pm$ 9, respectively. These values are derived from the spot catalogs we used in Papers I and II. A linear law was then used to fit the decrease of the sunspot appearance latitude during the cycle, and we added a latitude scattering to reproduce the butterfly diagram.\
We also had the possibility to create an asymmetry between the activity patterns in the northern and southern hemispheres, [[*i.e.*]{}]{} by adding more spots in one of them. As such a significant asymmetry has not been detected on the Sun during past solar cycles, we decided to put the same proportion of spots on each hemisphere.
Spot groups are finally well-known not to appear at random longitudes, but around the so-called active longitudes. Such preferred longitudes have been detected on the Sun as well as on several other stars [@berdyugina03; @ivanov07; @lanza09; @lanza10]. Their origin is still partly unexplained and is probably due to the dynamo process. Some trends seem to emerge from observations : there are generally two (seldom three) persistent active longitudes per hemisphere, shifted by $180\degr$. They form a rigid structure, although they cannot be fixed in a reference frame because of the differential rotation [@berdyugina03].
On the Sun, magnetic field and new activity seem to emerge at locations where activity is already present [see [[*e.g.*]{}]{} @harvey93]. Thus, instead of putting active longitudes with fixed longitude values in our model, we decided to build active longitudes from the position of already present sunspots. A fraction of new sunspots then appears in a restricted area in longitude around already present sunspots, while the remnant fraction is uniformly distributed in longitude. The fraction of new sunspots appearing in theses active areas and the longitude extension of the latter are reported in Table. \[parameters\].\
3. [*Large scale dynamics*]{}. We describe here the global motion of magnetic features at the solar surface during their lifetime. This motion can be decomposed in two components, namely a longitudinal and a latitudinal motion.\
The Sun is well known to exhibit a differential rotation in latitude: active structures rotate faster at the equator than at the poles. The longitudinal motion or rotation rate of the active structures $\omega$ (in degree per day) is therefore a function of the latitude $\theta$, according to the following equation [[[*e.g.*]{}]{} @ward66; @meunier05c]: $$\omega = \omega_{0} + \omega_{1}.sin^{2}(\theta) + \omega_{2}.sin^{4}(\theta)$$ The latitudinal motion or meridional flow $M$ of magnetic features is derived from [@komm93]. It is a poleward motion in each hemisphere, which is also a function of the latitude : $$M(\theta) = \alpha.sin(2\theta) + \beta.sin(4\theta)$$
Parameter Value Unit Reference
---------------------- -------------------------------------------- ---------------------------------- ---------------------------------- -----------------
Solar activity Reference cycle Solar Cycle 23
Cycle length 12.5 \[year\]
Time step 1 \[day\]
Time step random dispersion 4 \[hour\]
Wolf number normalization $n_{0}$ = 1.32
$n_{1}$ = 0.148
Wolf number random dispersion $d_{0}$ = 6.922
$d_{1}$ = 0.7594
$d_{2}$ = -0.00348
Spatio-temporal Mean start latitude $\pm 22$ \[$\degr$\]
distribution Mean end latitude $\pm 9$ \[$\degr$\]
Standard lat. dispersion 6 \[$\degr$\]
Max. lat. dispersion 20 \[$\degr$\]
North/South asymmetry 0.5 -
Active longitude spot fraction 0.4 -
Active longitude extension area $\pm 20$ \[$\degr$\]
Large scale dynamics Spot differential rotation $\omega_{0} = 14.523$ \[$\degr$/day\] [@ward66]
$\omega_{1} = -2.688$ \[$\degr$/day\] This paper.
$\omega_{2} = 0$ \[$\degr$/day\] This paper.
Facula and network $ \omega_{\mathrm{b0}} = 14.562$ \[$\degr$/day\] [@meunier05c]
differential rotation $ \omega_{\mathrm{b1}} = -2.04$ \[$\degr$/day\] This paper.
$\omega_{\mathrm{b2}} = -1.49$ \[$\degr$/day\] This paper.
Meridional flow, all structures $\alpha = 12.9$ \[[ms$^{\rm -1}$]{}\] [@komm93]
$ \beta = 1.4$ \[[ms$^{\rm -1}$]{}\] This paper.
Stellar radius $ 1 R_{\mathrm{\sun}} = 696400$ \[km\]
Spots properties
Isolated spots
Total fraction 0.4 - [@martinez93]
Mean initial size 46.51 \[$\mu$Hem\] [@baumann05]
Standard size deviation 2.14 \[$\mu$Hem\] This paper.
Max. size 1500 \[$\mu$Hem\] Papers I and II
Mean decay -18.9 \[$\mu$Hem/day\] [@martinez93]
Median decay -14.8 \[$\mu$Hem/day\] This paper.
Complex spot groups
Total fraction 0.6 - [@martinez93]
Mean initial size 90.24 \[$\mu$Hem\] [@baumann05]
Standard size deviation 2.49 \[$\mu$Hem\] This paper.
Max. size 5000 \[$\mu$Hem\] Papers I and II
Mean decay -41.3 \[$\mu$Hem/day\] [@martinez93]
Median decay -30.9 \[$\mu$Hem/day\] This paper.
Both spot types
Min. decay value -3 \[$\mu$Hem/day\]
Max. decay value -200 \[$\mu$Hem/day\]
Min. spot size 10 \[$\mu$Hem\] Papers I and II
Faculae properties [*q (facula-to-spot ratio)* ]{}
[*Mean log(q)* ]{} 0.8 -
[*Standard deviation (log(q))* ]{} 0.4 -
[*Min.- Max. log(q)* ]{} 0.1 – 5 -
Mean decay -27 \[$\mu$Hem/day\]
Median decay -20 \[$\mu$Hem/day\]
Min. facula size 3 \[$\mu$Hem\] Papers I and II
Network properties Diffusion coefficient 300 \[[km$^{\rm 2}$s$^{\rm -1}$]{}\] [@schrijver01]
[*Remainder fraction for decay*]{} 0.975 \[ /day\]
Min. size 3 \[$\mu$Hem\] Papers I and II
[*Facula fraction recovered in network*]{} 0.8 -
4. [*Spot properties*]{}. Spots appear preferentially with a given size, increase in area rapidly before slowly decreasing. The spot growing phase is very quick [in average 10 to 11 times faster than the decay phase, see @howard92a]. It is then often shorter than our one-day timestep for the small and mid-sized sunspots and of a few days for the largest ones. As we focus here on long-period planets, and as the active structure growing phase is smaller or of the order of our simulation time step, we decided not to include it in our model for the moment and assumed that spots appear with their maximal size and then decrease during all their lifetime. We will however include a description of the growing phase in future works to better describe the variations at small timescales. Thus the spot distribution can be described with only the initial size distribution and a decay law. According to the La Laguna classification [see @martinez93 and ref. therein], one can distinguish between isolated spots (La Laguna type 3) and spots belonging to complex groups (La Laguna type 2). In compliance with this classification, we built two distinct spot distributions. Both obey to similar evolution laws, but with different parameters. The initial spot size distribution can be fitted with a log-normal law, according to [@baumann05]. The two input parameters are therefore the mean spot size and the standard spot size deviation. We took the parameters used in our simulations from the snapshot model developed in [@baumann05]. The spot decay law has been parametrized by [@martinez93] as a log-normal law, with two input parameters : the mean and median decay values. We also put an upper size limit for the two types of spots, and a lower threshold of 10 $\mu$Hem for all spots, chosen to be in good agreement with the observed spots distribution used in Paper II.\
5. [*Facula properties*]{}. On the Sun, spots and faculae form active regions, spots being surrounded by large faculae. The ratio between the surface covered by faculae and spots (hereafter the size ratio) has been studied [@chapman97; @chapman01; @chapman11]. The authors found an average size ratio ranging between about 13 and 45 over solar cycles 22 and 23, depending on the method used to estimate the facular contrast. However, this corresponds to an average ratio measured for structures that can be at any state of evolution and any size. As for our simulation, we need instead an initial size ratio. Indeed, each time we add a spot, we add a facula at the same place. The facula size distribution then depends only on the initial size ratio. As there is little in the literature about this quantity, it is one of our few free parameters. We assume a log-normal law to describe the initial facula-to-spot size ratio. Facula decay is then described by two processes:
- First the “classical” decay is described as for dark spots, with a log-normal law. This decay corresponds to a certain surface being lost by the facula at each time step of the simulation.
- Second, a given proportion of this facula lost surface is converted into network (the rest being completely lost, for example by flux cancellation or submergence): facula fragments break away from active regions and diffuse over the surface.
As the facula distribution is based on the spot distribution, the facula growing phase is not reproduced for the moment in our model. The facula growth and decay are longer than the spot ones in average, due to their longer lifetimes [@howard92a], but the facula growth is still much shorter than its decay [see [[*e.g.*]{}]{} @howard91], of a few days then. We decided not to reproduce it to remain coherent with the spot distribution and as we focus mainly on timescales between a fraction of the rotational period and a complete activity cycle, and on long-period planets.\
6. [*Network properties*]{}. We chose to make the network originates in the decay of faculae (see point 5). Network behavior is then managed by its diffusion coefficient and by its decay rate. The network diffusion coefficient has been studied in many papers which provide a great range of values [@cadavid99 and ref.therein].
![[*Upper panel*]{}: observed (black line) and simulated (red line) spot sizes. [*Lower panel*]{}: same for bright feature sizes.[]{data-label="histo_size"}](comp_size.ps){width="1\hsize"}
As these authors pointed out, there is a large gap between the diffusion coefficient values found by modeling (which are usually higher than 500 [km$^{\rm 2}$s$^{\rm -1}$]{}) and the values found with magnetograms analysis (which span from a few tens of [km$^{\rm 2}$s$^{\rm -1}$]{} to less than 300 [km$^{\rm 2}$s$^{\rm -1}$]{}). They also raised out the possibility for the diffusion coefficient to be time-dependent.\
We finally decided to keep the value of 300 [km$^{\rm 2}$s$^{\rm -1}$]{} used by [@schrijver01] as it stands well between the two extreme groups of values. We assume the diffusion process to be isotropic [@cadavid99]. As for the decay rate, we simply described it by fixing the network remainder fraction at each time step (it is one of the model free parameters). We finally put a minimal size of 3 $\mu$Hem for all bright structures (faculae or network), again in compliance with Paper II.
Comparison with the observed activity pattern {#comps}
---------------------------------------------
Here we compare the structures generated with our parameterized model to the observed structures used in Paper II. To do this, we use the spot and bright feature lists over Cycle 23 that we retrieved from sunspot catalogs and MDI/SOHO magnetograms, respectively (Paper II). These data sets have some gaps, leading to a coverage of 3586 days, for a total duration of 4171 days, whereas our simulations have a length of 12.5 years with a daily sampling, [[*i.e.*]{}]{} 4566 days (length of Cycle 23). We therefore applied to our outputs the same calendar as in Paper II so as to make a relevant comparison. The full time series will be studied in Sect. \[inclin\].
### Size distributions {#sizedist}
{width="0.9\hsize" height="0.55\hsize"}
The size distributions (in $\mu$Hem) of our simulated structures are displayed in Fig. \[histo\_size\] and compared with the observed ones from Paper II. We want to highlight the fact that very few of the input parameters detailed in Table \[parameters\] are free parameters. For most of them, we retrieved the input laws and parameters from the literature so as to build a fully parameterized model of solar-like activity.\
For the dark spot properties, we have no free parameters ([[*i.e.*]{}]{}, they all originate from the literature). Remarkably, we find a good agreement between the spot size distribution simulated this way and the observed one. The lack of a few large spots (about 40 in all) for the simulated distribution is explained by the fact that the log-normal law usually used in the litterature slightly underestimates the number of very large spots [see also Figs. 2 and 3 in @baumann05]. The short-term differences reported above in the Wolf number distributions are then likely to originate in the time-to-time appearance of these very large spots (and not to a global lack of spots in our model, as the simulated distribution fits well the observed one). As for smaller spots, the observed distribution is discretized, which partly explained the gap (the remaining difference is again due to the fact that the observed distribution slightly differs from the log-normal law). This small discrepancy in the distribution for small spots will not influence significantly the RV signal as the RV effect of small spots is negligible compared to the larger ones.\
For the bright features ([[*i.e.*]{}]{}, faculae and network, hereafter), there are three free parameters (the facula-to-spot initial size ratio distribution, the facula fraction recovered in network and the network decay rate). These parameters are poorly documented, so we decided to adjust them to better fit the observed distribution. Overall, the simulated size distribution is in good agreement with the observed one (Fig. \[histo\_size\]). However, there are some remaining differences which are inherent to the model parameters. The size distribution for all observed bright features does not follow a log-normal law. For the simulated bright features, we remind that we chose to make the facula initial size distribution directly dependent on the spot distribution, and the network originate in the decrease of the faculae. This explains thus the “flattened-S” shape of the simulated distribution, which corresponds to the addition of the facula and network respective distributions. Reproducing even more accurately the shape of the observed distribution would require to make our model much more complex, for example by making the bright feature decay size-dependent or cycle-dependent. Given our goals, such a level of complexity is not necessary. As larger faculae have the largest influence on the RV, we decided to fit the distribution in priority for the highest sizes ([[*i.e.*]{}]{}, for sizes greater than 5000 $\mu$Hem) by adjusting our free parameters. There is then still a discrepancy between the observed and simulated distributions for the smallest sizes, but we consider that its effect on our observables will be mostly negligible at our level of precision.
### Latitude distributions
Another way to compare our simulated pattern to the observed one is to study the active structure latitude distributions. The latter are displayed in Fig. \[histo\_lat\]. By doing so we will be able to validate the large scale behavior of our model. The latitude distributions of observed and simulated structures are indeed in good agreement. The kind of oscillations that we can distinguish in the observed bright feature latitude distribution at high latitudes and with a 1-year period comes from a “seasonal” effect.
{width="0.9\hsize" height="0.52\hsize"}
The solar observations are not made exactly in the plane of the solar equator, and the active structure distribution seems to be slightly shifted towards one visible solar hemisphere or the other, depending on the observation time, due to the noise level in the observed magnetograms. This effect is also visible on the bright feature latitude histogram, where the observed distribution is spread a little more towards higher latitudes.
### Filling factors {#ffdist}
Moreover, we compare the filling factors of the projected active structures over the full solar cycle for the observed and simulated patterns in Fig. \[fillfactor\]. We find the observed and simulated filling factor time series to be in very good agreement. The main difference between the observed and simulated distributions is located during the high-activity period: the activity peak present in the observations between approximately JD 2452200 and JD 2452400 is not well reproduced in the simulations. This difference does not come from a global lack of active structures in the high activity period of our model because [*i*]{}), the simulated daily spot number comes from Wolf number observations and [*ii*]{}), the facula distribution directly depends on the spot one (Sect. \[listparam\]). It rather originates in an occasional concentration of a few great structures in the observed pattern that is not reproduced in our model. Thus, this discrepancy is not a bias of our model but can be rather considered more as statistical noise. In the case of the bright features only, we also note a slight discrepancy between the observed and the simulated filling factors during the low activity period (the simulated filling factor being higher than the observed one). We attribute this to the larger number of very small bright features injected in our model (see before).
Finally, we compare the ratio of the bright feature and spot filling factors over the solar cycle in Fig. \[size\_ratio\] to estimate the evolution of the facula-to-spot size ratio. We find the ratios for the observed and simulated time series to be in very good agreement. One can also note that the size ratio reaches high values (from 50 to 150) during the low activity period ([[*i.e.*]{}]{} at the beginning and at the end of the solar cycle), whereas it is much lower (around 20) during the major part of the cycle. This is because during periods of low activity, the number of dark spots is very low (equal or close to zero), whereas there is always a “background noise” due to bright features, and especially the network.
![Facula-to-spot size ratio averaged over 30 days, for observed ([*black*]{}) and simulated ([*red*]{}) structures. [*Blue solid line*]{}: average value over the cycle.[]{data-label="size_ratio"}](size-ratio.ps){width="0.9\hsize" height="0.65\hsize"}
We find the average value of the size ratios over the total cycle to be very close to each other, with values of 25.7 and 26.0 for the observed and simulated structures, respectively. These values are in agreement with the studies of [@chapman97; @chapman01; @chapman11]. We also remark that the size ratio time series displayed by [@chapman11] look very much like ours over cycle 23, with peaks above 100 during low activity periods and minima around 25 during high activity ones.
{width="0.9\hsize" height="0.25\hsize"}
Overall, we can conclude that our simulated solar activity pattern matches well the observed one and is thus a reliable model of a solar-like star magnetic activity. However, to definitely establish its validity, we compare the resulting RV and photometric time series. In particular, we check the impact of the differences between the two patterns on the time series.
Comparison with the Sun observations {#results}
====================================
In this section, we first describe the way we simulate the time series for the different observables and how we take into account the stellar properties (limb-darkening, active structures temperature contrasts, impact on convective blueshift, physical and geometrical stellar properties). We then compare the RV time series simulated in the case of an edge-on solar-like star to the RV time series obtained with the observed activity pattern described in Paper II to establish the validity of our model.
Description of the simulations
------------------------------
### Limb-darkening {#limbd}
In our previous papers, the center-to-limb darkening was linear with respect to $\mu$ [with $\mu = cos(\theta)$, $\theta$ being the angle to the center of the solar disc, @desort07], as follows: $$I(\mu) = 1 - \epsilon + \epsilon \ \mu$$ with $\epsilon = 0.6$. In this case, the limb-darkening was not temperature-dependent and therefore applied to both the inactive photosphere and the active structures indiscriminately. Here, we decided to change our limb-darkening law, so as to: [*i)*]{} have a more accurate one; and [*ii)*]{} use a temperature-dependent one, that could be adaptable to different types of stars. We took the non-linear limb-darkening law from [@claret03]: $$\frac{I(\mu)}{I(1)} = 1 - \sum_{k=1}^{4} a_{k}(1-\mu^{\frac{k}{2}})$$ where $I(1)$ stands for the intensity at the center of the stellar disc. The four Claret limb-darkening coefficients $a_{k}$ are the bolometric coefficients taken from ATLAS models [see @claret03]. These coefficients are temperature-dependent. That is why such a law is better adapted to our simulation tool, as the effective temperature is one of the input parameters, and such a law can be extrapolated not only for the Sun, but for any star for which the effective temperature is known. It also means that the limb-darkening coefficients applied to the spotted stellar surface will be slightly different from the ones applied to the inactive photosphere. We therefore use a different limb-darkening for the photosphere and for the dark spots, depending on their respective temperature. As faculae show a very strong contrast variation depending on their position on the disc due to more complex processes, we directly define their contrast $C_{pl}(\mu)$ with respect to the limb-darkened photosphere (see below) without using the Claret law according to their temperature.
### Spot and bright feature contrast and photometric time series
To estimate the contrast of an active structure (dark spot or bright feature) with respect to the quiet photosphere, we use the procedure described in Paper II. This procedure is an independent, preliminary step to our simulations ([[*i.e.*]{}]{}, to the generation of the spectra and of the RV and photometric time series), and is aimed at determining the active structure contrasts that we will use as input parameters. Starting from a given range of values for the spot and bright features contrasts, and based on observed structure patterns, we build (for each set of contrast values) two time series representing the respective relative contributions of dark and bright active structures to the stellar irradiance. A $\chi^{2}$ minimization between the sum of these contributions (quiet photosphere, dark and bright features) and the observed total solar irradiance (hereafter TSI) taken from [@frohlich98] is then performed over the solar cycle 23. We adopt the same value for the quiet Sun reference as in Paper II, [[*i.e.*]{}]{} $1365.46 \ W.m^{-2}$, which is very close to the average value found by [@crouch08] with their TSI model over twelve solar cycles. Contrary to Paper II, we include in the procedure the influence of the center-to-limb darkening, so as to take into account the Claret limb-darkening law we will now use in our simulations. We end up with a spot temperature deficit $\Delta T_{sp} = -605 K$ and a facula contrast $C_{pl} = 0.131618 - 0.218744\mu + 0.104757\mu^{2}$. We then use these contrasts as our input parameters in the simulations. As in Paper II, we finally compare the sum of the photometric contributions of spots and bright features obtained with our simulations with the observed TSI of [@frohlich98], so as to check the validity of our activity model. The comparison is done on 2263 points, ranging from 1996 to 2003.
The irradiance obtained with our simulations and the observed TSI of [@frohlich98] are displayed in Fig. \[phot\]. We match quite well the observed irradiance, excepted for some peaks in the high activity period that are not reproduced in the model. This can be explained by the fact that our model does not reproduce well the occasional appearance of very large active structures or large active structure clusters (see Sects. \[sizedist\] and \[ffdist\]). Consequently, the distribution of the simulated TSI is slightly narrower than the observed TSI one, and the averaged rms of the simulated TSI is about $15\%$ lower in comparison (see Table \[RVamp\]). However, the temporal evolution of the simulated TSI rms matches well the observed one. As already stated, we consider that these differences do not affect the significancy of our simulations as a model.
### Attenuation of the convective blueshift
For all following simulations, we adopt the same value for the attenuation of the convective blueshift as in Paper II, [[*i.e.*]{}]{} 190 [ms$^{\rm -1}$]{}. We refer to this paper for justification.
### Building the spectra
We build the spectra from the input structure lists as described in [@desort07] and in Papers I and II. We assume a stellar mass of 1 [M$_{\sun}$]{}, a temperature $T_{\rm eff}$ of 5800 K and a rotational velocity of 1.9 [kms$^{\rm -1}$]{} at the equator. We keep in this section the stellar inclination $i$ to 90, corresponding to a star seen edge-on. Briefly, we use a synthetic spectrum from Kurucz models [@kurucz93] corresponding to a G2V star and apply it to each cell of the visible stellar 3D hemisphere divided into a grid. The spectrum is shifted to the cell radial velocity. In case of the presence of an active structure, it is weighted with a black-body law, taking into account the active structure contrast with respect to the quiet photosphere. To include the effect of the attenuation of the convective blueshift, the cell spectrum is redshifted by 190 [ms$^{\rm -1}$]{}. This attenuation has other effects on the spectral lines, however [@dumusque14a] show that only considering the RV shift is sufficient to estimate the RV effect of active regions.
All cell spectra are then balanced by the cell’s projected surface and limb-darkening. We finally sum up all cell contributions to obtain the stellar spectrum.
### Computation of the RV time series
As in Paper II, we compute the RV using our Software for the Analysis of the Fourier Interspectrum Radial velocities [SAFIR, see @galland05] on the built spectra as if they were actual observed spectra. We use only the wavelength range corresponding to the order $\sharp~31$ of the [H[ARPS]{}]{} spectrograph, as done in Papers I and II. We obtain three distinct time series: two due to the respective photometric contributions of dark spots and bright features (hereafter the spot and facula time series, respectively); and the third due to the partial inhibition of the convective blueshift in the active structures (hereafter the convection time serie). For the latter, we consider only the inhibition of the convective blueshift ([[*i.e.*]{}]{} this effect is not weighted by the active region flux). We sum the three time series to obtain the total RV variations. Here we consider this sum to be a good approximation of the real RV variations given that the convective blueshift is dominant (see below).
![RV time series computed with Claret limb-darkening and corresponding structure contrasts versus RV time series computed with linear limb-darkening (as in Paper II). [*Top*]{}: contribution of spots and faculae. [*Bottom*]{}: sum of all contributions.[]{data-label="obsobsnew"}](rv_obs_vs_obsnew.ps){width="0.95\hsize"}
Validation of the activity model
--------------------------------
### RV time series based on the observed patterns {#sectcompobs}
In this section, we first check if the change of limb-darkening law and the corresponding change of the active structure contrasts have a significant impact on the resulting RV time series. To do so, we compute new RV time series corresponding to the observed solar activity patterns used in Paper II, but this time with the new limb-darkening law and structure contrasts. We then compare these new RV time series to the ones obtained in Paper II, using the same temporal sampling. We find the RV time series to be closely correlated and thus in good agreement. We display in Fig. \[obsobsnew\] the RV time series computed with the Claret limb-darkening versus the RV time series computed with the linear limb-darkening. We find both the slope of the fits and the correlation of the RV time series to be close to 1, as well in the case of the spot+facula RV time series as for the total RV ones. Despite the simpler limb-darkening law and a different active structure contrast, the results shown in Paper II are still valid for estimating the RV effect.
{width="0.95\hsize" height="0.95\hsize"}
Comparison between the RV time series based on the observed and simulated patterns {#comprvobssim}
----------------------------------------------------------------------------------
We compare here the RV time series based on the observed and simulated activity patterns (both being computed with the Claret limb-darkening) so as to assess the validity of our activity model. We compare the RV time series for the different contributions (spot, facula, convection and all). All time series, as well as their dispersion over the cycle and their histograms, are displayed in Fig. \[rv\]. We find the amplitudes of the observed and simulated RV time series to be in very good agreement for the spot and facula time series, with very similar histograms. As for the RV time series corresponding to the convective component, it is closely related to the facula filling factor (with a Pearson correlation coefficient of 0.97) and widely dominates the total RV signal, as in Paper II. The main visible difference comes from the activity peak at around JD 2452200-2452400 in the RV reconstructed from observations, which is not echoed in the simulated RV. We already discussed the origin of this difference in Sect. \[comps\]. In the low activity period, the averaged amplitude of the simulated convection time series is about 20% higher than for the observed one (see Table \[RVamp\]). We attribute this small discrepancy to the excess of very small bright features in the model that we discuss Sect \[comps\].
![Lomb-Scargle periodograms of the RV time series based respectively on the observed ([*black solid line*]{}) and simulated ([*red solid line*]{}) activity patterns, alongside with the 1$\%$ false-alarm probabilities (FAP, [*dashed lines*]{}). [*Top*]{}: spot+facula contribution. [*Bottom*]{}: sum of all contributions. Note that in both panels the periodograms corresponding to the simulations are vertically shifted for visibility. The solar rotation period and its two first harmonics are displayed in blue ([*solid*]{}, [*dashed*]{} and [*dashed-dotted blue lines*]{}).[]{data-label="comp_perio"}](COMP_perio_rv.ps){height="1.3\hsize"}
To have a better understanding of the RV signature of our simulated activity pattern, we also display the RV rms computed over 30-day intervals over the cycle in Fig. \[rv\]. This gives an idea of the temporal evolution of the RV time series dispersion. We find the evolution of the RV dispersion during the solar cycle to be in good agreement for the observed and simulated time series. We also provide the RV rms for the different components taken over the complete cycle and for low and high activity periods in Table \[RVrms\], and the RV amplitudes in Table \[RVamp\]. The low and high activity periods are the same as in Paper II. We note that:
- For the spot and facula simulated time series, the rms is slightly higher than for the observed time series in the low activity period (about two times higher in the case of bright features). However it is not the case for the convective time series.
- In the case of the convective component, the rms over the complete cycle is about 24% lower for the simulated data than for the observed data. When looking at the temporal evolution of the RV rms displayed Fig. \[rv\], we can see that this discrepancy mostly comes from a few high activity peaks in the observed time series.
We finally compare the Lomb-Scargle periodograms of the RV time series based on the observed and simulated activity patterns. The periodograms are displayed in Fig. \[comp\_perio\] in the case of the “photometric” contributions (spots and faculae) and of the sum of all contributions. The respective periodograms for the observed and simulated patterns remarkably present the same characteristics:
- For the “photometric” component, the power is mostly concentrated at the solar rotation period and its two first harmonics.
- For the sum of all contributions, there is still power at the rotation period, but there is much more power at longer periods (between 300 and 1000 days) due to the long-term activity cycle.
Simulated pattern spots faculae sp+fac conv. total
------------------- ------- --------- -------- ------- -------
All 0.34 0.32 0.33 1.98 2.00
Low 0.17 0.16 0.14 0.48 0.51
High 0.47 0.38 0.47 1.36 1.52
Observed pattern spots faculae sp+fac conv. total
All 0.37 0.25 0.32 2.59 2.62
Low 0.10 0.08 0.07 0.47 0.47
High 0.52 0.35 0.42 1.50 1.53
: []{data-label="RVrms"}
Simulated pattern rms RV ampl RV rms phot
------------------- -------- --------- ------------------
All 2.00 11.3 $2.83 \ 10^{-4}$
Low 0.51 2.5 $1.06 \ 10^{-4}$
High 1.52 7.7 $2.97 \ 10^{-4}$
Observed pattern rms RV ampl RV rms phot
All 2.62 11.4 $3.6 \ 10^{-4}$
Low 0.47 2.1 $1.2 \ 10^{-4}$
High 1.53 8.4 $4.5 \ 10^{-4}$
: []{data-label="RVamp"}
Nonetheless, we note a significant difference: for the spot+facula RV signal, the peaks at half the rotation period are emphasized compared to the peaks at the rotation period in the case of the observed pattern, whereas it is the contrary for our simulated pattern. We already noticed and discussed in Paper II the predominance of the power at half the rotation period in the RV time series derived from observed solar patterns. We explained it by the presence of two symmetrically active longitudes on the solar surface. Since here we do not impose two active longitudes separated by 180$\degr$ over the whole time series but rather variable active longitudes, this effect may not be very important. An alternative explanation could be that the difference between the periodograms corresponding to observed and simulated patterns may originate in the active structure quick growing phase (which is at present not reproduced in our model).
![[*Left*]{}: projected spot filling factor at maximum activity (in fraction of the stellar surface) versus stellar inclination [*i*]{}. [*Right*]{}: same for bright features.[]{data-label="varfp"}](var_fp.ps){width="0.7\hsize" height="1.2\hsize"}
This may also explain the smaller RV and photometric rms when compared to the observations.\
As detailed above, small discrepancies in amplitude and dispersion on short timescales are present both in the photometry and in the RV between the time series derived from observed and simulated activity patterns. We consider that these differences likely originate in two sources: first, the spot size distribution, where the few larger spots are hard to model; and second, the active structure growing phase. Overall, we consider that the temporal evolution over the complete cycle of both the amplitude and dispersion of the simulated time series match well the time series derived from the observed activity pattern. Hence, we conclude that: [*i*]{}) these short-term discrepancies are not significant enough to question our model reliability; and [*ii*]{}) the comparison between the RV time series derived from observed and simulated patterns then assess the overall validity of our model.
Photometric and RV time series of inclined solar-type stars {#inclin}
===========================================================
![[]{data-label="varphoto"}](var_rms_photo.ps){width="0.7\hsize" height="1.2\hsize"}
For solar-type stars of the same age as the Sun, it is commonly assumed that active structures are mostly concentrated in a belt around the stellar equator (even if bright features are much more dispersed in latitude). A different inclination of the stellar rotation axis should then have a significant effect on the various time series that needs to be investigated. In this section, we perform the same simulations as above, but for different inclinations of the stellar rotation axis. We consider inclinations between $i = 10\degr$ ([[*i.e.*]{}]{}, for a star seen nearly pole-on) and $i = 90\degr$ (star seen edge-on), with a sampling of 10. The time series are studied with their original temporal sampling, [[*i.e.*]{}]{} 4566 days and no gaps.
Photometric time series {#phot_serie}
-----------------------
The impact of the inclination [*i*]{} of the stellar rotation axis on the long-term solar irradiance variations was studied by [@knaack01]. The authors computed the solar irradiance corresponding to a 3-component model (quiet Sun, dark spot and bright facula) at activity extrema and and for a variable inclination. As for their active region distributions, they used simple active latitude belts with contrast corresponding to the given active structure ([[*i.e.*]{}]{} no individual structure was introduced in their model). According to them, the apparent active structure surface coverage decreases with a decreasing [*i*]{}. They nonetheless expected an increase of the TSI with a decreasing [*i*]{} since bright features are limb-brightened while the contrast of dark spots is roughly independent of the limb-darkening. Thus, when going from an edge-on toward a pole-on configuration, the impact of dark spots would decrease following their apparent covered surface. On the contrary, the decrease of the apparent surface covered by bright features would be at least compensated by their increased contrast since we would see them mainly on the limb. We display in Fig. \[varfp\] the evolution of the spot and facula projected filling factors with [*i*]{}. We find the projected spot filling factor to decrease by about $35\%$ when going from edge-on ([*i*]{}$= 90\degr$) to nearly pole-on ([*i*]{}$= 10\degr$), and the projected facula filling factor to decrease by about $43\%$. These results are quite similar to those found by [@knaack01]. We note that the filling factor does not tend towards 0 for $i \simeq 0\degr$.\
The relative variation of the TSI during the cycle (which is of the order of $0.1\%$ of the quiet Sun irradiance) vs. [*i*]{} is displayed in Fig. \[varphoto\] (upper panel). We find it to increase by only $14\%$ when going from [*i*]{}$= 90\degr$ to [*i*]{}$= 10\degr$. This is much smaller than the $40 \pm 10 \ \%$ increase predicted by [@knaack01] with comparable input parameters but a much simpler model. Yet, it points towards the same verdict as pulled through by [@knaack01], [[*i.e.*]{}]{} that photometric variations of seemingly inactive Sun-like stars cannot be explained by an inclination effect. We conclude that for a solar-like star with a similar activity configuration, the effect of an inclined rotation axis on the long-term variations of the total irradiance (of the order of the solar cycle length) is relatively small. On the contrary, the TSI short-term variations [which were indeed not studied by @knaack01] are strongly impacted by the inclination. As illustrated in Fig. \[varphoto\] (lower panel), we find the TSI short-term dispersion to be decreased by a factor $\sim 6$ when going from [*i*]{}$= 90\degr$ to [*i*]{}$= 10\degr$. A possible explanation for such a decrease is the following:
- First, for smaller inclinations, we mainly see the effect of bright features as they are more extended towards the higher latitudes than the dark spots.
- Then, due to the activity configuration (where active structures are mainly located on two belts on both sides of the solar equator), for small inclinations we see the same structures during all the rotation period and not during half a period.
Therefore, in the case of a star seen nearly pole-on, the short-term dispersion of the irradiance (of the order of a few rotation periods) originates no more in the structure crossing of the visible hemisphere for each half rotation period. It originates only in the structures appearance and decay. To confirm it, we display in Fig. \[planche\] the Lomb-Scargle periodograms of the simulated TSI for representative inclinations. For a star seen edge-on, the periodogram is dominated by power at the rotation period of the star. When going towards smaller inclinations, the signal at the rotation period gradually decreases until it disappears completely for a star seen nearly pole-on. On the contrary, the signal at a much longer period (which is likely induced by cycle-related long-term periodicities of the order of the cycle length) becomes increasingly preponderant for smaller inclinations. A reason for which the signals at long term periods (in the 600 to 1500-day range) become increasingly dominant with decreasing [*i*]{} in the TSI (and to a lesser degree in the total RV) periodograms could be the following: for nearly pole-on configurations, we see the active structures on one stellar hemisphere only, whereas we see them on two hemispheres for configurations closer to edge-on. This may induce a kind of an averaging effect on the long-term signal for the edge-on configuration and explain its increase with decreasing [*i*]{}. We finally display in Fig. \[photo1090\] the simulated TSI for representative configurations. As we found above, the long-term amplitudes of the two time series are nearly the same; on the contrary the dispersion is widely reduced with [*i*]{}.
RV time series {#RVINC}
--------------
We now study the impact of stellar inclination [*i*]{} on our simulated RV time series. Our first main result is that in contrast with the photometry, both the amplitude and the dispersion of the total RV decrease with a decreasing [*i*]{}. This is well illustrated in Fig. \[rv1090\] where we display the RV time series for all activity components and for significant inclinations. The amplitude as well as the dispersion (RV rms) of the total RV taken over the complete cycle are decreased by a factor $\sim 6$ when going from an edge-on to a nearly pole-on configuration.
We characterize both the long-term and short-term variations of the RV signals to investigate deeper the impact of inclination. Our results are illustrated in Fig. \[rvrms\]. The “long-term” variations simply correspond to the signal taken over the complete stellar cycle. To study the short-term variations of the signal, we perform a running average of the RV time series with a smoothing window of 30 days and substract it to the original data. Then the long-term variations should primarily be affected by the global activity cycle, whereas the short-term variations will come from rotation-related effects.
1. [*Peak-to-peak amplitude*]{}: For the total RV and when going from $i=90\degr$ to $i=10\degr$, the amplitude decreases by $\sim 80\%$ over the cycle and by $\sim 85\%$ on the short term. When taking only the “photometric contribution” ([[*i.e.*]{}]{}, spots and faculae) into account, the peak-to-peak amplitude is decreased by nearly $90\%$ on the long-term, and by $87\%$ on the short-term..
2. [*Dispersion*]{}: the total RV rms decreases by $70\%$ with inclination on the long term, and by $85\%$ on the short term. In the case of spots and faculae only, the decrease is the same on both timescales and is of nearly $85\%$. For a star with a solar-like activity pattern seen almost pole-on, a jitter of $\sim 0.5$ [ms$^{\rm -1}$]{} can be expected if the observation timescale is of the order of the activity cycle length, and a jitter of $\sim 0.2$ [ms$^{\rm -1}$]{} can be expected over a month.
3. [*Ratio between “photometric” and convective components*]{}: we also study the evolution of the relative contribution of the “photometric” ([[*i.e.*]{}]{}, due to spots and faculae) component to the total RV signal with the inclination. Over the cycle, the “photometric” fraction of the RV is reduced by a factor $\simeq 2$ when going from $i=90\degr$ to $i=10\degr$ ([[*i.e.*]{}]{}, the convective component is increasingly preponderant with a decreasing inclination). On the contrary, on the short-term the “photometric” relative contribution remains nearly constant with [*i*]{}, at a level of $\sim 0.3$. This value is at least two times larger than the value of the “photometric” fraction on the long-term ([[*i.e.*]{}]{}, $\sim 0.16$ in the edge-on configuration). This confirms that the spot+facula component has mostly a short-term effect on the RV signal, of the order of the rotation period.
We finally study the Lomb-Scargle periodograms of the RV time series for representative inclinations, first in the case of the sum of the flux contributions of dark and bright structures, and then when adding the convective component. The periodograms are displayed in Fig. \[planche\]. Starting from i $=90\degr$, the RV periodogram for the spot+facula signal is dominated by the stellar rotation. The power is concentrated at the rotation period and its two first harmonics, with about two times and six times less power for the first and second harmonics, respectively. The power is induced by the active structure crossing of the visible hemisphere during about half the rotation period (as the active structures are mainly located near and around the stellar equator). When [*i*]{} decreases, we begin to see the active structures on a time longer than half the rotation period, until we see them permanently for the nearly pole-on configuration. There is thus nearly no remaining component in the spot+facula RV periodogram for $i=10\degr$. This is in good agreement with the $90\%$ decrease in jitter already described.\
Starting from the edge-on configuration, the total RV periodogram is dominated: [*i*]{}) by a serie of peaks at long period (200 - 1000 days) due to the long-term activity cycle; [*ii*]{}) by peaks at the stellar rotation period and its three first harmonics.
![[]{data-label="photo1090"}](photo_1090.ps){width="0.9\hsize" height="1\hsize"}
Expectedly, the peaks corresponding to the rotation period and its harmonics gradually decrease when going towards smaller inclinations, until they nearly disappear for $i=10\degr$. We also note that the harmonics of the rotation signal decrease significantly faster than the rotation signal itself. On the contrary, the long-term signal corresponding to the activity cycle decreases slowly when $i$ decreases, but remains widely above the $1\%$ false-alarm probabilities (FAP) for $i=10\degr$. In the edge-on configuration, the power is equally distributed between the rotation and the long-term signals, but the latter become preponderant for smaller inclinations. This is in agreement with the decrease of the ratio of the RV “photometric” component over the total RV signal on the long-term we show above. We conclude that in the case of a star with a solar-type activity pattern, the stellar inclination strongly impacts both the observed stellar irradiance, RV variations and active regions filling factor (and hence the observed chromospheric activity). Indeed, the short-term variations of these various observables strongly decrease with a decreasing inclination. Then, on short to medium timescales (from a few days to a year, or well under the activity cycle length), it will probably be not possible to distinguish between a solar-like star seen nearly pole-on and an inactive star. Provided that the time baseline is sufficient, on timescales of the order of the cycle length, the activity cycle should however still be detectable for small inclinations, but with a reduced amplitude. It would then not be possible to distinguish it from a nearly inactive star with a low-amplitude activity cycle.
![Simulated total RV for significant inclinations. [*From top to bottom*]{}: $i=90\degr$ ([*black*]{}), $i=60\degr$ ([*purple*]{}), $i=30\degr$ ([*red*]{}) and $i=10\degr$ ([*orange*]{}). Vertical offsets have been introduced for visibility.[]{data-label="rv1090"}](rv10_90.ps){width="0.9\hsize" height="1\hsize"}
{width="0.8\hsize" height="1.2\hsize"}
Potential impact of stellar inclination on various correlations
---------------------------------------------------------------
### Correlation between total RV signal and facula filling factor
In Paper IV, we used the solar Calcium (Ca) index (S-index or log$(R^{'}_{\rm HK})$) as a tool to correct partially the activity-induced RV from its convective component (using the observed solar activity pattern). Indeed, the Ca index is correlated with the chromospheric plage filling factor and hence with the photospheric facula filling factor [@shapiro14 found a linear dependence between the Ca index and the facula filling factor in the case of the Sun].
As already explained, in the case of a solar-like star seen edge-on, the RV signal is closely correlated to the facula filling factor (as the convective component dominates the RV variations, see Sect. \[comprvobssim\] and Figs. \[fillfactor\], \[rv\]). When going toward smaller inclinations, we note that the correlation gets even stronger, with a Pearson correlation coefficient going from 0.97 for $i=90\degr$ to 0.99 for $i=10\degr$. This is in agreement with the fact that the relative contribution of the spot+facula component to the RV signal decreases with [*i*]{}. It would finally mean that the Ca index correction method can be used regardless of the stellar inclination.
![[*Top panel*]{}: Ratio of the mean facula filling factor over the total RV amplitude versus [*i*]{}. [*Bottom panel*]{}: Ratio of the mean facula filling factor over the total RV rms versus [*i*]{}.[]{data-label="ratiofacRV"}](ratio_facFF_RV.ps){height="1.1\hsize"}
### RV jitter, Ca index and facula filling factor
In recent studies [@wright05; @santos10; @isaacson10; @hillenbrand14], the average level of RV jitter was commonly taken as a proxy for the stellar magnetic activity level, along with the Ca index. For different samples of FGK stars, the authors found loose correlations between the average RV jitter and the mean Ca index, generally with a significant amount of dispersion. In our simulations, we find that the ratio of the mean facula filling factor over the RV rms (calculated over the complete activity cycle) is not constant over the range of inclinations [*i*]{} we explored. We display in Fig. \[ratiofacRV\] the facula filling factor to RV amplitude and to RV rms ratios. We find both ratios to increase towards smaller inclinations (in agreement with Sects. \[phot\_serie\] and \[RVINC\], where the decrease of the RV amplitude and rms with [*i*]{} is more pronounced that the decrease of the facula filling factor with [*i*]{}). Provided that the Ca index evolves in the same way as the facula filling factor with [*i*]{}, this could partly explain the large dispersion in the (RV rms, mean Ca index) relation found in the studies cited above (as the stellar inclination remains unknown for most of the observed targets). It also means that a clear (RV rms, Ca) relation will remain hard to establish for future studies, unless the uncertainty on the stellar inclination can be removed.
Detection limits {#thelimdets}
----------------
In this section, we compute the detection limits for the total RV time series over the range of stellar inclinations explored above.
### Approach
We use two different methods: the correlation-based method and the [*local power amplitude*]{} (hereafter LPA) method. Both were described in details and tested on real stellar RV data in [@meunier12]. In brief, the first method makes the correlation between the periodograms of a generated planetary RV signal (with the same temporal sampling as of the data) and of the actual RV data to which the planetary RV signal has been added ([[*i.e.*]{}]{}, it determines the correlation of the power of the (stellar data + fake planet) periodogram vs. the power of the fake planet periodogram). The detection limit corresponds then to the minimal mass and period for which the correlation values are all above a given threshold, for 100 realizations spanning all orbital phases. The threshold (spanning here from 0.003 for $i$ = 10$\degr$ to 0.05 for $i$ = 90$\degr$) corresponds to the maximum of the correlations obtained for a very low mass planet (here $\sim$0.6 [M$_{\rm Earth}$]{}. As for the LPA method, we compare the periodograms of the actual RV data and of a given generated planetary signal (for the same temporal sampling), but within a localized period range around the given planetary period. The detection limit at this period corresponds then to the minimal mass for which the maximum power of the planetary RV periodogram (within the limited period range) is always above the maximum power of the actual RV data periodogram in the period range.
We compute here our detection limits for only one period of 480.1 days, corresponding to a separation of about 1.2 au, as in Paper II. It is a representative value of the outer boundary of the HZ for a solar-type star. The total time span is always fixed to the complete simulation time span ([[*i.e.*]{}]{} 4566 days). As in Papers I and II, we assume the star to be observed eight months a year (meaning that we take into account only 2978 RV points instead of 4566). We assume different temporal samplings: all points ([[*i.e.*]{}]{} one-day sampling or 1:1), 1 point every 4 nights (1:4), 1 point every 8 nights (1:8) or 1 point every 20 nights (1:20); that is to say that the (1:20) series has 20 times less points that the (1:1). We also assume different precisions on the RV (no added noise, 0.01, 0.05 and 0.1[ms$^{\rm -1}$]{} noise levels), as we did in Paper I.
Finally, we compute the detection limits vs. inclination for two different cases:
1. In the first case, we compute the detection limits considering that the hypothetical planet is always seen orbiting edge-on for all stellar inclinations ([[*i.e.*]{}]{}, there is an increasing spin-orbit misalignment with decreasing inclination).
2. In the second case, we consider that the hypothetic planet always orbits in the stellar equatorial plane ([[*i.e.*]{}]{} there is always spin-orbit alignment).
Considering these two distinct cases is important as in most cases when searching for planets with RV, we do not know neither the stellar inclination nor the inclination of the planet orbit with respect to the line of sight. As for the inclination of the planet orbit, it is taken into account by giving minimal masses $m.sin(i)$ (with [*i*]{} denoting here the inclination of the planet orbit with respect to the line of sight) for detected planets. However, the detection limits are generally computed without knowing the stellar inclination itself, [[*i.e.*]{}]{} considering that the star is seen edge-on and that the activity-induced jitter is not reduced due to the projection effect. The detection limits we compute in case 1 correspond to this configuration and thus to the “best-possible” detection limits. As we go toward smaller stellar inclinations, they also correspond to more and more unlikely orbital configurations. Indeed, most of the exoplanets found so far indeed orbit in or near the stellar equatorial plane [even if in the specific case of Hot Jupiters, a significant fraction of systems show spin-orbit misalignments, [[*e.g.*]{}]{} @albrecht12; @triaud14].
{width="0.83\hsize" height="1.22\hsize"}
On the contrary, the detection limits that we compute in case 2, [[*i.e.*]{}]{} when considering a spin-orbit alignment, correspond to a more conservative but more realistic assumption. The detection limits for the correlation-based and LPA methods are displayed in Fig. \[limdets\].
### Results
#### Comparison with previous results –
For the edge-on configuration, we can compare the detection limits (for the 1:1 sampling) to the detection limits computed in Paper IV, [[*i.e.*]{}]{} on RV time series derived from observed solar activity patterns, as the latter time series had a similar time span, sampling and number of data points. In the case of the correlation method, they are in a fairly good agreement (5.5 [M$_{\rm Earth}$]{} for the present time series versus 6.8 [M$_{\rm Earth}$]{} for the time series from observed patterns), whereas there is a certain discrepancy for the LPA method (9.9 [M$_{\rm Earth}$]{} and 15.7 [M$_{\rm Earth}$]{} for the simulated and observed RV time series, respectively).
#### Comparison between the two methods –
The correlation-based method gives lower detection limits than the LPA method. This is also in agreement with our results from Paper IV. Note that we also compared the two methods in [@meunier12], this time on actual stellar data. Out of a 10 target sample, we found in most cases that the LPA method gave lower detection limits. However in these cases, the targets were massive A-F Main-Sequence stars with medium to high [$v\sin{i}$]{} (from 7 to $\geq$ 175 [kms$^{\rm -1}$]{}), some of them being young stars. They would then have a very different activity pattern than the solar one.
#### Impact of stellar inclination –
For the LPA method, the decrease of the detection limit with the stellar inclination [*i*]{} in case 1 (“best-possible” detection limits) is consistent with the decrease of the activity-induced RV rms when going toward a smaller [*i*]{}. In case 2, we observe that the detection limits remain around 10 [M$_{\rm Earth}$]{} for $i\geq 40\degr$ before increasing toward higher masses for smaller inclinations. This should mean that for $i\geq 40\degr$, the decrease of the planetary-induced RV amplitude for a decreasing inclination of the system is counterbalanced by the decrease in the same time of the activity-induced RV jitter. For smaller inclinations, the decrease of the activity RV jitter is less pronounced and hence it becomes more difficult to detect the planet. This is consistent with the fact that the behavior of the activity-induced RV jitter with respect to [*i*]{} is not sinusoidal.\
For the correlation method, the behavior of the detection limits is rather different. In case 1, the detection limits decrease with [*i*]{} until they reach a plateau at about 1.5 [M$_{\rm Earth}$]{} for $i\sim 40\degr$. When looking at case 2, surprisingly the detection limits begin to decrease with a decreasing [*i*]{}, to reach a minimum and best value of 2-2.5 [M$_{\rm Earth}$]{} for [*i*]{} $=40-50\degr$. For smaller inclinations, the detection limits increase with a decreasing [*i*]{}. Thus, the optimal configuration for the detection of the planet is an orbital plane inclined of $45\degr$ with respect to the line of sight (in the most probable case of spin-orbit alignment).
### Impact of the parameters
In this section, we study in more details the impact of the main parameters for the computation of the detection limits ([[*i.e.*]{}]{}, the added RV noise level and the temporal sampling. To do so, we first study their influence on the detection limits computed above (Fig. \[limdets\]). Then, we extend the study to a wider range of noise level or temporal sampling, but this time concentrating on one case ([*i*]{} $=$ 50$\degr$, case 2). This roughly corresponds to the inclination for which we get the better detection limits (in case 2).
![Tot. RV periodograms for [*i*]{} $=$ 90$\degr$, (1:1) sampling. [*Black*]{}: data taken eight months a year. [*Red*]{}: full data. The 1$\%$ FAP are displayed ([*dashed lines*]{}), as well as the solar rotational period and its two first harmonics ([*blue*]{}), and the detection limit period ([*green*]{}).[]{data-label="compperiocut"}](comp_perio_cut.ps){width="0.95\hsize" height="0.6\hsize"}
#### Temporal window –
Our detection limit computation is based on the periodograms of the RV time series taken eight months a year, so as to better mimic actual observations. The periodograms will thus be modified compared to the periodograms of the total RV time series taken over the full simulation time span, due to aliasing. We compare the periodograms of the total RV time series (in the [*i*]{} $=$ 90$\degr$ case) before and after removing 4 months a year in Fig. \[compperiocut\]. Removing four months a year indeed injects a large amount of power in the RV periodogram around one year ($\sim$ 360-400 days) as well as at lower harmonics (180 and 90 days). It will then impact our detection limit computation and deteriorate our detection limits, but it has to be taken into account to reproduce better the observations.
#### Temporal sampling –
We found in Paper IV that when taking into account all RV components, the detection limits got worse for the (1:20) sampling ([[*i.e.*]{}]{}, for a largely degraded sampling), while they did not vary significantly when going from a (1:1) to (1:4) or (1:8) samplings. Fig. \[limdets\] shows that the LPA detection limits do not vary significantly when going from the (1:1) to the (1:20) sampling. As for the correlation detection limits, they do not vary significantly when going from the (1:1) to the (1:8) sampling, but get worse for the (1:20) case.\
We now focus on the [*i*]{} $=$ 50$\degr$ case to study the sampling impact in more details. We compute the detection limits for an extended sampling range (from (1:1) to (1:70)) and display the results in Fig. \[limdetsamp\]. We find that both the correlation and the LPA detection limits are nearly independent from the temporal sampling up to the (1:10) sampling, and get worse for more degraded samplings (even if they show a large dispersion in the LPA case). The large dispersion at large temporal samplings probably reflects the larger uncertainty on the detection limits for a smaller number of points.
#### Added noise –
The detection limits computed with both the correlation and LPA methods are nearly independent from the added RV noise for a noise level up to 10 [cms$^{\rm -1}$]{} (Fig. \[limdets\]), [[*i.e.*]{}]{} for the best RV accuracy expected on future spectrographs. This is in agreement with the results of Paper IV. We consider that the small variations seen between the detection limits computed for the different noise levels in the case of the correlation method are not significant.\
![Detection limits vs. temporal sampling for [*i*]{} $=$ 50$\degr$, no noise. [*Squares*]{}: LPA method; [*Stars*]{}: correlation method.[]{data-label="limdetsamp"}](limdet_samp.ps)
![Detection limits vs. RV added noise for [*i*]{} $=$ 50$\degr$, (1:1) sampling. [*Squares*]{}: LPA method; [*Stars*]{}: correlation method.[]{data-label="limdetnoise"}](limdet_noise.ps)
We study then the impact of the added noise for an extended range of noise levels, up to 1 [ms$^{\rm -1}$]{} ([[*i.e.*]{}]{}, the current RV accuracy reached on the better spectrographs such as [H[ARPS]{}]{}), in the [*i*]{} $=$ 50$\degr$ case (Fig. \[limdetnoise\]). The LPA detection limits are constant up to 1 [ms$^{\rm -1}$]{}, while the correlation detection limits remain constant up to 0.7 [ms$^{\rm -1}$]{} with a slightly larger value at 1 [ms$^{\rm -1}$]{}.
Conclusion and perspectives {#conclu}
===========================
We built a fully parametrized model of the activity pattern of a solar-like star, including the dark spots and the bright features (large faculae and network). The model includes about 30 parameters that account for the different activity scales and the active structure behavior (most of them being well constrained by the litterature), and has a daily timescale over a complete activity cycle. Using the same approach as in the previous papers, we deduced the corresponding RV and photometric time series, taking the inhibition of the convective blueshift into account in the case of the RV. The simulated activity pattern, as well as the time series, were compared to the work done in Papers I, II and III with data from solar observations. We found our model to be in remarkably good agreement with the previous data, thus assessing its validity. We then study the case of a solar-like star seen under different configurations so as to estimate the impact of the stellar inclination on the activity-induced jitter. Our results are the following:
- For the stellar irradiance, the stellar inclination [*i*]{} has almost no effect on the amplitude of the long-term irradiance variations. The decrease of the active structure filling factor with a decreasing [*i*]{} mostly counterbalances the enhanced contrast of the structures, in agreement with some previous studies [@knaack01; @shapiro14]. On the contrary, the short-term jitter is reduced by a factor 6 when going from an edge-on to a nearly pole-on configuration.
- For the RV variations, the peak-to-peak amplitude as well as the rms are reduced by a factor 8 to 10 on both the long-term and the short-term. For a nearly pole-on configuration, the remaining RV jitter is about 0.6 [ms$^{\rm -1}$]{} when considering the whole cycle, and is about 0.2 [ms$^{\rm -1}$]{} on shorter timescales.
- When computed over the whole activity cycle, the activity-induced periodograms show power mainly in two period ranges: first at the stellar rotation period and its two or three first harmonics (induced by the crossing of active structures on the visible stellar hemisphere), and then at much longer periods, induced by more complex cycle effects. When going toward smaller inclinations, the power at the rotation period is reduced and nearly disappears for a pole-on configuration. The long-term cycle effects become preponderant.
- In the case of a solar-like star, the convective component widely dominates the RV variations. The “photometric” contribution of spots and bright features account for about $35\%$ of the RV jitter and decreases to about $15\%$ for a nearly pole-on configuration.
- For a solar-like star seen in a pole-on configuration, the photometric and RV (as well as presumably the chromospheric) activity-induced variations are most probably not distinguishable from those of a less active or almost inactive star.
- Finally, when considering a realistic orbital configuration ([[*i.e.*]{}]{},spin-orbit alignment), the optimal configuration for planet detection is a system inclined by about $45\degr$. In this case, the lowest detection limits reach planetary masses of about 2[M$_{\rm Earth}$]{} at 480 days, without applying a correction to the RV signal.
According to previous studies, solar-like stars show a great diversity of activity levels and properties [@schroder13]. The Sun itself is considered to be an average star, not particularly active but not quiet either. Being fully parametrized and validated for the solar case, our activity model now allows us to explore a wide range of activity parameters and stellar properties. In this paper, we have focused on solar-like activity and convection level stars; we expect stars with a lower activity level and/or stars with a lower convection level (such as K-type stars) to be much less affected and then to exhibit lower detection limits in the corresponding HZ. We also emphasize that activity correction methods (such as the Calcium index or the $H_{\alpha}$ line) are commonly applied to RV time series and already allow to reach detection limits at the [M$_{\rm Earth}$]{} level in the HZ (Paper IV). Optimized observation and reduction strategies are also promising [such as averaging, see [[*e.g.*]{}]{} @dumusque11b]. Despite not being the focus of our paper, improving such methods and strategies, as well as looking for new ones, is an extremely important question. Future instruments with very high RV accuracy may therefore be critical in implementing very efficient correction tools to extract low mass planet signals. Our goal is to be able to test and even reproduce the photometric and RV variations corresponding to each activity and stellar configuration. This should allow a better understanding of the magnetic stellar activity and open the way toward a better correction of the activity-induced stellar jitter.
In Paper III, we derived the astrometric time series corresponding to the observed solar activity pattern. Our activity model also allows us to produce the astrometric time series corresponding to the simulated activity pattern (apart of the photometric and RV ones). We will present our results in astrometry in a separate paper. This is quite justified as upcoming instruments such as the Nearby Earth Astrometric Telescope [NEAT, see [[*e.g.*]{}]{} @malbet12] should allow for the first time to search for low-mass planets around nearby FGK stars with astrometry.
We would like to thank our referee (X. Dumusque) for his very useful comments on the manuscript.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Antoine Lemenant\
Université Paris XI\
[email protected]
bibliography:
- 'biblio.bib'
title: 'On the homogeneity of global minimizers for the Mumford-Shah functional when $K$ is a smooth cone.'
---
[**Abstract.**]{} We show that if $(u,K)$ is a global minimizer for the Mumford-Shah functional in ${\ensuremath{\mathbb R}}^N$, and if $K$ is a smooth enough cone, then (modulo constants) $u$ is a homogenous function of degree $\frac{1}{2}$. We deduce some applications in ${\ensuremath{\mathbb R}}^3$ as for instance that an angular sector cannot be the singular set of a global minimizer, that if $K$ is a half-plane then $u$ is the corresponding cracktip function of two variables, or that if $K$ is a cone that meets $S^2$ with an union of $C^\infty$ curvilinear convex polygones, then it is a ${\ensuremath{\mathbb P}}$, ${\ensuremath{\mathbb Y}}$ or ${\ensuremath{\mathbb T}}$.
Introduction {#introduction .unnumbered}
============
The functional of D. Mumford and J. Shah [@ms] was introduced to solve an image segmentation problem. If $\Omega$ is an open subset of ${\ensuremath{\mathbb R}}^2$, for example a rectangle, and $g\in L^\infty(\Omega)$ is an image, one can get a segmentation by minimizing $$J(K,u):= \int_{\Omega \backslash K}|\nabla u|^2dx+\int_{\Omega\backslash K}(u-g)^2dx+H^{1}(K)$$ over all the admissible pairs $(u,K)\in
\mathcal{A}$ defined by $$\mathcal{A}:=\{(u,K); \; K \subset \Omega \text { is closed } , \; u \in W^{1,2}_{loc}(\Omega \backslash K) \}.$$
Any solution $(u,K)$ that minimizes $J$ represents a “smoother” version of the image and the set $K$ represents the edges of the image.
Existence of minimizers is a well known result (see for instance [@dcl]) using $SBV$ theory.
The question of regularity for the singular set $K$ of a minimizer is more difficult. The following conjecture is currently still open.
[[@ms] ]{} Let $(u,K)$ be a reduced minimizer for the functional $J$. Then $K$ is the finite union of $C^1$ arcs.
The term “reduced” just means that we cannot find another pair $(\tilde u, \tilde K)$ such that $K \subset \tilde K$ and $\tilde u$ is an extension of $u$ in $\Omega \backslash \tilde K$.
Some partial results are true for the conjecture. For instance it is known that $K$ is $C^1$ almost everywhere (see [@d1], [@b] and [@afp1]). The closest to the conjecture is probably the result of A. Bonnet [@b]. He proves that if $(u,K)$ is a minimizer, then every isolated connected component of $K$ is a finite union of $C^1$-arcs. The approach of A. Bonnet is to use blow up limits. If $(u,K)$ is a minimizer in $\Omega$ and $y$ is a fixed point, consider the sequences $(u_k,K_k)$ defined by $$u_k(x)=\frac{1}{\sqrt{t_k}}u(y+t_kx), \quad K_k=\frac{1}{t_k}(K-y), \quad \Omega_k=\frac{1}{t_k}(\Omega-y).$$ When $\{t_k\}$ tends to infinity, the sequence $(u_k,K_k)$ may tend to a pair $(u_\infty,K_\infty)$, and then $(u_\infty,K_\infty)$ is called a Global Minimizer. Moreover, A. Bonnet proves that if $K_\infty$ is connected, then $(u_\infty, K_\infty)$ is one of the list below :
$\bullet$[**1st case**]{}: $ K_\infty=\varnothing$ and $u_\infty$ is a constant.
$\bullet$[**2nd case**]{}: $K_\infty$ is a line and $u_\infty$ is locally constant.
$\bullet$[**3rd case**]{}: “*Propeller*”: $K_\infty$ is the union of 3 half-lines meeting with $120$ degrees and $u_\infty$ is locally constant.
$\bullet$[**4th case**]{}: “*Cracktip*”: $K_\infty=\{(x,0);x\leq 0\}$ and $u_\infty(r\cos(\theta),r\sin(\theta))=
\pm\sqrt{\frac{2}{\pi}}r^{1/2}\sin\frac{\theta}{2}+C$, for $r>0$ and $|\theta|<\pi$ ($C$ is a constant), or a similar pair obtained by translation and rotation.
We don’t know whether the list is complete without the hypothesis that $K_\infty$ is connected. This would give a positive answer to the Mumford-Shah conjecture.
The Mumford-Shah functional was initially given in dimension $2$ but there is no restriction to define Minimizers for the analogous functional in ${\ensuremath{\mathbb R}}^N$. Then we can also do some blow-up limits and try to think about what should be a global minimizer in ${\ensuremath{\mathbb R}}^N$. Almost nothing is known in this direction and this paper can be seen as a very preliminary step. Let state some definitions.
Let $\Omega \subset {\ensuremath{\mathbb R}}^N$, $(u,K) \in \mathcal{A}$ and $B$ be a ball such that $\bar B\subset \Omega$. A competitor for the pair $(u,K)$ in the ball $B$ is a pair $(v,L) \in \mathcal{A}$ such that $$\left.
\begin{array}{c}
u=v \\
K=L
\end{array}
\right\} \text{ in } \Omega \backslash B$$ and in addition if $x$ and $y$ are two points in $\Omega \backslash ( B \cup K) $ that are separated by $K$ then they are also separated by $L$.
The expression “be separated by $K$” means that $x$ and $y$ lie in different connected components of $\Omega \backslash K$.
\[defms\] A global minimizer in ${\ensuremath{\mathbb R}}^N$ is a pair $(u,K)\in \mathcal{A}$ (with $\Omega={\ensuremath{\mathbb R}}^N$) such that for every ball $B$ in ${\ensuremath{\mathbb R}}^N$ and every competitor $(v,L)$ in $B$ we have $$\int_{B \backslash K}|\nabla u|^2dx +H^{N-1}(K\cap B)\leq \int_{B\backslash L}|\nabla v|^2dx+H^{N-1}(L\cap B)$$ where $H^{N-1}$ denotes the Hausdorff measure of dimension $N-1$.
Proposition 9 on page 267 of [@d] ensures that any blow up limit of a minimizer for the Mumford-Shah functional in ${\ensuremath{\mathbb R}}^N$, is a global minimizer in the sense of Definition \[defms\]. As a beginning for the description of global minimizers in ${\ensuremath{\mathbb R}}^N$, we can firstly think about what should be a global minimizer in ${\ensuremath{\mathbb R}}^3$. If $u$ is locally constant, then $K$ is a minimal cone, that is, a set that locally minimizes the Hausdorff measure of dimension 2 in ${\ensuremath{\mathbb R}}^3$. Then by [@d3] we know that $K$ is a cone of type ${\ensuremath{\mathbb P}}$ (hyperplane), ${\ensuremath{\mathbb Y}}$ (three half-planes meeting with 120 degrees angles) or of type ${\ensuremath{\mathbb T}}$ (cone over the edges of a regular tetraedron centered at the origin). Those cones became famous by the theorem of J. Taylor [@ta] which says that any minimal surface in ${\ensuremath{\mathbb R}}^3$ is locally $C^1$ equivalent to a cone of type ${\ensuremath{\mathbb P}}$, ${\ensuremath{\mathbb Y}}$ or ${\ensuremath{\mathbb T}}$.
{width="4cm"} {width="4cm"}\
Cones of type ${\ensuremath{\mathbb Y}}$ and ${\ensuremath{\mathbb T}}$ in ${\ensuremath{\mathbb R}}^3$.
To be clearer, this is a more precise definition of ${\ensuremath{\mathbb Y}}$ and ${\ensuremath{\mathbb T}}$, as in [@ddpt].
\[prop\] Define $Prop\subset {\ensuremath{\mathbb R}}^2$ by $$Prop=\{(x_1,x_2);x_1 \geq 0, x_2=0\}$$ $$\hspace{4cm} \cup\{(x_1,x_2);x_1 \leq 0, x_2=-\sqrt{3}x_1\}$$ $$\hspace{6.5cm}\cup\{(x_1,x_2);x_1 \leq 0, x_2=\sqrt{3}x_1\}.$$ Then let $Y_0=Prop\times {\ensuremath{\mathbb R}}\subset {\ensuremath{\mathbb R}}^3.$ The spine of $Y_0$ is the line $L_0=\{x_1=x_2=0\}$. A cone of type ${\ensuremath{\mathbb Y}}$ is a set $Y=R(Y_0)$ where $R$ is the composition of a translation and a rotation. The spine of $Y$ is then the line $R(L_0)$.
\[defT\] Let $A_1=(1,0,0)$, $A_2=(-\frac{1}{3},\frac{2\sqrt{2}}{3},0)$, $A_3=(-\frac{1}{3},-\frac{\sqrt{2}}{3},\frac{\sqrt{6}}{3})$, and $A_4=(-\frac{1}{3},-\frac{\sqrt{2}}{3}, -\frac{\sqrt{6}}{3})$ the four vertices of a regular tetrahedron centered at $0$. Let $T_0$ be the cone over the union of the $6$ edges $[A_i,A_j]$ $i\not
=j$. The spine of $T_0$ is the union of the four half lines $[0,A_j[$. A cone of type ${\ensuremath{\mathbb T}}$ is a set $T=R(T_0)$ where $R$ is the composition of a translation and a rotation. The spine of $T$ is the image by $R$ of the spine of $T_0$.
So the pairs $(u,Z)$ where $u$ is locally constant and $Z$ is a minimal cone, are examples of global minimizers in ${\ensuremath{\mathbb R}}^3$. Another global minimizer can be obtained with $K_\infty$ a half-plane, by setting $u:= Craktip\times {\ensuremath{\mathbb R}}$ (see [@d] section 76). These examples are the only global minimizers in ${\ensuremath{\mathbb R}}^3$ that we know.
Note that if $(u,K)$ is a global minimizer in ${\ensuremath{\mathbb R}}^N$, then $u$ locally minimizes the Dirichlet integral in ${\ensuremath{\mathbb R}}^N\backslash K$. As a consequence, $u$ is harmonic in ${\ensuremath{\mathbb R}}^N \backslash K$. Moreover, if $B$ is a ball such that $K \cap B$ is regular enough, then the normal derivative of $u$ vanishes on $K \cap B$.
In this paper we wish to study global minimizers $(u,K)$ for which $K$ is a cone. It seems natural to think that any singular set of a global minimizer is a cone. But even if all known examples are cones, there is no proof of this fact. In addition, we will add some regularity on $K$. We denote by $S^{N-1}$ the unit sphere in ${\ensuremath{\mathbb R}}^N$ and, if $\Omega$ is a open set, $W^{1,2}(\Omega)$ is the Sobolev space. We will say that a domain $\Omega$ on $S^{N-1}$ has a piecewise $C^2$ boundary, if the topological boundary of $\Omega$, defined by $\partial \Omega=\bar \Omega \backslash \Omega$, consists of an union of $N-2$ dimensional hypersurfaces of class $C^2$. This allows some cracks, i.e. when $\Omega$ lies in each sides of its boundary. We will denote by $\tilde \Sigma$ the set of all the singular points of the boundary, that is $$\tilde \Sigma:=\{x \in \partial \Omega; \forall r >0, B(x,r)\cap \partial \Omega \text{ is not a } C^2 \text{ hypersurface } \}.$$
A smooth cone is a set $K$ of dimension $N-1$ in ${\ensuremath{\mathbb R}}^N$ such that $K$ is conical, centered at the origin, and such that $S^{N-1}\backslash K$ is a domain with piecewise $C^2$ boundary. Moreover we assume that the embedding $L^2(S^{N-1}\backslash K)\to W^{1,2}(S^{N-1}\backslash K)$ is compact. Finally we suppose that we can strongly integrate by parts in $B(0,1)\backslash K$. More precisely, denoting by $\Sigma$ the set of singularities $$\Sigma:=\{tx ; (t,x) \in {\ensuremath{\mathbb R}}^+ \times \tilde \Sigma \},$$ we want that $$\int_{B(0,1)\backslash K}\langle \nabla u,\nabla \varphi \rangle=0$$ for every harmonic function $u$ in $B(0,1)\backslash K$ with $\frac{\partial}{\partial n}u=0$ on $K \backslash \Sigma$, and for all $\varphi \in
W^{1,2}(B(0,1)\backslash K)$ with vanishing trace on $S^{N-1}\backslash K$.
For instance, the cone over a finite union of $C^2$-arcs on $S^2$ is a smooth cone in ${\ensuremath{\mathbb R}}^3$. Another example in ${\ensuremath{\mathbb R}}^N$ is given by the union of admissible set of faces (as in Definition (22.2) of [@dau1]).
Now this is the main result.
[\[mainth\].]{} Let $(u,K)$ be a global minimizer in ${\ensuremath{\mathbb R}}^N$. Assume that $K$ is a smooth cone. Then there is a $\frac{1}{2}$-homogenous function $u_1$ such that $u-u_1$ is locally constant.
As we shall see, this result implies that if $(u,K)$ is a global minimizer in ${\ensuremath{\mathbb R}}^N$, and if $K$ is a smooth cone other than a minimal cone, then $\frac{3-2N}{4}$ is an eigenvalue for the spherical Laplacian in $S^{N-1}\backslash K$ with Neumann boundary conditions. In section \[applications\] we will give some applications about global minimizers in ${\ensuremath{\mathbb R}}^3$, using the estimates on the first eigenvalue that can be found in [@da], [@dau1] and [@kmr]. More precisely, we have :
[\[app1\]]{} Let $(u,K)$ be a global Mumford-Shah minimizer in ${\ensuremath{\mathbb R}}^3$ such that $K$ is a smooth cone. Moreover, assume that $S^2\cap K $ is a union of convex curvilinear polygons with $C^\infty$ sides. Then $u$ is locally constant and $K$ is a cone of type ${\ensuremath{\mathbb P}}$, ${\ensuremath{\mathbb Y}}$ or ${\ensuremath{\mathbb T}}$.
Another consequence of the main result is the following.
[\[cracktip\]]{} Let $(u,K)$ be a global Mumford-Shah minimizer in ${\ensuremath{\mathbb R}}^3$ such that $K$ is a half plane. Then $u$ is equal to a function of type $cracktip\times \mathbb{R}$, that is, in cylindrical coordinates, $$u(r,\theta,z)=\pm \sqrt{\frac{2}{\pi}}r^{\frac{1}{2}}sin\frac{\theta}{2} +C$$ for $0<r< + \infty $, $-\pi< \theta < \pi$ where $C$ is a constant.
Finally, we deduce two other consequences from Theorem \[mainth\]. Let $(r,\theta,z)\in {\ensuremath{\mathbb R}}^+\times [-\pi,\pi]\times {\ensuremath{\mathbb R}}$ be the cylindrical coordinates in ${\ensuremath{\mathbb R}}^3$. For all $\omega \in
[0,\pi]$ set $$\begin{aligned}
\partial \Gamma_\omega:=\{(r,\theta,z) \in {\ensuremath{\mathbb R}}^3; \theta=-\omega \text{ or } \theta=\omega \}.\notag\end{aligned}$$ and $$\begin{aligned}
S_\omega:=\{(r,\theta,z) \in {\ensuremath{\mathbb R}}^3; z=0,\; r>0,\;\theta \in[-\omega, \omega]\;\}\end{aligned}$$
Observe that $S_{0}$ is a half line, $S_{\frac{\pi}{2}}$, $\partial \Gamma_{0}$ and $\partial \Gamma_{\pi}$ are half-planes, and that $S_{\pi}$ and $\partial \Gamma_{\frac{\pi}{2}}$ are planes.
[\[app3\]]{} There is no global Mumford-Shah minimizer in ${\ensuremath{\mathbb R}}^3$ such that $K$ is wing of type $\partial \Gamma_\omega$ with $\omega\not \in \{0, \frac{\pi}{2},\pi\}$.
[\[sect\]]{} There is no global Mumford-Shah minimizer in ${\ensuremath{\mathbb R}}^3$ such that $K$ is an angular sector of type $(u,S_\omega)$ for $\omega \not \in \{\frac{\pi}{2}, \pi\}$.
[**Acknowledgements**]{} : The author wishes to thank Guy David for having introduced him to the Mumford-Shah Functional, and for many helpful and interesting discussions on this subject.
If $K$ is a cone then $u$ is homogenous
=======================================
In this section we want to prove Theorem \[mainth\]. Notice that this result is only useful if the dimension $N\geq 3$. Indeed, in dimension 2, if $K$ is a cone then it is connected thus it is in the list described in the introduction.
Preliminary
-----------
Let us recall a standard uniqueness result about energy minimizers.
\[stam1\] Let $\Omega$ be an open and connected set of ${\ensuremath{\mathbb R}}^N$ and let $I\subset \partial \Omega$ be a hypersurface of class $C^\infty$. Suppose that $u$ and $v$ are two functions in $W^{1,2}(\Omega)$ such that $u=v$ a.e. on $I$ (in terms of trace), solving the minimizing problem $$\min E(w):=\int_{\Omega}|\nabla w(x)|^2dx$$ over all the functions $w\in W^{1,2}(\Omega)$ that are equal to $u$ and $v$ on $I$. Then $$u=v.$$
[**Proof :**]{} This comes from a simple convexity argument which can be found for instance in [@d], but let us write the proof since it is very short. By the parallelogram identity we have $$\begin{aligned}
E(\frac{u+v}{2})=\frac{1}{2}E(u)+\frac{1}{2}E(v)-\frac{1}{4}E(u-v). \label{paral}\end{aligned}$$ On the other hand, since $\frac{u+v}{2}$ is equal to $u$ and $v$ on $I$, and by minimality of $u$ and $v$ we have $$E(\frac{u+v}{2})\geq E(u)=E(v).$$ Now by we deduce that $E(u-v)=0$ and since $\Omega$ is connexe, this implies that $u-v$ is a constant. But $u-v$ is equal to $0$ on $I$ thus $u=v$.
The existence of a minimizer can also be proved using the convexity of $E(v)$.
Spectral decomposition
----------------------
The key ingredient to obtain the main result will be the spectral theory of the Laplacian on the unit sphere. Since $u$ is harmonic, we will decompose $u$ as a sum of homogeneous harmonic functions just like we usually use the classical spherical harmonics. The difficulty here comes from the lack of regularity of ${\ensuremath{\mathbb R}}^N\backslash K$.
It will be convenient to work with connected sets. So let $\Omega$ be a connected component of $S^{N-1}\backslash K$, and let $A(r)$ be $$A(r):=\{tx ; (x,t)\in \Omega \times [0,r[ \; \}.$$ We also set $$A(\infty):= \{tx ; (x,t)\in \Omega \times {\ensuremath{\mathbb R}}^+\; \}.$$ All the following results are using that the embedding $W^{1,2}(\Omega)$ in $L^2(\Omega)$ is compact. Recall that this is the case by definition, since $K$ is a smooth cone. Notice that for instance the cone property insures that the embedding is compact (see Theorem 6.2. p 144 of [@a]).
Consider the quadratic form $$Q(u)=\int_{\Omega}|\nabla u(x)|^2dx$$ of domain $W^{1,2}(\Omega)$ dense into the Hilbert space $L^{2}(\Omega)$. Since $Q$ is a positive and closed quadratic form (see for instance Proposition 10.61 p.129 of [@lb]) there exists a unique selfadjoint operator denoted by $-\Delta_n$ of domain $D(-\Delta_n)\subset W^{1,2}(\Omega)$ such that $$\forall u \in D(-\Delta_n), \; \forall v \in W^{1,2}(\Omega),\quad
\int_{\Omega}\langle \nabla u, \nabla v\rangle=\int_{\Omega}
\langle -\Delta_nu,v \rangle.$$
\[defin laplacien neumann\] The operator $-\Delta_n$ has a countably infinite discrete set of eigenvalues, whose eigenfunctions span $L^{2}(\Omega)$.
[**Proof :**]{} The proof is the same as if $\Omega$ was a regular domain. Consider the new quadratic form $$\tilde{Q}(u):= Q(u)+\|u\|_2^2$$ with the same domain $W^{1,2}(\Omega)$. The form $\tilde{Q}$ has the same properties than $Q$ and the associated operator is ${\rm
Id} - \Delta_n$. Moreover $\tilde{Q}$ is coercive. As a result, the operator ${ \rm Id}-\Delta_n$ is bijective and its inverse goes from $L^{2}(\Omega)$ to $D(-\Delta_n)\subset
W^{1,2}(\Omega)$. By hypothesis the embedding of $W^{1,2}(\Omega)$ into $L^2(\Omega)$ is compact. Thus the resolvant $({ \rm Id}-\Delta_n)^{-1}$ is a compact operator, and we conclude using the spectral theory of operators with a compact resolvant (see [@rs] Theorem XIII.64 p.245).
The domain of $-\Delta_n$ is not known in general. If $\Omega$ was smooth, then we could show that the domain is exactly $D(-\Delta_n)=\{u\in W^{2,2}(\Omega); \frac{\partial
u}{\partial n}=0 \; \rm{on}\;
\partial \Omega\}$. Here, the boundary of $\Omega$ has some singularities so this result doesn’t apply directly. But knowing exactly the domain of $-\Delta_n$ will not be necessary for us.
Now we want to study the link between the abstract operator $\Delta_n$ and the classical spherical Laplacian $\Delta_S$ on the unit sphere. Recall that if we compute the Laplacian in spherical coordinates, we obtain the following equality $$\begin{aligned}
\Delta = \frac{\partial^2 }{\partial
r}+\frac{N-1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\Delta_S.
\label{forumlelaplacien}\end{aligned}$$
\[regularite fonctions propres\] For every function $f\in D(-\Delta_n)$ such that $-\Delta_n
f=\lambda f$ we have $$\begin{aligned}
i) &\quad& f\in C^{\infty}(\Omega) \notag \\
ii)&\quad& -\Delta_S f=-\Delta_n f=\lambda f \text { in }
\Omega \notag\\
iii) &\quad& \frac{\partial f}{\partial n} \text{ exists and is
equal to } 0 \text{ on } K \cap \overline{\Omega} \backslash
\Sigma \notag\end{aligned}$$
[**Proof :**]{} Let $\varphi$ be a $C^{\infty}$ function with compact support in $\Omega$ and $f\in D(-\Delta_n)$. Then the Green formula in the distributional sense gives $$\int_{\Omega} \nabla f . \nabla \varphi = \langle -\Delta_S f , \varphi \rangle$$ where the left and right brackets mean the duality in the distributional sense. On the other hand, by definition of $-\Delta_n$ and since $f$ is in the domain $D(-\Delta_n)$, we also have $$\int_{\Omega} \nabla f . \nabla \varphi =\langle -\Delta_n
f ,\varphi \rangle$$ where this time the brackets mean the scalar product in $L^2$. Therefore $$\Delta_n f =\Delta_S f \quad \text{in}\;
\mathcal{D}'(\Omega).$$ In other words, $-\Delta_S f = \lambda f$ in $\mathcal{D}'(\Omega)$. But now since $f\in W^{1,2}(\Omega)$, by hypoellipticiy of the Laplacian we know that $f$ is $C^{\infty}$ and that $-\Delta_S f=\lambda f$ in the classical sense. That proves $i)$ and $ii)$. We even know by the elliptic theory that, since $K\backslash \Sigma$ is regular, $f$ is regular at the boundary on $K\backslash \Sigma$.\
Now consider a ball $B$ such that the intersection with $K\cap \overline{\Omega}$ does not meet $\Sigma$. Assume that $B$ is cut in two parts $B^+$ and $B^-$ by $K$, and that $B^+$ is one part in $\Omega$. Possibly by modifying $B$ in a neighborhood of the intersection with $K$, we can assume that the boundary of $B^+$ and $B^-$ is $C^2$. The definition of $\Delta_n$ implies that for all function $\varphi \in C^{2}(\bar \Omega)$ that vanishes out of $B^+$ we have $$\int_{B^+}\langle \nabla f , \nabla \varphi \rangle dx=
\int_{B^+}\langle-\Delta_n
f, \varphi \rangle dx=\lambda\int_{B^+}\langle f , \varphi \rangle dx.$$ On the other hand, integrating by parts, $$\begin{aligned}
\int_{B^+}\langle \nabla f , \nabla \varphi \rangle dx&=&
\int_{B^+}\langle-\Delta_S f, \varphi \rangle+\int_{\partial
B^+}\frac{\partial u}{\partial n} \varphi \notag \\
&=&
\lambda \int_{\partial B^+}\langle f, \varphi \rangle+\int_{\partial B^+}\frac{\partial f}{\partial n}
\varphi\notag\end{aligned}$$ thus $$\int_{\partial B^+}\frac{\partial f}{\partial n} \varphi=0.$$ In other words the function $f$ is a weak solution of the mixed boundary value problem $$\begin{aligned}
-\Delta_S u=\lambda f & \text{ in }& B^{+} \notag \\
u=f &\text{ on }& \partial B^{+}\backslash K \notag \\
\frac{\partial u}{\partial n}=0 &\text{ on }& K\cap\partial
B^{+}\notag
\end{aligned}$$ Therefore, some results from the elliptic theory imply that $f$ is smooth in $B$ and is a strong solution (see [@taylorm]).
Let us recapitulate what we have obtained. For all function $f\in
L^{2}(\Omega)$, there is a sequence of numbers $a_i$ such that $$\begin{aligned}
f=\sum_{i=0}^{+\infty}a_i f_i \label{serie}\end{aligned}$$ where the sum converges in $L^{2}$. The functions $f_i$ are in $C^{\infty}(\Omega)\cap W^{1,2}(\Omega)$, verify $-\Delta_S
f_i=\lambda_i f_i$ and $\frac{\partial f_i}{\partial n}=0$ on $K\cap \overline{\Omega}\backslash \Sigma$. Moreover, we can normalize the $f_i$ in order to obtain an orthonormal basis on $L^{2}(\Omega)$, in particular we have the following Parseval formula $$\|f\|_2^2=\sum_{i=0}^{+\infty}|a_i|^2.$$ Note that if $f$ belongs to the kernel of $-\Delta_n$ (i.e. is an eigenfunction with eigenvalue $0$), then $$\langle \nabla f,\nabla f \rangle = \langle -\Delta_n f, f \rangle =0$$ and since $\Omega$ is connected that means that $f$ is a constant. Thus $0$ is the first eigenvalue and the associated eigenspace has dimension $1$. Then we can suppose that $\lambda_0=0$ and that all the $\lambda_i$ for $i > 0$ are positive.
We define the scalar product in $W^{1,2}(\Omega)$ by $$\langle u,v \rangle_{W^{1,2}}:=\langle u,v \rangle_{L^2}+\langle \nabla u, \nabla v \rangle_{L^2}.$$
\[gradientsok\] The family $\{f_i\}$ is orthogonal in $W^{1,2}(\Omega)$. Moreover if $f\in W^{1,2}(\Omega)$ and if its decomposition in $L^2(\Omega)$ is $f=\sum_{i=0}^{+\infty} a_i f_i$, then the sum $\sum_{i=0}^{+\infty}|a_i|^2 \|\nabla f_i\|_{2}^{2}$ converges and $$\begin{aligned}
\sum_{i=0}^{+\infty}|a_i|^2 \|\nabla f_i\|_{2}^{2}= \|\nabla
f\|_{2}^{2}. \label{parseval}\end{aligned}$$
[**Proof :**]{} We know that $\{f_i\}$ is an orthogonal family in $L^2(\Omega)$. In addition if $i\not = j$ then $$\begin{aligned}
\int_{\Omega}\nabla f_i \nabla f_j&=&\int_{\Omega}-\Delta_n f_i
f_j \notag \\
&=& \lambda_i \int_{\Omega}f_i f_j \notag \\
&=& 0 \notag\end{aligned}$$ thus $\{f_i\}$ is also orthogonal in $W^{1,2}(\Omega)$ and $$\|f_i\|_{W^{1,2}}^2:=\|f_i\|_2^2+\|\nabla f_i\|_{2}^2=1+\lambda_i.$$ Consider now the orthogonal projection (for the scalar product of $L^2$) $$P_k:f \mapsto \sum_{i=0}^{k}a_i f_i.$$ The operator $P_k$ is the orthogonal projection on the closed subspace $A_k$ generated by $\{f_{0},...,f_k\}$. More precisely, we are interested in the restriction of $P_k$ to the subspace $W^{1,2}(\Omega)\subset L^2(\Omega)$. Also denote by $\tilde{P}_k:W^{1,2}\to A_k$ the orthogonal projection on the same subspace but for the scalar product of $W^{1,2}$. We want to show that $P_k=\tilde{P}_k$. To prove this, it suffice to show that for all sets of coefficients $\{a_i\}_{i=1..k}$ and $\{b_i\}_{i=1..k}$, $$\langle f-\sum_{i=0}^k a_i f_i , \sum_{i=0}^k b_i f_i\rangle_{W^{1,2}}=0.$$ Since we already have $$\langle f-\sum_{i=0}^k a_i f_i , \sum_{i=0}^k b_i f_i\rangle_{L^2}=0,$$ all we have to show is that $$\int_{\Omega}\langle \nabla f-\sum_{i=0}^k a_i \nabla f_i,\sum_{i=0}^k b_i \nabla f_i \rangle dx=0.$$ Now $$\begin{aligned}
\int_{\Omega}\langle \nabla f-\sum_{i=0}^k a_i \nabla
f_i,\sum_{i=0}^k b_i \nabla f_i \rangle &=& \int_{\Omega}\langle
\nabla f,\sum_{i=0}^k b_i \nabla f_i \rangle - \sum_{i=0}^k a_ib_i \|\nabla f_i\|_2^2\notag \\
&=&\sum_{i=0}^k b_i \langle -\Delta_n f_i,f
\rangle_{L^2}-\sum_{i=0}^{k}a_ib_i \lambda_i \notag \\
&=&\sum_{i=0}^{k}a_ib_i \lambda_i \notag-\sum_{i=0}^{k}a_ib_i
\lambda_i \notag \\
&=&0 \notag\end{aligned}$$ thus $P_k=\tilde{P}_k$ and therefore, by Pythagoras $$\|P_{k}(f)\|_{W^{1,2}}^{2}\leq \|f\|_{W^{1,2}}^2.$$ By letting $k$ tend to infinity we obtain $$\begin{aligned}
\sum_{i=0}^{+\infty}a_i^2 \|\nabla f_i\|_{2}^{2}\leq \|\nabla
f\|_{2}^{2}. \label{ineq}\end{aligned}$$ From this inequality we deduce that the sum is absolutely converging in $W^{1,2}(\Omega)$. Therefore, the sequence of partial sum $\sum_{i=0}^{K}a_i f_i$ is a Cauchy sequence for the norm $W^{1,2}(\Omega)$. Thus, since the sum $\sum a_i f_i $ already converges to $f$ in $L^2(\Omega)$, by uniqueness of the limit the sum converges to $f$ in $W^{1,2}(\Omega)$, so we deduce that is an equality and the prove is over.
Once we have a basis $\{f_i\}$ on $\Omega \subset S^{N-1}$, we consider for a certain $r_0>0$, the functions $$h_i(x)=r_0^{\alpha_i}f_i\left(\frac{x}{r_0}\right)$$ defined on $r_0\Omega$. The exponent $\alpha_i$ is defined by $$\begin{aligned}
\alpha_i=\frac{-(N-2)+\sqrt{(N-2)^2+4\lambda_i}}{2}. \label{defalphai}
\end{aligned}$$
The functions $h_i$ form a basis of $W^{1,2}(r_0\Omega)$. Indeed, if $f\in W^{1,2}(r_0\Omega)$, then $f(r_0x)\in
W^{1,2}(\Omega)$ thus applying the decomposition on $\Omega$ we obtain
$$f(r_0x)=\sum_{i=0}^{+\infty}b_i f_i(x)$$ thus $$f(x)=\sum_{i=0}^{+\infty}a_i h_i(x)$$ with $$\begin{aligned}
a_i=b_i r_0^{-\alpha_i}. \label{aibi}\end{aligned}$$ Notice that since $\|h_i\|_{2}^2=r_0^{2\alpha_i+N-1}$ we also have $$\begin{aligned}
\sum_{i=0}^{\infty}a_i^2
\|h_i\|^2_2=\sum_{i=0}^{\infty}a_i^2r_0^{2\alpha_i+N-1}
=\|f\|^2_{L^2(r_0\Omega)}<+\infty. \label{convhi}\end{aligned}$$ Moreover, applying Proposition \[gradientsok\] we have that $$\begin{aligned}
\sum_{i=0}^{\infty}b_i^2 \|\nabla f_i\|_2^2 = \|\nabla f(r_0
x)\|_{2}^2 <+\infty. \label{gradhi}\end{aligned}$$
We are now able to state our decomposition in $A(r_0)$.
\[decomp1\] Let $K$ be a smooth cone in ${\ensuremath{\mathbb R}}^N$, centered at the origin and let $\Omega$ be a connected component of $S^{N-1}\backslash K$. Then there exist some harmonic homogeneous functions $g_i$, orthogonal in $W^{1,2}(A(1))$, such that for every function $u\in W^{1,2}(A(1))$ harmonic in $A(1)$ with $\frac{\partial u}{\partial n}=0$ on $K\cap
A(1)\backslash \Sigma$, and for every $r_0\in ]0, 1[$, we have that $$u=\sum_{i=0}^{+\infty}a_i g_i \quad \text{ in } A(r_0)$$ where the $a_i$ do not depend on radius $r_0$ and are unique. The sum converges in $W^{1,2}(A(r_0))$ and uniformly on all compact sets of $A(1)$. Moreover $$\begin{aligned}
\|u\|_{W^{1,2}(A(r_0))}^2=\sum_{i=0}^{+\infty}a_i^2\|g_i\|^2_{W^{1,2}(A(r_0))}.
\label{etoile}\end{aligned}$$
[**Proof :**]{} Since $u\in W^{1,2}(A(1))$ then for almost every $r_0$ in $]0,1]$ we have that $$u|_{r_0\Omega}\in W^{1,2}(r_0\Omega).$$ Thus we can apply the decomposition on $r_0\Omega$ and say that $$u=\sum_{i=0}^{+\infty}a_i h_i \quad \text{ on }r_0\Omega.$$ Define $g_i$ by $$g_i(x):=\|x\|^{\alpha_i}f_i\left(\frac{x}{\|x\|}\right)$$ where $\alpha_i$ is defined by . Since the $f_i$ are eigenfunctions for $-\Delta_S$, we deduce from that $$\begin{aligned}
\Delta g_i&=&\frac{\partial ^2}{\partial
r}g_i+\frac{N-1}{r}\frac{\partial}{\partial
r}g_i+\frac{1}{r^2}\Delta_Sg_i \notag \\
&=&\alpha_i(\alpha_i -1)r^{\alpha_i-2}f_i+\frac{N-1}{r}\alpha_i
r^{\alpha_i-1}f_i-r^{\alpha_i-2}\lambda_if_i \notag \\
&=&(\alpha_i^2+(N-2)\alpha_i-\lambda_i)r^{\alpha_i-2}f_i\notag \\
&=&0 \notag\end{aligned}$$ by definition of $\alpha_i$, thus the $g_i$ are harmonic in $A(+\infty)$. Notice that the $g_i$ are orthogonal in $L^{2}(A(1))$ because they are homogeneous and orthogonal in $L^{2}(\Omega)$. Note also that $h_i$ is equal to $g_i$ on $r_0\Omega$. Moreover for all $0<r\leq
1$ we have $$\begin{aligned}
\|g_i\|_{L^2(A(r))}^2&=& \int_{A(r)}|g_i|^2=\int_{0}^r\int_{\partial B(t)\cap A(1)}|g_i(w)|^2dwdt \notag \\
&=&\int_{0}^{r}\int_{\Omega}t^{N-1}|g_i(ty)|^2dydt =
\int_{0}^rt^{2\alpha_i+N-1}\int_{\Omega}|g_i(y)|^2dydt
\notag \\
&=&
\frac{r^{2\alpha_i+N}}{2\alpha_i+N}\|f_i\|^2_{L^2(\Omega)}=\frac{r^{2\alpha_i
+N}}{2\alpha_i+N}\leq 1. \label{normegi}\end{aligned}$$
In the other hand, since the $f_i$ and their tangential gradients are orthogonal in $L^{2}(\Omega)$, we deduce that the gradients of $g_i$ are orthogonal in $A(1)$. Then, by a computation similar to we obtain for all $0<r\leq 1$ $$\begin{aligned}
\|\nabla g_i\|^2_{L^2(A(r))} &=& \int_{0}^r\int_{\partial B(t)\cap
A(1)}|\frac{\partial g_i}{\partial r}|^2
+|\nabla_\tau g_i|^2dwdt \notag \\
&=&\int_{0}^r\int_{\partial B(t)\cap A(1)}|\alpha_i
t^{\alpha_i-1}f_i(\frac{w}{t})|^2
+|t^{\alpha_i}\nabla_\tau f_i(\frac{w}{t})\frac{1}{t}|^2dwdt \notag \\
&=&\alpha_i^2\int_{0}^r t^{2(\alpha_i-1)}\int_{\partial B(t)\cap
A(1)}| f_i(\frac{w}{t})|^2dwdt+\int_{0}^r
t^{2(\alpha_i-1)}\int_{\partial B(t)\cap A(1)}
|\nabla_\tau f_i(\frac{w}{t})|^2dwdt \notag \\
&=&\alpha_i^2\int_{0}^r t^{2(\alpha_i-1)}\int_{\Omega}|
f_i(w)|^2t^{N-1}dw dt+\int_{0}^r t^{2(\alpha_i-1)}\int_{\Omega}
|\nabla_\tau f_i(w)|^2t^{N-1}dw dt \notag \\
&=&\alpha_i^2\frac{r^{2(\alpha_i-1)+N}}{2(\alpha_i-1)+N}\|f_i\|^2_{L^2(\Omega)} +
\frac{r^{2(\alpha_i-1)+N}}{2(\alpha_i-1)+N}\|\nabla_\tau f_i\|^2_{L^2(\Omega)} \notag \\
&=&\frac{r^{2(\alpha_i-1)+N}}{2(\alpha_i-1)+N}(\alpha_i^2+\lambda_i)\|f_i\|^2_{L^2(\Omega)}\notag \\
&\leq & Cr^{2\alpha_i}(\alpha_i^2+\lambda_i )
\label{estimgradients}\end{aligned}$$ because $\|\nabla_\tau f_i\|_2^2=\lambda_i \|f_i\|_2^2$, $r\leq 1$ and $\alpha_i\geq 0$. Moreover the constant $C$ depends on the dimension $N$ but does not depend on $i$.
We denote by $g$ the function defined in $A(\infty)$ by $$g:=\sum_{i=0}^{+\infty}a_i g_i.$$ Then $g$ lies in $L^{2}(A(r_0))$ because using and $$\|g\|_{L^2(A(r_0))}^2=\sum_{i=0}^{+\infty}|a_i|^{2}
\|g_i\|_{L^2(A(r_0))}^2\leq
\sum_{i=0}^{+\infty}|a_i|^2r_0^{2\alpha_i+N} <+\infty.$$ We want now to show that $g=u$.
$\bullet$ *First step :* We claim that $g$ is harmonic in $A(r_0)$. Indeed, since the $g_i$ are all harmonic in $A(r_0)$, the sequence of partial sums $s_k:=\sum_{i=0}^{k}a_i
g_i$ is a sequence of harmonic functions, uniformly bounded for the $L^2$ norm in each compact set of $A(r_0)$. By the Harnack inequality we deduce that the sequence of partial sums is uniformly bounded for the uniform norm in each compact set. Thus there is a subsequence that converges uniformly to a harmonic function, which in fact is equal to $g$ by uniqueness of the limit. Therefore, $g$ is harmonic in $A(r_0)$.
$\bullet$ *Second step :* We claim that $g$ belongs to $W^{1,2}(A(r_0))$. Firstly, since $u\in W^{1,2}(r_0\Omega)$, by and we have that $$\begin{aligned}
\sum_{i=0}^{+\infty}a_i^2r_0^{2\alpha_i} \| \nabla_\tau f_i
\|_{L^2(\partial B(0,1)\backslash K)}^2<+\infty
.\label{estimgradients2}\end{aligned}$$ In addition, since $\|\nabla_\tau f_i\|_2^2=\lambda_i\|f_i\|_{2}^2$ and $\|f_i\|_{2}=1$, we deduce $$\begin{aligned}
\sum_{i=0}^{+\infty}a_i^2 r_0^{2\alpha_i}\lambda_i<+\infty
\label{estimgradients3}\end{aligned}$$ and since $\alpha_i$ and $\lambda_i$ are linked by the formula we also have that $$\begin{aligned}
\sum_{i=0}^{+\infty}a_i^2 r_0^{2\alpha_i}\alpha_i^2<+\infty.
\label{estimgradients4}\end{aligned}$$
Now, since $\sum a_i g_i$ converges absolutely on every compact set, we can say that $$\nabla g = \sum_{i=0}^{+\infty}a_i \nabla g_i$$ thus using , , , and orthogonality, $$\begin{aligned}
\|\nabla g\|^2_{L^{2}(A(r_0))} &=& \sum_{i=0}^{+\infty}
a_i^2 \|\nabla g_i\|_{L^2}^{2} \notag \\
& \leq& C\sum_{i=0}^{+\infty}a_i^2r_0^{2\alpha_i}(
\alpha_i^2+\lambda_i)<+\infty \notag.\end{aligned}$$ Therefore, $g\in W^{1,2}(A(r_0))$.\
$\bullet$ *Third step :* We claim that $\frac{\partial
g}{\partial n}=0$ on $K\cap \overline{A(r_0)} \backslash \Sigma$. We already know that $\frac{\partial g_i}{\partial n}=0$ on $K\backslash
\Sigma$ (because the $f_i$ have this property). We want to show that $g$ is so regular that we can exchange the order of $\frac{\partial}{\partial n}$ and $\sum$. So let $x_0$ be a point of $K\cap \overline{A(r_0)}\backslash \Sigma$ and let $B$ be a neighborhood of $x_0$ in ${\ensuremath{\mathbb R}}^N$ that doesn’t meet $\Sigma$ and such that $K$ separates $B$ in two parts $B^+$ and $B^-$. Assume that $B^+$ is a part in $A(r_0)$. The sequence of partial sums $s_k:=\sum_{i=0}^k a_i g_i$ is a sequence of harmonic functions in $B^+$. Since $\partial B^+\cap K$ is $C^2$ we can do a reflection to extend $s_k$ in $B^-$. For all $k$, this new function $s_k$ is the solution of a certain elliptic equation whose operator become from the composition of the Laplacian with the application that makes $\partial B^+\cap K$ flat. Thus since $\sum a_i g_i$ converges absolutely for the $L^2$ norm, by the Harnack inequality $\sum a_i g_i$ converges absolutely for the uniform norm in a smaller neighborhood $B'\subset B$ that still contains $x_0$. Thus $s_k$ converges to a $C^1$ function denoted by $s$, which is equal to $g$ on $B^+$. And since $\frac{\partial s_k}{\partial n}(x_0)=0$, by the absolute convergence of the sum we can exchange the order of the derivative and the symbol $\sum$ so we deduce that $\frac{\partial s}{\partial
n}(x_0)=0$. Finally, since $s$ is equal to $g$ on $B^+$ we deduce that $g$ is $C^1$ at the boundary and $\frac{\partial
g}{\partial n}=0$ at $x_0$.
$\bullet$ *Fourth step :* we claim that $g$ is equal to $u$ on $r_0\Omega$. Let $r$ be a radius such that $r<r_0$. Then the function $x\mapsto g_r(x):= g(r\frac{x}{r_0})$ is well defined for $x\in r_0\Omega$, and since the $g_i$ are homogeneous we have $$g(r\frac{x}{r_0})=\sum_{i=0}^{+\infty}a_ig_i(r\frac{x}{r_0})=
\sum_{i=0}^{+\infty}\left(\frac{r}{r_0}\right)^{\alpha_i}a_ig_i(x)=
\sum_{i=0}^{+\infty}\left(\frac{r}{r_0}\right)^{\alpha_i}a_ih_i(x).$$ We deduce that the function $x\mapsto g(\frac{r}{r_0}x)$ is in $L^{2}(r_0\Omega)$ and its coefficients in the basis $\{h_i\}$ are $\{(\frac{r}{r_0})^{\alpha_i} a_i\}$. We want to show that $\|g_r-u\|_{L^2(r_0\Omega)}$ tend to $0$. Indeed, writing $u$ in the basis $\{h_i\}$ $$u=\sum_{i=0}^{+\infty}a_i h_i,$$ we obtain $$\begin{aligned}
\|g_r-u\|_2^2=
\sum_{i=0}^{+\infty}\left(\left(\frac{r}{r_0}\right)^{\alpha_i}-1\right)^2
a_i^2 \|h_i\|_2^2 \notag\end{aligned}$$ which tends to zero when $r$ tends to $r_0$ by the dominated convergence theorem because $\left(\left(\frac{r}{r_0}\right)^{\alpha_i}-1\right)^2\leq 1$. Therefore, there is a subsequence for which $g_r$ tends to $u$ almost everywhere. On the other hand, since $g$ is harmonic, the limit of $g_r$ exists and is equal to $g$. That means that $g$ tends to $u$ radially at almost every point of $r_0\Omega$.
$\bullet$ *Fifth step:* The functions $u$ and $g$ are harmonic functions in $A(r_0)$, with finite energy, with a normal derivative equal to zero on $K\cap \overline{A(r_0)}\backslash \Sigma$ and that coïncide on $\partial A(r_0)\backslash K$. To show that $u=g$ in $A(r_0)$ we shall prove that $g$ is an energy minimizer. Proposition \[stam1\] will then give the uniqueness.
Let $\varphi \in W^{1,2}(A(r_0))\backslash K)$ have a vanishing trace on $\partial B(0,r_0)$. Then, setting $J(v):=
\int_{A(r_0)}|\nabla v|^2$ for $v\in
W^{1,2}(A(r_0))$ we have $$J(g+\varphi)=J(g)+\int_{A(r_0)}\nabla g \nabla \varphi + J(\varphi).$$ Now since $g$ is harmonic with Neumann condition on $K\backslash \Sigma$ and since $\varphi$ vanishes on $r_0\Omega$, integrating by parts we obtain $$J(g+\varphi)=J(g)+J(\varphi).$$ Since $J$ is non negative and $g+\varphi$ describes all the functions in $W^{1,2}(A(r_0))$ with trace equal to $u$ on $r_0\Omega$, we deduce that $g$ minimizes $J$. We can do the same with $u$ thus $u$ and $g$ are two energy minimizers with same boundary conditions. Therefore, by Proposition \[stam1\] we know that $g=u$.
$\bullet$ *Sixth step :* The decomposition do not depends on $r_0$. Indeed, let $r_1$ be a second choice of radius. Then we can do the same work as before to obtain a decomposition
$$u(x):=\sum_{i=0}^{+\infty}b_i g_i(x) \quad \text{ in }
B(0,r_1)\backslash K.$$
Now by uniqueness of the decomposition in $B(0,min(r_0,r_1))$ we deduce that $b_i=a_i$ for all $i$.
In addition, $r_0$ was initially chosen almost everywhere in $]0,1[$. But since the decomposition does not depend on the choice of radius, $r_0$ can be chosen anywhere in $]0,1[$, by choosing a radius almost everywhere in $]r_0,1[$.
\[mainth\] Let $(u,K)$ be a global minimizer in ${\ensuremath{\mathbb R}}^N$ such that $K$ is a smooth cone. Then for each connected component of ${\ensuremath{\mathbb R}}^N\backslash K$ there is a constant $u_k$ such that $u-u_k$ is $\frac{1}{2}$-homogenous.
[**Proof :**]{} Let $\Omega$ be a connected component of ${\ensuremath{\mathbb R}}^N\backslash K$. We apply the preceding proposition to $u$. Thus $$u(x)=\sum_{i=0}^{+\infty}a_i g_i(x) \quad \text{ in } A(r_0).$$ for a certain radius $r_0$ chosen in $]0,1[$. Let us prove that the same decomposition is true in $A(\infty)$. Applying Proposition \[decomp1\] to the function $u_R(x)=u(R x)$ we know that there are some coefficients $a_i(R)$ such that $$u_R(x)=\sum_{i=0}^{+\infty}a_i(R)g_i(x) \text{ in } A(r_0).$$ Now since $u_R(\frac{x}{R})=u(x)$ we can use the homogeneity of the $g_i$ to identify the terms in $B(0,r_0)$ thus $a_i(R)=a_i
R^{\alpha_i}$. Now we fix $y=Rx$ and we obtain that $$u(y)=\sum_{i=0}^{+\infty}a_i g_i(y) \text{ in } A(Rr_0).$$ Since $R$ is arbitrary the decomposition is true in $A(\infty)$.
In addition for every radius $R$ we know that $$\begin{aligned}
\|\nabla u\|^2_{L^{2}(A(R))}=
\sum_{i=0}^{+\infty}a_i^2 \|\nabla g_i\|^2_{L^2(A(R))} \label{som}\end{aligned}$$ and since $g_i$ is $\alpha_i$-homogenous, $$\|\nabla g_i\|_{L^2(A(R))}^2=R^{2(\alpha_i-1)+N}\|\nabla g_i\|_{L^2(A(1))}^2.$$ Now, since $u$ is a global minimizer, a classical estimate on the gradient obtained by comparing $(u,K)$ with $(v,L)$ where $v={\mathbf{1}}_{B(0,R)^c}u$ and $L=\partial B(0,R)\cup (K\backslash B(0,R))$ gives that there is a constant $C$ such that for all radius $R$ $$\|\nabla u\|_{L^2(B(0,R)\backslash K)}^2\leq C R^{N-1}.$$ We deduce $$\sum_{i=0}^{+\infty}a_i^2 R^{2(\alpha_i-1)+N}\|\nabla g_i\|^2_{L^2(A(1))}\leq CR^{N-1}.$$ Thus $$\sum_{i=0}^{+\infty}a_i^2 R^{2\alpha_i-1}\|\nabla g_i\|^2_{L^2(A(1))}\leq C.$$ This last quantity is bounded when $R$ goes to infinity if and only if $a_i=0$ whenever $\alpha_i> 1/2$. On the other hand, this quantity is bounded when $R$ goes to $0$, if and only if $a_i=0$ whenever $0<\alpha_i< 1/2$. Therefore, $u-a_0$ is a finite sum of terms of degree $\frac{1}{2}$.
In Chapter 65 of [@d], we can find a variational argument that leads to a formula in dimension $2$ that links the radial and tangential derivatives of $u$. For all $\xi
\in K\cap
\partial B(0,r)$, we call $\theta_\xi \in [0,\frac{\pi}{2}]$ the non oriented angle between the tangent to $K$ at point $\xi$ and the radius $[0,\xi]$. Then we have the following formula $$\int_{\partial B(0,r)\backslash K}\left(\frac{\partial u}{\partial r}\right)^2dH^1=
\int_{\partial B(0,r)\backslash K}\left(\frac{\partial u}{\partial
\tau}\right)^2dH^1+\sum_{\xi \in K \cap \partial B(0,r)}\cos
\theta_\xi -\frac{1}{r}H^1(K\cap B(0,r)).$$ Notice that for a global minimizer in ${\ensuremath{\mathbb R}}^2$ with $K$ a centered cone we find $$\begin{aligned}
\int_{\partial B(0,r)\backslash K}\left(\frac{\partial u}{\partial
r}\right)^2dH^1= \int_{\partial B(0,r)\backslash
K}\left(\frac{\partial u}{\partial \tau}\right)^2dH^1.
\label{deriv}\end{aligned}$$ Now suppose that $(u,K)$ is a global minimizer in ${\ensuremath{\mathbb R}}^N$ with $K$ a smooth cone centered at $0$. Then by Theorem \[mainth\] we know that $u$ is harmonic and $\frac{1}{2}$-homogenous. Its restriction to the unit sphere is an eigenfunction for the spherical Laplacian with Neumann boundary condition and associated to the eigenvalue $\frac{2N-3}{4}$. We deduce that $$\|\nabla_\tau u\|_{L^2(\partial B(0,1))}^2=\frac{2N-3}{4}\|u\|_{L^2(\partial
B(0,1))}^2.$$ On the other hand $$\frac{\partial u}{\partial
r}(x)=\frac{1}{2}\|x\|^{-\frac{1}{2}}u(\frac{x}{\|x\|})$$ thus $$\|\frac{\partial u}{\partial r}\|_{L^2(\partial B(0,1))}^2=\frac{1}{4}\|u\|_{L^2(\partial B(0,1))}^2.$$ So $$\|\nabla_\tau u\|_{L^2(\partial B(0,1))}^2=(2N-3)\|\frac{\partial u}{\partial r}\|_{L^2(\partial B(0,1))}^2.$$ In particular, for $N=2$ we have the same formula as .
Some applications {#applications}
=================
As it was claimed in the introduction, here is some few applications of Theorem \[mainth\].
\[app1\] Let $(u,K)$ be a global minimizer in ${\ensuremath{\mathbb R}}^3$ such that $K$ is a smooth cone. Moreover, assume that $S^2\cap K $ is a union of convex curvilinear polygons with $C^\infty$ sides. Then $u$ is locally constant and $K$ is a cone of type ${\ensuremath{\mathbb P}}$, ${\ensuremath{\mathbb Y}}$ or ${\ensuremath{\mathbb T}}$.
[**Proof :**]{} In each polygon we know by Proposition 4.5. of [@da] that the smallest positive eigenvalue for the operator minus Laplacian with Neumann boundary conditions is greater than or equal to $1$. Thus it cannot be $\frac{3}{4}$ and $u$ is locally constant. Then $K$ is a minimal cone in ${\ensuremath{\mathbb R}}^3$ and we know from [@d3] that it is a cone of type ${\ensuremath{\mathbb P}}$, ${\ensuremath{\mathbb Y}}$ or ${\ensuremath{\mathbb T}}$.
Let $(r,\theta,z)\in {\ensuremath{\mathbb R}}^+\times [-\pi,\pi]\times {\ensuremath{\mathbb R}}$ be the cylindrical coordinates in ${\ensuremath{\mathbb R}}^3$. For every $\omega \in
[0,\pi]$ set $$\Gamma_\omega:=\{(r,\theta,z) \in {\ensuremath{\mathbb R}}^3; -\omega<\theta<\omega \}$$ of boundary $$\partial \Gamma_\omega:=\{(r,\theta,z)\in {\ensuremath{\mathbb R}}^3; \theta=-\omega \text{ or } \theta=\omega \}.$$ Consider $\Omega_\omega= \Gamma_\omega \cap S^2$ and let $\lambda_1$ be the smallest positive eigenvalue of $-\Delta_S$ in $\Omega_\omega$ with Neumann conditions on $\partial
\Omega_\omega$. Then by Lemma 4.1. of [@da] we have that $$\lambda_1=\min(2,\lambda_\omega)$$ where $$\lambda_\omega=\left(\frac{\pi}{2\omega}+\frac{1}{2}\right)^2-\frac{1}{4}.$$
In particular for the cone of type ${\ensuremath{\mathbb Y}}$, $\omega=\frac{\pi}{3}$ thus $\lambda_1=2$.
Observe that for $\omega\not = \pi $, $\lambda_\omega\not = \frac{3}{4}$. So we get this following proposition.
\[app3\] There is no global Mumford-Shah minimizer in ${\ensuremath{\mathbb R}}^3$ such that $K$ is wing of type $\partial \Gamma_\omega$ with $\omega\not \in \{0,\frac{\pi}{2},\pi \}$.
Another consequence of Theorem \[mainth\] is the following. Let $P$ be the half plane $$P:=\{(r,\theta, z) \in {\ensuremath{\mathbb R}}^3; \theta=\pi\}.$$
\[cracktip\] Let $(u,K)$ be a global Mumford-Shah minimizer in ${\ensuremath{\mathbb R}}^3$ such that $K=P$. Then $u$ is equal to $cracktip\times \mathbb{R}$, that is in cylindrical coordinates $$u(r,\theta,z)=\pm \sqrt{\frac{2}{\pi}}r^{\frac{1}{2}}sin\frac{\theta}{2} +C$$ for $0<r< + \infty $ and $-\pi< \theta < \pi$.
In Section \[ann\] we will give a second proof of Proposition \[cracktip\].
We already know that $u=cracktip\times {\ensuremath{\mathbb R}}$ is a global minimizer in ${\ensuremath{\mathbb R}}^3$ (see [@d]).
To prove Proposition \[cracktip\] we will use the following well known result.
\[cracktip2\] The smallest positive eigenvalue for $-\Delta_n$ in $S^{2}\backslash P$ is $\frac{3}{4}$, the corresponding eigenspace is of dimension 1 generated by the restriction on $S^2$ of the following function in cylindrical coordinates $$u(r,\theta,z)=r^{\frac{1}{2}}sin\frac{\theta}{2}$$ for $0<r< + \infty $ and $-\pi< \theta < \pi$.
Now the proof of Proposition \[cracktip\] can be easily deduce from Proposition \[cracktip2\] and Theorem \[mainth\].
[**Proof of Proposition \[cracktip\]:**]{} If $(u,P)$ is a global minimizer, we know that after removing a constant the restriction of $u$ to the unit sphere is an eigenfunction for $-\Delta_n$ in $S^{2}\backslash P$ associated to the eigenvalue $\frac{3}{4}$. Therefore, from Proposition \[cracktip2\] we know that $$u(r,\theta,z)=Cr^{\frac{1}{2}}sin\frac{\theta}{2}$$ so we just have to determinate the constant $C$. But by a well known argument about Mumford-Shah minimizers we prove that $C$ must be equal to $\pm \sqrt{\frac{2}{\pi}}$ (see [@d] Section 61 for more details).
Now set $$S_\omega:=\{(r,\theta,0); r>0,\theta \in[-\omega,\omega]\}$$
\[sect\] There is no global Mumford-Shah minimizer in ${\ensuremath{\mathbb R}}^3$ such that $K$ is an angular sector of type $(u,
S_\omega)$ for $0<\omega<\frac{\pi}{2}$ or $\frac{\pi}{2}<\omega<\pi$.
[**Proof :**]{} According to Theorem \[mainth\], if $(u, S_\omega)$ is a global minimizer, then $u-u_0$ is a homogenous harmonic function of degree $\frac{1}{2}$, thus its restriction to $S^2\backslash S_\omega$ is an eigenfunction for $-\Delta_n$ associated to the eigenvalue $\frac{3}{4}$. Now if $\lambda(\omega)$ denotes the smallest eigenvalue on $\partial B(0,1)\backslash S_\omega$, we know by Theorem 2.3.2. p.47 of [@kmr] that $\lambda(\omega)$ is non decreasing with respect to $\omega$. Since $\lambda(\frac{\pi}{2})=\frac{3}{4}$, we deduce that for $\omega < \frac{\pi}{2}$, we have $$\begin{aligned}
\lambda(\omega)\geq \frac{3}{4}. \label{inega}\end{aligned}$$ In [@kmr] page 53 we can find the following asymptotic formula near $\omega=\frac{\pi}{2}$ $$\begin{aligned}
\lambda(\omega)=\frac{3}{4}+\frac{2}{\pi}\cos\omega+O(\cos^2\omega).
\label{dev}\end{aligned}$$ this proves that the case when is a equality only arises when $\omega=\frac{\pi}{2}$. Thus such eigenfunction $u$ doesn’t exist.
Consider now the case $\omega> \frac{\pi}{2}$. For $\omega=\pi$ there are tow connected components. Thus $0$ is an eigenvalue of multiplicity 2. The second eigenvalue is equal to $2$. Therefore, for $\omega=\pi$ the spectrum is $$0 \leq 0 \leq 2 \leq \lambda_3\leq ... \quad \quad \omega= \pi$$ By monotonicity, when $\omega$ decreases, the eigenvalues increase. Since the domain becomes connexe, $0$ become of multiplicity 1 thus the second eigenvalue become positive. The spectrum is now $$0 \leq \lambda_1 \leq \lambda_2 \leq ... \quad \quad \omega < \pi$$ with $\lambda_i \geq 2$ for $i \geq 2$. Thus the only eigenvalue that could be equal to $\frac{3}{4}$ is $\lambda_2$ which is increasing from from $0$ to $\frac{3}{4}$, reached for $\omega=\frac{\pi}{2}$. Now says that the increasing is strict near $\omega=\frac{\pi}{2}$. Therefore there is no eigenvalue equal to $3/4$ for $\omega > \frac{\pi}{2}$ and there is no possible global minimizer.
Second proof of Propositions \[cracktip\] and \[cracktip2\] {#ann}
===========================================================
Here we want to give a second proof of Proposition \[cracktip\], without using Theorem \[mainth\], and which do not use Proposition \[cracktip2\]. In a remark at the end of this section, we will briefly explain how to use this proof of Proposition \[cracktip\] in order to obtain a new proof of Proposition \[cracktip2\] as well.
Let assume that $K$ is a half plane in ${\ensuremath{\mathbb R}}^3$. We can suppose for instance that $$\begin{aligned}
K=P:=\{x_2=0\}\cap\{x_1\leq 0\} \label{definp}\end{aligned}$$ We begin by studying the harmonic measure in ${\ensuremath{\mathbb R}}^3\backslash P$.
Let $B:=B(0,R)$ be a ball of radius $R$ and let $\gamma$ be the trace operator on $\partial B(0,R) \backslash P$. We denote by $T$ the image of $W^{1,2}(B\backslash K)$ by $\gamma$. We also denote by $C_{b}^{0}(\partial B \backslash K)$ the set of continuous and bounded functions on $\partial B(0,1)\backslash P$. Finally set $A:=T\cap C_{b}^{0} $. Obviously $A$ is not empty. To every function $f\in A $, Proposition 15.6. of [@d] associates a unique energy minimizing function $u\in
W^{1,2}(B\backslash K)$ such that $\gamma(u)=f$ on $\partial B\backslash P$. Since $u$ is harmonic we know that it is $C^\infty$ in $B\backslash K$. Let $y\in B\backslash K$ be a fixed point and consider the linear form $\mu_y$ defined by $$\begin{aligned}
\mu_y: A &\to&{\ensuremath{\mathbb R}}\label{defmuy} \\
f&\mapsto&u(y). \notag\end{aligned}$$
By the maximum principle for energy minimizers, we know that for all $f\in A$ we have $$| \mu_y(f)|\leq \|f\|_{\infty}$$ thus $\mu_y$ is a continuous linear form on $A$ for the norm $\|\;\|_\infty$. We identify $\mu_y$ with its representant in the dual space of $A$ and we call it *harmonic measure*.
Moreover, the harmonic measure is positive. That is, if $f\in
A$ is a non negative function, then (by the maximum principle) $\mu_y(f)$ is non negative. By positivity of $\mu_y$, if $f \in A$ is a non negative function and $g \in A$ is such that $fg\in A$, then since $(\|g\|_\infty+g)f$ and $(\|g\|_\infty-g)f$ are two non negative functions of $A$ we deduce that $$\begin{aligned}
|\langle fg , \mu_y \rangle|\leq \|g\|_{\infty}\langle f,\mu_y
\rangle . \label{majoration}\end{aligned}$$
Now here is an estimate on the measure $\mu^R_y$.
\[rylem1\] There is a dimensional constant $C_N$ such that the following holds. Let $R$ be a positive radius. For $0<\lambda<\frac{R}{2}$ consider the spherical domain $$\begin{aligned}
\mathcal{C}_\lambda:=\{x \in \mathbb{R}^{3}\; ; \;|x|=R \textrm{ and } d(x,P)\leq \lambda\}. \notag\end{aligned}$$ Let $\varphi_{\lambda} \in C^\infty(\partial B(0,R))$ be a function between $0$ and $1$, that is equal to $1$ on $\mathcal{C}_\lambda$ and $0$ on $\partial B(0,R)\backslash \mathcal{C}_{2\lambda}$ and that is symmetrical with respect to $P$. Then for every $y \in B(0,\frac{R}{2})\backslash P$ we have $$\mu^R_y(\varphi_{\lambda})\leq C\frac{\lambda}{R}.$$
[**Proof :**]{} Since $\varphi_{\lambda}$ is continuous and symmetrical with respect to $P$, by the reflection principle, its harmonic extension $\varphi$ in $B(0,R)$ has a normal derivative equal to zero on $P$ in the interior of $B(0,R)$. Moreover $\varphi_\lambda$ is clearly in the space $A$. Thus by definition of $\mu_y$, $$\varphi(y)=\langle \varphi_{\lambda} , \mu^R_y \rangle .$$ On the other hand, since $ \varphi_{\lambda}$ is continuous on the entire sphere, we also have the formula with the classical Poisson kernel $$\begin{aligned}
\varphi(y)=\frac{R^2-|y|^2}{N\omega_N R}\int_{\partial B_R}\frac{
\varphi_\lambda(x)}{|x-y|^{3}}ds(x) \notag\end{aligned}$$ with $\omega_N$ equal to the measure of the unit sphere. In other words $$\mu^R_y(\varphi_\lambda)=\frac{R^2-|y|^2}{N\omega_N R}\int_{\partial B_R}\frac{
\varphi_\lambda(x)}{|x-y|^{3}}ds(x).$$ For $x\in \partial B_R$ we have $$\frac{1}{2}R\leq |x|-|y|\leq |x-y|\leq |x|+|y|\leq \frac{3}{2}R.$$ We deduce that $$\mu^R_y(\varphi_\lambda)\leq C_N \frac{1}{R^2}\int_{\mathcal{C}_{2\lambda}}ds.$$ Now integrating by parts, $$\begin{aligned}
\int_{\mathcal{C}_\lambda}ds&=&2\int_{0}^\lambda 2\pi \sqrt{R^2-w^2}dw \notag \\
&=&4\pi\frac{\lambda}{2}\sqrt{R^2-\lambda^2}+R^2 \arcsin (\frac{\lambda}{R}) \notag\\
&\leq& C R\lambda \notag\end{aligned}$$ because $\arcsin(x)\leq \frac{\pi}{2} x$. The proposition follows.
Now we can prove the uniqueness of $cracktip \times {\ensuremath{\mathbb R}}$.
[**Second Proof of Proposition \[cracktip\] :**]{} Let us show that $u$ is vertically constant. Let $t$ be a positive real. For $x=(x_1,x_2,x_3)\in
{\ensuremath{\mathbb R}}^3$ set $x_t:=(x_1,x_2,x_3+t)$. We also set $$u_t(x):=u(x)-u(x_t).$$ Since $u$ is a function associated to a global minimizer, and since $K$ is regular, we know that for all $R>0$, the restriction of $u$ to the sphere $\partial B(0,R)\backslash K$ is continuous and bounded on $\partial B(0,R)\backslash K$ with finite limits on each sides of $K$. It is the same for $u_t$. Thus for all $x\in {\ensuremath{\mathbb R}}^3\backslash P$ and for all $R>2\|x\|$ we can write $$u_t(x):= \langle u_t|_{\partial B(0,R)\backslash P}, \mu^R_x \rangle$$ where $\mu_x$ is the harmonic measure defined in . We want to prove that for $x \in {\ensuremath{\mathbb R}}^3 \backslash P$, $\langle u_t|_{\partial B(0,R)\backslash P}, \mu^R_x \rangle$ tends to $0$ when $R$ goes to infinity. This will prove that $u_t=0$.
So let $x\in {\ensuremath{\mathbb R}}^3\backslash P$ be fixed. We can suppose that $R>100(\|x\|+t)$. Let $\mathcal{C}_\lambda$ and $\varphi_\lambda$ be as in Lemma \[rylem1\]. Then write $$\begin{aligned}
u_t(x)= \langle u_t|_{\partial B(0,R)\backslash
P}\varphi_\lambda, \mu^R_x \rangle + \langle u_t|_{\partial
B(0,R)\backslash P}(1-\varphi_\lambda), \mu^R_x \rangle .\notag\end{aligned}$$ Now by a standard estimate on Mumford-Shah minimizers (that comes from Campanato’s Theorem, see [@afp] p. 371) we have for all $x \in {\ensuremath{\mathbb R}}^N
\backslash P$, $$|u_t(x)|\leq C\sqrt{t}.$$ Then, using Lemma \[rylem1\] we obtain $$\begin{aligned}
| \langle u_t|_{\partial B(0,R)\backslash P}\;\varphi_\lambda\;,\;
\mu^R_x \rangle| \leq C\sqrt{t}\frac{\lambda }{R} \notag.\end{aligned}$$ On the other hand, for the points $y$ such that $d(y,P)\geq \lambda$, since $\tilde u: u(.)-u(y)$ is harmonic in $B(y,d(y,P))$ we have, by a classical estimation on harmonic functions (see the introduction of [@gt]) $$|\nabla \tilde u(y)|\leq C\frac{1}{d(y,P)}\|\tilde u\|_{L^\infty(\partial B(y,\frac{1}{2}d(y,P)))}.$$ Now using Campanato’s Theorem again we know that $$\|\tilde u\|_{L^\infty(\partial B(y,\frac{1}{2}d(y,P)))}\leq
Cd(y,P)^{\frac{1}{2}}$$ thus $$|\nabla u(y)|\leq
C\frac{1}{d(y,P)^{\frac{1}{2}}}$$ and finally by the mean value theorem we deduce that for all the points $y$ such that $d(y,P)\geq \lambda$, $$|u_t(y)| \leq C\sup_{z \in [y,y_t]}|\nabla u(z)|.|y-y_t|\leq t\frac{1}{\lambda^\frac{1}{2}}.$$ Therefore, $$\begin{aligned}
| \langle u_t|_{\partial B(0,R)\backslash P}(1-\varphi_\lambda),
\mu^R_x \rangle| \leq Ct\frac{1 }{\lambda^{\frac{1}{2}}}. \notag\end{aligned}$$ So $$|u_t(x)|\leq C\sqrt{t}\frac{\lambda}{R}+Ct\frac{1 }{\lambda^{\frac{1}{2}}}$$ thus by setting $\lambda= R^{\frac{1}{2}}$ and by letting $R$ go to $+\infty$ we deduce that $u_t(x)=0$ thus $z\mapsto
u(x,y,z)$ is constant.
Now we fix $z_0=0$ and we introduce $P_0:=P\cap \{z=0\}$. We want to show that $(u(x,y,0),P_0)$ is a global minimizer in $\mathbb{R}^2$. Let $(v(x,y),\Gamma)$ be a competitor for $u(x,y,0)$ in the 2-dimensional ball $B$ of radius $\rho$. Let $\mathcal{C}$ be the cylinder $\mathcal{C}:=B\times[-R,R]$. Define $\tilde v$ and $\tilde
\Gamma$ in $\mathbb{R}^3$ by $$\tilde v(x,y,z)= \left\{
\begin{array}{cc}
v(x,y) &\text{ if } (x,y,z)\in \mathcal{C} \\
u(x,y,z) &\text{ if } (x,y,z) \not \in \mathcal{C}
\end{array}\right.$$
$$\tilde \Gamma := (\mathcal{C}\cap[\Gamma \times [-R,R]])\cup
(P\backslash \mathcal{C})\cup (B\times \{\pm R\}).$$ It is a topological competitor because ${\ensuremath{\mathbb R}}^3\backslash P$ is connected (thus $P$ doesn’t separate any points). Now finally let $\tilde B$ be a ball that contains $\mathcal{C}$. Then $(\tilde v, \tilde \Gamma)$ is a competitor for $(u,P)$ in $\tilde B$. By minimality we have : $$\int_{\tilde B}|\nabla u|^{2}+H^{2}(P \cap \tilde B) \leq \int_{\tilde B}|\nabla \tilde v|^{2}
+H^{2}(\tilde \Gamma \cap \tilde B).$$ In the other hand $u$ is equal to $\tilde
v$ in $\tilde B \backslash \mathcal{C}$ and it is the same for $\Gamma$ and $\tilde \Gamma$. We deduce $$\int_{\mathcal{C}}|\nabla u|^{2}dxdydz+H^{2}(P \cap \mathcal{C}) \leq \int_{\mathcal{C}}|\nabla \tilde v|^{2}
dxdydz+H^{2}(\tilde \Gamma \cap \mathcal{C}).$$ Now, since $u$ and $\tilde v$ are vertically constant, $\nabla_z u =
\nabla_z \tilde v =0$, and $\nabla_x u$, $\nabla_y u$ are also constant with respect to the variable $z$ (as for $\tilde v$). Thus $$2R\int_{B}|\nabla u(x,y,0)|^{2}dxdy+H^{2}(P \cap \mathcal{C}) \leq 2R\int_{B}|\nabla v(x,y)|^{2}dxdy +H^{2}(\tilde \Gamma \cap \mathcal{C}).$$ To conclude we will use the following lemma.
\[rotyi\] If $\Gamma$ is rectifiable and contained in a plane $Q$ then $$H^{2}(\Gamma \times [-R,R])=2R H^{1}(\Gamma).$$
[**Proof :**]{} We will use the coarea formula (see Theorem 2.93 of [@afp]). We take $f:\mathbb{R}^{3}\to \mathbb{R}$ the orthogonal projection on the coordinate orthogonal to $Q$. By this way, if $E:=\Gamma \times [-R,R]$, we have $E\cap
f^{-1}(t)=\Gamma $ for all $t\in [-R,R]$. $E$ is rectifiable (because $\Gamma$ is by hypothesis). So we can apply the coarea formula. To do this we have to calculate the jacobian $c_k d^{E}f_x$. By construction, the approximate tangente plane in each point of $E$ is orthogonal to $Q$. We deduce that if $T_x$ is a tangent plane, then there is a basis of $T_x$ $(\overrightarrow{b_1},\overrightarrow{b_2})$ such that $\overrightarrow{b_1}$ is orthogonal to $Q$. Since the function $f$ is the projection on $\overrightarrow{b_1}$, and its derivative as well (because $f$ is linear ) we obtain that the matrix of $d^{E}f_x:T_x \to
\mathbb{R}$ in the basis $(\overrightarrow{b_1},\overrightarrow{b_2})$ is $$d^{E}f_x=(1,0)$$ thus $$c_k d^{E}f_x=\sqrt{det[(1,0).^t(1,0)]}=1.$$ Therefore $$H^{2}(E)=\int_{-R}^{R}H^{1}(\Gamma)=2RH^{1}(\Gamma).\qed$$
Here we can suppose that $\Gamma$ is rectifiable. Indeed, the definition of Mumford-Shah minimizers is equivalent if we only allow rectifiables competitors. This is because the jump set of a $SBV$ function is rectifiable and in [@dcl] it is proved that the relaxed functional on the $SBV$ space has same minimizers.
So we have $$2R\int_{B}|\nabla u(x,y,0)|^{2}dxdy+2R H^{1}(P \cap B) \leq 2R\int_{B}|\nabla v(x,y)|^{2}dxdy +
2RH^{1}(\Gamma \cap B)+H^2(B\times\{\pm R\}).$$ Then, dividing by $2R$, $$\int_{B}|\nabla u(x,y,0)|^{2}dxdy+ H^{1}(P \cap B) \leq \int_{B}|\nabla v(x,y)|^{2}dxdy +
H^{1}(\Gamma \cap B)+\pi \frac{\rho^2}{R}$$ thus, letting $R$ go to infinity, $$\int_{B}|\nabla u(x,y,0)|^{2}dxdy+ H^{1}(P \cap B) \leq \int_{B}|\nabla v(x,y)|^{2}dxdy +
H^{1}(\Gamma \cap B).$$ This last inequality proves that $(u(x,y,0),P_0)$ is a global minimizer in $\mathbb{R}^{2}$, and since $P_0$ is a half-line, $u$ is a $cracktip$.
Using a similar argument as the preceding proof, we can show that the first eigenvalue for $-\Delta$ in $S^2\backslash P$ with Neumann boundary conditions (where $P$ is still a half-plane), is equal to $\frac{3}{4}$. Moreover we can prove that the eigenspace is of dimension $1$, generated by a function of type $cracktip\times {\ensuremath{\mathbb R}}$, thus we have a new proof of Proposition \[cracktip2\]. The argument is to take an eigenfunction $f$ in $S^2 \backslash P$, then to consider $u(x):=\|x\|^{\alpha}f(\frac{x}{\|x\|})$ with a good coefficient $\alpha\in ]0,\frac{1}{2}]$ that makes $u$ harmonic. Finally we use the same sort of estimates on the harmonic measure to prove that $u$ is vertically constant. Thus we have reduced the problem in dimension 2 and we conclude using that we know the eigenfunctions on the circle. A detailed proof is done in [@l1].
Open questions
==============
As it is said in the introduction, this paper is a very short step in the discovering of all the global minimizers in ${\ensuremath{\mathbb R}}^N$. This final goal seems rather far but nevertheless some open questions might be accessible in a more reasonable time. All the following questions were pointed out by Guy David in [@d], and unfortunately they are still open after this paper.
$\bullet$ Suppose that $(u,K)$ is a global minimizer in ${\ensuremath{\mathbb R}}^N$. Is it true that $K$ is conical ?
$\bullet$ Suppose that $(u,K)$ is a global minimizer in ${\ensuremath{\mathbb R}}^N$, and $K$ is a cone. Is it true that $\frac{3-2N}{4}$ is the smallest eigenvalue of the Laplacian on $S^{N-1}\backslash K$ ?
$\bullet$ Suppose that $(u,K)$ is a global minimizer in ${\ensuremath{\mathbb R}}^3$, and suppose that $K$ is contained in a plan (and not empty). Is it true that $K$ is a plane or a half-plane ?
$\bullet$ Could one found an extra global minimizer in ${\ensuremath{\mathbb R}}^3$ by blowing up the minimizer described in section 76.c. of [@d] (see also [@mer])?
One can find other open questions on global minimizers in the last page of [@d].
ADDRESS :
Antoine LEMENANT\
e-mail : [email protected]
Université Paris XI\
Bureau 15 Bâtiment 430\
ORSAY 91400 FRANCE
Tél: 00 33 169157951
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The dynamical exchange-correlation kernel $f_{xc}$ of a non-uniform electron gas is an essential input for the time-dependent density functional theory of electronic systems. The long-wavelength behavior of this kernel is known to be of the form $f_{xc}= \alpha/q^2$ where $q$ is the wave vector and $\alpha$ is a frequency-dependent coefficient. We show that in the limit of weak non-uniformity the coefficient $\alpha$ has a simple and exact expression in terms of the ground-state density and the frequency-dependent kernel of a [*uniform*]{} electron gas at the average density. We present an approximate evaluation of this expression for Si and discuss its implications for the theory of excitonic effects.'
author:
- 'V. U. Nazarov'
- 'G. Vignale'
- 'Y.-C. Chang'
title: 'Exact dynamical exchange-correlation kernel of a weakly inhomogeneous electron gas'
---
Since its introduction in works of Runge, Gross, and Kohn [@Runge-84; @Gross-85], the time-dependent density-functional theory (TDDFT) has evolved into a powerful tool of investigation of systems ranging from isolated atoms to bulk solids. In the important linear-response regime, the key quantity of TDDFT is the dynamical exchange-correlation (xc) kernel $f_{xc}$ defined as the functional derivative $$\begin{aligned}
f_{xc}[n_0({\bf r})]({\bf r},{\bf r}',\omega)=\left. \frac{\delta V_{xc}[n]({\bf r},\omega)}{\delta n({\bf r}',\omega)}\right|_{n=n_0({\bf r})}\end{aligned}$$ of the dynamical exchange and correlation potential $V_{xc}$ with respect to the dynamical electron density $n$, taken at the ground-state value $n_0$ of the latter. With this definition, the density-response function $\chi$ can be represented in operator notation as [@Gross-85] $$\begin{aligned}
\chi(\rv,\rv',\omega)=\left\{[1-\chi_{KS}(C+f_{xc})]^{-1} \chi_{KS}\right\}(\rv,\rv',\omega)\,,
\label{oper}\end{aligned}$$ where $\chi_{KS}$ is the Kohn-Sham (KS) density-response function of independent electrons, $C=e^2/|{\bf r}-{\bf r}'|$ is the Coulomb interaction, and $e$ is the absolute value of the electron charge. While the density-response function of non-interacting electrons $\chi_{KS}$ can be straightforwardly calculated in many cases of interest (e.g., for homogeneous electron gases in three and two dimensions it is given by the analytical Lindhard’s [@Lindhard-54] and Stern’s [@Stern-67] formulas, respectively), the construction of $f_{xc}$, whose role is to account for dynamical many-body correlations, is not straightforward.
As an instructive specific case, let us consider the excitonic effect in a semiconductor, which would manifest itself as an enhancement of the imaginary part of $\chi$ for frequencies close to the fundamental absorption edge. We neglect for a moment local-field effects and write down the diagonal elements of the density response in momentum space as of Eq. (\[oper\]) $$\begin{aligned}
\chi(\qv,\qv,\omega)=\frac{\chi_{KS}(\qv,\qv,\omega)}{1-\chi_{KS}(\qv,\qv,\omega)[\frac{4\pi e^2}{q^2}+f_{xc}(\qv,\qv,\omega)]}\,
\label{opers}\end{aligned}$$ where $4 \pi e^2/q^2$ is the Fourier transform of the Coulomb interaction. On the one hand, the excitonic enhancement of $\chi$ is a many-body effect and, therefore, it needs a nonzero $f_{xc}$ to be accounted for within TDDFT. On the other hand, because of the divergent Coulomb part $4\pi e^2/q^2$ in Eq. (\[opers\]), any $f_{xc}(q,q,\omega)$ that remained finite at $q=0$ would give no contribution in the long-wave limit $q\rightarrow 0$. This simple observation shows that in order to include the exciton, $f_{xc}(\qv,\qv,\omega)$ must be divergent in the long-wave limit at least as strongly as the Coulomb term. And indeed, when the $q^{-2}$ divergence has been introduced empirically in papers dealing with the optical absorption spectrum of semiconductors [@Reining-02; @Del_Sole-03; @Botti-04], it has yielded a good TDDFT description of the excitonic effect.
Clearly it would be highly desirable to have a first-principle theory of the small-$\qv$ behavior of the xc kernel, rather than relying on empirical parametrizations. In this Letter we take a step in this direction. We first show that the asymptotic relation $$\begin{aligned}
\lim_{\qv\rightarrow 0} f_{xc}(\qv,\qv,\omega)=\frac{e^2 \alpha(\omega)}{q^2}
\label{as}\end{aligned}$$ (we introduce the $e^2$ so that $\alpha$ is dimensionless) is a rigorous consequence of exact sum rules for the current density response function. Then, in the limit of weak non-uniformity we obtain a simple and exact expression for $\alpha(\omega)$ in terms of the ground-state density and the dynamical xc kernel of a homogeneous electron gas at the average density[^1].
We start by noting that the local density approximation (LDA) to $f_{xc}$ is unable to produce the divergence. Indeed, within LDA [@Gross-85] $$\begin{aligned}
f_{xc}({\bf r},{\bf r}',\omega)= f_{xc}^h[n({\bf r}),\omega] \, \delta({\bf r}-{\bf r}'),
\label{LDA}\end{aligned}$$ where $f_{xc}^h(n,\omega)$ is the long-wave limit of the xc kernel of the homogeneous electron gas of density $n$. The latter is known [@Gross-85; @Nifosi-98; @Conti-99; @Qian-02] to be finite, no divergence arising, therefore, in the Fourier transform of Eq. (\[LDA\]).
To obtain an accurate non-local $f_{xc}$, we resort to the recently proposed general method [@Nazarov-07] derived from the time-dependent [*current*]{}-density functional theory (TDCDFT) [@Vignale-96]. This method is based on the exact relation that holds between the scalar density-response function $\chi$ (density response to a scalar potential) and the tensor current-density-response function $\hat{\chi}$ (current-density response to a vector potential): $$\label{GaugeRelation}
\chi(\qv,\qv',\omega)=\frac{c}{e\omega^2}\qv \cdot \hat \chi(\qv,\qv',\omega) \cdot \qv'.
\label{rel}$$ Both response functions are expressed in terms of the corresponding Kohn-Sham response functions and xc kernels in the following manner: $$\label{Chifxc1}
\chi^{-1}(\qv,\qv',\omega)=\chi_{KS}^{-1}(\qv,\qv',\omega)-f_{xc}(\qv,\qv',\omega)-\frac{4\pi e^2}{q^2}\delta_{\qv\qv'}$$ and $$\label{Chifxc2}
\hat\chi^{-1}(\qv,\qv',\omega)=\hat\chi_{KS}^{-1}(\qv,\qv',\omega)-\hat f_{xc}(\qv,\qv',\omega)-\frac{4\pi e c}{\omega^2}\hat L_{\qv}\delta_{\qv\qv'}\,,$$ where $\hat L_{\qv,ij} \equiv q_iq_j/q^2$, $i$ and $j$ are cartesian indices.
Equations (\[GaugeRelation\]-\[Chifxc2\]) establish a connection between $f_{xc}$ and its tensor counterpart $\hat f_{xc}$. The usefulness of this connection stems from the fact that the tensor quantities $\hat \chi_{KS}$ and $\hat f_{xc}$ satisfy a broader set of exact sum rules than the corresponding scalar quantities. These sum rules were derived in Ref. [@Vignale-B]. Specializing to the case of a periodic systems, the two most important sum rules for our purposes are $$\begin{aligned}
\label{SumRule1}
&&\hat{\chi}_{KS,ij}({\bf G},0,\omega) = \frac{e}{m c} n_0({\bf G}) \delta_{ij}\cr
&&- \frac{1}{m \omega^2} \sum\limits_{{\bf G}',k} \hat{\chi}_{KS,i,k}({\bf G},{\bf G}',\omega) G'_k G'_j V_{KS}({\bf G}')\,,\end{aligned}$$ and $$\begin{aligned}
\label{SumRule2}
\sum\limits_{ {\bf G}'}\hat{f}_{xc,ij}({\bf G},{\bf G}',\omega) n_0({\bf G}') = \frac{c}{e \omega^2} \, G_i G_j V_{xc}({\bf G})\,,\end{aligned}$$ where $\Gv$ are reciprocal lattice vectors. These sum rules connect three different types of components of, say, $\chi_{KS}(\Gv,\Gv',\omega)$: the $({\bf 0},{\bf 0})$ component, the $({\bf 0},{\Gv\neq{\bf 0}})$ and $({\Gv\neq{\bf 0}},{\bf 0})$ components, and the $({\Gv\neq{\bf 0}},{\Gv'\neq{\bf 0}})$ components.
Let us further restrict our attention to the case of a weakly inhomogeneous system: $|n_0(\Gv)|\ll n_0({\bf 0}) \equiv \bar n_0$, and $|V_{KS}(\Gv)| \ll \hbar^2G^2/2m$ for $\Gv \neq {\bf 0}$. Then it is easily shown that the homogenous electron gas approximation for the $({\Gv\neq{\bf 0}},{\Gv'\neq{\bf 0}})$ components completely determines the $({\bf 0},{\Gv\neq{\bf 0}})$ and $({\Gv\neq{\bf 0}},{\bf 0})$ components to [*first order*]{} in $n_0(\Gv \ne {\bf 0})$, which in turn completely determines the $({\bf 0},{\bf 0})$ component to [*second order*]{} in $n_0(\Gv \ne {\bf 0})$. Thus, for $\hat{\chi}_{KS}$ we obtain
$$\begin{aligned}
\hat{\chi}_{KS,ij}({\bf G}\ne \0v,{\bf G}'\ne \0v,\omega)&=&
\left[ \frac{e \omega^2}{c\, G^2} L_{{\bf G},ij} \, \chi^{hL}_{KS}(G,\omega)+T_{{\bf G},ij} \, \chi^{hT}_{KS}(G,\omega)
\right] \delta_{{\bf G G}'}\,,\cr\cr
\hat{\chi}_{KS,ij}({\bf G} \ne \0v,\0v,\omega)&=&\hat{\chi}_{KS,ij}(\0v,-{\bf G},\omega)=
\frac{e}{m c} \left[n_0({\bf G}) \delta_{ij}
- L_{{\bf G},ij} \chi^{hL}_{KS}(G,\omega) V_{KS}({\bf G})\right]\,,\cr\cr
\hat{\chi}_{KS,ij}(\0v,\0v,\omega)&=& \frac{e \bar n_0}{m c } \delta_{ij} + \frac{e}{m^2 \omega^2 c} \sum\limits_{{\bf G}\ne \0v} G^2 L_{{\bf G},ij} |V_{KS}({\bf G})|^2
\left[\chi^{hL}_{KS}(G,\omega) -\chi^{hL}_{KS}(G,0)\right]\,.
\label{chiKS}\end{aligned}$$
to the zero-th, first, and second order in $V_{KS}({\bf G})$, respectively. Here $\chi^{hL}_{KS}$ and$\chi^{hT}_{KS}$ are, respectively, the longitudinal and transverse KS density-response functions of the homogeneous electron gas of density $\bar n_0$, and $T_{{\bf G},ij}=\delta_{ij} -L_{{\bf G},ij}$. Similarly, for $\hat f_{xc}$ we have $$\begin{aligned}
&&\hat{f}_{xc,ij}({\bf G}\ne 0,{\bf G}' \ne 0,\omega)= \frac{c}{e \omega^2 } \, G^2 \! \left[ f_{xc}^{hL}(G,\omega) L_{{\bf G},ij} \! + \! f_{xc}^{hT}(G,\omega) T_{{\bf G},ij} \right] \delta_{{\bf G G}'}\,,
%\label{fxct3}
\nonumber
\\
&&\hat{f}_{xc,ij}({\bf G}\ne \0v,\0v,\omega) \! = \! \hat{f}_{xc,ji}(\0v,-{\bf G},\omega) \! = \!
-\frac{c \, G^2}{e \omega^2 \bar n_0} n_0({\bf G}) \! \left\{ [f_{xc}^{hL}(G,\omega) \! - \! f_{xc}^{hL}(G,0)] L_{{\bf G},ij} \! + \! f_{xc}^{hT}(G,\omega) T_{{\bf G},ij} \! \right\}\,,
%\label{fxct2}
\nonumber
\\
&&\hat{f}_{xc,ij}(\0v,\0v,\omega)=
\frac{c}{e \omega^2 \bar n_0^2} \sum\limits_{ {\bf G}\ne \0v} G^2 |n_0({\bf G})|^2
\left\{ \left[f_{xc}^{hL}(G,\omega) -f_{xc}^{hL}(G,0) \right]L_{{\bf G},ij} + f_{xc}^{hT}(G,\omega) T_{{\bf G},ij} \right\},
%\label{fxct1}
\label{FXC}\end{aligned}$$
where $f_{xc}^{hL}$ and $f_{xc}^{hT}$ are the longitudinal and transverse, respectively, xc kernels of the homogeneous electron gas of density $\bar n_0$.
![\[Flowchart\]Scheme of the procedure for calculating the xc kernel $f_{xc}$ starting from the expressions (\[chiKS\]) and (\[FXC\]) for $\hat \chi_{KS}$ and $\hat f_{xc}$, respectively. ](flowchart.eps){width="40.00000%" height="30.00000%"}
The following steps, which involve repeated inversions of infinite matrices, rely on the mathematical fact that to find the $({\bf 0},{\bf 0})$, $({\bf 0},{\bf G}\ne {\bf 0})/({\bf G}\ne {\bf 0},{\bf 0})$, and $({\bf G}\ne {\bf 0},{\bf G}'\ne {\bf 0})$ elements of the inverse matrix to the second, first, and zeroth order in the inhomogeneity, respectively, it is sufficient to know the corresponding elements of the original matrix to the same accuracy, and then the inversion can be performed in a closed form [@Sturm-82].
The complete procedure is schematically illustrated in Fig. (\[Flowchart\]). Starting from Eqs. (\[chiKS\]) and (\[FXC\]) for $\hat \chi_{KS}$ and $\hat f_{xc}$ we (i) Invert Eqs. (\[chiKS\]) to get $\hat{\chi}_{KS}^{-1}$; (ii) Combine $\hat{\chi}_{KS}^{-1}$ and $\hat f_{xc}$ to get $\hat{\chi}^{-1}$ by virtue of Eq. (\[Chifxc2\]); (iii) Invert $\hat{\chi}^{-1}$ to get $\hat \chi$ ; (iv) Use Eq. (\[GaugeRelation\]) and its KS analogue to find the scalar response function $\chi$ from $\hat{\chi}$ and $\chi_{KS}$ from $\hat{\chi}_{KS}$; (v) Invert $\chi$ and $\chi_{KS}$ to get $\chi^{-1}$ and $\chi_{KS}^{-1}$; and, (vi) Apply Eq. (\[Chifxc2\]) to find $f_{xc}$. The final result of this procedure is
$$\begin{aligned}
&& \lim_{q \rightarrow 0} f_{xc}({\bf G}\ne \0v,{\bf G}' \ne \0v,\omega) = f^h_{xc,L}(G,\omega) \delta_{{\bf G G}'},
\label{fxc3}\\
&& \lim_{q \rightarrow 0} f_{xc}({\bf G} \ne \0v,{\bf q},\omega) = f_{xc}(-{\bf q},-{\bf G},\omega)= - \frac{({\bf G}\cdot {\hat{\bf q}})}{\bar n_0 q}
\left[f_{xc}^{hL}(G,\omega) -f_{xc}^{hL}(G,0) \right] n_0({\bf G}),
\label{fxc2} \\
&& \lim_{q \rightarrow 0} f_{xc}({\bf q},{\bf q},\omega) =
\frac{1}{\bar n_0^2 q^2}
\sum\limits_{{\bf G}\ne \0v} ({\bf G}\cdot {\hat{\bf q}})^2 \left[f_{xc}^{hL}(G,\omega) -f_{xc}^{hL}(G,0) \right] |n_0({\bf G})|^2,
\label{fxc}\end{aligned}$$
where $\hat{\bf q}$ is the unit vector parallel to ${\bf q}$. It should be noted at this point that the above expression for the scalar kernel $f_{xc}({\bf q},{\bf q},\omega)$ differs from what one would get by simply taking the longitudinal component of $\hat f_{xc,ij}({\bf q},{\bf q},\omega)$, i.e. $f_{xc}({\bf q},{\bf q},\omega) \neq \frac{e\omega^2}{c q^2}\sum_{i,j} \hat q_i \hat{f}_{xc,ij}({\bf q},{\bf q},\omega) \hat q_j$. The implication is that the scalar xc potential ($V_{xc}$) of time-dependent DFT is [*not*]{} equivalent to the longitudinal component of the vector potential ($\Av_{xc}$) of time-dependent CDFT: rather, it should be constructed through the careful inversion procedure described above. A recent interesting attempt to construct $V_{xc}$ from $\Av_{xc}$ [@Maitra-07] should be re-examined in the light of this result.
From the result of the step (iv) for $\chi$ and making use of the relation $$\begin{aligned}
\frac{1}{\epsilon_M(\omega)}= 1+\lim_{q\rightarrow 0} \frac{4\pi e^2}{q^2} \chi({\bf q},{\bf q},\omega),
\label{eMM}\end{aligned}$$ we obtain a formula for the macroscopic dielectric function of a crystal $$\begin{aligned}
&& \! \! \! \! \! \! \! \! \! \! \! \epsilon_M(\omega) = 1 -
\frac{4\pi e^2 \bar n_0}{ m \omega^2 } -
\frac{ e^2} { m^2 \omega^4 } \cr\cr
&& \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \times \sum\limits_{{\bf G}\ne 0} |V_{0}({\bf G})|^2 G^2
(\hat{{\bf q}} \cdot {\bf G})^2
\left[ \frac{1}{\epsilon^{hL}(G,\omega)}- \frac{1}{\epsilon^{hL}(G,0)}\right] \! \! ,
\label{eM}\end{aligned}$$ where $V_0$ is the bare crystalline potential and $$\begin{aligned}
\epsilon^{hL}(q,\omega)=1-\frac{4\pi e^2}{q^2} \frac{\chi^{hL}_{KS}(q,\omega)}{1-\chi^{hL}_{KS}(q,\omega) f^h_{xc,L}(q,\omega)}\end{aligned}$$ is the longitudinal dielectric function of the homogeneous electron liquid. Equation (\[eM\]) is in agreement with the Hopfield’s formula for optical conductivity [@Hopfield-65], while in the RPA \[$f^h_{xc,L}(G,\omega)=0$\] it coincides with the corresponding result of Ref. .
Equations (\[fxc3\]-\[fxc\]) are the main result of this paper. They replace the grossly inaccurate LDA formula $$\begin{aligned}
\lim_{q \rightarrow 0} f_{xc}({\bf G}+{\bf q},{\bf G}'+{\bf q},\omega) = f^h_{xc,L}(G,\omega) \delta_{{\bf G G}'},
\label{fxc33}\end{aligned}$$ which does not contain the singularity in $q$. Identifying the ($\0v,\0v$) element of the microscopic matrix of the xc kernel in Eqs. (\[fxc\]) as the averaged $f_{xc}$, we see that $f_{xc}$ diverges for $q\rightarrow 0$ as described by Eq. (\[as\]), wherein $\alpha(\omega)$ is given by $$\begin{aligned}
\alpha(\omega) \! \! = \! \!
\! \! \sum\limits_{{\bf G}\ne 0} \! \! \! \frac{({\bf G} \! \cdot \! \hat{{\bf q}})^2 }{ \bar n_0^2} \! \left[f_{xc}^{hL}(G,\omega) \! - \! \! f_{xc}^{hL}(G,0) \! \right] \! |n_0({\bf G})|^2.
\label{alpha}\end{aligned}$$ Notice that $\alpha(\omega)=0$ in the uniform limit and $\alpha(0)=0$ up to second order in $n_0(\Gv \ne {\bf 0})$ [^2].
In order to calculate $\alpha(\omega)$ from Eq. (\[alpha\]) we need the Fourier amplitudes of the ground-state electron density and the wave vector and frequency-dependent $f_{xc}^{hL}$ of the homogeneous electron gas, evaluated at reciprocal lattice vectors. The first ingredient is straightforwardly obtained from standard electronic structure calculations. Unfortunately, the same cannot be said of the second ingredient $f_{xc}^{hL}(q,\omega)$, for which we do not have reliable expressions. The best that can be done, at this time, is either to disregard the wave vector dependence, or to make use of the interpolation formula proposed in Ref. , which however fails to reproduce, at small $q$, what is presently believed to be the qualitatively correct form of the frequency dependence. In spite of these difficulties, it must be emphasized that the calculation of $f_{xc}^{hL}(q,\omega)$ is still a much simpler problem than the calculation of the dynamical xc kernel of the non-uniform system. Thus, our Eq. (\[alpha\]) does not simply express an unknown quantity in terms of another unknown quantity, but actually opens the way to systematic calculations of $\alpha$ based on the many-body theory of the homogeneous electron gas. Further, Eqs. (\[fxc3\])-(\[fxc\]) for $f_{xc}$ offer a promising alternative to the widespread practice of treating the dynamical exchange and correlations effects in the LDA.
In Fig. \[Fig\] we plot $\alpha(\omega)$ from Eq. (\[alpha\]) vs frequency for crystalline silicon. The Fourier coefficients of the electron density were calculated with the code FHI98MD [@Bockstedte-97], and we approximated $f_{xc}^{hL}(q,\omega) \simeq f_{xc}^{hL}(0,\omega)$, taking the latter from Ref. . In the range 0-22 eV, the real part of $\alpha(\omega)$ is negative, changing sign for positive above 22 eV. It reaches its minimum of $\alpha \approx$ -0.1 at $\omega\approx$ 14 eV. In the range 3-5 eV of the main absorption in silicon, ${\rm Re} \, \alpha$ changes from -0.01 to -0.03, which is an order of magnitude smaller than the empirical value of $\alpha \approx$ -0.2 found as the best fit to the experimental spectrum in Ref. . This large difference may simply indicate that the nearly free electron model, while being adequate for simple metals and even for semiconductors in the high-frequency regime [@Sturm-82], is not sufficiently accurate for semiconductors at frequency lower than or comparable to the band gap (see also footnote \[\]). Another probable source of discrepancy is that our approach is a pure TDDFT, whereas the value of $\alpha \approx -0.2$ was obtained in Refs. and with the use of self-energies incorporated in the Green’s function via the $GW$ approximation.
![\[Fig\] The frequency dependence of the real (upper panel) and image (lower panel) parts of the $\alpha$ coefficient in Eq. (\[as\]) for silicon calculated by Eq. (\[alpha\]). ](alpha.eps){width="47.50000%" height="35.00000%"}
In conclusion, within the nearly free electron approximation, we have constructed the otherwise exact exchange-correlation kernel for time-dependent density-functional theory. This kernel is nonlocal in space, exhibiting the $q^{-2}$ singularity in the reciprocal space. The strength of this singularity, which is frequency-dependent, has been directly related to the magnitude of the non-uniformity of the density of valence electrons, and this singularity disappears in the limiting case of the homogeneous electron liquid. We are proposing an improvement over the conventional LDA scheme of including the dynamical exchange and correlation effects into [*ab initio*]{} calculations of the linear response of crystalline solids which consistently accounts for the long-wave divergence in the exchange-correlation kernel.
GV acknowledges support from DOE Award No. DE-FG02-05ER46203.
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[^1]: The existence of a divergence in the [*off-diagonal*]{} components $f_{xc}(\qv,\kv+\qv,\omega)$ for $\qv\to 0$ and $\kv$ finite was first pointed out in Ref. .
[^2]: This is not to say that $f_{xc}(\qv,\qv,0)$ is always free of the $q^{-2}$ singularity, but it means that such a singularity, if present, cannot be reached by a perturbative expansion about the homogeneous ground-state. Indeed, there is evidence that the $f_{xc}(\qv,\qv,0)$ of band insulators has a $q^{-2}$ singularity, which is missed in the present approach.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Parameswaran Kamalaruban[^1]\
École Polytechnique Fédérale de Lausanne\
`[email protected]`\
Victor Perrier\
ISAE-SUPAERO & Data61, CSIRO\
`[email protected]`\
Hassan Jameel Asghar\
Macquarie University & Data61, CSIRO\
`[email protected]`\
Mohamed Ali Kaafar\
Macquarie University & Data61, CSIRO\
`[email protected]`\
title: '$d_{\mathcal{X}}$-Private Mechanisms for Linear Queries'
---
[^1]: Work done while Kamalaruban was a Postgraduate Researcher at Data61, CSIRO.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the capacitance-coupled Josephson junction array beyond the small-coupling limit. We find that, when the scale of the system is large, its Hamiltonian can be obtained without the small-coupling approximation and the system can be used to simulate strongly frustrated one-dimensional Ising spin problems. To engineer the system Hamiltonian for an ideal theoretical model, we apply a dynamical decoupling technique to eliminate undesirable couplings in the system. Using a 6-site junction array as an example, we numerically evaluate the system to show that it exhibits important characteristics of the frustrated spin model.'
author:
- 'Liang-Hui Du'
- Xingxiang Zhou
- 'Yong-Jian Han'
- 'Guang-Can Guo'
- 'Zheng-Wei Zhou'
title: 'Strongly-coupled Josephson junction array for simulation of frustrated one-dimensional spin models'
---
Introduction
============
As an important application of quantum information science, quantum simulation of difficult physics problems has received much attention in recent years. Theoretically, there have been many proposals of quantum simulators based on various physical systems [@ref:TheoQuanSim1; @ref:TheoQuanSim2; @ref:TheoQuanSim3; @ref:TheoQuanSim4; @ref:TheoQuanSim5]. Experimentally, simulations of some important physics models have been demonstrated [@ref:QuSimOptLatt1; @ref:QuSimOptLatt2; @ref:QuSimOptLattMag1; @ref:QuSimOptLattMag2; @ref:QuSimTrapIon1; @ref:QuSimTrapIon2].
Quantum simulation is most valuable for studying strongly correlated problems since there are no generally-applicable theoretical methods to solve them. The strong interactions involved in these problems usually translate into strong couplings between entities in a quantum simulator. This requirement of strong couplings often poses a challenge for the design and construction of a quantum simulator, since it can be difficult to engineer such couplings in a simulation system. Even when strong couplings are available, it can still be nontrivial to tailor the couplings in an well-controlled manner that is required for the problems to be simulated. As one such example, it is usually difficult to obtain the system Hamiltonian for a Josephson junction array when the couplings between the junctions are strong. Consequently, Josephson device based quantum simulation systems often operate in the small coupling limit in which the system Hamiltonian can be obtained by treating the coupling as perturbation [@ref:smlCapApprox1; @ref:smlCapApprox2; @ref:smlCapApprox3].
In order to go beyond the small-coupling limit and construct a system useful for simulating strongly correlated physics, in this paper we investigate a one-dimensional Josephson junction array which is coupled by large capacitances that cannot be treated perturbatively. Interestingly, the system Hamiltonian can be obtained exactly without the small coupling approximation. It is found that, in the large coupling limit, the interaction strength between next nearest neighbors can become comparable with that between the nearest neighbors. Because of this, we can use the system to study the important problem of one-dimensional frustrated spin models whose phase diagrams and properties have not been completely resolved [@ref:FrusModPhaDiaAgr1; @ref:PhsDiaInteF; @ref:PhsDiaFinitSiz; @ref:FrusModPhaDiaAgr2; @ref:FrusModPhaDiaAgr3]. In order to control the system Hamiltonian to match that of the ideal theoretical model, we use a dynamical decoupling technique to suppress interactions between neighbors that are three site locations or farther apart.
System Hamiltonian of the Josephson-junction array
==================================================
The system we study is an $N$-site Josephson junction array as shown in Fig. \[JoseJunArr\]. Each site consists of a charge island biased by a voltage source $V_{g_i}$ through a gate capacitance $C_{g_i}$, where $i=1,...,N$. The charge on the island can tunnel through a SQUID device whose total capacitance is $C_J$ and whose effective Josephson energy $\mathcal {E}_J$ can be adjusted by a flux bias. Adjacent charge islands, as well as those at the ends of the array, are coupled by a capacitance $C_c$. We take the average phase $\varphi_i$ of the SQUID on site $i$ as the generalized coordinates for the system. Its rate of change is determined by the voltage $V_j$ across the Josephson junction according to the Josephson relation $V_j=(\hbar/2e)\dot{\varphi}_i$ [@ref:SupConduct]. The charging energy $T$ of the capacitances and the Josephson energy $V$ of the junctions are $$\begin{aligned}
T&=&\frac{1}{2}\sum\limits_{i=1}^{N}[(\frac{\hbar}{2e})^2C_{J}\dot{\varphi}_i^2+C_{gi}(V_{gi}-\frac{\hbar}{2e}\dot{\varphi}_i)^2\nonumber\\
&+&C_c(\frac{\hbar}{2e})^2(\dot{\varphi}_i-\dot{\varphi}_{i+1})^2].\label{kiEn}\\
V&=-&\sum_i\mathcal {E}_J\cos\varphi_i.\label{poEn}\end{aligned}$$
[![(Color online) The capacitance-coupled Josephson junction array. $C_c $ is the coupling capacitance between neighboring junctions. $V_{gi}$ is the bias voltage and $C_{gi}$ is the bias capacitance of the Josephson junction at site $i$. The total capacitance and Josephson energy are $C_{ji}$ and $\mathcal
{E}_J$.\[JoseJunArr\]](fig1_JoseJunArr.eps "fig:"){width="0.8\columnwidth"}]{}
From the Lagrangian $\mathcal {L}=T-V$ of the system, we can derive the generalized momentum which is related to the charge number on the islands $$P_i=\partial\mathcal {L}/\partial\dot{\varphi_i} \equiv -\hbar
n_i.\label{generalmomentum}$$
Denoting the charge number on the $i$th island $n_i$ and the bias charge number $n_{gi}=C_{g_i}V_{g_i}/2e$, we can write the system’s equation of motion $$\frac{\hbar}{(2e)^2}\bm{M}\vec{\dot{\bm{\varphi}}}=\vec{\bm{n}},$$ where $\vec{\dot{\bm{\varphi}}}=[\dot{\varphi}_1,\dot{\varphi}_2,...,
\dot{\varphi}_i,...,\dot{\varphi}_N]^T$, $\vec{\bm{n}}=[n_1-n_{g1},n_2-n_{g2},...,n_i-n_{gi},...,n_N-n_{gN}]^T$, and $\bm{M}$ is the matrix $$\bm{M}=\left(\begin{array}[c]{cccccc}%
C_{\Sigma} & -C_c & \cdots&\cdots&\cdots& -C_c\\
-C_c & C_{\Sigma} & -C_c & \cdots &\cdots& 0\\
0 & -C_c & C_{\Sigma} & -C_c & \cdots & 0\\
\multicolumn{6}{c}{\dotfill}\\
\cdots&\cdots&-C_c&C_{\Sigma }&-C_c&\cdots\\
-C_c&\cdots&\cdots&\cdots&\cdots&\vdots
\end{array}\right),$$ where $C_{\Sigma}=C_{J}+C_{g_i}+2C_c$ is the total capacitance of each island.
The Hamiltonian of this Josephson junction array is: $$\begin{aligned}
H&=&\sum\limits_{i} P_i\dot{\varphi}_i-\mathcal {L}\nonumber\\
&=&\frac{1}{2}(\frac{\hbar}{2e})^2\vec{\dot{\bm{\varphi}}}^T\bm{M}\vec{\dot{\bm{\varphi}}}+V\nonumber\\
&=&\frac{1}{2}(2e)^2\vec{\bm{n}}^T\bm{M}^{-1}\vec{\bm{n}}+V.\label{Hamivecn}\end{aligned}$$
In order to obtain the system Hamiltonian, the inverse matrix of $\bm{M}$ must be calculated. Since it is nontrivial to exactly solve for $\bm{M}^{-1}$, most previous studies [@ref:smlCapApprox1; @ref:smlCapApprox2] have assumed a small coupling capacitance $C_c$ so the system Hamiltonian can be obtained by treating the coupling as perturbation. Unfortunately, this precludes the system from being used to simulate strongly-correlated problems in which strong couplings are required. Here, we assume that $C_c$ is not necessarily small and try to solve for $\bm{M}^{-1}$ exactly.
Considering the translational symmetry of the problem, we see the inverse matrix $\bm{M}^{-1}$ must be in the form $$\bm{M}^{-1}=\left(\begin{array}[c]{ccccccc}%
a_1 & a_2 & \cdots& a_{N/2+1}&\cdots& a_3 &a_2\\
a_2 & a_1 & & & & & a_3\\
\vdots & &\ddots & & & & \\
a_{N/2+1}& & & & & &\vdots\\
\vdots & & & &\ddots & &\\
a_3 & & & & &a_1&a_2\\
a_2 & & & \cdots& & a_2 &a_1
\end{array}\right),$$
Using some mathematic techniques for solving polynomials, we can calculate the values of the matrix elements in $\bm{M}^{-1}$ exactly: $$a_i=\frac{1}{C_{\Sigma}}(\lambda^{i-1}A_0+\frac{1}{\lambda^{i-1}}B_0),\label{elemofInvM}$$ where $$\begin{aligned}
A_0&=&\frac{1}{1-2\beta\lambda+(1-\frac{2\beta}{\lambda})\frac{2\beta-\lambda}{1-2\beta\lambda}\lambda^{N-1}},\label{paraofInvM1}\\
B_0&=&\frac{(2\beta-\lambda)\lambda^{N-1}}{1-2\beta\lambda}A_0,\label{paraofInvM2}\\
\lambda&=&\frac{1-\sqrt{1-4\beta^2}}{2\beta},\label{paraofInvM3}\\
\beta&=&\frac{C_c}{C_{\Sigma}}=\frac{C_c}{2C_c+C_{gi}+C_{J}}<\frac{1}{2}.\label{paraofInvM4}\end{aligned}$$
[![(Color online) The relationship between $\lambda$ and $\beta$. The dashed red line corresponds to $\lambda=1/2$. \[lambdabeta\]](fig2_lambdabeta.eps "fig:"){width="0.8\columnwidth"}]{}
When $N\rightarrow \infty$, the above results simplify to $A_0\rightarrow1/(1-2\beta\lambda)$, $B_0\rightarrow0$, and $a_i=\lambda^{i-1}A_0/C_{\Sigma}$.
As can be seen in Eq. (\[Hamivecn\]), $\lambda$ characterizes the ratio between adjacent and non-adjacent interaction strengths in our system. According to Eqs. (\[paraofInvM3\]) and (\[paraofInvM4\]), $\lambda$ is determined by the coupling capacitance $C_c$. In the weak coupling limit $C_c \ll
C_{\Sigma}$, $\beta\ll 1$ and $\lambda$ is nearly equal to $\beta$. However, when the coupling is strong, $\lambda$ increases quickly, as shown in Fig. \[lambdabeta\]. In particular, when the coupling capacitance $C_c$ dominates, $\lambda$ can approach 1, and the Hamiltonian in Eq. (\[Hamivecn\]) describes a deeply frustrated system with appreciable non-adjacent interactions. Meanwhile, according to the results of Ref. [@ref:LargeCapCoup], there exist many close energy levels as $\beta$ approaches 1/2 too closely, which can fail the two-level approximation. Therefore we only pay our attention to a large $\beta$, but not closely approaching 1/2 in the rest of our paper.
Under proper conditions, if we bias the charge islands at the vicinity of $n_{gi}=1/2$, we can use the two-level approximation for the charge qubits with $n_i=0$ and $n_i=1$ as the basis states. We can then write the system Hamiltonian in the following Pauli matrix representation $$H_{JJA}=\sum_i\sum_j(-1)^j(\lambda)^{j-1}\sigma_i^z\sigma_{i+j}
^z-B\sum_i\sigma_i^x,
\label{eq:Spin-H}$$ where the spin up and down states represent the $n_i=0$ and $n_i=1$ states, and the transverse magnetic field $B\!\!=\!\!-\mathcal{E}_jC_{\Sigma}/(2\lambda e^2 A_0)$. $B$ can be adjusted by tuning the magnetic flux of the SQUIDs. Here we applied a canonical $\sigma^x$ transformation on even sites for the convenience of our following discussion about phase diagram without changing the physics of the system. Usually, two-level approximation works very well for a single Josephson charge island [@ref:twoLevelAppro1; @ref:twoLevelAppro2; @ref:twoLevelAppro3]. In our system of Josephson junction array, more careful analysis is necessary. Appendix A provides a detailed discussion about the applicability of the two-level approximation in our system.
Simulation of the ANNNI model using dynamical decoupling
========================================================
Our circuit is useful for quantum simulation of frustrated spin problems since it exhibits strong non-adjacent spin interactions in the large coupling limit. However, the Hamiltonian in Eq. (\[eq:Spin-H\]) does not correspond to an ideal theoretical model with limited-range interactions yet. We intend to further engineer it for quantum simulation of well known frustrated models. As an example, we show how to simulate the one dimensional axial next-nearest-neighbor Ising (ANNNI) model in external fields. Its Hamiltonian is given by $$H_{AI}=-\sum_i\sigma_i^z\sigma_{i+1}^z+\lambda\sum_i\sigma_i^z\sigma_{i+2}^z-B\sum_i\sigma_i^x.
\label{eq:ANNNI-H}$$ The ANNNI problem is an important model for studying frustrated physics due to competition between adjacent and non-adjacent neighbor interactions [@ref:ANNNI1; @ref:ANNNI2]. Despite years of research, its phase diagrams and physical properties have not been completely resolved [@ref:FrusModPhaDiaAgr1; @ref:PhsDiaInteF; @ref:PhsDiaFinitSiz; @ref:FrusModPhaDiaAgr2; @ref:FrusModPhaDiaAgr3].
[![(Color online) The dynamical decoupling control scheme to eliminate interactions between spins that are 3 sites apart. The ellipses in groups of 3 along the vertical direction represent simultaneous $\pi$ or $3\pi$ pulses applied to the qubits. \[dd\]](fig3_dd.eps "fig:"){width="0.8\columnwidth"}]{}
Comparing our circuit Hamiltonian in Eq. (\[eq:Spin-H\]) and target Hamiltonian in Eq. (\[eq:ANNNI-H\]), we find that there are extra terms that describe interactions between spins that are 3 or more sites apart. We eliminate these extra terms by using techniques of dynamical decoupling [@ref:DDtech1; @ref:DDtech2]. In this practice, we apply carefully designed sequences of fast short pulses to individual qubits to engineer a Hamiltonian that can be very different than the original Hamiltonian.
In the following, we will demonstrate how to eliminate interactions between spins that are 3 sites apart for a spin chain with the Hamiltonian in Eq. (\[eq:Spin-H\]). For clarification and ease of illustration, we discuss the details of our scheme on a 6-site spin chain with periodic boundary condition. The same technique applies in a spin chain with arbitrary length.
When the number of qubits in the system is 6, the Hamiltonian in Eq. (\[eq:Spin-H\]) reads $$\begin{aligned}
H_{s}&=&J_1(\sigma_1^z\sigma_2^z+\sigma_2^z\sigma_3^z+\sigma_3^z\sigma_4^z+\sigma_4^z\sigma_5^z+\sigma_5^z\sigma_6^z+\sigma_6^z\sigma_1^z)\nonumber\\
&
&+J_2(\sigma_1^z\sigma_3^z+\sigma_2^z\sigma_4^z+\sigma_3^z\sigma_5^z+\sigma_4^z
\sigma_6^z+\sigma_5^z\sigma_1^z+\sigma_6^z\sigma_2^z)\nonumber\\ &
&+J_3(\sigma_1^z\sigma_4^z+\sigma_2^z\sigma_5^z+\sigma_3^z\sigma_6^z)
-B\sum\limits_{i=1}^6\sigma_i^x
\label{HGenl_six}\end{aligned}$$ where $J_1$ and $J_2$ are interaction strengths between the nearest and next-nearest neighbors and $J_3$ is that between spins that are 3 sites apart.
Our scheme to engineer the desired Hamiltonian involves applying rapid pulsed operations $R_x(\theta )=\exp\{-(i\theta \sigma _x)/2\}$ on individual qubits. When $\theta=\pi$ ($3\pi$) the corresponding unitary operation is $R_x(\pi)=-i\sigma^x$ ($R_x(3\pi)=R_x^\dagger(\pi)=i\sigma^x$). We have the following commutation relations: $$\begin{aligned}
R^\dagger_x(\pi)\sigma^zR_x(\pi)&=&-\sigma^z,\label{commut_relation1}\\
R^\dagger_x(\pi)\sigma^xR_x(\pi)&=&\sigma^x,\label{commut_relation2}\\
R^\dagger_x(\pi)e^{-i\hat{H}t}R_x(\pi)&=&e^{-i[R^\dagger_x(\pi)\hat{H}R_x(\pi)]t}.\label{quan_Alg}\end{aligned}$$
Our procedure contains four phases as shown in Fig. \[dd\]. Each phase is completed in an interval of $2\delta t$ where $\delta t$ is a short time. At the beginning of each phase, the operations $R^{(i,j,k)}_x(\pi)\!=\!R^i_x(\pi)R^j_x(\pi)R^k_x(\pi)$ ($i,j,k=1,2,...,6$) are applied to the qubits on sites $i,j$, and $k$ simultaneously (see Fig. \[dd\]). After a time interval of $\delta t$, the conjugate operations $R^{\dagger(1,2,3)}_x(\pi)$ are applied in the second half of the phase. The evolution of the system at the end of phase 1 is given by $$U_1=e^{-iH_s\delta t}R^{\dagger(1,2,3)}_x(\pi)e^{-iH_s\delta
t}R^{(1,2,3)}_x(\pi).\label{U_1}$$ For a short time duration $\delta t$, we have $e^{-iH_1\delta
t}e^{-iH_2\delta t}=e^{-i(H_1+H_2)\delta t}+O(\delta t^2)$. To first order in $\delta t$, the unitary operator $U_1=\exp(-iH_1^{eff}2\delta t)$, where the effective Hamiltonian in phase 1 is $$\begin{aligned}
H_1^{eff}&=&H_s+R^{\dagger(1,2,3)}_x(\pi)H_sR^{(1,2,3)}_x(\pi)\nonumber\\
&=&J_1(\sigma_1^z\sigma_2^z+\sigma_2^z\sigma_3^z+\sigma_4^z\sigma_5^z+\sigma_5^z\sigma_6^z)\nonumber\\
& &+J_2(\sigma_1^z\sigma_3^z+\sigma_4^z\sigma_6^z)-B\sum\limits_{i=1}^6\sigma_i^x.\label{H1_eff}\end{aligned}$$
From Eq. (\[H1\_eff\]), we can see that all couplings between qubits that are 3 sites apart have been eliminated. However, some terms that we want to keep, such as the nearest-neighbor coupling $\sigma_3^z\sigma_4^z$ and next-nearest coupling $\sigma_3^z\sigma_5^z$, are also eliminated. In order to make up for this problem, in phase 2 and 3 we use the same technique but shift the target qubits one site a time as shown in Fig. \[dd\]. This gives us the following effective Hamiltonian for these two phases $$\begin{aligned}
H_2^{eff}
&=&J_1(\sigma_2^z\sigma_3^z+\sigma_3^z\sigma_4^z+\sigma_5^z\sigma_6^z+\sigma_6^z
\sigma_1^z)\nonumber\\
&&+J_2(\sigma_2^z\sigma_4^z+\sigma_5^z\sigma_1^z)-B\sum\limits_{i=1}
^6\sigma_i^x;\label{H2_eff}\\
H_3^{eff}
&=&J_1(\sigma_1^z\sigma_2^z+\sigma_3^z\sigma_4^z+\sigma_4^z\sigma_5^z+\sigma_6^z
\sigma_1^z)\nonumber\\
&
&+J_2(\sigma_3^z\sigma_5^z+\sigma_6^z\sigma_2^z)
-B\sum\limits_{i=1}^6\sigma_i^x.\label{H3_eff}\end{aligned}$$ Obviously, the missing terms for nearest and next-nearest neighbor couplings in $H_1^{eff}$ in Eq. (\[H1\_eff\]) are compensated by remaining terms in Eqs. (\[H2\_eff\]) and (\[H3\_eff\]). Similarly, missing terms in $H_2^{eff}$ and $H_3^{eff}$ are compensated. Nevertheless, some next-nearest neighbor coupling terms are still missing from the sum of $H_1^{eff}$, $H_2^{eff}$ and $H_3^{eff}$, since in each phase 2/3 of the nearest-neighbor couplings remain but only 1/3 of the next-nearest-neighbor couplings survive. In order to obtain all the nearest-neighbor and the next-nearest-neighbor couplings, we need a phase 4 as shown in Fig. \[dd\]. By keeping the next-nearest-neighbor interactions and eliminating the nearest-neighbor interactions, it gives us the effective Hamiltonian $$\begin{aligned}
H_4^{eff}&=&J_2(\sigma_1^z\sigma_3^z+\sigma_2^z\sigma_4^z+\sigma_3^z\sigma_5^z+\sigma_4^z\sigma_6^z+\sigma_5^z\sigma_1^z+\sigma_6^z\sigma_2^z))\nonumber\\
& &-B\sum\limits_{i=1}^6\sigma_i^x.\label{H4_eff}\end{aligned}$$
The combined evolution of the system for the 4 phases is $$\begin{aligned}
U\!\!&=&\!\!U_4U_3U_2U_1\!\approx \!e^{-iH_4^{eff}2\delta
t}e^{-iH_3^{eff}2\delta
t}e^{-iH_2^{eff}2\delta t}e^{-iH_1^{eff}2\delta t}\nonumber\\
&\approx&\!\!e^{-i[H_4^{eff}+H_3^{eff}+H_2^{eff}+H_1^{eff}]2\delta
t}\nonumber\\
&\!\!\triangleq&e^{-iH_{eff}8\delta t}\end{aligned}$$ where the effective average Hamiltonian is $$\begin{aligned}
H_{eff}&=&\frac{1}{2}[J_1(\sigma_1^z\sigma_2^z+\sigma_2^z\sigma_3^z+\sigma_3^z\sigma_4^z+\sigma_4^z\sigma_5^z+\sigma_5^z\sigma_6^z+\sigma_6^z\sigma_1^z)\nonumber\\
& &+J_2(\sigma_1^z\sigma_3^z+\sigma_2^z\sigma_4^z+\sigma_3^z\sigma_5^z+\sigma_4^z\sigma_6^z+\sigma_5^z\sigma_1^z+\sigma_6^z\sigma_2^z)]\nonumber\\
& &-B\sum\limits_{i=1}^6\sigma_i^x.\label{H_eff}\end{aligned}$$ This is exactly the ANNNI model with periodic boundary condition.
Though we have used a 6-site chain to demonstrate how to eliminate couplings between qubits that are 3 sites apart, it is obvious that, by applying the operations $R^{(i,j,k)}_x(\pi)$ to groups of 3 qubits in the chain and shifting the target qubits by 1 site at a time in each phase, we can use the same technique to eliminate couplings between qubits 3 sites apart in an infinite-length spin chain. Notice that in Eq. (\[eq:Spin-H\]) there are couplings between qubits 4 or more sites apart. These couplings are weaker since the interaction strengths $\lambda^{j-1}$ in Eq. (\[eq:Spin-H\]) decreases with the distance between qubits, but the error caused by them may still be unacceptable depending on the required accuracy of the simulation. By using a nested dynamical decoupling scheme, it can be shown that all couplings between qubits that are separated by 3 or more sites can be eliminated [@Du]. Therefore, given a required accuracy, we can in principle achieve the ANNNI Hamiltonian in Eq. (\[eq:ANNNI-H\]).
phase diagram of the ANNNI model
================================
Now that we can simulate the ANNNI model, we perform some analysis on its phase diagram. When $B=0$ and $\lambda$ is small, the nearest-neighbor interactions dominate and the ground state is the ferromagnetic state $|\downarrow\downarrow\cdots\downarrow\downarrow\cdots\rangle_z$ (or $|\uparrow\uparrow\cdots\uparrow\uparrow\cdots\rangle_z$). As $\lambda$ increases, the next-nearest-neighbor interactions become important. When $\lambda$ reaches some critical value, they become the dominating factor and the antiphase $|\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\cdots\rangle_z$ (or $|\downarrow\downarrow\uparrow\uparrow\downarrow\downarrow\cdots\rangle_z$) which minimizes the next-nearest neighbor interaction energy becomes the ground state. In the limit of large transverse field $B$, the ground state will be the paramagnetic phase $|\uparrow\uparrow\cdots\uparrow\uparrow\cdots\rangle_x$ to minimize the Zeeman energies.
From the above analysis, we see that there should be a ferromagnetic phase, a paramagnetic phase and an antiphase in the ANNNI model. However, there could be more subtle regimes in the phase diagram. Studies have shown (inconclusively) that there could be a unique floating phase in the deeply frustrated regime [@ref:FrusModPhaDiaAgr2; @ref:FrusModPhaDiaAgr3]. This phase is characterized by the fact that the $n$th-neighbor spin-spin correlation function in the longitudinal direction decays algebraically. The exact origin and range of the floating phase is still an open question and therefore a good subject for quantum simulation. Since the floating phase is located in the deeply frustrated regime, our circuit with strong couplings offers a good system for its simulation.
Many numerical recipes such as finite-size scaling method [@ref:PhsDiaFinitSiz] and the interface approach [@ref:PhsDiaInteF] have been used to calculate the phase diagram of frustrated Ising model. We use the new method of time-evolving block decimation (TEBD) algorithm [@ref:TEBDMeth1; @ref:TEBDMeth2] to calculate the ground state energy of ANNNI model. TEBD is an powerful algorithm to simulate quantum evolution process based on matrix product state and Trotter expansion [@ref:TEBDMeth1; @ref:TEBDMeth2]. By making the evolution time imaginary, we get the so called i-TEBD which can be used to determine the ground state of a system efficiently. The TEBD method is based on the following matrix representation of a quantum state $$|\Psi\rangle=\sum_{i_1=1}^d\cdots\sum_{i_n=1}^dc_{i_1\cdots
i_n}|i_1\rangle\otimes\cdots\otimes|i_n\rangle
\label{MPS}$$ where $d$ is the number of local energy levels on every site. The coefficients $$c_{i_1\cdots
i_n}\!=\!\!\!\sum_{\alpha_1,\cdots,\alpha_{n-1}}^\chi\!\!\!
\Gamma_{\alpha_1}^{[1]i_1}\xi
_{\alpha_1}^{[1]}\Gamma_{\alpha_1\alpha_2}^{[2]i_2}
\xi_{\alpha_2}^{[2]}\Gamma_{\alpha_2\alpha_3}^{[3]i_3}\cdots\Gamma_{\alpha_{n-1}}^{[n]i_n}.\label{coefMPS}$$ are defined with the help of $n$ tensors $\{\Gamma^{[1]},\cdots,\Gamma^{[n]}\}$ and $n-1$ vectors $\{\xi^{[1]},\cdots,\xi^{[n-1]}\}$, where $\chi$ is the maximal number of two-party Schmidt decomposition coefficients. In practice, $\chi$ does not need to be very large, because the Schmidt coefficients roughly decay exponentially with $\alpha$. Any single-site operation or adjacent-site joined operation on the state can be achieved by updating the corresponding tensors and vectors. Here, we use the second order Trotter expansion for i-TEBD (see reference [@ref:TEBDMeth2]).
[![(Color online) The finite-size scaling of second order derivative of ground state energy with $\lambda\doteq0$ (solid lines) and $\lambda=0.2$ (dashed lines) \[ProPT\]](fig4_ProPT.eps "fig:"){width="0.8\columnwidth"}]{}
[![(Color online) (a) Fine structures and small dips in the $dE_g^2/dB^2$ curve with the parameters $\lambda=0.7$ and $N=60$. (b) The phase diagram of the ANNNI model. (c) The spin-spin correlation function $c^2_s(d)$ versus the spin separation $d$ at point A ($\lambda=0.7$ and $B=0.2$) in the phase diagram. (d) $c^2_s(d)$ at point B ($\lambda=0.7$ and $B=0.3$) in the phase diagram. (e) $c^2_s(d)$ at point C ($\lambda=0.7$ and $B=0.6$) in the phase diagram. \[fp&phase\_a&phase\_b&phase\_c&phasediag\]](fig5_fpevi_phsdiag_Cs.eps "fig:"){width="0.7\columnwidth"}]{}
Using the TEBD algorithm, we calculate the ground state energy of ANNNI model in the external field $B$. How the ground state energy changes with the parameters in the Hamiltonian is of great significance because it gives us important clues on the quantum phase transition points. Considering this, we plot the second derivative of the ground state energy $d^2E/dB^2$ in Fig. \[ProPT\], for different coupling strength $\lambda$ and system size $N$. Notice that there are dips in the curves in Fig. \[ProPT\]. Their positions nearly do not change with the system size $N$ when $N$ is large enough. These dips are where quantum phase transitions occur, and we can read from their positions the corresponding critical field strengths at the phase transition points. These critical parameter values allow us to construct the system’s phase diagram which is shown in Fig. 5(b). The phase diagram is consistent with earlier results [@ref:FrusModPhaDiaAgr3] obtained using different numerical methods.
Interestingly, as shown in Fig. 5(a) we find that in the strongly frustrated regime there can be a segment in the curve of $d^2E_g/dB^2$ where there are multiple extra small dips in addition to the main dip. This region is labeled by the green area in the phase diagram in Fig. 5(b). It is roughly at the location of the floating phase obtained in earlier work [@ref:FrusModPhaDiaAgr3]. To further clarify the characteristics of the system in this region, we calculate the spin-spin correlation function $$c_s(d)\!=\!\left\langle \!\sigma _{N/2+1}^z\sigma
_{N/2+1+d}^z\!\right\rangle \!-\!\left\langle\! \sigma
_{N/2+1}^z\right\rangle\! \left\langle\! \sigma
_{N/2+1+d}^z\!\right\rangle.$$ in this region (point B in Fig. 5(b)) and compare it to the results in the antiphase and paramagnetic phase (point A and C in Fig. 5(b)). The results are plotted in Figs. 5(c), 5(d) and 5(e). As can be seen in the plots, the spin-spin correlation function at point $A$ (Fig. 5(c)) exhibits perfect long-range order which is characteristic of the antiphase. At point C in the paramagnetic phase, the spin-spin correlation function (Fig. 5(e)) decays exponentially with spin separation. At point B in the green area in the phase diagram, the spin-spin correlation function (Fig. 5(d)) appears to decay algebraically which is indicative of the floating phase.
simulating the ANNNI model in the six-junction array system
===========================================================
The achievable scale of an experimental simulation system is limited, by both decoherence and the requirement for the two-level approximation to be valid (see appendix A). In order to study the feasibility of our Josephson circuit system for simulation of frustrated physics, we examine a small system to see how close its ground state is to certain phases in Fig. 5(b). This information will help us determine if it is possible to study the essential characteristics of a frustrated spin system using a quantum simulator of limited size.
Taking a 6-site Josephson junction array as an example, we obtain the ground state $|\psi\rangle_g$ of the corresponding $N=6$ ANNNI model using exact diagonalization. Such a short chain is insufficient to exhibit the characteristics of the floating phase, therefore we will focus on the ferromagnetic, paramagnetic and antiphase phases. We calculate the probabilities of $|\psi\rangle_g$ being the ferromagnetic and paramagnetic states, $$\begin{aligned}
P(FM)&=&|_g\langle\psi|\downarrow\downarrow\cdots\downarrow\downarrow\rangle_z|^2\!\!+\!\!|_g\langle\psi|\downarrow\downarrow\cdots\downarrow\downarrow\rangle_z|^2,\label{FMOccu}\\
P(PM)&=&|_g\langle\psi|\uparrow\uparrow\cdots\uparrow\uparrow\rangle_x|^2.\label{PMOccu}\end{aligned}$$ The results are plotted in Fig. \[site6Occu\] for different values of $B$ and $\lambda$.
[![(Color online) (a) The probability of the ground state $|\psi\rangle_g$ of the 6-site Josephson junction array being the ferromagnetic state. (b) The probability of $|\psi\rangle_g$ being the paramagnetic state. \[site6Occu\]](fig6_Neq6FM_PM.eps "fig:"){width="0.8\columnwidth"}]{}
We can see in Fig. \[site6Occu\] that there exists a clear junction point $\lambda=0.5$, which agrees well with the result in Fig. 5(b). If we associate the high probability regime with the corresponding phase, we can see that the phase regimes are nearly the same as well. This indicates that the 6-site example can already reveal some essential properties of the ANNNI model.
[![ The fidelity of the system state to the state of the corresponding ANNNI model, for different number of dynamical decoupling control sequences. \[ddctrleff\]](fig7_dd1.eps "fig:"){width="0.8\columnwidth"}]{}
The results in Fig. \[site6Occu\] are based on exact diagonalization of the $N=6$ ANNNI Hamiltonian. In order to obtain the ANNNI model from the Hamiltonian of the Josephson junction array, dynamical decoupling pulse sequences need to be applied to the qubits as shown in Fig. \[dd\]. To study the error of the pulse engineered ANNNI model, we evaluate the evolution of the Josephson junction array system in a time of $T=\pi$ under the control pulses, and compare it to that of a strict ANNNI model in the same amount of time. The initial state of the system is set to the maximal superposition state $|\Psi_0\rangle=\bigotimes\limits_i(|\uparrow\rangle_i+|\downarrow\rangle_i)
/\sqrt{2}$. We divide the total time $T$ into $m$ identical time interval: $T=m \delta t$. In each interval $\delta t$, the dynamical decoupling sequences shown in Fig. \[dd\] are applied. In Fig. \[ddctrleff\], the fidelities of the Josephson junction array system state compared to the state of the corresponding ANNNI model is plotted. We see that the fidelities increase with the number $m$ of decoupling sequences, and they already reach $95\%$ within a few sequences.
We can further evaluate the effectiveness of our pulse control scheme by comparing how the ground state changes with the external field $B$ in a strict 6-site ANNNI model and in a pulse sequence controlled 6-site Josephson junction array. We simulate this process by adiabatically changing $B$ in time and calculating the probability of the system ground state being in the ferromagnetic phase. Here, we set $\lambda=0.4$. In Fig. 8(a), we calculate the state evolution of a 6-site ANNNI system initially in the ferromagnetic state $|\downarrow\downarrow\cdots\rangle_z$ when $B=0$. We gradually increase the magnetic field $B$ from 0 with velocity $v=0.002$, and plot the probability that the system remains in the original ferromagnetic state at different values of $B$. We can see that when $B$ becomes large, the system has deviated from the ferromagnetic state, indicating that it has changed to a different phase. In Fig. 8(b), we carry out the same study on the 6-site Josepshon junction array under the control pulse scheme in Fig. \[dd\]. As can be seen, the probability of the system remains in the ferromagnetic state is almost identical to that in the corresponding ANNNI model when the number of control sequences is sufficient. These studies show that our control pulse based scheme can be used to accurately simulate the frustrated ANNNI model.
[![(Color online) (a) Phase Transition process of the ANNNI model with $\lambda=0.4$. (b) Phase transition process of the Josephson junction array system with and without the dynamical-decoupling control sequence. The black dashed curve is the result without dynamical-decoupling control sequences. The solid blue and dotted red curves are the results with 1 and 4 control sequences in unit time. \[PT&DDPT\]](fig8_DDPT.eps "fig:"){width="0.8\columnwidth"}]{}
conclusion
==========
In conclusion, we have shown how to simulate frustrated spin models using strongly coupled Josephson junction array. We find that the system Hamiltonian is exactly solvable beyond the small coupling limit, and we design a dynamical decoupling scheme to engineer the Hamiltonian for quantum simulation of the ANNNI model. We calculate the phase diagram of the system numerically using the TEBD method, and demonstrate that our control pulse based scheme can be used to simulate the corresponding ANNNI model accurately.
Acknowledgment
==============
This work was funded by National Natural Science Foundation of China (Grant Nos. 11174270, 60836001, 60921091), National Basic Research Program of China (Grant Nos. 2011CB921204, 2011CBA00200), the Fundamental Research Funds for the Central Universities (Grant No. WK2470000006), and Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20103402110024). L. -H. Du and Z. -W. Zhou thank Man-Hong Yung for fruitful discussion. Z. -W. Zhou gratefully acknowledges the support of the K. C. Wong Education Foundation, Hong Kong.
Conditions for two-level approximation
======================================
In deriving the spin system Hamiltonian in Eq. (\[eq:Spin-H\]), we used the two-level approximation for the Josephson qubits which kept only the $n_i=0,1$ states. We study the conditions for the two-level approximation to remain valid in this Appendix.
The Hamiltonian for the Josephson junction array system considering contributions from all charge states reads $$\begin{aligned}
H_n&=&\frac{1}{\lambda}\sum\limits_{i=1}^{N}(n_i-1/2)^2\nonumber\\
& &+\sum_i\sum\limits_{j=1}^2(\lambda)^{j-1}(n_i-1/2)(n_{i+j}-1/2)\nonumber\\
& &-B\sum\limits_i\sum\limits_n(|n\rangle_i\langle n+1|+h.c.),
\label{eq:A1}\end{aligned}$$ where terms in the first line are the on-site charging energies of the Josephson qubits, terms in the second line are coupling energies between qubits, and terms in the third line are the Josephson tunneling energies. When the effective magnetic field $B$ is nonzero, the Josephson tunneling energies can potentially cause leakage out of the $n_i=0,1$ states and invalidate the two-level approximation.
We take the ferromagnetic phase of the ANNNI model as an example to estimate the probability for the qubits in the system to escape the $n_i=0,1$ states. Initially, assume $B=0$ and the system is in the ferromagnetic state $|\Psi\rangle_g=|0101\ldots\rangle$ (recall that there has been a canonical transformation applied on the even sites.). When $B$ increases , the qubits can make transitions out of the $n_i=0,1$ states. We examine the system states that result when one of the qubits originally in the $n=1$ state changes to the $n=2$ state, because they are closest to $|\Psi\rangle_g$ in energy among all states that violate the two-level approximation. We denote such states $|\Psi\rangle_n$. Their energies differ from that of $|\Psi\rangle_g$ by $\Delta
E=2(1/\lambda-1+\lambda)$ according to Eq. (\[eq:A1\]).
According to the first order perturbation theory, the ground state with a nonzero magnetic filed is $$\begin{aligned}
|\Psi'\rangle_g&=&|\Psi\rangle_g+\sum_n\!'\frac{
_n\langle\Psi|H'|\Psi\rangle_g}{E_g-E_n}|\Psi\rangle_n\nonumber\\
&=&|\Psi\rangle_g+\frac{B}{\Delta
E}(\Sigma'_n|\Psi\rangle_n).\label{newgrdst}\end{aligned}$$
With the form of $|\Psi'\rangle_g$, the total escaping probability out of the $n_i=0,1$ Hilbert subspace can be estimated to be $$P_{esc}= (\frac{B}{\Delta E})^2\frac{N}{2},\label{escprob}$$ where the factor of $N$ is due to the translational symmetry.
From the Eq. (\[escprob\]), we find that the escaping probability is proportional to $N$. Therefore the allowable system size $N$ is limited by the tolerable escaping probability and the amplitude of the magnetic field. For example, if an escaping probability of $5\%$ is acceptable, and $B=0.2$, the allowable size of the Josephson junction array is about $N=10$.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We provide a systematic coset construction of the effective field theories governing the low-energy dynamics of relativistic fluids and solids, and of their ‘super’ counterparts. These effective theories agree with those previously derived via different techniques. As an application of our methods, we re-derive the Wess-Zumino term relevant for anomalous charge-carrying fluids in (1+1) dimensions.'
author:
- Alberto Nicolis
- Riccardo Penco
- 'Rachel A. Rosen'
bibliography:
- 'fluids.bib'
title: 'Relativistic Fluids, Superfluids, Solids and Supersolids from a Coset Construction'
---
Introduction
============
Hydrodynamics is usually studied at the level of its equations of motion, which are nothing but the local conservation laws for energy, momentum, and possibly additional conserved charges carried by the fluid in question. Recently, however, it has been realized that an effective field theory treatment in terms of a local action functional might be more convenient for certain applications. This is partially due to the fact that long-wavelength hydrodynamical modes can be thought of as the Goldstone excitations associated with certain spontaneously broken spacetime symmetries, and effective field theory is at present the most efficient tool we have to characterize systematically the low-energy dynamics of Goldstone excitations. In fact, this logic was probably first explored for the dynamics of phonons in solids [@Leutwyler:1996er], and only later generalized to zero-temperature superfluids [@Son:2002zn], supersolids [@Son:2005ak], ordinary fluids [@Dubovsky:2005xd; @Dubovsky:2011sj], finite-temperature superfluids [@Nicolis:2011cs], and supersymmetric fluid systems [@Hoyos:2012dh; @Andrianopoli:2013dya].
The power of this new approach lies partially in its economy. One functional of the fields, i.e., the action, encodes all the information about the theory: the equations of motion, the stress-energy tensor, the other conserved currents, quantum phenomena, etc. The power of this approach also lies in how systematic it is. The action one writes down should be the most general local functional compatible with the symmetries, organized at low-energies as a perturbative expansion in the fields’ derivatives.
For Goldstones associated with standard spontaneous symmetry breaking in particle physics, the celebrated coset construction [@Callan:1969sn; @Coleman:1969sm] is the most exhaustive technique we have to write down such low-energy effective actions. Such a technique has been generalized to spontaneously broken [*spacetime*]{} symmetries in [@Volkov:1973vd; @Ogievetsky:1974ab], but so far it has not been applied directly to the systems of our interests: fluids, solids, and variations thereof.
Our perhaps modest goal in this paper is to carry out this application. The motivation is twofold. On the one hand, we want to confirm that the effective theories that have been written down so far for these systems are indeed the most general ones compatible with the appropriate symmetries to lowest order in the derivative expansion (it turns out that they are). On the other hand, we hope that the formalism we develop here will turn out to be useful in overcoming certain stumbling blocks that have been encountered in trying to extend these effective theories to higher orders—regarding for instance the inclusion of Wess-Zumino terms in dimensions higher than $(1+1)$ [@Dubovsky:2011sk], and the inclusion of dissipative effects to non-linear order in the hydrodynamical modes [@Endlich:2012vt].
We will start with the coset construction for an ordinary fluid that carries a conserved charge. In a sense that will be clear in what follows, this is the most complicated system among those in our title. Then, by gradually removing symmetries, we will be able to generalize such a construction to the other systems as well. For simplicity we will not consider the supersymmetric versions of our systems. Moreover, when covering solids and supersolids, we will only consider isotropic systems that realize the full $SO(3)$ symmetry rather than one of its discrete subgroups, as would be more appropriate for actual crystals.
To describe a generic system featuring spontaneous symmetry breaking, the only input needed by the coset construction is the symmetry breaking pattern. For the systems that we are interested in, the corresponding symmetry breaking patterns have been discussed extensively in the literature cited above. We will thus not re-derive them here, but rather build on these previous results. It is worth mentioning that all these symmetry breaking patterns feature a crucial interplay between spacetime symmetries and internal ones, whereby [*all*]{} the unbroken symmetries are linear combinations of both types. This further complicates the already subtle coset construction for broken spacetime symmetries.
Symmetries of Perfect Fluids
============================
In $D=d+1$ space-time dimensions, the low-energy behavior of perfect fluids can be described using $d$ scalar fields $\phi^I(\vec{x},t)$ [@Dubovsky:2011sk; @Dubovsky:2011sj]. These fields give the comoving (Lagrangian) coordinates $\phi^I$ of the fluid as a function of the physical spatial coordinates $\vec{x}$ and of time $t$. The action for these scalars is invariant under *internal* volume-preserving diffeomorphisms, i.e., \[vdiff\] \^I \^I(\^J), (\^I/\^J) =1 . Physically, such a large internal symmetry group encodes the insensitivity of a perfect fluid’s dynamics to adiabatically slow deformations that do not change the volume of the individual fluid elements. Among an infinite number of symmetries, this group includes most notably shifts and rotations of the comoving coordinates $\phi^I$.
To describe a perfect fluid with a conserved charge, we introduce an additional scalar field $\phi^0$ that shifts under the associated $U(1)$ symmetry. We demand that the charge be separately conserved within each volume element of the fluid, which is equivalent to demanding that the charge “flow with the fluid", $j^\mu = n u^\mu$. This corresponds to requiring that the action be invariant under shifts of $\phi^0$ that depend on the comoving coordinates $\phi^I$, i.e. \[chemshift\] \^0 \^0+f(\^I) , where $f$ is an arbitrary function [@Dubovsky:2011sj; @sibiryakov]. This symmetry is referred to as the “chemical shift" symmetry. We call this field $\phi^0$ for notational convenience, but [*no*]{} internal Lorentz invariance rotating $\phi^0$ into the $\phi^I$’s is implied.
When the fluid is in equilibrium, its comoving coordinates can be chosen to coincide with the physical coordinates, and $\phi^0$ with time, up to proportionality constants which we omit for simplicity: \^i = x\^i , \^0 = t . Such a field configuration spontaneously breaks all of the above internal symmetries as well as boosts, spacetime translations, and spatial rotations, and is left invariant only by a linear combination of internal and space-time translations and rotations. At sufficiently low energies, the only relevant excitations around the equilibrium configuration (\[vacuum\]) are the Goldstone bosons associated with this symmetry breaking. In what follows, we will derive a low-energy effective action for these Goldstone bosons. In order to successfully carry out such a construction, we will use an approximation. The internal volume-preserving diffeomorphisms and the chemical shift are potentially unwieldy as they are described by an infinite number of generators. To handle this, we restrict the transformation to constant shifts, generated by $Q_I$, and special linear transformations (i.e., unit determinant $3\times3$ matrices), generated by $M_{IJ}$. The antisymmetric part of $M_{IJ}$ generates internal rotations and we will denote it by $L_{IJ}$. Similarly, we will restrict the chemical shift symmetry to a constant shift, generated by $Q_0$, as well as a shift linear in $\phi^I$, generated by $F_I$. As we will see, by demanding invariance under this restricted set of symmetries, our action will be accidentally invariant under the full transformations (\[vdiff\]) and (\[chemshift\]) to lowest order in derivatives. In fact, at present it is not clear yet whether the infinite-dimensional symmetries postulated above should survive beyond lowest order in the derivative expansion. For instance, dissipative effects associated with shear viscosity and heat conduction apparently violate and [@Endlich:2012vt]. This further indicates that these symmetries might not be fundamental, but only accidental, in which case they should not enter the coset construction.
Finally, we will denote the generators of space-time translations, rotations, and Lorentz boosts by $P_\mu$, $J_{ij}$ and $K_i$ respectively. Therefore, the pattern of symmetry breaking that we will consider can be summarized as follows: \[pattern\]
[lcl]{} &=& {
[ll]{} [|[P]{}]{}\_P\_+Q\_&\
[|[J]{}]{}\_[ij]{} J\_[ij]{}+L\_[ij]{}&
.\
&&\
&=& {
[ll]{} K\_i &\
Q\_&\
F\_i &\
M\_[ij]{} &
.
In what follows, we will denote the full symmetry group by $G$ and the unbroken subgroup by $H$. In $D=4$ dimensions there are 25 generators in total, 18 of which are broken by the field configuration (\[vacuum\]).
Notice that since Lorentz invariance is spontaneously broken, the $\mu=0$ and $\mu=i$ components should be treated as independent here and in what follows. Moreover, we have stopped differentiating between the internal $I,J, \dots$ indices and the spatial $i,j, \dots$ ones, since the only index contractions that make sense at the level of the coset construction are those associated with the unbroken symmetries. So, from here on out all quantities carrying $i,j, \dots$ indices transform as tensors under the unbroken rotations generated by $\bar J_{ij}$.
Coset Construction for Fluids {#fluids}
=============================
As per the usual construction [@Callan:1969sn; @Coleman:1969sm; @Volkov:1973vd; @Ogievetsky:1974ab], we parameterize the space of (left) cosets $G/H$ by introducing one Goldstone field for each broken generator: \[Om\] (x) = e\^[i x\^[|[P]{}]{}\_]{} e\^[i \^i (x) K\_i]{} e\^[i \^(x) Q\_]{} e\^[i \^i(x) F\_i]{} e\^[i \^[ij]{}(x) M\_[ij]{}]{} . In order to construct an effective action that is invariant under the full symmetry group $G$, one considers the Maurer-Cartan form expanded in the basis of generators
[lcl]{} \^[-1]{} d&=& i \_P\^[|[P]{}]{}\_+i \_J\^[ij]{} [|[J]{}]{}\_[ij]{}+i \_K\^i K\_i\
&&+i \_Q\^Q\_+i \_F\^i F\_i +i \_M\^[ij]{} M\_[ij]{} .
The one-forms $\omega_P^\mu$ are related to the spacetime vielbeins, \_P\^= e\_\^ dx\^ . The one-forms associated with the broken generators, $\omega_K^i$, $\omega_Q^\mu$, $\omega_F^i$ and $\omega_M^{ij}$, are related to the covariant derivatives of the Goldstone fields,
[lcl]{} \_Q\^&=& e\_\^ [[D]{}]{}\_\^ dx\^ ,\
\_F\^i &=& e\_\^ [[D]{}]{}\_\^i dx\^ ,\
\_M\^[ij]{} &=& e\_\^ [[D]{}]{}\_\^[ij]{} dx\^ ,\
\_K\^i &=& e\_\^ [[D]{}]{}\_\^i dx\^ .
These forms transform covariantly and can thus be used as building blocks of the invariant Lagrangian. An action that is constructed to be manifestly invariant under the unbroken group $H$, will automatically be invariant under the full group $G$.
Given the coset parametrization , we can use the Poincaré algebra [@Weinberg:1995mt] as well as the commutation relations
&=& i (\_[jk]{} M\_[il]{} - \_[il]{} M\_[kj]{}) ,\
ł\[M\_[ij]{},Q\_k\] &=& i (\_[ik]{} Q\_j - \_[ij]{} Q\_k ) ,\
ł\[M\_[ij]{},F\_k\] &=& i ( \_[ij]{} F\_k - \_[jk]{} F\_i) ,\
ł\[F\_i,Q\_j\] &=& i \_[ij]{} Q\_0 ,
to calculate the Maurer-Cartan form (\[MC\]). We obtain
[ccl]{} \^[-1]{}\_&=&i \_\^[|[P]{}]{}\_+e\^[-i\^iK\_i]{}\_e\^[i\^jK\_j]{}\
&&+i\[\_\^[0]{}+\_\^0+(\_\^[i]{}+\_\^i)\_i-\_\^[0]{}\]Q\_0\
&&+i\[(\_\^[i]{}+\_\^i)\_i\^[j]{}-\_\^[j]{}\]Q\_j\
&&+i\_\^i \_i\^[j]{} F\_j+e\^[-i\^[ij]{}M\_[ij]{}]{}\_e\^[i\^[kl]{}M\_[kl]{}]{} ,
where $\xi_i{}^{j} = \xi_i{}^{j}(\alpha)$ is a special linear transformation, and $ \Lambda_\mu{}^{\nu} = \Lambda_\mu{}^{\nu} (\eta)$ is a Lorentz transformation of rapidity $\vec \eta$. We have not expanded the terms involving the $K$ and $M$ generators. Since the nested $K$ commutators only generate $K$’s and $J$’s, and the nested $M$ commutators only generate $M$’s, these will not contribute to the coefficients of the generators $Q_\mu$, which are our primary interest at the moment for reasons that we will explain towards the end of this section. These coefficients give the following covariant derivatives for the $\pi^\mu$ Goldstones:
\[Ds\] [[D]{}]{}\_\^0 &=& -\_\^[0]{}+\^\_(\_\^0+\_\^i \_i ) ,\
[[D]{}]{}\_\^i &=& -\_\^[i]{}+\^\_\_\^j\_j\^[i]{} ,
where we have simplified the notation by introducing the fields $\phi^\mu = x^\mu+\pi^\mu$. It will turn out that these fields are exactly the $\phi^i, \phi^0$ fields described in the previous section.
Not all of the Goldstone bosons we have introduced necessarily describe independent degrees of freedom [@Volkov:1973vd; @Nielsen:1975hm; @Ivanov:1975zq; @Low:2001bw]. Depending on the symmetry breaking mechanism, there may be some gauge transformations acting on the Goldstones that do not affect the physical fluctuations of the order parameter [@Nicolis:2013sga]. When that is the case, one can remove the redundant Goldstone fields by imposing gauge fixing conditions that respect all the global symmetries[^1].
There is a simple rule of thumb to determine whether such gauge redundancies may exist in the first place [@Ivanov:1975zq]. One simply needs to consider the commutators of the unbroken translation generators with a broken symmetry generator. In our case, the relevant commutators are
\[commutators\] &=&-i([|[P]{}]{}\_i-Q\_i) ,\
&=&-i\_[ij]{}([|[P]{}]{}\_0-Q\_0) ,\
&=&-i(\_[ik]{}Q\_j-\_[ij]{}Q\_k) .
Since the broken generators on the RHS’s of these equations are independent of the broken generators on the LHS’s, the Goldstone fields associated with the latter—namely $\eta^i, \theta^i$ and $\alpha^{ij}$—[*may*]{} be redundant. Whether they [*are*]{}, is a question that cannot be answered without further information on the symmetry breaking mechanism [@Nicolis:2013sga]. To proceed we will assume that they are, and we will thus construct the effective theory for the minimal set of Goldstones required to realize all the symmetries. Non-minimal choices where not all the potentially redundant Goldstones are redundant will be studied elsewhere [@framids].
The gauge fixing conditions that eliminate the redundant Goldstones while preserving all the symmetries are
&[[D]{}]{}\_0\^i = 0 ,&\
&[[D]{}]{}\_i\^0 = 0 ,&\
&[[D]{}]{}\_i \_j-\_[ij]{}[[D]{}]{}\_k\^k = 0 .&
In $(3+1)$ dimensions, these gauge fixing conditions allow one to eliminate 14 out of 18 Goldstone fields.\
$\boldsymbol{{{\mathcal D}}_0\pi^i=0}$: We can use the first gauge fixing condition to eliminate the $\eta$ Goldstones in favor of the $\pi$ Goldstones or, equivalently, in favor of the $\phi$’s. To do so, we use the covariant derivatives given in eqs. . It is convenient to parameterize $\Lambda_\mu{}^{\nu}$ in terms of the usual velocity vector $\beta^i$, and solve for $\beta^i$ rather for $\eta^i$. Defining the 4-vector $\beta^\mu \equiv (1,-\beta^i)$, we find that ${{\mathcal D}}_0\pi^i = 0$ implies $\beta^\mu \d_\mu \phi^i =0$. This equation is solved by \^= J\^/J\^0 , where $J^\mu \equiv \epsilon^{\mu \alpha_1 \ldots \alpha_d}\partial_{\alpha_1}\phi^1\ldots\partial_{\alpha_d}\phi^d$ is the identically conserved current of [@Dubovsky:2005xd; @Dubovsky:2011sj].\
$\boldsymbol{{{\mathcal D}}_i\pi^0=0}$: The second gauge fixing condition can be used to eliminate the $\theta$ Goldstones. Again using equations , we find \_i = - \_i\^j \^\_j \_\^0. where $\Lambda \partial \phi$ stands for $ \Lambda^\mu{}_i \partial_\mu \phi^j$.\
$\boldsymbol{{{\mathcal D}}_i\pi^j-\tfrac{1}{d}\ \delta_i^j{{\mathcal D}}_k\pi^k= 0}$: The remaining gauge fixing condition can be used to eliminate the $\alpha$ Goldstones. In this case, it is easier (and sufficient) to solve for $\xi_j^{~i} (\alpha)$. The gauge fixing condition (\[gf3\]) implies that ${{\mathcal D}}_k \pi^i \propto \delta_k^i$. Hence, \[X\] [[D]{}]{}\_k \^i =-\_k\^[i]{}+\^\_[k]{}\_\^j\_j\^[i]{} = C \_k\^[i]{} , for some function of the fields $C$. Solving for $\xi$ gives (\^[-1]{})\_[k]{}\^j = . Finally, using that, by definition, $\det \xi = 1$, we can solve for $C$, \[X2\] C = -1+(\^\_[ k]{}\_\^j )\^[1/d]{} . By combining equations (\[solbeta\]), (\[solxi\]) and (\[X2\]), the $\xi_j^{~i} (\alpha)$ can now be expressed solely in terms of the $\phi$’s.\
Returning to the covariant derivatives for the $\pi$ Goldstones, we can now write them in terms of the $\pi$ (or $\phi$) fields alone. After imposing the gauge fixing conditions (\[gf\]), the only non-zero components are ${{\mathcal D}}_0 \pi^0$ and ${{\mathcal D}}_1\pi^1=\ldots={{\mathcal D}}_d\pi^d =C$. The latter can be simplified using that for any matrix $A$ one has $\det A= \sqrt{\det A^T A}$, which, combined with the properties of $\beta^i$, yields [[D]{}]{}\_1 \^1 = (\_\^i \^\^j)\^[1/(2d)]{} -1 b\^[1/d]{} -1. By combining equations (\[dmupi0\]), (\[solbeta\]) and (\[soltheta\]), we can rewrite also ${{\mathcal D}}_0 \pi^ 0$ in a fairly compact form: [[D]{}]{}\_0 \^ 0 = -1 = -1 y -1. One may in principle consider also the covariant derivatives of the $\eta$, $\theta$, and $\alpha$ Goldstones, expressed in terms of the $\pi$’s. However, the solutions (\[solbeta\]), (\[soltheta\]) and (\[solxi\]) show that these fields necessarily start at first order in derivatives of the $\pi$’s, which means that their covariant derivatives can only enter the action at higher orders in the derivative expansion.
Thus, at lowest order in derivatives, the covariant derivatives (\[d1p1\]) and (\[d0p0\]) are the only invariant building blocks of the low-energy action for the perfect fluid, which therefore can be written as \[S\] [S]{} = d\^D x F(b,y) , where $F$ is a generic function. Notice that for notational convenience we have removed the fractional $1/d$ power in , as well as the $-1$ offsets in and . This is consistent since $F$ is a completely generic function anyway. We should keep in mind however that this action should be interpreted as a perturbative series about $b=y=1$, and we have no guarantee that the same effective field theory holds for background values of $b$ and $y$ that are much different than this[^2]. In many physical systems one will encounter phase transitions at critical values for these quantities. Identical considerations apply to the generalizations that we will analyze below.
This action coincides with that extensively studied in [@Dubovsky:2011sk; @Dubovsky:2011sj]. This supports our earlier claim that the fields $\phi^i$ are indeed the comoving coordinates of the fluid. Also, notice that the action is invariant under the full group of volume preserving diffeomorphisms as well as the full chemical shift symmetry , even though only the linearized version of these symmetries was imposed to carry out the coset construction. The reason is that at this order the action only involves the first derivatives of the $\phi$ fields, which—unlike the $\phi$ fields themselves—transform covariantly under and , that is, linearly.
Superfluids
===========
The coset construction for perfect fluids carried out in the previous section can be easily modified to describe relativistic superfluids. It is well known that a superfluid at finite temperatures can be thought of as an admixture of an ordinary perfect fluid and a zero temperature superfluid [@landau:1987bo]. From our field-theoretic perspective, this means that a finite-temperature superfluid in $D=d+1$ space-time dimensions can once again be described using $D$ scalars $\phi^\mu (\vec{x},t)$: the $d$ fields $\phi^I$ describe the ordinary component whereas $\phi^0$ describes the superfluid one [@Nicolis:2011cs]. The action must still be invariant under volume preserving diffeomorphisms (\[vdiff\]), but since the two components do not need to “flow” together, the action is only invariant under constant shifts \^0 \^0+c\^0 , and not under the full chemical shift (\[chemshift\]). The coset construction carried out in the previous section can be easily modified to take this into account. Setting $\theta_i = 0$ in equation (\[Om\]) or, equivalently, directly in the covariant derivatives (\[Ds\]) is formally equivalent to not having introduced $F_i$ initially.
The only gauge-fixing conditions we can now impose are those in equations (\[gf1\]) and (\[gf3\]). Since both equations involve only the covariant derivatives ${{\mathcal D}}_\mu \pi^i$, which did not depend on the Goldstones $\theta_i$, their solutions remain the same. After eliminating the $\beta^i$, the expression for ${{\mathcal D}}_0 \pi^0$ in terms of the $\pi$ Goldstones also remains unchanged. Thus the Lagrangian for a finite-temperature superfluid is still a function of the quantities $b$ and $y$ defined in equations (\[d1p1\]) and (\[d0p0\]). In addition, there is one more invariant that we can write down [[D]{}]{}\_i \^0 [[D]{}]{}\^i \^0 = \_\^0 \^\^0 - = X + y\^2 , where $X \equiv \d_\mu \phi^0 \d^\mu \phi^0$. Therefore, the low-energy effective action for a finite-temperature superfluid is = d\^D x F(X,b,y) , in agreement with [@Nicolis:2011cs].
One can recover the effective action for a superfluid at $T=0$ by neglecting the ordinary component of a finite-temperature superfluid. Following the same logic as above, this amounts to setting $\phi^i =0$ everywhere, in addition to $\theta^i$. Now there is one covariant derivative of the form \[D0\] [[D]{}]{}\_\^0 &=& -\_\^[0]{}+\^\_\_\^0 . None of our previous commutators can now be used to derive gauge-fixing conditions. Instead, we note that \[new commutator\] &=&-i\_[ij]{}([|[P]{}]{}\_0-Q\_0) . Accordingly, we can set ${{\mathcal D}}_i\pi^0=0$ in order to eliminate the boost Goldstones $\beta^i$ from the spectrum [^3].
We can solve this equation to get: \_i = . By plugging the relation (\[solbeta2\]) into the expression for ${{\mathcal D}}_0 \pi^0$ given above in eq. , we get [[D]{}]{}\_0 \^0 = 1 - 1- . Thus the effective action for the zero-temperature superfluid can be written as = d\^D x F(X) . This agrees with the effective action derived with alternative methods in [@Son:2002zn]. Notice that the coset construction for superfluids at $T=0$ can be trusted to all orders in the derivative expansion. This is because the pattern of symmetry breaking involves only a finite number of generators and hence we no longer need to restrict to a smaller subset of generators for the coset construction.
Solids and Supersolids
======================
The coset construction for perfect fluids can also be adapted to describe relativistic solids and supersolids. Let us start with supersolids. In $D$ space-time dimensions, their low-energy behavior is once again described by $D$ scalar fields $\phi^\mu(\vec{x},t)$ [@Son:2005ak]. However, the action for supersolids is invariant only under a subset of the symmetries (\[vdiff\]) and (\[chemshift\]), namely (constant) internal shifts and rotations: \^\^+ c\^, \^i R\^i\_j \^j. This changes the pattern of symmetry breaking associated with the equilibrium field configuration (\[vacuum\]). However, after setting $F_i = 0$ and replacing $M_{ij}$ with $L_{ij}$ in equation (\[Om\]), it is straightforward to repeat the construction carried out in Section \[fluids\]. The covariant derivatives for the $\pi$ fields are now
\[Ds2\] [[D]{}]{}\_\^0 &=& -\_\^[0]{}+\^\_ \_\^0 ,\
[[D]{}]{}\_\^i &=& -\_\^[i]{}+\^\_\_\^jR\_j\^[i]{} ,
where $R_j{}^{i} =R_j{}^{i} (\alpha)$ is a $d$-dimensional rotation. The relevant commutation relations to determine potential gauge redundancies are
&=&-i([|[P]{}]{}\_i-Q\_i) ,\
&=&i(\_[ik]{}Q\_j-\_[jk]{}Q\_i) ,
and the corresponding gauge fixing conditions are
&[[D]{}]{}\_0\^i = 0 ,\
&[[D]{}]{}\_i \_j - [[D]{}]{}\_j \_i = 0 .
$\boldsymbol{{{\mathcal D}}_0\pi^i=0}$: This condition still implies $\beta^\mu \d_\mu \phi^i =0$ and therefore the solution (\[solbeta\]) remains valid even for supersolids. We can then easily express the covariant derivatives of $\pi^0$ as a function of the fields $\phi^\mu$ only: [[D]{}]{}\_0 \^ 0 = y -1 , [[D]{}]{}\_i \^ 0 = \^\_i \_\^0 . $\boldsymbol{{{\mathcal D}}_{[i}\pi_{j]}=0}$: In order to solve this equation to eliminate the $\alpha$ Goldstones, it is convenient to introduce the matrix $N_i{}^j \equiv \Lambda^\mu{}_i \d_\mu \phi^j$. Then, from equation (\[Ds2-2\]) we see that the condition ${{\mathcal D}}_{[i}\pi_{j]}=0$ is tantamount to requiring that $N R = S$ with $S$ some symmetric matrix. It follows that $R N = R S R^T \equiv S'$ must also be a symmetric matrix and that $S'^2 = N^TN$. After a few algebraic manipulations, we find that $(S^{\prime 2})_{ij} =\d_\mu \phi_i \d^\mu \phi_j$, which implies: [[D]{}]{}\_[(i]{}\_[j)]{} = - \_[ij]{} + (N N\^[-1]{})\_[ij]{} . The low-energy effective action for supersolids is therefore a generic functional of the building blocks (\[dss\]) and (\[D(IPIJ)\]) that is manifestly invariant under all unbroken symmetries. We note that [[D]{}]{}\_i \^0 &=& (N\^[-1]{})\^j\_i \_\^0 \^\_j . Since all the indices must be contracted to preserve rotational invariance, we can equivalently use as our building blocks
[lclcl]{} [[D]{}]{}\_i \^0& &A\_i & &\_\^0 \^\_i ,\
[[D]{}]{}\_[(i]{}\_[j)]{} && B\_[ij]{}&&\_\_i \^\_j .
Therefore, the low-energy effective action for supersolids is = d\^D x F (A\_i, B\_[jk]{}, y) . This action is the straightforward relativistic generalization of that derived by Son in [@Son:2005ak].
The coset construction for solids is even simpler. The low-energy effective action can be obtained directly by setting $\phi^0 =0$ in the action (\[ssaction\]). The only remaining building block is then $B_{ij}$, which should be contracted with itself as to preserve rotational invariance. In $D=3$ spatial dimensions, there are only three invariants one can write down using a symmetric matrix such as $B_{ij}$. Following [@Endlich:2012pz], we can choose them to be W = \[B\], Y = , Z = , where the brackets $[ \cdots]$ stand for the trace of the matrix within. The low-energy effective action for solids is therefore: = d\^D x F (W,Y,Z) .
The $(1+1)D$ Anomaly
====================
The coset construction performed thus far has the limitation that it only generates terms that are [*exactly*]{} invariant under the chosen symmetries, as opposed to terms that are invariant up to total derivatives. The latter are known as Wess-Zumino terms and are necessary if one wishes to consider anomalous symmetries. The logic is that, upon gauging, the Wess-Zumino terms may no longer be invariant, thus indicating an anomaly.
There is a straightforward prescription for constructing such terms using the building blocks obtained from the coset construction [@DHoker:1994ti; @Goon:2012dy]. For a $D$-dimensional Wess-Zumino term, one constructs an exact, invariant $(D+1)$-form in $D+1$ dimensions, say $\alpha = d \beta$. Now, the $D$-form $\beta$ itself is not necessarily invariant but it can shift by a total derivative since $\alpha$ is invariant, and can thus be used as a Wess-Zumino term in the $D$-dimensional action.
As an example, let us consider the Maurer-Cartan form for a perfect fluid in $(1+1)$ dimensions before imposing any gauge fixing conditions. For now, we will also treat the Goldstone bosons as independent from the space-time coordinates so that we can construct an invariant $3$-form. The Maurer-Cartan form is given by \^[-1]{}d= \_[[|[P]{}]{}]{}\^[|[P]{}]{}\_+\_K K+\_Q\^Q\_+\_F F , where
\_[[|[P]{}]{}]{}\^&=& \_\^[ ]{}dx\^ ,\
\_Q\^0 &=& d\^0+d\^1+(\_\^[ 0]{}- \_\^[ 0]{}+\_\^[ 1]{} )dx\^ ,\
\_Q\^1 &=& d\^1+(\_\^[ 1]{}-\_\^[ 1]{})dx\^ ,\
\_F &=& d ,\
\_K &=& d ,
with $ \Lambda_\nu^{~\mu}\equiv \Lambda_\nu^{~\mu}(\eta)$. By combining the forms (\[1+1forms\]) we can write down 20 different 3-forms that are manifestly invariant under the unbroken symmetries, which in $(1+1)$ dimensions are just translations. Out of these 3-forms, we were able to identify 11 linear combinations which are exact. However, only 3 of them give rise to Wess-Zumino terms that have at most one-derivative per field and are exactly Lorentz invariant, rather than invariant up to a total derivative[^4]. These are
&&\_[[|[P]{}]{}]{}\^0 \_[[|[P]{}]{}]{}\^1 \_Q\^1 = d ( \^1 dx\^0 dx\^1 )\
&&\_[[|[P]{}]{}]{}\^0 \_[[|[P]{}]{}]{}\^1 \_F = d ( dx\^0 dx\^1 )\
&&\_[[|[P]{}]{}]{}\^0 \_Q\^1 \_F + \_Q\^0 \_[[|[P]{}]{}]{}\^1 \_F + \_Q\^0 \_Q\^1 \_F =\
&& = d \[ ( dx\^0d\^1+ d\^0dx\^1 +d\^0d\^1)\].
After pulling back to the space-time manifold and imposing the gauge fixing conditions to express $\theta$ in terms of the $\pi$ Goldstones, we find that the corresponding Wess-Zumino terms are:
S\_[WZ]{}\^[(1)]{} &=& d\^2 x (\^1-1)\
S\_[WZ]{}\^[(2)]{} &=& d\^2 x = - d\^2 x\
S\_[WZ]{}\^[(3)]{} &=& - d\^2 x \[\^ \_\^0 \_\^1 -1 \] .
The first term is just a tadpole and therefore we will neglect it. The second and third terms are instead more interesting. By design, they are invariant under all the symmetries we assumed as a starting point of our coset construction. However, they are not invariant under the full chemical shift (\[chemshift\]). From the point of view of our construction, this symmetry arises accidentally at lowest order in the derivative expansion. Since the Wess-Zumino terms (\[WZ\]) follow from the forms (\[exact\_forms\]) which have more than one derivative per field, it is not surprising that in general they do not respect this accidental symmetry. What instead is perhaps surprising, is that there still exists a linear combination of $S_{WZ}^{(2)}$ and $S_{WZ}^{(3)}$ that is invariant under the full chemical shift, namely \[SWZ\] S\_[WZ]{}\^[(2)]{} + S\_[WZ]{}\^[(3)]{}= - d\^2x\^\_\^0 \_\^1 . This term was extensively discussed in [@Dubovsky:2011sj]. Our analysis shows that this is the only Wess-Zumino term for a perfect fluid in $(1+1)$ dimensions that is exactly invariant under Lorentz transformations as well as invariant under the full chemical shift, up to a total derivative.
This term arises from an exact 3-form of the form \_ \^\^\_F , where $ \omega^\mu \equiv \omega_{{\bar{P}}}^\mu+\omega_Q^\mu$. While it is not straightforward to generalize, this structure may give some hint as to the form of the Wess-Zumino term in higher dimensions, which is still an open question [@Dubovsky:2011sj].
Outlook
=======
We have applied the coset construction to the spacetime symmetry breaking patterns characterizing fluid and solid systems. We thus confirm that the effective field theories that have been considered so far for these systems indeed describe the most general low-energy Goldstone dynamics that are invariant under all the symmetries.
We plan to apply this formalism to two problems that so far have resisted a satisfactory resolution at the effective field theory level. The first is how to include hydrodynamical dissipative effects systematically. Ref. [@Endlich:2012vt] argues that this should be done by coupling the Goldstones to another sector that “lives in the fluid." This sector stands for the microscopic degrees of freedom that are averaged over by the hydrodynamical description, and which are physically responsible for dissipation. The lowest-order couplings have been written down, and they successfully reproduce the standard dissipative effects due to shear viscosity, bulk viscosity, and heat conduction, as well as the associated Kubo relations. It is not obvious, however, how to go beyond linear order in the Goldstones and, more importantly, how to systematically implement the symmetries, given that these “successful" lowest order couplings [*violate*]{} some of the symmetries [@Endlich:2012vt]. We believe that the coset construction—which provides systematic rules for how to couple the Goldstones to other degrees of freedom in all ways allowed by symmetries and to all orders in perturbation theory—will shed light on these issues.
The second problem we have in mind is how to incorporate in the Goldstone effective theory Wess-Zumino terms that correctly describe quantum anomalies in (3+1)-dimensional hydrodynamics [@Son:2009tf]. Ref. [@Dubovsky:2011sj] constructed a Wess-Zumino term for (1+1)-dimensional fluids carrying an anomalous charge. We reproduced that term above via the coset construction, and showed that in fact it is the only possible Wess-Zumino term consistent with all the symmetries. However, the $(3+1)$-dimensional case is qualitatively more complicated, because it requires moving to one higher order in the derivative expansion, since the anomaly is expected to manifest itself at the one-derivative level beyond the perfect fluid approximation [@Son:2009tf]. Notice that this is the same order in the derivative expansion at which dissipative effects appear. It might well be that the two problems are related—in particular, that the insistence that has been placed so far on the infinite-dimensional symmetries and is misguided. On the one hand, the linear couplings of [@Endlich:2012vt] correctly reproduce the Kubo relations for first order transport coefficients, yet they violate and . On the other hand, one can show that the symmetry implies the existence of a conservation law that seems to be incompatible with anomalous hydrodynamics [@son]. It is thus entirely possible that these symmetries have to be abandoned beyond the lowest order in the derivative expansion. However, one should make sure that there are enough symmetries left that make them re-emerge as accidental symmetries at low enough momenta, for instance as is the case for our “linearized" symmetries $M_{IJ}$ and $F_I$.
From this viewpoint superfluids, solids, and supersolids are cleaner systems: they realize finite-dimensional symmetries, and we expect all of these to survive to all orders in the derivative expansion. It would be interesting per se—and useful as a warmup for the fluid case—to use our coset construction to characterize dissipation and anomalies in these systems. We plan to carry out all these projects in the near future.
[**Acknowledgments:**]{} This work was supported by NASA under contract NNX10AH14G and by the DOE under contract DE-FG02-11ER41743.
[^1]: These covariant gauge-fixing conditions usually go under the name of “inverse Higgs constraints". We will use instead the terminology and the interpretation introduced in [@Nicolis:2013sga], which emphasizes their being, in general, optional gauge choices.
[^2]: See, however, [@Nicolis:2013sga] for certain regularity conditions that can be imposed on the Lagrangian of superfluid systems in order to keep the theory consistent and weakly coupled all the way to zero density.
[^3]: Notice that in the ordinary fluid case we were getting the same ${{\mathcal D}}_i\pi^0=0$ gauge-fixing condition from the $[\bar P_i, F_j]$ commutator, and we used it to eliminate the $\theta_i$ Goldstones, while the $\beta_i$ Goldstones had been eliminated via the $[\bar P_0, K_i]$ gauge-fixing condition. So, in the ordinary fluid case we did not need the commutator , because it can be thought of as an [*alternative*]{} gauge fixing condition for $\beta^i$ with respect to the one we used.
[^4]: We want the Lagrangian to be exactly Lorentz invariant rather than only up to total derivatives, because the latter option would probably entail gravitational anomalies, i.e. Lorentz-invariance violations in the presence of gravitational fields, which would be inconsistent with dynamical gravity. Since in the real world gravity [*is*]{} dynamical, we find it safer to retain exact Lorentz invariance.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Yi-Xin Chen, Yi-Jian Du and Qian Ma\
Zhejiang Institute of Modern Physics, Zhejiang University\
Hangzhou 310027, P. R. China\
E-mail:, ,
title: 'Relations Between Closed String Amplitudes at Higher-order Tree Level and Open String Amplitudes'
---
Introduction
============
Superstring theories are theories without ultraviolet divergences. They contain both gravitational and gauge interactions as low energy limits[@1; @2]. Thus they offer a possible solution to the problem of unifying all of the fundamental interactions in a consistent quantum theory. In string theory, gravitons are massless states of closed strings and gauge particles are massless states of open strings. To study the relations between gravity and gauge field, we should explore the relations between closed and open strings. The duality between open and closed strings[@3; @4; @5; @6; @7; @8] also motivates us to explore the relations between closed and open strings.
The most simple relation is any excited mode of a free closed string $\left|N_L, N_R\right>\otimes \left|p\right>$ can be factorized by left- and right- moving open string excited modes: $$\left|N_L\right>\otimes\left|N_R\right>\otimes\left|p\right>.$$ However, when we consider the interactions among strings, there are nontrivial relations between closed and open string amplitudes. The first nontrivial relation was given by Kawai, Lewellen and Tye[@9]. They express an amplitude for $N$ closed strings on sphere($S_2$) by the following equation[^1]: $$\label{KLT relations}
\mathscr{A}_{S_2}^{(N)}=\epsilon_{\alpha\beta}\mathscr{A}_{S_2}^{(N)\alpha\beta}=\left(\frac{i}{2}\right)^{N-3}\kappa^{N-2}\epsilon_{\alpha\beta}
\sum\limits_{P,
P'}\mathscr{M}^{(N)\alpha}(P)\mathscr{\bar{M}}^{(N)\beta}(P')e^{i\pi
F(P, P')}.$$ Here $\mathscr{A}_{S_2}^{N}$ is the amplitude for $N$ closed strings on $S_2$ and $\mathscr{A}_{S_2}^{(N)\alpha\beta}$ is the closed string amplitude without polarization tensors. $\mathscr{M}^{(N)\alpha}(P)$ and $\mathscr{\bar{M}}^{(N)\beta}(P')$ are the open string partial amplitudes on $D_2$ corresponding to the left- and right-moving sectors respectively. They are dependent on the orderings of the external legs. If we sum over the orderings $P$ and $P'$, we get the total amplitudes $\sum\limits_P\mathscr{M}^{(N)\alpha}(P)$ and $\sum\limits_{P'}\mathscr{\bar{M}}^{(N)\beta}(P')$ for the left- and the right-moving open strings respectively. Then we can see, except for a phase factor, a closed string amplitude on $S_2$ can be factorized by two open string tree amplitudes corresponding to the left- and right-moving sectors(see fig. \[fig1\].(a)). There is no interaction between left- and right-moving open strings. Any closed string polarization tensor has left and right indices, they correspond to the left- and the right-moving modes respectively. The left and right indices of polarization tensors must contract with the indices in the amplitude for left- and right-moving open strings respectively. The phase factor is entirely independent of which open and closed string theories we are considering. It only depends on $P$ and $P'$. Contour deformations can be used to reduce the number of the terms in eq.. The number of the terms can be reduced to $$\label{KLT term reduction1}
(N-3)!(\frac{1}{2}(N-3))!(\frac{1}{2}(N-3))!, \text{$N$ odd},$$ and $$\label{KLT term reduction2}
(N-3)!(\frac{1}{2}(N-2))!(\frac{1}{2}(N-4))!, \text{$N$ even}.$$
![(a) A closed string amplitude on $S_2$ can be factorize by two open string tree amplitudes corresponding to the left- and right-sectors. (b) A closed string amplitude on $D_2$ can be given by connecting the open string world-sheets for the two sectors with a time reverse in the right-moving sector. (c) A closed string amplitude on $RP_2$ can be given by connecting the open string world-sheets for the two sectors with a time reverse and a twist in the right-moving sector.[]{data-label="fig1"}](relations.eps){width="70.00000%"}
In the low energy limits, the massive modes decouple. Only massless states are left. Then KLT relations can be used to factorize the amplitudes for gravitons into products of two amplitudes for gauge particles. Gauge theory has a better ultraviolet behavior than gravity. Then KLT relations can be used to investigate the ultraviolet properties of gravity. Researches with KLT relations support that $N=8$ supergravity may be finite[@11; @12; @13; @14; @15; @16]. However, a question arises: Do KLT factorization relations hold for any gravity amplitude? In string theory, to calculate the S-matrix, we should sum over all the topologies of world-sheets. $S_2$ is just the most simple topology. If we consider other topologies, we should reconsider the relations between closed and open strings. Then the question becomes: Do the factorization relations hold for any topology?
Earlier works[@17; @25; @26; @27] have given some insights into the relations on Disk($D_2$). In [@17], some examples of the relations on $D_2$ are given. In [@25; @26; @27], the most simple process of D-brane and closed string interactions are discussed. In the paper by Garousi and Myers[@25], they found that the two-point scattering amplitudes of closed strings from a D-brane in Type II theory is identical with the four-point open string amplitudes upon a certain identification between the momenta and polarizations. In [@26; @27], The amplitude for one closed string and two open strings attached to a D-brane are calculated. They shown that this amplitude are also identical with the four-point open string amplitude. In these examples, The KLT factorization relations do not hold. Then they imply the KLT factorization relations may not hold for general amplitudes on $D_2$. The amplitudes on real projective plane($RP_2$) have similar structures with open string amplitudes[@30]. In fact, Both $D_2$ and $RP_2$ can be obtained by a sphere with a $\mathbb{Z}_2$ identification. Then the KLT factorization relations may also not hold in the $RP_2$ case.
In this paper, we consider the general amplitudes on $D_2$ and $RP_2$. These two cases contribute to the higher-order tree amplitude[@1] for closed strings. We find that the factorization relations do not hold on $D_2$ and $RP_2$. The amplitudes with closed strings on $D_2$ and $RP_2$ can not be factorized by the left- and the right-moving open string amplitudes. The amplitudes satisfy new relations. Particularly, an amplitude for $N$ closed strings on $D_2$ can be given by an amplitude for $2N$ open strings: $$\label{A_(D_2)^(N,0)}
\mathscr{A}_{D_2}^{(N)}=\epsilon_{\alpha\beta}\mathscr{A}_{D_2}^{(N)\alpha\beta}=\left(\frac{i}{4}\right)^{N-1}\kappa^{N-1}\epsilon_{\alpha\beta}\sum\limits_P\mathscr{M}^{(2N)\alpha\beta}(P)e^{i\pi\Theta(P)}.$$ In this equation, $\mathscr{M}^{(2N)\alpha\beta}(P)$ is the tree amplitude for $2N$ open strings. $N$ open strings come from the left-moving sector and the other $N$ open strings come from the right-moving sector. The left- and the right-moving sectors are not independent of each other. The two sectors are connected into a single sector. Then the left indices contract with the right indices. The reason is that the left-(right-)moving waves must be reflected at the boundary of $D_2$ and then become right-(left-)moving waves. Then the interactions between the left-(right-)moving waves and their reflected waves become interactions between the two sectors. If there are open strings on the boundary of $D_2$, the left- and the right-moving sectors of closed strings also interact with the open strings, then an amplitude for $N$ closed strings and $M$ open strings on $D_2$ can be given by a tree amplitude for $2N+M$ open strings except for a phase factor: $$\label{A_(D_2)^(N,M)} \mathscr{A}_{D_2}^{(N,
M)}=\epsilon_{\alpha\beta\gamma}\mathscr{A}_{D_2}^{(N,
M)\alpha\beta\gamma}=\left(\frac{i}{4}\right)^{N-1}\kappa^{N-1}g^M\epsilon_{\alpha\beta\gamma}
\sum\limits_P\mathscr{M}^{(2N,M)\alpha\beta\gamma}(P)e^{i\pi\Theta'(P)}.$$
The amplitudes on $RP_2$ can also be factorized by one amplitude for open strings: $$\label{A_RP2^(N)}
\mathscr{A}_{RP_2}^{(N)}=\epsilon_{\alpha\beta}\mathscr{A}_{RP_2}^{(N)\alpha\beta}=-\left(\frac{i}{4}\right)^{N-1}\kappa^{N-1}\epsilon_{\alpha\beta}\sum\limits_P\mathscr{M}^{(2N)\alpha\beta}(P)e^{i\pi\Theta(P)}.$$ In this case, there is a crosscap but not a boundary here. However, the left-(right-) moving waves are also reflected at the crosscap and turn into the right-(left-)moving waves. Then there are also interactions between left- and right-moving sectors of closed strings. The two sectors are connected into one single sector again. The phase factors in and are complicated in concrete calculations. By considering the contour deformations, the number of the terms can be reduced[@9; @29]. It is noticed that the relations on $D_2$ are same with on $RP_2$ except for a minus. In a theory containing both $D_2$ and $RP_2$, the two amplitudes cancel out. Under a T-duality, the relation gives the amplitude for $N$ closed strings and $M$ open strings attached to a D-brane by pure open string amplitudes while the relation gives the amplitude for $N$ closed strings scattering from an O-plane by pure open string amplitudes. In this case, the amplitudes on $D_2$ and $RP_2$ can not cancel out.
An important fact will be used in our paper is that the amplitudes with closed strings are invariant under conformal transformations in each single sector. This allows us to transform the form of the interactions between left- and right-moving sectors. After some appropriate transformation in one sector, the interactions between left- and the right-moving sectors have the same form with interactions between open strings in a same sector. Then we can treat the two sectors of $N$ closed strings as a single sector with $2N$ open strings.
In the low energy limit of an unoriented open string theory, the amplitudes for $N$ closed strings on $D_2$, $RP_2$ and $S_2$ contribute to the tree amplitudes for $N$ gravitons. In this case, we can not only use KLT factorization relations on $S_2$ but also use the relations on $D_2$ and $RP_2$ to calculate the tree amplitudes for gravitons. The amplitudes for $N$ closed strings and $M$ open strings on $D_2$ become tree amplitudes for $N$ gravitons and $M$ gauge particles. Then the gauge-gravity interactions can be given by pure gauge interactions.
The structure of this paper is as follows. In section \[relations on D\_2\] we will consider the correlation functions and the amplitudes on $D_2$. We will show KLT factorization relations do not hold on $D_2$. We will give the relations between closed string amplitudes on $D_2$ and open string tree amplitudes. We will also give the relations in the case of $N$ closed strings and $M$ open strings inserted on $D_2$. In section \[relations on RP\_2\] we will consider $RP_2$. We will show KLT factorization relations do not hold on $RP_2$. The relations between amplitudes on $RP_2$ and open string amplitudes will be given in this section. Our conclusion will be given in section\[conclusion\].
Relations between amplitudes on $D_2$ and open string tree amplitudes {#relations on D_2}
=====================================================================
In this section, we will show the correlation functions on $D_2$ can not be factorized into the left- and the right-moving sectors. The two sectors are connected together. Then we will give the relations between amplitudes on $D_2$[@1; @2; @18; @19; @22] and open string tree amplitudes[@1; @2; @22; @23; @24; @25; @26; @27; @28].
In string theory, vertex operator for any closed string can be given as $$\mathscr{V}(\omega,
\bar{\omega})=\mathscr{V}_L(\omega)\tilde{\mathscr{V}}_R(\bar{\omega})\mathscr{V}_0(\omega,
\bar{\omega}),$$ where $\omega=\tau+i\sigma$. $\mathscr{V}_L$ and $\mathscr{V}_R$ are nonzero modes of open string vertex operators. They correspond to the left- and the right-moving sectors. $\mathscr{V}_0(\omega,
\bar{\omega})$ correspond to the zero modes. Thus, the closed string vertex operators can be factorized by two open string vertex operators corresponding to the left- and the right-moving sectors(except for the zero modes).
![Only the annihilation modes are reflected at the boundary of $D_2$.[]{data-label="fig2"}](2.eps){width="50.00000%"}
Now we consider the correlation function of vertex operators. On $S_2$ the left- and the right-moving waves are independent of each other. Then a correlation function on $S_2$ can be factorized by the left- and the right-moving sectors[@1; @2]. However, when we add a boundary to $S_2$, we get $D_2$. The left- and the right-moving waves must be reflected at the boundary of $D_2$. The reflection waves of the left-moving waves are in the right-moving sector and the reflection waves of the right-moving waves are in the left-moving sector. Waves must interact with their reflection waves, then their must be interactions between the two sectors. To see this, we should use the boundary state[@20; @21] to give the correlation functions on $D_2$. The correlation function for N closed strings on $D_2$ is $$\label{D2 correlation}
\left<0\mid\mathscr V_N(\omega,
\tilde{\omega})...\mathscr{V}_1(\omega, \tilde{\omega})\mid
B\right>,$$ where $\mid B\rangle\equiv B\mid 0\rangle$ is the boundary state for $D_2$. In this paper, for convenience, we use the bosonized vertex operators[^2] $$\label{bosonized vertex operator}
\begin{split}
\mathscr{V}(\omega,\bar{\omega})=&:\exp{(q\phi_6+\tilde{q}\tilde{\phi}_6)}
\\&\exp{(i\lambda\circ\phi+i\sum\limits_{i=1}^{m}\varepsilon^i\circ\partial\phi_i
+i\tilde{\lambda}\circ\tilde{\phi}+i\sum\limits_{i=1}^{\tilde{m}}\bar{\varepsilon}^i\circ\bar{\partial}\tilde{\phi_i}
)}
\\&\exp{(ik\cdot
X+i\sum\limits_{i=1}^n\epsilon^i\cdot\partial
X+i\sum\limits_{j=1}^{\tilde{n}}\bar{\epsilon}^j\cdot\bar{\partial}X)}(\omega,\bar{\omega}):|_{multilinear}
\end{split}$$ With the definition of normal ordering, we have $$\mathscr{V}(\omega,\bar{\omega})=\mathscr{V}^{(+)}_L(\omega)\mathscr{V}^{(-)}_L(\omega)\tilde{\mathscr{V}}^{(+)}_L(\bar{\omega})\tilde{\mathscr{V}}^{(-)}_R(\bar{\omega})\mathscr{V}_0(\omega,
\bar{\omega}),$$ where $(+)$ and $(-)$ correspond to the creation modes and the annihilation modes respectively. In $\mathscr{V}_0$, we consider $x$ as creation operator and $p$ as annihilation operator. Then in the normal ordering, $x$ must on the left of $p$. The bosonized boundary operator is[^3] $$\begin{split}
B&=\exp{(\sum\limits_{n=1}^\infty
a_n^{\dagger}\cdot\tilde{a}_{n}^{\dagger})}\otimes\exp{(\sum\limits_{n=1}^\infty
b_n^{\dagger}\circ\tilde{b}_{n}^{\dagger})}\otimes\exp{(\sum\limits_{n=1}^\infty
c_n^{\dagger}\tilde{c}_{n}^{\dagger})},
\end{split}$$ where $a^{\dag}$ and $\tilde{a}^{\dag}$ are creation modes of $X$, $b^{\dag}$ and $\tilde{b}^{\dag}$ are creation modes of $\phi_i$ and $\tilde{\phi_i}$ respectively, $c^{\dag}$ and $\tilde{c}^{\dag}$ are creation modes of $\phi_6$ and $\tilde{\phi}_6$ respectively. To get the correlation function on $D_2$ we substitute the bosonized vertex operators and the bosonized boundary operators into . We can move the boundary operator $B$ to the left of all the vertex operators. Then use the creation operators in $B$ to annihilate the state $\langle 0 \mid$. Because $B$ is constructed by creation operators, it commutes with the creation modes and the zero modes of the vertex operators and does not commute with the annihilation modes of the vertex operators. It means only the annihilation modes are reflected at the boundary(see fig. \[fig2\]). When we move $B$ to the left of the annihilation modes of the vertex operators $\mathscr{V}^{(-)}_L(\omega)$ and $\tilde{\mathscr{V}}^{(-)}_R(\bar{\omega})$, the ”images” of the annihilation modes $\tilde{\mathscr{V}}^{(+)}_L(-\omega)$ and $\mathscr{V}^{(+)}_R(-\bar{\omega})$ are created respectively. Though $\tilde{\mathscr{V}}^{(+)}_L(-\omega)$ is depend on $\omega$, it is constructed by $\tilde{a}^{\dag}$, $\tilde{b}^{\dag}$ and $\tilde{c}^{\dag}$. It must interact with operators constructed by $\tilde{a}$, $\tilde{b}$ and $\tilde{c}$. In a similar way, $\mathscr{V}^{(+)}_R(-\bar{\omega})$ must interact with operators constructed by $a$, $b$ and $c$. Then the correlation function can be factorized as $$\label{correlation function on D2}
\begin{split}
&\left<\mathscr{V}^{(+)}_L(\omega_N)\mathscr{V}^{(-)}_L(\omega_N)\mathscr{V}^{(+)}_R(-\bar{\omega}_N)...\mathscr{V}^{(+)}_L(\omega_1)\mathscr{V}^{(-)}_L(\omega_1)\mathscr{V}^{(+)}_R(-\bar{\omega}_1)\right>
\\\times &\left<\tilde{\mathscr{V}}^{(+)}_R(\bar{\omega}_N)\tilde{\mathscr{V}}^{(-)}_R(\bar{\omega}_N)\tilde{\mathscr{V}}^{(+)}_L(-\omega_N)...\tilde{\mathscr{V}}^{(+)}_R(\bar{\omega}_1)\tilde{\mathscr{V}}^{(-)}_R(\bar{\omega}_1)\tilde{\mathscr{V}}^{(+)}_L(-\omega_1)\right>
\\\times &\left<\mathscr{V}_0(\omega_N, \tilde{\omega}_N)...\mathscr{V}_0(\omega_1,
\tilde{\omega}_1)\right>.
\end{split}$$ Here, the first correlation function only contain operators constructed by $a$, $b$, $c$ and $a^{\dag}$, $b^{\dag}$, $c^{\dag}$, the second correlation function only contain operators constructed by $\tilde{a}$, $\tilde{b}$, $\tilde{c}$ and $\tilde{a}^{\dag}$, $\tilde{b}^{\dag}$, $\tilde{c}^{\dag}$, the third correlation function only contain operators constructed by zero modes. Though the nonzero modes are factorized into two correlation functions, both of them contain the interactions between left- and the right-moving sectors. Actually, in the first correlation function, if we move $\mathscr{V}^{(-)}_L(\omega_i)$ to the right of $\mathscr{V}^{(+)}_L(\omega_j)$, we get the interaction in the left-moving sector and if we move $\mathscr{V}^{(-)}_L(\omega_i)$ to the right of $\mathscr{V}^{(+)}_R(-\bar{\omega}_j)$, we get the interactions between the two sectors. In the same way, the second correlation function contain interactions in the right-moving sector and the interactions between left- and right-moving sectors. The interactions between the two sectors are just the interactions between vertex operators and their images. Then the correlation function on $D_2$ can not be factorized by the the two sectors, interactions between the two sectors connect them together.
To get the relations between amplitudes, we should calculate the correlation function, then integral over the fundamental region and divide the integrals by the volume of conformal Killing group[@1; @2; @22; @23]. for convenience, we use the $z$ coordinate instead of $\omega$ coordinate. They are connected by a conformal transfermation: $$z=e^{\omega}.$$ Then the amplitude for $N$ closed strings on $D_2$ becomes $$\begin{aligned}
\label{AD2superstring}
\nonumber\mathscr{A}_{D_2}^{(N,0)}&=\kappa^{N-1}\int_{|z|<1}
\prod\limits_{i=1}^Nd^2z_i\frac{|1-z_o\bar{z_o}|^2}{2\pi d^2z_o}
\\&\nonumber\times\prod\limits_{s>r}(z_s-z_r)^{\frac{\alpha'}{2}k_r\cdot
k_s+\lambda_r\circ\lambda_s-q_rq_s}(\bar{z}_r-\bar{z}_s)^{\frac{\alpha'}{2}k_r\cdot
k_s+\tilde{\lambda}_r\circ\tilde{\lambda}_s-\tilde{q}_r\tilde{q}_s}
\prod\limits_{r,
s}(1-(z_r\bar{z}_s)^{-1})^{\frac{\alpha'}{2}k_r\cdot
k_s+\lambda_r\circ\tilde{\lambda}_s-q_r\tilde{q}_s}
\\&\nonumber\times \exp{\sum\limits_{r=1}^N\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{\tilde{n}_s}\left(-\frac{\alpha'}{2}\right)\epsilon_r^{(i)}\cdot\bar{\epsilon}_r^{(j)}
-\sum\limits_{i=1}^{m_r}\sum\limits_{j=1}^{\tilde{m}_s}\varepsilon_r^{(i)}\circ\bar{\varepsilon}_r^{(j)}\right)(1-|z_r|^2)^{-2}}
\\&\times \exp{\sum\limits_{s>r}\left[\left(\sum\limits_{i=1}^{\tilde{n}_r}\sum\limits_{j=1}^{n_s}\left(-\frac{\alpha'}{2}\right)\bar{\epsilon}_r^{(i)}\cdot\epsilon_s^{(j)}
-\sum\limits_{i=1}^{\tilde{m}_r}\sum\limits_{j=1}^{m_s}\bar{\varepsilon}_r^{(i)}\circ\varepsilon_s^{(j)}\right)(1-\bar{z}_rz_s)^{-2}+c.c.\right]}
\\&\nonumber\times\exp{\left[-\sum\limits_{s>r}\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{n_s}\left(-\frac{\alpha'}{2}\right)\epsilon_r^{(i)}\cdot\epsilon_s^{(j)}-\sum\limits_{i=1}^{m_r}\sum\limits_{j=1}^{m_s}\varepsilon_r^{(i)}\circ\varepsilon_s^{(j)}\right)(z_s-z_r)^{-2}+c.c.
\right]}
\\&\nonumber\times \exp{\sum\limits_{r\neq s}\left[\left(\sum\limits_{i=1}^{n_s}\left(-\frac{\alpha'}{2}\right)k_r\cdot\epsilon_s^{(i)}
-\sum\limits_{i=1}^{m_s}\lambda_r\circ\varepsilon_s^{(i)}\right)((z_r-z_s)^{-1}+(\bar{z_r}^{-1}-z_s)^{-1})+c.c.\right]}
\\&\nonumber\times
\exp{\sum\limits_{r=1}^N\left[\left(\left(-\frac{\alpha'}{2}\right)k_r\cdot\sum\limits_{i=1}^{n_r}\epsilon_r^{(i)}
-\lambda_r\circ\sum\limits_{i=1}^{m_r}\varepsilon_r^{(i)}\right)((\bar{z_r}^{-1}-z_r)^{-1}+{z_r}^{-1})+c.c.\right]}|_{multilinear},\end{aligned}$$ where we have $\sum\limits_{r=1}^N\lambda_r=\sum\limits_{r=1}^N\tilde{\lambda}_r=0$, $\sum\limits_{r=1}^Nk_r=0$ and $\sum\limits_{r=1}^N(q_r+\tilde{q}_r)=-2$ correspond to the conservation of fermion number, the conservation of momentum and the fact that background superghost number is $-2$. $\frac{2\pi
d^2z_o}{|1-z_o\bar{z_o}|^2}$ is the volume element of the CKG[^4], it can be used to fix one complex coordinate.
An integral over the fundamental region $|z|<1$ is equal to an integral over the other fundamental region $|z|>1$. So we can use ${\left(\frac{1}{2}\right)}^{N-1}\int\limits_{\mathbb{C}}\prod\limits^N_{i=1}d^2z_i$ instead of the integrals over the unit disk. For any $z_r=x_r+iy_r$, the $z_r$ integral can be given by $\int\limits_{-\infty}^{\infty}dx_r\int\limits_{-\infty}^{\infty}dy_r$. We then follow the same steps as in[@9]. We rotate the contour of the $y$ integrals along the real axis to pure imaginary axis. The fixed point should be transformed simultaneously to guarantee the conformal invariance. Define the new variables: $$\label{variable redefine}
\begin{split}
\xi_1=\xi_o=x_o+iy_o &, \eta_1=\eta_o=x_o-iy_o, \\\xi_r\equiv
x_r+iy_r &, \eta_r\equiv x_r-iy_r\text{ }(r>1).
\end{split}$$ Then the integrals become real integrals: $$\begin{aligned}
\label{AD2real integrals}
\nonumber\mathscr{A}_{D_2}^{(N,0)}&=\kappa^{N-1}\left(\frac{1}{2}\right)^{N-1}\int
\prod\limits_{i=1}^Nd\xi_id\eta_i\frac{|1-\xi_o\eta_o|^2}{2\pi
d\xi_od\eta_o}
\\&\nonumber\times\prod\limits_{s>r}(\xi_s-\xi_r)^{\frac{\alpha'}{2}k_r\cdot
k_s+\lambda_r\circ\lambda_s-q_rq_s}(\eta_r-\eta_s)^{\frac{\alpha'}{2}k_r\cdot
k_s+\tilde{\lambda}_r\circ\tilde{\lambda}_s-\tilde{q}_r\tilde{q}_s}
\prod\limits_{r,
s}(1-(\xi_r\eta_s)^{-1})^{\frac{\alpha'}{2}k_r\cdot
k_s+\lambda_r\circ\tilde{\lambda}_s-q_r\tilde{q}_s}
\\&\nonumber\times \exp{\sum\limits_{r=1}^N\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{\tilde{n}_s}\left(-\frac{\alpha'}{2}\right)\epsilon_r^{(i)}\cdot\bar{\epsilon}_r^{(j)}
-\sum\limits_{i=1}^{m_r}\sum\limits_{j=1}^{\tilde{m}_s}\varepsilon_r^{(i)}\circ\bar{\varepsilon}_r^{(j)}\right)(1-\xi_r\eta_r)^{-2}}
\\&\times \exp{\sum\limits_{s>r}\left[\left(\sum\limits_{i=1}^{\tilde{n}_r}\sum\limits_{j=1}^{n_s}\left(-\frac{\alpha'}{2}\right)\bar{\epsilon}_r^{(i)}\cdot\epsilon_s^{(j)}
-\sum\limits_{i=1}^{\tilde{m}_r}\sum\limits_{j=1}^{m_s}\bar{\varepsilon}_r^{(i)}\circ\varepsilon_s^{(j)}\right)(1-\eta_r\xi_s)^{-2}+c.c.\right]}
\\&\nonumber\times\exp{\left[-\sum\limits_{s>r}\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{n_s}\left(-\frac{\alpha'}{2}\right)\epsilon_r^{(i)}\cdot\epsilon_s^{(j)}-\sum\limits_{i=1}^{m_r}\sum\limits_{j=1}^{m_s}\varepsilon_r^{(i)}\circ\varepsilon_s^{(j)}\right)(\xi_s-\xi_r)^{-2}+c.c.
\right]}
\\&\nonumber\times \exp{\sum\limits_{r\neq s}\left[\left(\sum\limits_{i=1}^{n_s}\left(-\frac{\alpha'}{2}\right)k_r\cdot\epsilon_s^{(i)}
-\sum\limits_{i=1}^{m_s}\lambda_r\circ\varepsilon_s^{(i)}\right)((\xi_r-\xi_s)^{-1}+(\eta_r^{-1}-\xi_s)^{-1})+c.c.\right]}
\\&\nonumber\times
\exp{\sum\limits_{r=1}^N\left[\left(\left(-\frac{\alpha'}{2}\right)k_r\cdot\sum\limits_{i=1}^{n_r}\epsilon_r^{(i)}
-\lambda_r\circ\sum\limits_{i=1}^{m_r}\varepsilon_r^{(i)}\right)((\eta_r^{-1}-\xi_r)^{-1}+{\xi_r}^{-1})+c.c.\right]}|_{multilinear},\end{aligned}$$ The real variables $\xi_r$ correspond to the left-moving sector and $\eta_r$ correspond to the left-moving sector. An open string tree amplitude for $M$ bosonized vertices has the form $$\label{open string amplitude}
\begin{split}
\mathscr{M}_{D_2}^{(N)}&=(g)^{M-2}\int \prod\limits_{i=1}^{M}d
x_i\frac{|x_a-x_b||x_b-x_c||x_c-x_a|}{d x_ad x_bd
x_c}\prod\limits_{s>r}|x_s-x_r|^{2\alpha'k_r\cdot
k_s}(x_s-x_r)^{\lambda_r\circ\lambda_s-q_rq_s}
\\&\times\exp{\left[\sum\limits_{s>r}\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{n_s}\left(2\alpha'\right)\epsilon_r^{(i)}\cdot\epsilon_s^{(j)}+\sum\limits_{i=1}^{m_r}\sum\limits_{j=1}^{m_s}\varepsilon_r^{(i)}\circ\varepsilon_s^{(j)}\right)(x_s-x_r)^{-2}
\right]}
\\&\times \exp{\left[\sum\limits_{r\neq s}\left(\sum\limits_{i=1}^{n_s}\left(-2\alpha'\right)k_r\cdot\epsilon_s^{(i)}
-\sum\limits_{i=1}^{m_s}\lambda_r\circ\varepsilon_s^{(i)}\right)(x_r-x_s)^{-1}\right]}|_{multilinear},
\end{split}$$ where $g$ is the coupling constant for open strings, it can be related with closed string coupling constant by $\kappa\sim g^2$. By comparing with the open string amplitude, we can see, the interactions in one sector can be considered as interactions between open strings. The interactions between left- and right-moving sectors look like those between open strings inserted at $\xi_r$ and $(\eta_s)^{-1}$. $\eta_s$ can be considered as the coordinates of the right-moving open string. Then in the $(\tau, \sigma)$ coordinate, ${\eta_s}^{-1}=e^{-\tau}$ can be considered as a time reverse in the right-moving sector. Thus the interactions between the two sectors can be regarded as interactions between left- and right-moving open strings with a time reverse in the right moving sector(see fig. \[fig1\].(b)). The amplitude then can be considered as an amplitude for $2N$ open strings. $N$ of them correspond to the left-moving sector and the other $N$ of them correspond to the right-moving sector. In the amplitude we have a time reverse in the right-moving sector.
From fig. \[fig1\].(b), we can see, if we reverse the time in the right-moving sector, we will get an open string tree amplitude. In fact, we can replace all the ${\eta_r}^{-1}$ by $\eta_r$. By using the mass-shell condition[@18; @19] which is determined by the conformal invariance in one sector, the interactions between the two sectors as well as the interactions in one sector become those between open strings. Define $$\label{variable redefine}
\xi_{r+N}\equiv\eta_r, k_{r+N}\equiv k_r,
\tilde{\lambda}_{r+N}\equiv\lambda_r,
\bar{\epsilon}_{r+N}\equiv\epsilon_r,
\bar{\varepsilon}_{r+N}\equiv\varepsilon_r.$$ After the simultaneous transformations, the volume of CKG becomes $\frac{1}{2\pi}\int\frac{d\xi_od\eta_o}{|\xi_o-\eta_o|^2}$. The fixed points become $\xi_1=\xi_o$ and $\xi_{1+N}=\xi_o$. The conformal Killing volume has another form $\int\frac{dx_adx_bdx_c}{|x_a-x_b||x_b-x_c||x_c-x_a|}$, it can be used to fix three real variables. We reset the fixed points at: $$\xi_1=x_a=0, \xi_2=x_b=1, \xi_{2N}=x_c=\infty.$$ The amplitude for $N$ closed strings on $D_2$ then becomes $$\label{AD_2N1without phase factor}
\begin{split}
\mathscr{A}_{D_2}^{(N,0)}&=\kappa^{N-1}{\left(\frac{i}{4}\right)}^{N-1}\int
\prod\limits_{i=1}^{2N}d\xi_i\frac{|\xi_a-\xi_b||\xi_b-\xi_c||\xi_c-\xi_a|}{d\xi_ad\xi_bd\xi_c}\prod\limits_{s>r}(\xi_s-\xi_r)^{\frac{\alpha'}{2}k_r\cdot
k_s}(\xi_s-\xi_r)^{\lambda_r\circ\lambda_s-q_rq_s}
\\&\times\exp{\left[\sum\limits_{s>r}\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{n_s}\left(2\alpha'\right)\epsilon_r^{(i)}\cdot\epsilon_s^{(j)}+\sum\limits_{i=1}^{m_r}\sum\limits_{j=1}^{m_s}\varepsilon_r^{(i)}\circ\varepsilon_s^{(j)}\right)(\xi_s-\xi_r)^{-2}
\right]}
\\&\times \exp{\left[\sum\limits_{r\neq s}\left(\sum\limits_{i=1}^{n_s}\left(-2\alpha'\right)k_r\cdot\epsilon_s^{(i)}
-\sum\limits_{i=1}^{m_s}\lambda_r\circ\varepsilon_s^{(i)}\right)(\xi_r-\xi_s)^{-1}\right]}|_{multilinear}e^{i\pi\Theta(P)}
.
\end{split}$$ After taking an appropriate phase factor out, we get $$\label{AD_2N1}
\begin{split}
\mathscr{A}_{D_2}^{(N,0)}&=\kappa^{N-1}{\left(\frac{i}{4}\right)}^{N-1}\int
\prod\limits_{i=1}^{2N}d\xi_i\frac{|\xi_a-\xi_b||\xi_b-\xi_c||\xi_c-\xi_a|}{d\xi_ad\xi_bd\xi_c}\prod\limits_{s>r}|\xi_s-\xi_r|^{\frac{\alpha'}{2}k_r\cdot
k_s}(\xi_s-\xi_r)^{\lambda_r\circ\lambda_s-q_rq_s}
\\&\times\exp{\left[\sum\limits_{s>r}\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{n_s}\left(2\alpha'\right)\epsilon_r^{(i)}\cdot\epsilon_s^{(j)}+\sum\limits_{i=1}^{m_r}\sum\limits_{j=1}^{m_s}\varepsilon_r^{(i)}\circ\varepsilon_s^{(j)}\right)(\xi_s-\xi_r)^{-2}
\right]}
\\&\times \exp{\left[\sum\limits_{r\neq s}\left(\sum\limits_{i=1}^{n_s}\left(-2\alpha'\right)k_r\cdot\epsilon_s^{(i)}
-\sum\limits_{i=1}^{m_s}\lambda_r\circ\varepsilon_s^{(i)}\right)(\xi_r-\xi_s)^{-1}\right]}|_{multilinear}e^{i\pi\Theta(P)}
,
\end{split}$$ where we have absorbed a factor $\frac{1}{2}$ into each $\epsilon$. $\Theta(P)$ is defined as $$\Theta(P)=\sum\limits_{s>r}2\alpha'k'_s\cdot
k'_r\theta(\xi_s-\xi_r),$$ where $k'^{\mu}_r=\frac{1}{2}k^{\mu}_r$ is the momentum of the open string and $$\begin{aligned}
\theta(\xi_s-\xi_r)=
\biggl\{
\begin{array}{l}
0 ( \xi_s>\xi_r) \\
1 ( \xi_s<\xi_r) \\
\end{array}.\end{aligned}$$ From and we can see amplitudes for $N$ closed strings on $D_2$ can be given by one open string tree amplitude for $2N$ open strings except for a phase factor. The phase factor is caused by taking absolute number of $(\xi_s-\xi_r)$ in $(\xi_s-\xi_r)^{\frac{\alpha'}{2}k_r\cdot k_s}$. It is used to guarantee the integrals in the right branch cut. It only depend on the the orderings of the open strings. For a certain order $P$, the phase factor decouple from the integrals. So we can break the integrals into pieces, take the multilinear terms in $\epsilon$, $\bar{\epsilon}$, $\varepsilon$ and $\bar{\varepsilon}$, replace the polarization vectors by the polarization tensors of closed strings. Then we get the relation between closed string amplitudes and partial amplitudes for open strings on $D_2$: $$\label{D2relations}
\mathscr{A}_{D_2}^{(N,0)}=\kappa^{N-1}\epsilon_{\alpha\beta}\mathscr{A}_{D_2}^{(N)\alpha\beta}=\left(\frac{i}{4}\right)^{N-1}\kappa^{N-1}\epsilon_{\alpha\beta}\sum\limits_P\mathscr{M}^{(2N)\alpha\beta}(P)e^{i\pi\Theta(P)},$$ where $\mathscr{M}$ is the open string amplitude without the coupling constant $g$, and we sum over all the orderings $P$ of the open strings.
If there are open strings on the boundary of $D_2$, we can insert the open string vertices into the amplitude . Because is already an amplitude for open strings on the real axis except for a phase factor, we just increase the number of the open strings on the boundary of $D_2$ and adjust the phase factor to make the integrals in the right branch cuts. The phase factor should be adjusted because we must consider the interactions between closed and open strings. Then we have $$\label{(N, M)D2relations}
\mathscr{A}_{D_2}^{(N,
M)}=\epsilon_{\alpha\beta\gamma}\mathscr{A}_{D_2}^{(N,
M)\alpha\beta\gamma}=\left(\frac{i}{4}\right)^{N-1}\kappa^{N-1}g^M\epsilon_{\alpha\beta\gamma}
\sum\limits_P\mathscr{M}^{(2N,M)\alpha\beta\gamma}(P)e^{i\pi\Theta'(P)},$$ where we have defined the coordinates of the left-moving open strings are $\xi_1,...,\xi_N$, those of right-moving open strings are $\xi_{1+N+M},...,\xi_{2N+M}$ and the coordinates of other open strings are $\xi_{1+N},...,\xi_{M+N}$. $$\Theta'(P)=\sum\limits_{s>r}2\alpha'k'_s\cdot
k'_r\theta'(\xi_s-\xi_r),$$ where $k'_r$ are the momentums of the open strings. If $\xi_s>\xi_r$, $\theta'(\xi_s-\xi_r)=0$, else if $\xi_s<\xi_r$ but $N<s,r<N+M+1$, $\theta'(\xi_s-\xi_r)=0$, otherwise $\theta'(\xi_s-\xi_r)=1$. This relation can also be derived by choosing the fundamental region as the upper half-plane, then repeat the similar steps in the case of $N$ closed strings on $D_2$. We can see if $M=0$, gives the relation for $N$ closed strings on $D_2$ and if $N=0$ it gives the open string tree amplitude.
By comparing the relations with KLT factorization relations , we can see, in , the left- and the right-moving sectors are independent of each other. In , they are not independent of each other. The interactions connect the two open string amplitudes into a single one. because the interactions between the two sectors are just the open string interactions, the amplitudes for $N$ closed strings then can be given by tree amplitudes for $2N$ open strings.
We can consider the relations on $D_2$ as any closed strings can be splitted into two open strings. Each open string catch half of the momentum of the closed string. Move the open strings corresponding to the two sectors of closed strings onto the boundary of $D_2$. Then an amplitude for $N$ closed strings and $2M$ open strings on $D_2$ is given by an amplitude for $N+2M$ open strings.
In , the $D_2$ amplitudes have been given by the a sum of open string partial amplitudes with $2N+M$ external legs correspondingly. We have to sum over all the orderings of the open strings in this relation. However, as in the case of $S_2$[@9], the contour treatment[@29] can reduce the number of the terms in this relation. The main points of the treatment of[@29] is there are relations among open string partial amplitudes. Then any open string partial amplitudes can be expressed in terms of a minimal basis. All the $M$-point open string partial amplitude can be expressed in terms of the minimal basis of $(M-3)!$ independent partial amplitudes. Then for the $(N, M)$ case, the amplitude can be given by $(2N+M-3)!$ open string partial amplitudes.
If we consider the interactions between open strings attached to a D$p$-brane and closed strings, we should do appropriate replacements in the right-moving sector. For example, if the external legs are gravitons, we just need to replace the momenta $\frac{1}{2}k^{\mu}_r$ corresponding to the right-moving sector by $\frac{1}{2}D^{\mu}_{\nu}\cdot k^{\nu}_r$ and replace the polarization tensor $\epsilon_{\mu\nu}$ by $\epsilon_{\mu\lambda}D^{\lambda}_{\nu}$ in the relation[@25; @26; @27]. Where $D^{\mu}_{\nu}$ is defined as
$$\left(
\begin{array}{cccccc}
1 & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\
\text{ } & \ddots & \text{ } & \text{ } & \text{ } & \text{ } \\
\text{ } & \text{ } & 1 & \text{ } & \text{ } & \text{ }\\
\text{ } & \text{ } & \text{ } & -1 & \text{ } & \text{ }\\
\text{ } & \text{ } & \text{ } & \text{ } & \ddots &\text{ }\\
\text{ } & \text{ } & \text{ } & \text{ } & \text{ } & -1
\end{array}
\right).$$
Then this relation reveals the amplitudes between $N$ closed string and $M$ open strings on a D$p$-brane can be given by $2N+M$-point open string partial amplitudes. Though in [@25; @26; @27], $(2,
0)$ amplitude and $(1, 2)$ amplitude are four-point open string amplitudes upon a certain identification between the momenta and polarizations, in general case, there is a phase factor in the relations.
In the low energy limit of an open string theory, gravitons are closed string states and gauge particles are open string states. Then in this case, the KLT factorization relations do not hold. We should use one amplitude for $2N$ gauge particles instead of the product of two amplitudes for $N$ gauge particles to give an amplitude for $N$ gravitons.
Relations between amplitudes on $RP_2$ and open string tree amplitudes {#relations on RP_2}
======================================================================
In this section, we will explore the amplitudes on $RP_2$[@1; @2; @28]. We first show the correlation functions on $RP_2$ can not be factorized by the left- and the right-moving sectors. The two sectors are connected together. Then we will give the relations between amplitude on $RP_2$ and tree amplitude for open strings.
$RP_2$ is an unoriented surface, it can be derived by identifying the diametrically opposite points on $S_2$. It can be considered as a sphere with a crosscap. With this equivalence, the waves must be reflected at the crosscap. the reflection waves of the left-moving waves are in the right-moving sector and the reflection waves of the right-moving waves are in the left-moving sector. The waves must interact with their reflection waves, thus the two sectors must interact with each other. This is similar with the case of $D_2$.
![Only the annihilation modes are reflected at the crosscap of $RP_2$.[]{data-label="fig3"}](3.eps){width="50.00000%"}
Particularly, the correlation function on $RP_2$ is given as $$\label{RP2 correlation}
\left<0\mid\mathscr V_N(\omega,
\tilde{\omega})...\mathscr{V}_1(\omega, \tilde{\omega})\mid
C\right>,$$ where $\mid C\rangle=C\mid0\rangle$ is the boundary sate for $RP_2$[@20; @21]. The bosonized boundary operator $C$ is $$C=\exp{(\sum\limits_{n=1}^\infty
(-1)^na_n^{\dagger}\cdot\tilde{a}_{n}^{\dagger})}\left|0\right>_{X}
\otimes\exp{(\sum\limits_{n=1}^\infty
(-1)^nb_n^{\dagger}\circ\tilde{b}_{n}^{\dagger})}\left|0\right>_{\phi}
\otimes\exp{(\sum\limits_{n=1}^\infty
(-1)^nc_n^{\dagger}\tilde{c}_{n}^{\dagger})}\left|0\right>_{\phi_6}.$$ In this case we can see, the image point of $\omega$ is $-\bar{\omega}+i\pi$. When we move $C$ to the left of a vertex operator it commute with the creation modes and the zero modes of the vertex operator. It does not commute with the annihilation modes $\mathscr{V}^{(-)}_L(\omega)$ and $\tilde{\mathscr{V}^{(-)}_R}(\bar{\omega})$. This means only the annihilation modes can be reflected at the crosscap(see fig\[fig3\]). After moving the boundary operator to the left of the annihilation modes $\mathscr{V}^{(-)}_L(\omega)$ and $\tilde{\mathscr{V}^{(-)}_R}(\bar{\omega})$, the images $\tilde{\mathscr{V}}^{(+)}_L(-\omega-i\pi)$ and $\mathscr{V}^{(+)}_R(-\bar{\omega}+i\pi)$ are created respectively. We move the boundary operator to the left of all the vertex operators. Then use the creation operators in the boundary operator to annihilate the state $\langle0\mid$. The correlation becomes $$\label{correlation function on RP2}
\begin{split}
&\left<\mathscr{V}^{(+)}_L(\omega_N)\mathscr{V}^{(-)}_L(\omega_N)\mathscr{V}^{(+)}_R(-\bar{\omega}_N+i\pi)...\mathscr{V}^{(+)}_L(\omega_1)\mathscr{V}^{(-)}_L(\omega_1)\mathscr{V}^{(+)}_R(-\bar{\omega}_1+i\pi)\right>
\\\times &\left<\tilde{\mathscr{V}}^{(+)}_R(\bar{\omega}_N)\tilde{\mathscr{V}}^{(-)}_R(\bar{\omega}_N)\tilde{\mathscr{V}}^{(+)}_L(-\omega_N-i\pi)...\tilde{\mathscr{V}}^{(+)}_R(\bar{\omega}_1)\tilde{\mathscr{V}}^{(-)}_R(\bar{\omega}_1)\tilde{\mathscr{V}}^{(+)}_L(-\omega_1-i\pi)\right>
\\\times &\left<\mathscr{V}_0(\omega_N, \tilde{\omega}_N)...\mathscr{V}_0(\omega_1,
\tilde{\omega}_1)\right>.
\end{split}$$ As in the case of $D_2$, the first correlation function in only contain $a$, $b$, $c$ and $a^{\dag}$, $b^{\dag}$, $c^{\dag}$. When we move the left-moving modes of a vertex operator $\mathscr{V}^{(-)}_L(\omega_r)$ to the right of the operator $\mathscr{V}^{(+)}_L(\omega_s)$, we get the interaction in the left-moving sector. When we move $\mathscr{V}^{(-)}_L(\omega_r)$ to the right of $\mathscr{V}^{(+)}_R(-\bar{\omega}_s+i\pi)$, we get the interaction between the left- and the right-moving sectors. In the same way, the second correlation function in gives the interactions in the right-moving sector and those between the two sectors. Thus the correlation function can not be factorized by the two sectors. Interactions connect the two sectors together.
Now we consider the amplitude for $N$ closed strings on $RP_2$. We calculate the correlation function, integral over the fundamental region and divide the integrals by the volume of the CKG[@1; @2; @22; @23] on $RP_2$. As we have done in the case of $D_2$, we also extend the integral region to the complex pane, rotate the $y$ integrals to the real axis and redefine the integral variables. The amplitude for $N$ closed strings on $RP_2$ can be given as $$\begin{aligned}
\label{ARP2superstring}
\nonumber\mathscr{A}_{RP_2}^{(N)}&=\kappa^{N-1}\left(\frac{1}{2}\right)^{N-1}\int
\prod\limits_{i=1}^Nd\xi_id\eta_i\frac{|1+\xi_o\eta_o|^2}{2\pi
d\xi_o\eta_o}
\\&\nonumber\times\prod\limits_{s>r}(\xi_s-\xi_r)^{\frac{\alpha'}{2}k_r\cdot
k_s+\lambda_r\circ\lambda_s-q_rq_s}(\eta_r-\eta_s)^{\frac{\alpha'}{2}k_r\cdot
k_s+\tilde{\lambda}_r\circ\tilde{\lambda}_s-\tilde{q}_r\tilde{q}_s}
\prod\limits_{r,
s}(1+(\xi_r\eta_s)^{-1})^{\frac{\alpha'}{2}k_r\cdot
k_s+\lambda_r\circ\tilde{\lambda}_s-q_r\tilde{q}_s}
\\&\nonumber\times \exp{\sum\limits_{r=1}^N\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{\tilde{n}_s}\left(-\frac{\alpha'}{2}\right)\epsilon_r^{i}\cdot\bar{\epsilon}_r^{j}
-\sum\limits_{i=1}^{m_r}\sum\limits_{j=1}^{\tilde{m}_s}\varepsilon_r^{i}\circ\bar{\varepsilon}_r^{j}\right)(1+\xi_r\eta_r)^{-2}}
\\&\times \exp{\sum\limits_{s>r}\left[\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{n_s}\left(-\frac{\alpha'}{2}\right)\bar{\epsilon}_r^{(i)}\cdot\epsilon_s^{(j)}
-\sum\limits_{i=1}^{\tilde{m}_r}\sum\limits_{j=1}^{m_s}\bar{\varepsilon}_r^{(i)}\circ\varepsilon_s^{(j)}\right)(1+\eta_r\xi_s)^{-2}+c.c.\right]}
\\&\nonumber\times\exp{\left[-\sum\limits_{s>r}\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{n_s}\left(-\frac{\alpha'}{2}\right)\epsilon_r^{(i)}\cdot\epsilon_s^{(j)}-\sum\limits_{i=1}^{m_r}\sum\limits_{j=1}^{m_s}\varepsilon_r^{(i)}\circ\varepsilon_s^{(j)}\right)(\xi_s-\xi_r)^{-2}+c.c.
\right]}
\\&\nonumber\times \exp{\sum\limits_{r\neq s}\left[\left(\sum\limits_{i=1}^{n_s}\left(-\frac{\alpha'}{2}\right)k_r\cdot\epsilon_s^{(i)}
-\sum\limits_{i=1}^{m_s}\lambda_r\circ\varepsilon_s^{(i)}\right)((\xi_r-\xi_s)^{-1}+(-\eta_r^{-1}-\xi_s)^{-1})+c.c.\right]}
\\&\nonumber\times
\exp{\sum\limits_{r=1}^N\left[\left(\left(-\frac{\alpha'}{2}\right)k_r\cdot\sum\limits_{i=1}^{n_r}\epsilon_r^{(i)}
-\lambda_r\circ\sum\limits_{i=1}^{m_r}\varepsilon_r^{(i)}\right)((-\eta_r^{-1}-\xi_r)^{-1}+{\xi_r}^{-1})+c.c.\right]}|_{multilinear}.\end{aligned}$$ Then the amplitude has been given by real integrals. The interactions in one sector are the open string interactions. The interaction between left- and right-moving sectors can be considered as interactions between open strings inserted at $\xi_r$ and $(-\eta_s)^{-1}$. $\frac{1}{-\bar{\eta_s}}$ can be considered as a time reverse and a twist in the right-moving sector. Then the interactions between left- and right-moving sectors can be regarded as interactions between left- and right-moving open strings with a time reverse and a twist in the right-moving sector(see fig\[fig1\].(c)).
From fig. \[fig1\].(c), we can see, if we twist the right-moving sector and reverse the time in the right-moving sector, we will get an open string tree amplitude. In fact, we can replace all the $\eta_r$ by $-\frac{1}{\eta_r}$. Then by using the mass-shell condition, the interactions between the two different sectors as well as in one sector become the interactions between open strings. Redefine the variables in the right-moving sector by Eq.. The amplitude on $RP_2$ then becomes $$\label{ARP_2N1}
\begin{split}
\mathscr{A}_{RP_2}^{(N)}&=-{\left(\frac{i}{4}\right)}^{N-1}\kappa^{N-1}\int
\prod\limits_{i=1}^{2N}d\xi_i\frac{|\xi_a-\xi_b||\xi_b-\xi_c||\xi_c-\xi_a|}{d\xi_ad\xi_bd\xi_c}\prod\limits_{s>r}|\xi_s-\xi_r|^{\frac{\alpha'}{2}k_r\cdot
k_s}(\xi_s-\xi_r)^{\lambda_r\circ\lambda_s-q_rq_s}
\\&\times\exp{\left[\sum\limits_{s>r}\left(\sum\limits_{i=1}^{n_r}\sum\limits_{j=1}^{n_s}\left(2\alpha'\right)\epsilon_r^{(i)}\cdot\epsilon_s^{(j)}+\sum\limits_{i=1}^{m_r}\sum\limits_{j=1}^{m_s}\varepsilon_r^{(i)}\circ\varepsilon_s^{(j)}\right)(\xi_s-\xi_r)^{-2}
\right]}
\\&\times \exp{\left[\sum\limits_{r\neq s}\left(\sum\limits_{i=1}^{n_s}\left(-2\alpha'\right)k_r\cdot\epsilon_s^{(i)}
-\sum\limits_{i=1}^{m_s}\lambda_r\circ\varepsilon_s^{(i)}\right)(\xi_r-\xi_s)^{-1}\right]}|_{multilinear}e^{i\pi\Theta(P)}
.
\end{split}$$ This amplitude is different from $D_2$ amplitude by a factor $-1$. It is caused by the difference between the measure of the CKG on $RP_2$ and $D_2$. When we change the topology, this $-1$ appears. The phase factor only depends on the ordering of the open strings. We can break the integrals into pieces as in the case of $D_2$ and keep the multilinear terms of the polarization tensers. Then Eq. becomes $$\label{RP2relations}
\mathscr{A}_{RP_2}^{(N)}=\epsilon_{\alpha\beta}\mathscr{A}_{RP_2}^{(N)\alpha\beta}=-\left(\frac{i}{4}\right)^{N-1}\kappa^{N-1}\epsilon_{\alpha\beta}\sum\limits_P\mathscr{M}^{(2N)\alpha\beta}(P)e^{i\pi\Theta(P)}.$$
As in the case of $D_2$, KLT factorization relations do not hold on $RP_2$. The left- and the right-moving sectors are not independent of each other again. The interactions between the two sectors connect them into a single sector. Since the interactions between the two sectors are just the those between open strings, the two open string tree amplitude in the case of $S_2$ are connected into one amplitude for open strings. In the relation, we also sum over all the orderings of the external legs of the open strings. By using the same method in [@29], the relations on $RP_2$ for $N$ closed strings can be reduced to $(2N-3)!$ terms.
From the relations and we can see, the amplitudes on $D_2$ and $RP_2$ with same external closed string states are equal except for a factor $-1$. In fact, after we transform the complex variables into real ones, the image of a point $\xi_r$ in the left-moving sector becomes $\frac{1}{\eta_r}$ on $D_2$ and $-\frac{1}{\eta_r}$ on $RP_2$. The minus means a twist in the right-moving sector. After this twist, the amplitude on $RP_2$ becomes that on $D_2$ except for a factor $-1$. Then if we consider a theory containing both $D_2$ and $RP_2$, the amplitudes with same external states cancel out. However, if we consider T-duality, the interactions on $D_2$ becomes interactions between closed strings and D-brane, while the interactions on $RP_2$ becomes interactions between closed strings and O-plane. As we have seen in section \[relations on D\_2\], we should make appropriate replacement on the momenta and polarizations in the right-moving sector to give the relations in $D_2$ case. Under the T-duality, we also need to replace the vertex operators on $RP_2$ by new ones[@30]. We take the massless NS-NS vertex operator as an example. The vertex operator after T-duality becomes $$\label{RP_2 vertex}
\begin{split}
&\mathscr{V}^{RP_2}(\epsilon, k, z, \bar{z}) \\&=\frac{1}{2}\left(
\epsilon_{\mu\nu}:\mathscr{V}^\mu_{\alpha}(k,
z)::\tilde{\mathscr{V}}^{\mu}_{\beta}(k,
\bar{z}):+(D\cdot\epsilon^T\cdot
D)_{\mu\nu}:\mathscr{V}^{\mu}_{\alpha}(k\cdot D,
z)::\tilde{\mathscr{V}}^{\nu}_{\beta}(k\cdot D, \bar{z}):\right).
\end{split}$$ Here $\mathscr{V}^{\mu}_{\alpha}(p, \epsilon)$ in $0$ and $-1$ picture are $$\begin{split}
\mathscr{V}^{\mu}_{-1}(k, z)&=e^{-\phi(z)}\psi^{\mu}(z)e^{ik\cdot
X(z)}
\\\mathscr{V}^{\mu}_0(k, z)&=(\partial X^{\mu}(z)+ik\cdot
\psi(z)\psi^{\mu}(z))e^{ik\cdot X}.
\end{split}$$ The second term in can also be given by replacing $\epsilon_{\mu\nu}$ and $k^{\mu}$ in the original vertex by $(D\cdot\epsilon^T\cdot D)_{\mu\nu}$ and $(k\cdot D)^{\mu}$ respectively. Now we consider the amplitudes for $N$ NS-NS strings. Each vertex operator have two terms, each term can be considered as a vertex operator on $RP_2$ under appropriate replacement. The amplitude then is given by $2^N$ terms, each term can be obtained from the amplitude before T-duality by appropriate replacements. Then each term can be given by partial amplitudes for $2N$ open strings again. So the $RP_2$ relation gives the amplitudes for closed strings scattering from an O-plane by open string amplitudes. Generally, under the T-duality, the $D_2$ amplitudes can not be canceled by the $RP_2$ amplitudes[@30].
In the low energy limit of an unoriented string theory, the amplitudes for closed strings on $RP_2$ contribute to the amplitudes for gravitons. Then the KLT factorization relations do not hold in this case as in the case of $D_2$. The amplitudes for $N$ gravitons can not be factorized by two amplitudes for $N$ gauge particles. They can be given by an amplitude for $2N$ gauge particles.
Conclusion
==========
In this paper, we investigated the relations between closed and open strings on $D_2$ and $RP_2$. We have shown that the KLT factorization relations do not hold for these two topologies. The closed string amplitudes can not be factorized by tree amplitudes for left- and right-moving open strings. However, the two sectors are connected into a single sector. We can give the amplitudes with closed strings in these two cases by amplitudes in this single sector. The terms in the relations on $D_2$ and $RP_2$ can be reduced by contour deformations.
Under the T-duality, the relations on $D_2$ and $RP_2$ give the amplitudes between closed strings scattering from D-brane and O-plane respectively by open string partial amplitudes.
In the low energy limits of these two cases, we can not use KLT relations to factorize amplitudes for gravitons into products of two amplitudes for gauge particles. Interactions between the “left-” and the “right-” moving gauge fields connect the two amplitudes into one. Then an graviton amplitude in these two cases can be given by one amplitude for both left- and right-moving gauge particles.
The relations for other topologies have not been given. However, we expect there are also some relations between closed and open string amplitudes. If there are more boundaries and crosscaps on the world-sheet, the boundaries and the crosscaps also connect left- and the right-moving sectors, then in these cases, KLT factorization relations do not hold.
Acknowledgement {#acknowledgement .unnumbered}
===============
We would like to thank C.Cao, J.L.Li, Y.Q.Wang and Y.Xiao for helpful discussions. We would like to thank the referee for many helpful suggestions. The work is supported in part by the NNSF of China Grant No. 90503009, No. 10775116, and 973 Program Grant No. 2005CB724508.
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[^1]: We use $\epsilon_{\alpha\beta}$ to denote all the polarization tensors for convenience. $\alpha$ correspond to the left indices and $\beta$ correspond to the right indices. If there are open strings on the boundary of $D_2$, we use $\epsilon_{\alpha\beta\gamma}$ to denote all the polarization tensors for convinience. $\gamma$ correspond to the indices of open strings.
[^2]: Here $\phi_i(z) (i=1...5)$ and $\tilde{\phi}_i(\bar{z}) (i=1...5)$ are bosonic fields. They are used to bosonize holomorphic and antiholomorphic fermionic fields and spinor fields. $\phi_6(z)$ and $\tilde{\phi}_6(\bar{z})$ are used to bosonize the holomorphic and antiholomorphic superconformal ghost respectively. $\epsilon$ and $\bar{\epsilon}$ correspond to the components of polarization tensors contracting with bosonic fields $\partial X$ and $\bar{\partial}X$ respectively. $\varepsilon$ and $\bar{\varepsilon}$ correspond to the components contracting with $\partial\phi$ and $\bar{\partial}\tilde{\phi}$ respectively. We pick up the pieces multilinear in $\epsilon$, $\bar{\epsilon}$ and $\varepsilon$, $\bar{\varepsilon}$, then replace these polarization vectors by the polarization tensor of the vertex operator. $\lambda_i'=i\lambda_i$ and $\tilde{\lambda}_i'=i\tilde{\lambda}_i$ (i=1...5) are vectors in the weight lattice[@18; @19] of the left- and right-moving sectors respectively. $q$ and $\tilde{q}$ are the $\gamma$ ghost number in the left- and right-moving sectors respectively. We use $\circ$ to denote the inner product in the five dimensional weight space and use $\cdot$ to denote the inner product in the space-time.\
Physical vertices containing higher derivatives can be transformed into the vertices with only first derivatives. In fact we can do partial integrals to reduce the order of the derivatives. After the integrals on the world-sheet, the surface terms turn to zero. Redefine the polarization tensor, the vertices then turn to those only contain first derivatives.
[^3]: Here, we only consider Neumann boundary condition for convenience. The case with Dirichlet boundary conditions has similar relations.
[^4]: To divide the amplitude by the volume of CKG, we can fix three real coordinate. We can also fix two real coordinate or one complex coordinate, then divide the amplitude by volume of the one-parameter subgroup left. The two method are equivalence. Here, we use the second method to fix $z_1=z_o$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present numerical results for a chemical reaction of colloidal particles which are transported by a laminar fluid and are focused by periodic obstacles in such a way that the two components are well mixed and consequently the chemical reaction is speeded up. The roles of the various system parameters (diffusion coefficients, reaction rate, obstacles sizes) are studied. We show that focusing speeds up the reaction from the diffusion limited rate $\sim t^{-1/2}$ to very close to the perfect mixing rate, $\sim t^{-1}$.'
author:
- 'A. M. Lacasta$^1$, L. Ramírez–Piscina$^1$, J. M. Sancho$^2$ and K. Lindenberg$^3$'
title: Speeding chemical reactions by focusing
---
Introduction
============
The motion of colloidal particles on modulated surfaces has attracted a great deal of attention in the past decade [@macdonald03; @grier03; @korda02; @huang04; @Morton; @Xuan]. The interest in this subject has mainly been directed at sorting phenomena, and a considerable portion of the work has been experimental. The work has focused on mixtures of particles which are sorted into separate streams of different species when the mixture is transported under laminar conditions along a structured or modulated medium with periodic obstacles or traps. In this scenario it is possible to control the transport of materials such as DNA fragments or functionalized biological colloidal particles. The modulated surfaces are specifically designed to present periodic arrays of traps [@grier03; @korda02] or microfabricated obstacles [@huang04] among other configurations. This technique can be applied not only to solid spherical particles but also to other objects such as cells, proteins, DNA, and droplets in inmimiscible fluids [@Champagne]. Theoretical studies complemented with stochastic simulations have received considerable attention [@katja2; @ana; @James]. Other sorting methods based on inertia and hydrodynamics have also been explored [@DiCarlo]. The sorting phenomenon consists of a lateral or orthogonal displacement of the particles with respect to the driving force or velocity direction of the fluid mixture. The deviation of the particles in a mixture from the direction of flow of the mixture depends on some property or group of properties of the particles such as, for instance, size, mass, or charge, causing the particles with different values of these properties to flow in different directions.
The same principle can be used to achieve the converse effect, namely, to focus particles coming from different directions and mix them if the modulated structure of obstacles or traps is prepared accordingly [@Morton]. Focusing of particles is useful in a number of different scenarios such as counting, detecting, and mixing [@Xuan]. This property has special relevance in the laminar regime, where slow molecular diffusion makes it difficult to concentrate and mix particles.
Our particular focus in this work lies in using this methodology to mix reactants in order to speed their reaction. It is our objective to show that using an appropriate modulated or structured surface of obstacles one can concentrate two reactants in a very small domain, thus favoring their chemical reaction. Our main result is that reactants that arise from non-homogeneous distributions can be efficiently mixed and as a result are able to reach the classical law of mass action reaction regime characterized by the reactant concentration decay law $ \sim t^{-1}$ much sooner than they would in the absence of a mixing mechanism. We present results on the efficiency of some obstacle geometries toward this purpose. We discuss the roles of the different control parameters and of the densities, the diffusion coefficients of the different species, the reaction rates, and the particle and obstacle sizes.
Our presentation proceeds as follows. First in Sec. \[sec2\] we begin with the description of the continuous dynamical scenario and present the associated dynamical equations which contain diffusion, advection and reaction terms. In Sec. \[sec3\] we present numerical results from the simulation of the equations and thereby analyze the roles of the different parameters. Finally we close with some conclusions and perspectives in Sec. \[sec4\].
Dynamical model {#sec2}
===============
The theoretical scenario is the advective reaction–diffusion model of chemical kinetics corresponding to the simple irreversible reaction $A+B \rightarrow 0$. The dynamical equations for the two reactants are
\[eq:1\] $$\begin{aligned}
\label{eq:1a}
\frac{\partial}{\partial t} c_a(x,y;t) & = -\nabla {\mathbf J}_a(x,y;t) - k c_a c_b, \\
\frac{\partial}{\partial t} c_b(x,y;t) & = -\nabla {\mathbf J}_b(x,y;t) - k c_a c_b,
\label{eq:1b} \end{aligned}$$
where $c_a$ and $c_b$ are the time dependent local concentrations of the reactants $A$ and $B$, $k$ is the reaction rate constant, and ${\bf J}_a$, ${\bf J}_b$ are the fluxes of the reactants. The latter are given by
$$\begin{aligned}
{\mathbf J}_a(x,y;t) &= c_a(x,y;t) {\mathbf v}(x,y;t)
-D_a \nabla c_a
-U_0 c_a\nabla U, \\
{\mathbf J}_b(x,y;t) &= c_b(x,y;t) {\mathbf v}(x,y;t)
-D_b \nabla c_b
-U_0 c_b\nabla U.\end{aligned}$$
Here $D_a$ and $D_b$ are the diffusion coeficients, $U_0 U (x/\lambda, y/\lambda)$ is the modulated potential interaction due to obstacles, and ${\mathbf v}( x/\lambda, y/\lambda)$ is the local velocity responsible for the advective flux, which we assume to be a Hele-Shaw flow (that is, a flow between two very close parallel plates). We have explicitly extracted the amplitude $U_0$ of the potential so that $U(x/\lambda,y/\lambda)$ is the potential of unit amplitude. This potential is modeled by placing a circular tower at each obstacle, which changes from unit value at the center of the disk defining the base of the obstacle to a zero value outside the range of the interaction. Specifically, it is modeled by the expression $$U({\bf r}) =
\sum_{k=1}^N \frac{1}{2}\left(1 - \tanh{\frac{\left|{\bf r}-{\bf R}_k \right| - d}{\delta}} \right),
%=
%\sum_{k=1}^N \left(1+\exp{2\frac{\left|{\bf r}-{\bf R}_k \right| - d}{\delta}}\right) ^{-1},$$ where ${\bf R}_k$, $k=1 \dots N$, are the positions of the centers of the $N$ obstacles of base radius $a$, $d>a$ is the radius of the interaction, and $\delta$ is the (small) scale that characterizes a substantial change in the value of the potential. To take into account the finite size of the advected particles in a model with a continuous concentration field we have introduced an interaction potential with a radius $d$ larger than the obstacle radius $a$. The distance $d-a$ then represents the particle radius (see Fig. \[lines\]). The potential range $d$ corresponds to the minimum distance between the centers of the colloidal particle and the obstacle, and hence is the sum of their radii. The flow field is obtained by solving the Laplace equation with boundary conditions that reflect the presence of the $N$ circular obstacles of radius $a$, but neglecting any effects that the advected particles might have on the flow. Even this is a rather arduous task that we have moved to the Appendix.
Equations (\[eq:1\]) can be simplified with a change to the new variables $\tau$, $x'$, and $y'$, $$t = t_0 \tau, \qquad x= \lambda x', \qquad y= \lambda y',$$ where $\lambda$ and $t_0$ are characteristic length and time scales. This transforms Eqs. (\[eq:1\]) to
$$\begin{aligned}
\frac{\partial}{\partial \tau} c_a(x',y',\tau)& = {\hat D}_a \nabla^2 c_a +{\hat U}_0 \nabla (c_a\nabla U) + \nabla ({\mathbf v} c_a) - {\hat k} c_a c_b, \\
\frac{\partial}{\partial \tau} c_b(x',y',\tau)& = {\hat D}_b \nabla^2 c_b +{\hat U}_0 \nabla (c_b\nabla U) + \nabla ({\mathbf v} c_b) - {\hat k} c_a c_b. \end{aligned}$$
\[eq:1-b\]
The dimensionless parameters are given by $${\hat D_i} = \frac{D_i t_0}{\lambda^2}, \qquad {\hat U}_0= \frac{U_0 t_0}{\lambda^2}, \qquad
{\hat k} = \frac{k t_0}{\lambda^3},$$ and the concentrations and derivative operators are also dimensionless. We fix the parameters ${\hat D}_i=0.01$ and ${\hat U}_0=0.01$ throughout the paper. From now on for simplicity of notation we drop the primes on $x'$ and $y'$ and alert the reader not to confuse $x$ and $y$ as used henceforth with the original variables.
The equations are simulated on a two-dimensional lattice of $N_x=1000, N_y=300$ cell centers and cell dimensions (dimensionless quantities) $\Delta x= \Delta y= 0.05$, which corresponds to system dimensions $l_x=50$ and $l_y=15$. Our integration time step is $\Delta \tau=5 \times10^{-4}$. The dynamical evolution is reported every $5$ time units up to a final time $\tau =60$.
Figure \[lines\] shows the role of one obstacle, the flow lines and the finite size of a colloidal particle. When the particle following a flux line is close to the obstacle its trajectory changes to a different flow line pointing away from the obstacle. The result is a lateral deviation of the particle from the initial flow line as it circumnavigates the obstacle. In the presence of several obstacles, and depending on their specific spatial distribution, these deviations can favor focusing and hence accelerate the reaction. To check this hypothesis we have employed two obstacle patterns, one with 293 obstacles in a tilted periodic pattern (PP), as shown in Fig. \[1evol\_I\], and a second one with the same number of obstacles but distributed randomly over the same area (random pattern, RP). We have also performed some simulations with no pattern (NP). We have mainly used obstacles of radius $a=0.15$ and an interaction potential radius $d=0.25$, which corresponds to a radius $d-a=0.10$ of reacting particles. We have also used the values $a=0.10$, $0.05$ and $d=0.20$ in some cases in order to study the dependence of the results on these parameters. We have also performed some simulations in which the flow is not affected by the obstacles and thus the particles move at a constant velocity, [*i.e.*]{}, ${\bf v} =$ constant ($a=0)$, see below. This is done because in previous calculations concerning flows of particle mixtures over surfaces with obstacles we did not take the effect of the obstacles on the flow field into account [@katja2; @ana; @James]. Here we have the opportunity to assess the importance of doing so (see below).
\
We are interested in a number of observables along the flow direction, and not in a direction perpendicular to the flow, which we integrate over. The observables we focus on are the local reaction rate $R(x)$, $$R(x) = k \int dy \, c_a(x, y) c_b(x, y),
\label{Rx}$$ and the total flux as a function of the position $x$, $$J(x,t) = \int dy \left[ J_{x,a}(x,y, t) + J_{x,b}(x, y;t) \right].$$ From this quantity the reaction efficiency at a long time (here taken as $\tau=60$) is evaluated comparing the fluxes of non reacted particles at two points, $x={0.5}$ and $x={40}$, $$\eta(k) = 1 - \frac{J(x=40)}{J(x=0.5)}.
\label{flux}$$
In Fig. \[1evol\_I\] we present the two-dimensional landscape in which the advection, diffusion, and reaction of the two components $A$ and $B$ takes place. One can see the periodic and tilted structure of the circular obstacles. Figure \[2evol\_I\] (top) shows the distribution of both concentrations at time $\tau=60$. It is clear that the obstacles focus the concentrations toward the center line, where most of the reaction process takes place. This situation should be compared with the case of randomly distributed obstacles, Fig. \[2evol\_I\] (bottom).
Analysis of numerical results {#sec3}
=============================
In Fig. \[prof\_k0\] we present initial and final concentration profiles. The left column shows the effect of obstacles on the flow of reactants in the absence of a reaction (${\hat k}=0$). From top to bottom we see the reactant flow patterns when the obstacles are placed in the tilted periodic pattern (PP), in the random placement (RP), and without obstacles (NP). We clearly see the strong focusing effect of the PP geometry compared to the other cases. In the right column we see the same three cases but now the two species are allowed to react with rate coefficient ${\hat k}=0.2$. The reaction clearly proceeds much more rapidly when the reactants are focused by the obstacles. The reaction in the presence of random obstacles and of no obstacles occurs more slowly and at comparable speeds in the two cases, the former slightly more rapidly than the latter. In Fig. \[R\] the local reaction rate $R(x)$ given in Eq. (\[Rx\]) and the flux $J(x)$ of Eq. (\[flux\]) in the steady state are plotted for the focusing geometry (PP) along the flow direction. We observe repeated positions at which the reaction is more efficient. These points correspond to rows where the innermost focusing obstacles are closest together. This result opens the possibility of alternative patterns with more active reaction domains. The flux is shown for the focusing obstacle geometry for a number of rate coefficients and is seen to be a monotonically decaying function of position. For the focusing obstacle geometry with ${\hat k} =0.2$, for instance, we calculate a reaction efficiency of $\sim 85 \%$, a fairly high value for such a relatively small system. In Fig. \[efficiency\] we explore the dependence of the efficiency on the reaction parameter ${\hat k}$. We see that the efficiency comes close to a maximum value around ${\hat k}=0.2$, with little improvement above that. This almost-independence of the rate coefficient is an interesting unanticipated feature.
\
\
\
It is informative to compare this efficiency with that obtained in the most effective case, that is, that of perfect (totally homogeneous) mixing. In this case, assuming equal initial concentrations $c_a=c_b=c$. the reaction equation is $$\frac{ d c}{d \tau} = -{\hat k} c^2.$$ Its solution is $$\frac{c(\tau)}{c(0)} = \frac{1}{1 + c(0) {\hat k} \tau }.
\label{pmixing}$$
We choose the comparison time $\tau=40$. If we further choose $c(0)=1$ and ${\hat k}=0.25$, we find that $c(40)/c(0) = 1/11$ which gives an efficiency $\eta =0.91$. This an upper bound for this setup. The focusing configuration thus leads to almost perfect mixing.
We can use this result to obtain an analytic estimate for the efficiency. If we take the velocity of the flowing components $\sim 1$ we can substitute time for space, $\tau=x/v$. We expect the flux to have a functional form similar to that of the concentration in Eq. (\[pmixing\]), and thus we propose the following expresion for the efficiency at $x_F$: $$\eta(x_F,{\hat k})= \frac{a {\hat k} x_F}{1+ b {\hat k} x_F },
\label{fit}$$ with parameters $a$ and $b$ to be fitted. A nonlinear fit to our numerical data yields $a=1.085$ and $b=1.195$ (PP), $a=0.358$ and $b=0.788$ (RP) and $a=0.105$ and $b=0.325$ (NP). In Fig. \[efficiency\] we have plotted this function for $x_F=40$. This numerical result implies that most of the reacted matter was in the perfect mixing regime characterized by the decay $\sim \tau^{-1}$. This is an unexpected result because of the inhomogeneous concentrations in this system. It tells us that the reaction is dominated by the small domains near the center of the array where there is quasi-perfect mixing, as is seen in Fig. \[2evol\_I\].
To complete our analysis, we will explore the effects on these results of varying the diffusion coefficient (assumed to be the same for both species), the focusing pattern geometry, and the sizes of obstacles and particles. In the left panel of Fig. \[diff\] we present the effect of the diffusion parameter. We indicate the focusing domain for the case of our focusing obstacle pattern by dashed lines, that is, the focusing takes place within this domain. We see that inside this region the reaction is dominated by focusing, but outside of this region the reaction is dominated by diffusion.
In the right panel of Fig. \[diff\] we present three fluxes that correspond to different geometries. We see that the flux with the focusing pattern decays faster than the random pattern which is also more effective than no pattern at all.
\
Finally we discuss the effect of the parameters $d$ (radius of potential interaction) and $a$ (radius of the obstacles) on the reaction efficiency. From Fig. \[radius\] we can see that the stronger effect comes from the change in the parameter $d$. A comparison of pairs of curves with a common value of $a$ shows that the reactant flux is strongly reduced by increasing $d$ from 0.20 to 0.25. This reduction in the flux implies an increment in the efficiency (see inset of Fig. \[radius\]), as follows. Returning to Fig. \[lines\], we see that larger $d$ means that more stream lines of the liquid flow are affected by the potential, and as a consequence more particles traveling along the stream lines are deviated. Moreover, the deviation of each particle will be greater, that is, particles are moved to further stream lines. An increase of $d$ then clearly produces a larger focussing effect of the reacting particles, and as a consequence the reaction is enhanced. Note that such a change in the parameter $d$ with fixed $a$ corresponds to a change in the particle radius $d-a$ (a change in the reactants) while maintaining the same pattern of obstacles.
Next we fix the parameter $d$ and explore the role of the parameter $a$. In Fig. \[radius\] we see that an increase in $a$ reduces the efficiency (see inset). The explanation again lies in the fact that the obstacles lead to a deformation of the stream lines (see Fig. \[lines\]), and this effect is stronger for larger $a$. With a stronger deformation the stream lines are displaced further from the obstacles, and as a result more particles can escape from the action of the interaction potential, whose range $d$ is now held fixed. As a consequence the increase of $a$ at fixed $d$ reduces the focussing effect of the pattern and hence the reaction efficiency decreases.
Perhaps most interesting from an experimental point of view is to analyze the effect of varying both $a$ and $d$ while maintaining their difference constant. This corresponds to changing the dimensions of the obstacles, easily modified in an experiment, while keeping the particle radius $d-a$ constant. This would be the scenario in which one attempts to optimize the reaction process for given reactive species. In Fig. \[radius\] we can compare the case $d=0.20, a=0.10$ with the case $d=0.25, a=0.15$, which corresponds to changing the obstacle size with a common particle radius of $d-a=0.10$. Results in the figure show that the increase of obstacle size in this example clearly reduces the reactant flux and hence improves the reaction efficiency.
As a last test, we have compared these results to those of approximation in which the deformation of the fluid flow is neglected, [*i.e.*]{}, in which the flow velocity is taken to be constant and only the scale $d$ of the interaction potential is taken into account. In Fig. \[lines\] this situation would correspond to straight, horizontal stream lines. This approximation was used in our previous work on particle sorting [@katja2; @ana; @James], and in our present scheme corresponds to the limit $a=0$. In this situation the stream lines are not deformed and hence a larger number of particles are deviated by the potential. In Fig. \[radius\] we see the expected result, namely, that this approximation overestimates the reaction efficiency when compared to a finite $a$ value (for a given value of $d$). For the smallest values of $a$ (obstacles smaller than the particles) this approximation does not qualitatively change the focussing scenario, but as $a$ increases the importance of including the effects of advection clearly increases.
Conclusions {#sec4}
===========
Obstacles placed in carefully selected geometrical patterns have been used in the past to effectively separate colloidal mixtures that are caused to flow over a surface containing such obstacles. In this paper we have explored the converse, namely, the possibility of speeding up the mixing of components that undergo advective diffusion over a surface containing carefully situated obstacles. We have illustrated the effects of this mixing by considering the reaction of two species and comparing the reaction rates when the species are allowed to mix by ordinary advective diffusion in the absence and presence of these obstacles. We have shown that a periodic pattern of tilted obstacles, as opposed to a random placement, is able to effectively focus the streams of reactive species. We have furthermore shown that this focusing mechanism leads rather rapidly to reaction rates comparable to those obtained with perfect mixing. We have studied the dependence of the reaction efficiency on the different parameters of the problem. We interpret the focusing mechanism as a consequence of the finite size of the reacting particles. These results could be useful for the design and interpretation of new experiments.
This work is supported by Ministerio de Economía y Competitividad (Spain) through project FIS2012-37655, by the Generalitat de Catalunya through project 2009SGR-878, and by the US National Science Foundation through Grant No. PHY-0855471.
Numerical solution of the Hele-Shaw problem with circular obstacles
===================================================================
We will consider a fluid contained between two parallel plates, separated by a gap $d$ (Hele-Shaw cell). In the limit of small $d$ the velocity field $\mathbf u$ can be considered as two-dimensional, and is given by $${\mathbf u} = -\frac{d^2}{12\mu} \nabla p,
\label{eq-darcy}$$ which is identical to the Darcy’s law for the flow in a porous medium. $\mu$ is the viscosity of the fluid and $p$ is the pressure, which satisfies $\nabla^2 p =
0$. We place $N$ circular obstacles or disks of radius $a$ centered at positions ${\mathbf R}_k$, $k=1\dots N$. The velocity of the fluid far from the obstacles is ${\mathbf u}_\infty$. At the rigid boundaries the velocity satisfies the condition of zero normal component, but not the no-slip boundary condition. The stream lines passing the obstacles are identical to those of a two dimensional inviscid fluid with the same geometry [@batchelor].
The velocity field can be written in terms of the velocity potential $\phi$ as $${\mathbf u} = \nabla \phi, \label{eq-grad}$$ which obeys the Laplace equation $$\nabla^2 \phi = 0. \label{eq-laplace}$$
The general solution of the 2D Laplace equation (\[eq-laplace\]) can be written in polar coordinates as $$\phi(r,\theta) = \sum_{\lambda=1}^\infty
\left(a_{\lambda 1} \left(\frac{r}{a} \right)^\lambda +
a_{\lambda 2}\left(\frac{r}{a} \right)^{-\lambda} \right)
\left(b_{\lambda 1} e^{\mathrm{i} \lambda \theta} + b_{\lambda 2} e^{-\mathrm{i}
\lambda \theta} \right),
\label{eq-genlaplace}$$ where $a_{\lambda i}, b_{\lambda i}$ are constants that depend on the boundary conditions. It is easy to establish that for real $\phi$ these constants should satisfy $$\begin{aligned}
a_{\lambda 1} b_{\lambda 1} = (a_{\lambda 1} b_{\lambda 2})^*
\label{eq-condition1}\\
a_{\lambda 2} b_{\lambda 2} = (a_{\lambda 2} b_{\lambda
1})^*.\label{eq-condition2}\end{aligned}$$
We will use the multipole expansion (\[eq-genlaplace\]) for the velocity field in the neighbhood of each disk, with the polar coordinates centered at the disk. Hence we will have $N$ such expansions. The advantage of these expansions is that the boundary conditions at the disks are easy to formulate. In particular, the normal velocity vanishes, that is, $$\left. \frac{\partial \phi}{\partial r}\right|_{r=a} = 0,$$ and then, taking into account the conditions Eqs. (\[eq-condition1\]) and (\[eq-condition2\]) we get $$\begin{aligned}
a_{\lambda 1} = a_{\lambda 2} = 1 \\
b_{\lambda 1} = b_{\lambda 2}^* \equiv b_\lambda.\end{aligned}$$ We still have to find one set of $b_\lambda$ constants for each disk $k$ ($k=1\dots N$), which we will denote as $b_\lambda (k)$.
We next consider the ensemble of $N$ disks. All multipole expansions should simultaneously satisfy boundary conditions on all the discs and at infinity. It is convenient to employ a complex variable for the position, $\it i.e.,$ we define the complex position $\tilde r$ as $$\tilde r = x + y\mathrm{i}.$$ Henceforth a tilde on any vector will denote a similar definition as a complex variable. By employing this notation the general solution Eq. (\[eq-genlaplace\]) can be expressed more compactly as a series of powers of $\tilde r$.
Note that each expansion (\[eq-genlaplace\]) is valid near the corresponding disk, but not far from it. In particular the expansions diverge at infinity. The way to manage an expression for the potential valid at arbitrary distances is to use only the decreasing powers of the multipole expansions, adding the terms corresponding to all the disks. Taking into account the boundary condition at infinity we can then write $$\phi({\mathbf r}) = {\mathbf r}\cdot {\mathbf u}_\infty + \sum_{k=1}^N \phi_k ({\mathbf r}),
\label{eq-expN}$$ where $\phi_k ({\mathbf r})$ is the distortion of the potential produced by the disc $k$, which will have the form $$\phi_k({\mathbf r}) = \sum_{\lambda=1}^\infty \left[c_\lambda (k) \left(
\frac{a}{\tilde r_k} \right)^\lambda + c.c. \right].
\label{eq-expNi}$$ Here $\tilde r_k = \tilde r - \tilde R_k$ is the position relative to the center of disk $k$, and the $c_\lambda (k)$ are constants. With this expression the boundary condition at infinity has already been taken into account. To find the unknowns $c_\lambda(k)$ we should to make this expansion compatible with those of each disk, Eq. (\[eq-genlaplace\]), which incorporating boundary contitions can be written as $$\phi({\mathbf r}) = \sum_{\lambda=0}^\infty
\left(b_{\lambda} (j) \left(\frac{a}{\tilde r_j} \right)^\lambda +
b_{\lambda}^*(j)\left(\frac{\tilde r_j}{a} \right)^{\lambda} + c.c. \right),
\,\,\, j= 1 \dots N.
\label{eq-exp1}$$
To relate the expansion Eq. (\[eq-expN\]) with Eq. (\[eq-expNi\]) to those of Eq. (\[eq-exp1\]), it is convenient to use the Taylor expansion of the negative powers of $\tilde r_k$ (position relative to disk $k$) in terms of the positive powers of $\tilde r_j$ (position relative to any other disk $j\neq k$). The idea is that the positive powers in the local expansion Eq. (\[eq-exp1\]) of any disk should correspond to the terms coming from the rest of the disks in Eq. (\[eq-expN\]). That is, we write $$\left(\frac{a}{\tilde r_k}\right)^{\lambda'}=
\sum_{\lambda=0}^{\infty}q_{\lambda`\,\lambda}
\left(\frac{a}{\tilde R_{k j}}\right)^{\lambda+\lambda'}
\left(\frac{\tilde r_j}{a}\right)^{\lambda},\,\,\,j\neq k,
\label{eq-expanr}$$ where $\tilde R_{k j}=\tilde R_j - \tilde R_k$ so that $\tilde r_k = \tilde R_{k j} + \tilde r_j$, and the constants $q_{\lambda`\,\lambda}$ can be obtained from the recurrence relation $$\begin{aligned}
q_{\lambda'\,\lambda}&=&
\frac{1-\lambda-\lambda'}{\lambda}q_{\lambda'\,\lambda-1},\nonumber\\
q_{\lambda'\,0}&=&1.\end{aligned}$$ We now substitute the expansion Eq. (\[eq-expanr\]) into Eq. (\[eq-expNi\]), to obtain $$\begin{aligned}
\phi({\mathbf r}) & = & \frac{1}{2} a \tilde u_\infty^*\frac{\tilde r}{a} +
\sum_{\lambda=1}^{\infty}\left[c_\lambda(j)\left(\frac{a}{\tilde
r_j}\right)^\lambda + c.c. \right] \nonumber \\
& & +\sum_{k\neq j} \sum_{\lambda'=1}^\infty \left[c_{\lambda'}(k)
\sum_{\lambda=0}^\infty q_{\lambda'\, \lambda}
\left(\frac{a}{\tilde R_{kj}}\right)^{\lambda+\lambda'}
\left(\frac{\tilde r_j}{a}\right)^\lambda + c.c. \right],\,\,\,j=1\dots N.
\label{eq-exptot}\end{aligned}$$
By equating terms of the same power of $\tilde r$ in Eqs. (\[eq-exp1\]) and (\[eq-exptot\]) we find $$\begin{aligned}
b_\lambda(j) & = & c_\lambda(j), \,\,\, \lambda=1\dots N, \label{eq-res00}\\
b_0^*(j) & = & \sum_{k\neq j} \sum_{\lambda'=1}^\infty c_{\lambda'}(k)
\left(\frac{a}{\tilde R_{kj}}\right)^{\lambda'} , \label{eq-res0}\\
b_1^*(j) & = & \frac{a}{2}\tilde u_\infty^* -
\sum_{k\neq j} \sum_{\lambda'=1}^\infty c_{\lambda'}(k) \lambda'
\left(\frac{a}{\tilde R_{kj}}\right)^{\lambda'+1} , \label{eq-res1}\\
b_\lambda^*(j) & = & \sum_{k\neq j} \sum_{\lambda'=1}^\infty c_{\lambda'}(k)
q_{\lambda'\,\lambda}\left(\frac{a}{\tilde R_{kj}}\right)^{\lambda'+\lambda},
\,\,\,\lambda>1. \label{eq-reslambda}\end{aligned}$$
These equation can be solved to obtain the constants $c_\lambda(j)$ with which the potential and the velocity field can be calculated through Eq. (\[eq-expN\]). In practice one can use a few multipole terms, $\lambda = 1\dots \lambda_{max}$, for a moderate value of $\lambda_{max}$ since the series converge rapidly (more terms are necessary as the distances between disks are decreased). The procedure involves finding $c_\lambda(j)$ only once, and permits us to find fluid velocity at any position by only summing a few contributions from each disk.
A dipolar approximation ($\lambda_{max}=1$) provides reasonably good results if the disks are not very close to each other, and its solution can be used as an initial step of an iterative solution of the complete problem. This solution reads: $$c_1(j) \simeq \frac{a}{2} \tilde u_\infty -
\frac{a}{2} \tilde u_\infty^* \sum_{k\neq j}
\left(\frac{a}{\tilde R_{kj}}\right)^2 \,\,\,\lambda=1\dots N.
\label{eq-dipolar}$$ Then the velocity field in this approximation is $$\begin{aligned}
u_x({\mathbf r}) \simeq u_{x\, \infty} -
a \sum_{j=1}^N \left( \frac{c_1(j)}{\tilde r_j^2} + c.c. \right)
\nonumber \\
u_y({\mathbf r}) \simeq u_{y\, \infty} -
a \sum_{j=1}^N \left( \frac{c_1(j)}{\tilde r_j^2}\mathrm{i} + c.c. \right).\end{aligned}$$
Finally, the system of equations (\[eq-res0\])-(\[eq-reslambda\]) takes the following form: $$\begin{aligned}
c_1(j) & = & \frac{a}{2}\tilde u_\infty -
\sum_{k\neq j} \sum_{\lambda'=1}^\infty c_{\lambda'}^*(k) \lambda'
\left(\frac{a}{\tilde R_{kj}^*}\right)^{\lambda'+1} , \label{eq-sist1}\\
c_\lambda(j) & = & \sum_{k\neq j} \sum_{\lambda'=1}^\infty c_{\lambda'}^*(k)
q_{\lambda'\,\lambda}\left(\frac{a}{\tilde R_{kj}^*}\right)^{\lambda'+\lambda},
\,\,\,\lambda>1. \label{eq-sistlambda}\end{aligned}$$ With these, the velocity field is given by $$\begin{aligned}
u_x({\mathbf r}) = u_{x\,\infty} - \sum_{j=1}^N \sum_{\lambda'=1}^\infty
\left( \lambda c_\lambda(j) \frac{a^\lambda}{(\tilde r - \tilde R_j)^{\lambda+1}} + c.c \right) \nonumber \\
u_y({\mathbf r}) = u_{y\,\infty} - \sum_{j=1}^N \sum_{\lambda'=1}^\infty
\left( \lambda c_\lambda(j) \frac{a^\lambda}{(\tilde r - \tilde R_j)^{\lambda+1}}\mathrm{i} + c.c \right).\end{aligned}$$
To solve Eqs. (\[eq-sist1\]) and (\[eq-sistlambda\]) an iterative method can be used. These equations can be formulated as a matrix relation, $$C = A C^* + B,$$ where $C$ is a vector containing all the $c_\lambda(k)$ coefficients. Then we can apply the following iteration, which can be shown to be convergent: $$C_i = A C_{i-1}^* + B,$$ with the initial term $C_0$ given by the dipolar approximation Eq. (\[eq-dipolar\]).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We generalize the mixed tori which appear in the second author’s JSJ-type decomposition theorem for symplectic fillings of contact manifolds. Mixed tori are convex surfaces in contact manifolds which may be used to decompose symplectic fillings. We call our more general surfaces splitting surfaces, and show that the decomposition of symplectic fillings continues to hold. Specifically, given a strong or exact symplectic filling of a contact manifold which admits a splitting surface, we produce a new symplectic manifold which strongly or exactly fills its boundary, and which is related to the original filling by Liouville surgery.'
author:
- Austin Christian and Michael Menke
bibliography:
- '../../references.bib'
title: Splitting symplectic fillings
---
=1
Introduction
============
Contact geometry is a close relative of symplectic geometry, and one manifestation of this relationship is the tendency for symplectic manifolds-with-boundary to endow their boundaries with contact structures. For instance, suppose $(W,\omega)$ is a compact symplectic manifold which admits a Liouville vector field near its boundary. That is, there is a vector field $Z$ on $W$ pointing out of $\partial W$ with the property that $\mathcal{L}_Z\omega=\omega$ in some neighborhood of $M=\partial W$. Then $M$ inherits an orientation from $W$ and $\lambda:=\iota_Z\omega$ determines a co-oriented contact structure $\xi:=\ker(\lambda|_M)$ on $M$. In this case say that $(W,\omega)$ is a *strong symplectic filling* of the contact manifold $(M,\xi)$.\
It is natural to wonder about the extent to which this construction is reversible. That is, we begin with a fixed contact manifold $(M,\xi)$ and ask existence and uniqueness questions about the strong symplectic fillings of this manifold. Eliashberg and Gromov showed in [@eliashberg1991convex] that a fillable contact manifold must be tight, so the overtwisted contact manifolds immediately give a large class of manifolds which are not symplectically fillable. In [@etnyre2002tight] Etnyre and Honda showed that while tightness is necessary for fillability, it is not sufficient. Another early result, due to Eliashberg ([@eliashberg1990filling]) and Gromov ([@gromov1985pseudo]), says that symplectic fillings of the standard 3-sphere $(S^3,\xi_{std})$ are unique up to symplectic deformation equivalence and blowup. If we further require the filling to be *exact*, meaning that $\mathcal{L}_Z\omega=\omega$ on all of $W$, then $(S^3,\xi_{std})$ in fact has a unique filling up to symplectomorphism.\
A number of contact 3-manifolds have seen their exact fillings classified up to symplectomorphism, symplectic deformation equivalence, or diffeomorphism. Wendl showed in [@wendl2010strongly] that $(\mathbb{T}^3,\xi_1)$ has a unique exact filling up to symplectomorphism, where $\xi_1$ is the canonical contact structure on $ST^*\mathbb{T}^2$, and work of McDuff ([@mcduff1990structure]) and Lisca ([@lisca2008symplectic]) classified the exact fillings of lens spaces $(L(p,q),\xi_{std})$ up to diffeomorphism. Some classification results also exist for higher-dimensional contact manifolds, but giving precise symplecto-geometric descriptions of higher-dimensional fillings is difficult. The most famous result in high dimensions is probably the Eliashberg-Floer-McDuff theorem ([@mcduff1991symplectic]), which says that, up to diffeomorphism, $(S^{2n-1},\xi_{std})$ has a unique symplectically aspherical strong symplectic filling, for all $n\geq 3$.\
In [@menke2018jsj], the second author introduced the notion of a *mixed torus* — a special kind of convex torus — in a contact 3-manifold, and showed that if $(M,\xi)$ admits a mixed torus, then we may construct from any strong symplectic filling $(W,\omega)$ of $(M,\xi)$ another symplectic manifold $(W',\omega')$ which strongly fills its boundary $(M',\xi')$. Moreover, the relationship between $(W,\omega)$ and $(W',\omega')$ may be stated rather explicitly, with $(W,\omega)$ obtained from $(W',\omega')$ by Liouville surgery in a prescribed manner. This allows us to leverage an understanding of the fillings of $(M',\xi')$ into information about the fillings of $(M,\xi)$.\
In this note we consider higher-genus analogues of mixed tori, which we call *splitting surfaces*. We will give a precise definition of splitting surfaces in Section \[sec:background\], but a splitting surface of genus 1 is simply a mixed torus. The purpose of this note is to show that the main theorem of [@menke2018jsj] continues to hold in any genus.
\[main-theorem\] Let $(M,\xi)$ be a closed, co-oriented 3-dimensional contact manifold and let $(W,\omega)$ be a strong (respectively, exact) filling of $(M,\xi)$. If $(M,\xi)$ admits a splitting surface $\Sigma$ of genus $g$, then there exists a symplectic manifold $(W',\omega')$ such that
1. $(W',\omega')$ is a strong (respectively, exact) filling of its boundary $(M',\xi')$;
2. there are Legendrian graphs $\Lambda_1,\Lambda_2\subset\partial W'$ with standard neighborhoods $N(\Lambda_1),N(\Lambda_2)$ such that $$M \simeq \left(\partial W' - \bigcup_{i=1}^{2}\operatorname{int}(N(\Lambda_i))\right)/(\partial N(\Lambda_1)\sim \partial N(\Lambda_2)),$$ where the boundaries $\partial N(\Lambda_i)$ are glued in such a way that their dividing sets and meridians are identified;\[main-thm:decomp\]
3. $(W,\omega)$ can be recovered from $(W',\omega')$ by attaching a symplectic handle $(H_{R_+(\Sigma)},\omega_\beta)$ constructed from the positive region of $\Sigma$.
The first use of mixed tori to classify symplectic fillings came in the form of [@menke2018jsj Theorem 1.2], where it is shown that if $(M,\xi)$ is obtained from $(M_0,\xi_0)$ by Legendrian surgery along a Legendrian knot which has been stabilized both positively and negatively, then every exact filling of $(M,\xi)$ is obtained from an exact filling of $(M_0,\xi_0)$ by attaching a round symplectic 1-handle along the Legendrian knot. In particular, this means that contact manifolds obtained from $(S^3,\xi_{std})$ by Legendrian surgery along twice-stabilized Legendrian knots have unique exact fillings. The following is then obtained by repeatedly applying [@menke2018jsj Theorem 1.2]:
\[corollary:links\] Let $\Lambda\subset(S^3,\xi_{std})$ be a linear chain of Legendrian unknots, so that Legendrian surgery along $\Lambda$ produces a tight lens space. If each unknot has been stabilized both positively and negatively, then this lens space admits a unique exact filling up to symplectomorphism.
We mention this result here because the methods that were used to prove [@menke2018jsj Theorem 1.2] from [@menke2018jsj Theorem 1.1] could also be used to prove Corollary \[corollary:links\] from Theorem \[main-theorem\]. An interesting (if vague) question is then the following: let $\Lambda\subset(S^3,\xi_{std})$ be a Legendrian link, and let $(M,\xi)$ be the result of Legendrian surgery along $\Lambda$. Other than the condition listed in Corollary \[corollary:links\], are there topological properties of $\Lambda$ or configurations of stabilizations on its components which force $(M,\xi)$ to admit a splitting surface? Under what circumstances does this yield a classification of the fillings of $(M,\xi)$?\
Our strategy of proof for the main theorem follows in the tradition of Eliashberg’s “filling by holomorphic disks,” initiated in [@eliashberg1990filling]. A splitting surface $\Sigma\subset(M,\xi)$ of genus $g$ gives us two surfaces in $M$ with genus 0 and $g+1$ boundary components, each of which can be lifted to a family of $J$-holomorphic curves in the symplectization of $M$. If we have a filling $(W,\omega)$ of $(M,\xi)$, these families can be extended to a single 1-dimensional family of $J$-holomorphic curves in the completion $(\widehat{W},\widehat{\omega})$, and the geometric conditions on $\Sigma$ will control the topology of this family. Removing a neighborhood of this family will lead us to the new symplectic manifold $(W',\omega')$.\
In Section \[sec:background\] we recall some useful definitions and results from contact geometry and give a definition of our splitting surfaces. Section \[sec:main-theorem\] contains the proof of Theorem \[main-theorem\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors would like to thank Ko Honda for a number of helpful conversations and suggestions during the completion of this project.
Background {#sec:background}
==========
Throughout this section we fix a closed contact 3-manifold $(M,\xi)$.
Fillings of contact manifolds {#subsec:fillings}
-----------------------------
As mentioned above, many symplectic manifolds endow their boundaries with contact structures, and there are various levels of compatibility between the symplectic and contact structures. In the other direction, we say that our contact manifold $(M,\xi)$ is *fillable* if it can be realized as the boundary of such a symplectic manifold. We have the following definitions.
Fix a co-oriented contact manifold $(M,\xi)$ and suppose $(W,\omega)$ is a symplectic manifold with $\partial W=M$ as oriented manifolds. We say that $(W,\omega)$ is
- a *weak symplectic filling* of $(M,\xi)$ if $\omega|_\xi>0$;
- a *strong symplectic filling* of $(M,\xi)$ if there is a 1-form $\lambda$ on $W$ such that $\omega=d\lambda$ on some neighborhood of $\partial W$ and $\xi=\ker(\lambda|_{\partial W})$;
- an *exact filling* of $(M,\xi)$ if there is a 1-form $\lambda$ on $W$ such that $\omega=d\lambda$ on all of $W$ and $\xi=\ker(\lambda|_{\partial W})$.
We say that $(M,\xi)$ is *weakly symplectically fillable*, *strongly symplectically fillable*, or *exactly fillable* if it admits a weak symplectic, strong symplectic, or exact fillling, respectively.\
Certainly every exact filling is a strong filling and every strong filling is a weak filling, so we have inclusions $$\{\text{exactly fillable}\}
\subseteq
\{\text{strongly symplectically fillable}\}
\subseteq
\{\text{weakly symplectically fillable}\}.$$ One hypothesis of Theorem \[main-theorem\] is that our contact manifold $(M,\xi)$ admits a strong or exact filling $(W,\omega)$, so our manifolds will always be at least strongly fillable.
Convex surfaces {#subsec:convex-surfaces}
---------------
We quickly recall the notion of convexity in contact topology, as explored by Giroux in [@giroux1991convexite]. First, a *contact vector field* on a contact 3-manifold $(M,\xi)$ is a vector field whose flow preserves $\xi$. Notice that if $\lambda$ is a contact form for $\xi$ and $X$ is a contact vector field, then $$\mathcal{L}_X\lambda = g\lambda$$ for some positive smooth function $g$, so flowing along $X$ produces conformal dilations of the contact form. For this reason we say that a surface $\Sigma\subset (M,\xi)$ is *convex* if there is a contact vector field for $(M,\xi)$ which is transverse to $\Sigma$. An important observation is that convex surfaces exist in abundance.
Any closed surface in a contact manifold $(M,\xi)$ is $C^\infty$-close to a convex surface.
If $\Sigma\subset(M,\xi)$ is convex and $X$ is a contact vector field transverse to $\Sigma$, then the *dividing set* of $\Sigma$ is $$\Gamma_\Sigma = \{p\in\Sigma~|~X(p)\in\xi_p\}.$$ Three important observations about the multi-curve $\Gamma_\Sigma$ are
1. $\Gamma_\Sigma$ divides $\Sigma$ into positive and negative regions: $\Sigma\setminus\Gamma_\Sigma=R_+(\Sigma)\sqcup R_-(\Sigma)$;
2. $\Gamma_\Sigma$ is transverse to the characteristic foliation $\Sigma_\xi$ of $\Sigma$;
3. $\Sigma$ admits a volume form $\omega$ and a vector field $Y$ so that $Y$ points transversely out of $R_+(\Sigma)$ along $\Gamma_\Sigma$, directs $\Sigma_\xi$, and dilates $\omega$ in the sense that $\pm\mathcal{L}_Y\omega>0$ on $R_{\pm}(\Sigma)$.
These three characteristics determine $\Gamma_\Sigma$ up to isotopy, so we will refer to the dividing set $\Gamma_\Sigma$ and the regions $R_{\pm}(\Sigma)$ of a convex surface $\Sigma$ without reference to a particular contact vector field.
Bypasses and stabilizations {#subsec:bypasses}
---------------------------
If $\Sigma\subset M$ is a convex surface, recall that a *bypass* for $\Sigma$ is an oriented embedded half-disk $D$ such that
1. $\partial D$ is the union of two Legendrian arcs $\alpha_1,\alpha_2$ which intersect at their endpoints;
2. $D$ intersects $\Sigma$ transversely along $\alpha_1$;
3. $D$ has positive elliptic tangencies at $\alpha_1\cap\alpha_2$, one negative elliptic tangency on the interior of $\alpha_1$, and only positive tangencies along $\alpha_2$, alternating between elliptic and hyperbolic;
4. $\alpha_1$ intersects the dividing set $\Gamma_\Sigma$ exactly at the elliptic points of $\alpha_1$.
We will refer to $\alpha_1\subset D$ as the *attaching arc* for the bypass $D$, and we say that $D$ *straddles* the component $c\subset\Gamma_\Sigma$ containing the negative elliptic tangency.\
When a bypass $D$ for $\Sigma$ exists it is known that there is a neighborhood of $\Sigma\cup D$, diffeomorphic to $\Sigma\times[0,1]$, such that $\Sigma_i=\Sigma\times\{i\}$, $i=0,1$, are convex and the dividing set $\Gamma_{\Sigma_1}$ is obtained from $\Gamma_{\Sigma_0}$ by Honda’s *bypass attachment* operation, depicted in Figure \[fig:bypass-attachment\]. A bypass which does not change the dividing set is said to be *trivial*. The effect of bypass attachment on the dividing set of $\Sigma$ can also be seen through Giroux’s contact handle decompositions. The surface $\Sigma_1$ is obtained from $\Sigma$ by attaching a contact 1-handle and then a contact 2-handle in topologically canceling manner. A detailed description of this process can be found in [@ozbagci2011contact Section 3].\
We are now prepared to define our splitting surfaces.
We call a closed, connected, oriented, convex surface $\Sigma\subset(M,\xi)$ of genus $g$ a *splitting surface* if
1. the regions $R_{\pm}(\Sigma)$ are planar, with $g+1$ boundary components $c_1,\ldots,c_{g+1}$;
2. there exist bypasses $D^{\pm}_1,\ldots,D^\pm_g\subset(M,\xi)$, attached to $\Sigma$ along Legendrian arcs $\alpha^{\pm}_1,\ldots,\alpha^\pm_g$, with $\alpha^\pm_i$ straddling $c_i$ and having its endpoints on $c_{g+1}$;
3. for $i=1,\ldots,g$, there is an arc $a_i\subset c_{g+1}$ which contains the endpoints of $\alpha_i^+$ and $\alpha_i^-$, and contains no endpoints of $\alpha_j^\pm$ for $j\neq i$;
4. the bypasses $D^+_1,\ldots,D^+_g$ are attached from one side of $\Sigma$ and the bypasses $D^-_1,\ldots,D^-_g$ are attached from the other side.
Liouville hypersurfaces {#subsec:liouville}
-----------------------
The last statement of Theorem \[main-theorem\] says that we can obtain our original symplectic filling $(W,\omega)$ from our new filling $(W',\omega')$ by attaching a symplectic handle. The construction of the handle in question begins with the positive region $R_+(\Sigma)$ of our splitting surface, which is a *Liouville hypersurface*. In this section we want to review Avdek’s definition ([@avdek2012liouville]) of Liouville hypersurfaces and produce the corresponding symplectic handle.\
A *Liouville domain* is a pair $(\Sigma_L,\beta)$, where
1. $\Sigma_L$ is a smooth, compact manifold with boundary;
2. $d\beta$ is a symplectic form on $\Sigma_L$;
3. the vector field $X_\beta$ defined by $\iota_{X_\beta}d\beta=\beta$ points out of $\partial\Sigma_L$ transversely.
We call $X_\beta$ the *Liouville vector field* for $(\Sigma_L,\beta)$.\
Let $(M,\xi)$ be a contact 3-manifold and let $(\Sigma_L,\beta)$ be a 2-dimensional Liouville domain. A *Liouville embedding* $i\colon(\Sigma_L,\beta)\hookrightarrow(M,\xi)$ is an embedding for which there exists a contact form $\lambda$ on $(M,\xi)$ satisfying $i^*\lambda=\beta$. We call the image of a Liouville embedding a *Liouville hypersurface* and denote it by $(\Sigma_L,\beta)\subset(M,\xi)$.\
The standard example of a Liouville hypersurface is the positive region of a convex surface. The following result says that these regions are in fact the source of all Liouville hypersurfaces.
A hypersurface $\Sigma_L\subset(M,\xi)$ is Liouville if and only if there is a convex hypersurface $\Sigma\subset(M,\xi)$ for which $\Sigma_L$ is $R_+(\Sigma)$ minus some collar neighborhood of $\partial R_+(\Sigma)$.
Given a Liouville hypersurface $(\Sigma_L,\beta)$, Avdek constructs a symplectic handle $(H_{\Sigma_L},\omega_\beta)$, and we summarize this construction here. For full details see [@avdek2012liouville].\
The construction begins with a standard neighborhood $\mathcal{N}(\Sigma_L)$ of $(\Sigma_L,\beta)$ in $(M,\xi)$. If $\lambda$ is a contact form for $(M,\xi)$ satisfying $\lambda|_{T\Sigma_L}=\beta$, then there is a neighborhood $N(\Sigma_L) = [-\epsilon,\epsilon]\times\Sigma_L$ with $\lambda|_{N(\Sigma_L)}=dz+\beta$, for some sufficiently small $\epsilon$. This neighborhood will have corners at $\{\pm\epsilon\}\times\partial\Sigma_L$, but an edge-rounding process produces $\mathcal{N}(\Sigma_L)$, a neighborhood of $(\Sigma_L,\beta)$ with smooth, convex boundary.\
With an abstract copy of this standard neighborhood in hand, consider the symplectic manifold $$(H_{\Sigma_L},\omega_\beta) = ([-1,1]\times\mathcal{N}(\Sigma),d\theta\wedge dz+d\beta),$$ where $\theta$ and $z$ are the coordinates on $[-1,1]$ and $[-\epsilon,\epsilon]$, respectively. This is the symplectic handle constructed from $(\Sigma_L,\beta)$. There is a vector field $V_\beta=z\partial_z+X_\beta$ which points transversely out of $\partial H_{\Sigma_L}$ along $[-1,1]\times\partial\mathcal{N}(\Sigma_L)$ and whose flow dilates $\omega_\beta$. This vector field can be perturbed so that it also points into $\partial H_{\Sigma_L}$ along $\{\pm 1\}\times\mathcal{N}(\Sigma_L)$, making this portion of $\partial H(\Sigma_L)$ concave while $[-1,1]\times\partial\mathcal{N}(\Sigma_L)$ is convex.\
Let us also describe how Avdek attaches the symplectic handle $(H_{\Sigma_L},\omega_\beta)$ to a strong symplectic filling $(W,\omega)$. For this attachment to be possible there must exist a pair of disjoint Liouville embeddings $$i_1\colon(\Sigma_L,\beta)\hookrightarrow (M,\xi)
\quad\text{and}\quad
i_2\colon(\Sigma_L,\beta)\hookrightarrow (M,\xi),$$ where $(M,\xi)$ is the boundary of $(W,\omega)$. These embeddings admit standard neighborhoods $\mathcal{N}(i_1(\Sigma_L))$ and $\mathcal{N}(i_2(\Sigma_L))$, each contactomorphic to $\mathcal{N}(\Sigma_L)$. We form a sort of symplectic-filling-with-corners $W^\square$ by removing $\mathcal{N}(i_1(\Sigma_L))$ and $\mathcal{N}(i_2(\Sigma_L))$ from $(W,\omega)$ and attaching $H_{\Sigma_L}$ along $\{\pm 1\}\times\mathcal{N}(\Sigma_L)$. Because $(W,\omega)$ is a strong filling of $(M,\xi)$, there is a Liouville vector field on $W$ pointing out of $\partial W$. We glue $(H_{\Sigma_L},\omega_\beta)$ to $(W\setminus(\mathcal{N}(i_1(\Sigma_L))\cup\mathcal{N}(i_2(\Sigma_L))),\omega)$ in such a way that this vector field agrees with $V_\beta$ along $\{\pm 1\}\times \mathcal{N}(\Sigma_L)$. The edges of $W^\square$ are then rounded to produce a new symplectic filling $(W',\omega')$. This new filling is the result of attaching the handle $(H_{\Sigma_L},\omega_\beta)$ to $(W,\omega)$ along $i_1(\Sigma_L)$ and $i_2(\Sigma_L)$.
Proof of Theorem \[main-theorem\] {#sec:main-theorem}
=================================
Throughout this section we take $(M,\xi)$ to be a contact manifold satisfying the hypotheses of Theorem \[main-theorem\]. Let $(W,\omega)$ be a strong filling of $(M,\xi)$ and $\Sigma_g$ a splitting surface of genus $g$, with dividing set $\Gamma_{\Sigma_g}=c_1\cup\cdots\cup c_{g+1}$. There are attaching arcs $\alpha^{\pm}_1,\ldots,\alpha^{\pm}_g$ and associated bypasses $D^{\pm}_1,\ldots,D^{\pm}_g$ as described in the definition of splitting surfaces.\
We will denote by $(\widehat{W},\widehat{\omega})$ the completion of $(W,\omega)$, obtained by attaching the positive end $([0,\infty)\times M,d(e^t\alpha))$ of the symplectization of $M$. We take $J$ to be an almost complex structure on $\widehat{W}$ adapted to the contact form $\alpha$ for $(M,\xi)$. That is, $J$ is translation invariant, $J\xi=\xi$, and $J\partial_t=R_\alpha$, where $t$ is the $[0,\infty)$-coordinate on the symplectization and $R_\alpha$ is the Reeb vector field for $\alpha$.\
We will prove Theorem \[main-theorem\] by adapting the proof of [@menke2018jsj Theorem 1.1]. Specifically, our goal is to use $\Sigma_g$ to construct a 1-parameter family $\mathcal{S}$ which sweeps out a properly embedded handlebody in $(\widehat{W},\widehat{\omega})$. Removing this handlebody from $(W,\omega)$ will leave us with the desired manifold $(W',\omega')$.\
Because our proof is adapted from [@menke2018jsj], many of our lemmas are arbitrary-genus analogues of lemmas found there. Some of these require new proofs, while others, such as the following standardization of the contact form on $M$, are genus-independent and therefore survive unaltered.
\[lemma:dividing-orbits\] There is a choice of contact form on a neighborhood of $\Sigma_g$ such that the components $\Gamma_{\Sigma_g}$ are non-degenerate elliptic Reeb orbits of Conley-Zehnder index 1 with respect to the framing induced by $\Sigma_g$.
Denote the Reeb orbits constructed in Lemma \[lemma:dividing-orbits\] by $e_1,\ldots,e_{g+1}$, with $e_{g+1}$ containing the endpoints of $\alpha^\pm_1,\ldots,\alpha^\pm_g$ and $e_i$ the dividing curve straddled by $\alpha^\pm_i$. Menke’s proof of Lemma \[lemma:dividing-orbits\] produces an explicit model for $\Sigma_g$ with these orbits comprising the dividing set, and this model is depicted in Figure \[fig:dividing-curves\].
\[lemma:description-of-orbits\] Let $\Sigma_g\subset (M,\xi)$ be a splitting surface of genus $g>1$, with dividing set $e_1\cup\cdots\cup e_{g+1}$ and bypasses $D^\pm_1,\ldots,D^\pm_g$ as described above. There is a one-sided neighborhood $$N = N(\Sigma_g\cup D^+_1\cup \cdots\cup D^+_g)$$ and an extension of the contact form $\alpha$ chosen in Lemma \[lemma:dividing-orbits\] to $N$. This neighborhood contains contact 1-handles $N_1^i$, contact 2-handles $N_2^i$, and surfaces with corners $\Sigma^{i-1}_g,\Sigma^i_{g+1}$ of genus $g$ and $(g+1)$, respectively, for $i=1,\ldots,g$. Moreover,
1. the boundary $\partial N$ is given by $\Sigma_g$ and $\tilde{\Sigma}$, where $\tilde{\Sigma}$ is another convex surface of genus $g$, with dividing set given by elliptic orbits $\tilde{e}_1,\ldots,\tilde{e}_{g+1}$;
2. $\Sigma^0_g=\Sigma_g$, and for $i=1,\ldots,g-1$, $\Sigma^i_g$ meets $\tilde{\Sigma}$ in the orbits $\tilde{e}_1,\ldots,\tilde{e}_i$, meets $\Sigma_g$ in the orbits $e_{i+1},\ldots,e_g$, and has dividing set given by these orbits, along with an elliptic orbit $e^{i+1}_{g+1}$;
3. for $i=1,\ldots,g$ we have a neighborhood $$N(\Sigma^{i-1}_g\cup D^+_i) = N_1^i \cup_{\Sigma_{g+1}^i} N_2^i,$$ with $\Sigma^i_{g+1}$ containing the orbits $\tilde{e}_1,\ldots,\tilde{e}_{i-1},e_i,\ldots,e_g,e^{i+1}_{g+1}$, as well as the elliptic orbit $\overline{e}_i$;
4. all of the elliptic orbits listed have Conley-Zehnder index 1;
5. the Reeb vector field $R_\alpha$ is positively (negatively) transverse to the positive (negative) region of each of the surfaces listed;
6. there are hyperbolic orbits $h^i_{g+1},\tilde{h}_i$ in $N_1^i$ and $N_2^i$, respectively, which have Conley-Zehnder index 0 with respect to $\Sigma_g$;
7. if $\gamma$ is any other Reeb orbit in $N$ and $\bar{\gamma}$ is any of $e_i,h_{g+1}^i$, or $\tilde{h}_i$, then\[part:action-bound\] $$\mathcal{A}(\bar{\gamma}) < \mathcal{A}(\bar{e}_j),\mathcal{A}(\tilde{e}_j) \ll\mathcal{A}(\gamma),$$ for all $j$. In particular, $\mathcal{A}(\gamma)$ is sufficiently large as to prohibit the existence of a pseudoholomorphic curve in the symplectization of $M$ from having $\gamma$ among its negative ends while its positive ends form a subset of the curves listed.\[vaugon-action-bounds\]
As in the proof of [@menke2018jsj Lemma 3.2], we obtain the neighborhood $N$ by successively attaching the contact handles $N_1^i$ and $N_2^i$, and we extend $\alpha$ to $N$ by extending this contact form to each of these handles.\
The first contact handle we attach, $N_1^1$, corresponds to the bypass $D^+_1$, which has its endpoints on $e_{g+1}$. Attaching this handle requires a convex-to-sutured boundary modification, which introduces the hyperbolic orbit $h^1_{g+1}$. We then apply a sutured-to-convex boundary modification before attaching $N_2^1$. The result is an extension of $\alpha$ to the neighborhood $N(\Sigma_g\cup D^+_1)$ as described, and we repeat this process inductively to obtain $N$. We choose our extension of $\alpha$ across each 1-handle so that the actions of $\overline{e}_i$ and $e_{g+1}^{i+1}$ are much larger than those of $e_i, e^i_{g+1},$ and $h^i_{g+1}$. The fact that all other Reeb orbits intersecting $N$ have sufficiently large action as to be irrelevant follows from [@vaugon2015reeb Theorem 2.1].
A schematic of the neighborhood $N(\Sigma^{i-1}_g\cup D^+_i)$ is depicted in Figure \[fig:neighborhood-orbits\].
As stated above, we will build a 1-parameter family of holomorphic curves in $\widehat{W}$ that will sweep out a handlebody of genus $g$. The splitting surface $\Sigma_g$ will help us do this by providing targets $R_{\pm}(\Sigma_g)$ for which our family can aim at its ends. That is, our 1-parameter family will have its ends in the symplectization part $[0,\infty)\times M$ of $\widehat{W}$, and we want the projection $\pi\colon[0,\infty)\times M\to M$ to take the ends of our family to the regions $R_{\pm}(\Sigma_g)$. The first step towards building our 1-parameter family is then to lift $R_{\pm}(\Sigma_g)$ to embedded holomorphic curves $$u_{\pm}\colon S^2\setminus\{p_1,\ldots,p_{g+1}\}\to[0,\infty)\times M.$$ We can obtain these lifts by employing the following strategy: for each $1\leq i\leq g+1$ we construct a holomorphic half-cylinder $$u_i\colon[0,\infty)\times S^1\to \mathbb{R}\times M$$ which is positively asymptotic to $e_i$. These half-cylinders project under $\pi$ to collar neighborhoods of $e_1,\ldots,e_{g+1}$ in $R_{\pm}(\Sigma_g)$, the deletion of which leaves $R'_{\pm}$, a 2-dimensional Weinstein domain. Our lifting problem is then solved if we can lift $R'_{\pm}$ to a holomorphic curve in $\mathbb{R}\times M$ and then glue the holomorphic half-cylinders $u_1,\ldots,u_{g+1}$ to the boundary. The following lemma, proved in [@menke2018jsj], allows us to lift $R'_{\pm}$.
\[lemma:weinstein-domains\] Let $(B,\beta=-df\circ J)$ be a 2-dimensional Weinstein domain, where $f\colon B\to\mathbb{R}$ is a Morse function such that $\partial B$ is a level set of $f$, and let $\alpha=dt+\beta$ be a contact form on $[-\epsilon,\epsilon]\times B$, where $t$ is the coordinate on $[-\epsilon,\epsilon]$. Then there is an adapted almost complex structure on $\mathbb{R}\times[-\epsilon,\epsilon]\times B$ such that we can lift $B$ to a holomorphic curve by the map $u(\mathbf{x})=(f(\mathbf{x}),0,\mathbf{x})$.
The construction of the holomorphic half-cylinders $u_1,\ldots,u_{g+1}$ and the gluing of these to our lifts is also carried out in [@menke2018jsj]; this establishes the following result.
\[lemma:lifts\] There are embedded holomorphic curves $$u_{\pm}\colon S^2\setminus\{p_1,\ldots,p_{g+1}\}\to[0,\infty)\times M$$ such that
1. both are Fredholm regular with index 2 and positively asymptotic to $e_1,\ldots,e_{g+1}$;
2. under the projection $\pi\colon[0,\infty)\times M\to M$ we have $\operatorname{im}(\pi\circ u_{\pm})=R_\pm(\Sigma_g)$.
The same holomorphic half-cylinder strategy is used in [@menke2018jsj] to prove the next result that we will need. Because $\Sigma_g$ is a splitting surface, it admits collections of bypasses ${\mathbf{D}}_+$ and ${\mathbf{D}}_-$ from opposite sides, and Lemma \[lemma:description-of-orbits\] describes the orbits that appear in a neighborhood $N(\Sigma_g\cup {\mathbf{D}}_+\cup {\mathbf{D}}_-)$. Specifically, Lemma \[lemma:description-of-orbits\] gives a list of relevant orbits in $N(\Sigma_g\cup{\mathbf{D}}_+)$, and produces a corresponding list in $N(\Sigma_g\cup{\mathbf{D}}_-)$. We distinguish the orbits in $N(\Sigma_g\cup{\mathbf{D}}_-)$ from those in $N(\Sigma_g\cup{\mathbf{D}}_+)$ with a prime (e.g., $\overline{e}_i'$ instead of $\overline{e}_i$). Some of these orbits are represented diagrammatically in Figure \[fig:double-neighborhood-orbits\]. In Lemma \[lemma:description-of-orbits\], the attachment of the bypass $D^+_i$ was accomplished by attaching the contact handles $N_1^i$ and $N_2^i$; we use the handles $(N_2^i)'$ and $(N_1^i)'$ to attach $D^-_i$. The same approach used to prove Lemma \[lemma:lifts\] produces a collection of holomorphic curves which project to $N(\Sigma_g\cup {\mathbf{D}}_+\cup {\mathbf{D}}_-)$ and will be useful to us in constructing our 1-parameter family.
\[lemma:walls\] For $i=1,\ldots,g$, there are embedded holomorphic curves $$\bar{u}_{\pm,i},\bar{u}'_{\pm,i}\colon S^2\setminus\{p_1,\ldots,p_{g+2}\}\to [0,\infty)\times N(\Sigma_g\cup {\mathbf{D}}_+\cup {\mathbf{D}}_-)$$ and $$\tilde{u}_{\pm,i},\tilde{u}'_{\pm,i}\colon S^2\setminus\{p_1,\ldots,p_{g+1}\}\to [0,\infty)\times N(\Sigma_g\cup {\mathbf{D}}_+\cup {\mathbf{D}}_-),$$ all Fredholm regular of index 2, all positively asymptotic to $\tilde{e}_1,\ldots,\tilde{e}_{i-1},e_{i+1},\ldots,e_g$, and additionally
1. $\bar{u}_{\pm,i}$ is positively asymptotic to $e_i,\bar{e}_i$, and $e_{g+1}^{i+1}$;
2. $\bar{u}'_{\pm,i}$ is positively asymptotic to $e_i,\bar{e}'_i$, and $(e_{g+1}^{i+1})'$;
3. $\tilde{u}_{\pm,i}$ is positively asymptotic to $\tilde{e}_i$ and $e_{g+1}^{i+1}$;
4. $\tilde{u}'_{\pm,i}$ is positively asymptotic to $\tilde{e}'_i$ and $(e_{g+1}^{i+1})'$.
Curves with the same asymptotic ends are distinguished by whether their projections to $\Sigma_g\subset M$ agree with that of $R_+(\Sigma_g)$ or $R_-(\Sigma_g)$.
The holomorphic curves given by Lemma \[lemma:walls\] serve as “walls" between the contact handles that have been attached to $\Sigma_g$ and will be used to enumerate certain holomorphic curves appearing in the symplectization $\mathbb{R}\times M$. Some of these walls are depicted as heavily shaded curves in in Figure \[fig:neighborhood-orbits\].\
Let $\mathcal{M}(e_1,\ldots,e_{g+1})$ be the index-2 moduli space of curves $u\colon S^2\setminus\{p_1,\ldots,p_{g+1}\}\to\mathbb{R}\times M$ which are positively asymptotic to $e_1,\ldots,e_{g+1}$ and homologous to either $u_+$ or $u_-$. This space admits an obvious translation action by $\mathbb{R}$, and the following lemma describes the compactification of $\mathcal{M}(e_1,\ldots,e_{g+1})/\mathbb{R}$.
\[lemma:boundary-curves\] The compactification $\overline{\mathcal{M}(e_1,\ldots,e_{g+1})/\mathbb{R}}$ contains a pair of closed intervals $\mathcal{N}_{\pm}$ such that
1. $\mathcal{N}_{\pm}$ contains the equivalence class of $u_{\pm}$;
2. the boundary $\partial\mathcal{N}_{\pm}$ contains a two-level holomorphic building with top level $v_{1,\pm}$ a cylinder positively asymptotic to $e_{g+1}$ and negatively asymptotic to $h^1_{g+1}$, and with bottom level $v_{0,\pm}$ positively asymptotic to $e_1,\ldots,e_g,h^1_{g+1}$;
3. the other boundary element of $\partial\mathcal{N}_{\pm}$ is a two-level holomorphic building with top level $v'_{1,\pm}$ a cylinder positively asymptotic to $e_{g+1}$ and negatively asymptotic to $(h^1_{g+1})'$, and with bottom level $v'_{0,\pm}$ positively asymptotic to $e_1,\ldots,e_g,(h^1_{g+1})'$.
We assume that $\mathcal{A}(e_1)=\mathcal{A}(e_2)=\cdots=\mathcal{A}(e_{g+1})$; we will use this action information as well as a description of the homology classes of the relevant curves to determine $\partial\mathcal{N}_{\pm}$. Consider $$H_1(N(\Sigma_g\cup \mathbf{D}_+\cup \mathbf{D}_-)) \simeq H_1(\Sigma_g) \simeq \mathbb{Z}^{2g},$$ and notice that we may choose curves $b_1,\ldots,b_g\subset\Sigma_g$ so that $[e_1],\ldots,[e_g],[b_1],\ldots,[b_g]$ forms a basis for $H_1(\Sigma_g)$. Moreover, the curve $b_i$ is chosen so that if the attaching arc $\alpha_i^\pm$ is joined with (a subarc of) the arc $a_i$ identified in the definition of a splitting surface, then the resulting closed curve is homologous to $b_i$. After orienting the curves $b_1,\ldots,b_g$, we compute the following homology classes[^1] $$\label{eq:new-homology-classes}
[\tilde{e}_i] = [e_i]-\sum_{k=1}^{g-i}[b_{i+k}],
\quad
[e^i_{g+1}] = [e^{i-1}_{g+1}] + \sum_{k=1}^{g-i}[b_{i+k}],
\quad\text{and}\quad
[\overline{e}_i]=[b_i],$$ where $e^1_{g+1}:=e_{g+1}$. The equation on the left is valid for $1\leq i\leq g$, the right is valid for $2\leq i\leq g+1$, and we recall that $\tilde{e}_{g+1}=e^{g+1}_{g+1}$. Similarly, $$[\tilde{e}'_i] = [e_i]+\sum_{k=1}^{g-i}[b_{i+k}],
\quad
[(e^i_{g+1})'] = [(e^{i-1}_{g+1})'] + \sum_{k=1}^{g-i}[b_{i+k}],
\quad\text{and}\quad
[\overline{e}'_i]=-[b_i],$$ with the same conventions. Of course $[h^1_{g+1}]=[(h^1_{g+1})']=[e_{g+1}]$, while $[h^i_{g+1}]=[e^i_{g+1}]$ and $[(h^i_{g+1})']=[(e^i_{g+1})']$ for $2\leq i\leq g$. We also have $[\tilde{h}_i]=[\tilde{e}_i]$ for $1\leq i\leq g$. Now suppose we have a $(k+1)$-level holomorphic building $w_k\cup w_{k-1}\cup\cdots\cup w_0$ in $\partial\mathcal{N}_{\pm}$, with top level $w_k$ and bottom level $w_0$. Let $w^+_i$ and $w^-_i$ denote the sets of Reeb orbits to which $w_i$ is positively and negatively asymptotic, respectively. We denote by $\mathcal{A}(w^{\pm}_i)$ the sum of the $\alpha$-actions of the Reeb orbits in $w^{\pm}_i$ and by $[w^{\pm}_i]$ the sum of their homology classes. Of course we must have $\mathcal{A}(w^-_i)<\mathcal{A}(w^+_i)$ and $[w^-_i]=[w^+_i]$. We also point out that the curves $\overline{u}_{\pm,i}, \overline{u}'_{\pm,i}, \tilde{u}_{\pm,i}$, and $\tilde{u}'_{\pm,i}$ are all disjoint from the curves $u_{\pm}$ and hence, by the positivity of intersections, from our holomorphic building. In particular, these curves are disjoint from each level $w_i$. Moreover, the projections of these curves to $M$ remain disjoint, so for each $i$, the image of $\pi\circ w_i$ is contained in a neighborhood $N_1^j$ or $(N_2^j)'$, for some $j$.\
Now because $u_{\pm}$ is positively asymptotic to $e_1,e_2,\ldots,e_{g+1}$ we know that $$w^+_k \subseteq \{e_1,e_2,\ldots,e_{g+1}\}.$$ We now consider the neighborhoods $N_1^j$ or $(N_2^j)'$ in which $\pi\circ w_k$ might land. First, suppose that $\pi\circ w_k\subset N_1^j$ for some $j>2$. Because $\Sigma^{j-1}_g$ meets $\Sigma_g$ in the curves $e_j,\ldots,e_g$, we have $$w_k^+ \subset \{e_j,\ldots,e_g\}
\quad\text{and}\quad
w_k^- \subset \{e_j,\ldots,e_g,\overline{e}_j,e^j_{g+1},h^j_{g+1},e^{j+1}_{g+1}\}.$$ The action bounds of Lemma \[lemma:description-of-orbits\] allow us to exclude other curves from $w_k^-$. From equation \[eq:new-homology-classes\] we see that the homological requirement $[w_k^+]=[w_k^-]$ can only be satisfied if we have $w_k^+=w_k^-$, and this of course violates the action requirement $\mathcal{A}(w_k^+)>\mathcal{A}(w_k^-)$. We conclude that $\pi\circ w_k$ cannot be contained in $N_1^j$ if $j>1$. A completely analogous argument shows that $\pi\circ w_k$ cannot be contained in $(N_2^j)'$ when $j>1$.\
So $\pi\circ w_k$ is contained in either $N_1^1$ or $(N_2^1)'$. In the first case we see that $$w_k^+ \subset \{e_1,\ldots,e_{g+1}\}
\quad\text{and}\quad
w_k^- \subset \{e_1,\ldots,e_{g+1},\overline{e}_1,h^1_{g+1},e^2_{g+1}\},$$ again using the action bounds of Lemma \[lemma:description-of-orbits\]. The homological requirement $[w_k^+]=[w_k^-]$ then leads us to $$w_k^- = (w_k^+\setminus\{e_{g+1}\})\cup\{h^1_{g+1}\}
\quad\text{or}\quad
w_k^- = (w_k^+\setminus\{e_{g+1}\})\cup\{\overline{e}_1,e^2_{g+1}\}.$$ The latter case is ruled out by part \[part:action-bound\] of Lemma \[lemma:description-of-orbits\] and the fact that $\mathcal{A}(w_k^-)<\mathcal{A}(w_k^+)$. So if $\pi\circ w_k$ is contained in $N_1^1$, then $w_k^-=(w_k^+\setminus\{e_{g+1}\})\cup\{h^1_{g+1}\}$, and similarly if $\pi\circ w_k$ is contained in $(N_2^j)'$, then $w_k^-=(w_k^+\setminus\{e_{g+1}\})\cup\{(h^1_{g+1})'\}$.\
An important observation at this point is that $w_k^-$ contains either $h^1_{g+1}$ or $(h^1_{g+1})'$, and thus so does $w_{k-1}^+$. As with $\pi\circ w_k$, $\pi\circ w_{k-1}$ must be contained in a neighborhood of the form $N_1^j$ or $(N_2^j)'$. Indeed, if $h^1_{g+1}\in w_{k-1}^+$, then $\pi\circ w_{k-1}\subset N_1^1$ and if $(h^1_{g+1})'\in w_{k-1}^+$, then $\pi\circ w_{k-1}\subset (N_2^1)'$. We now consider these two cases.\
If $\pi\circ w_{k-1}$ is contained in $N_1^1$ then $$w^-_{k-1}\subseteq \{e_1,\ldots,e_{g+1},\overline{e}_1,h^1_{g+1},e^2_{g+1}\}.$$ Now $w^+_{k-1}$ must contain $h^1_{g+1}$, must be homologous to $w^-_{k-1}$, and must satisfy $\mathcal{A}(w^-_{k-1})<\mathcal{A}(w^+_{k-1})$. The first two conditions are satisfied if $$w^+_{k-1} = \{e_1,\ldots,e_g,h^1_{g+1}\}
\quad\text{and}\quad
w^-_{k-1} = \emptyset$$ or if $$w^+_{k-1} = \{h^1_{g+1}\} \cup \bar{w}
\quad\text{and}\quad
w^-_{k-1} = \{\bar{e}_1,e^2_{g+1}\} \cup \bar{w}$$ for some $\bar{w}\subseteq \{e_1,\ldots,e_{g}\}$. However, the latter case is prohibited by the action bound, so we conclude that $w^-_{k-1}=\emptyset$, meaning that our building has height two. All that remains is to verify that the top level of our building is a cylinder. To see that this is the case, notice that $w_{k-1}$ must be connected, since $w_{k-1}^+=\{e_1,\ldots,e_g,h^1_{g+1}\}$ and the only null-homologous combination of these positive ends is $e_1+\cdots+e_g+h^1_{g+1}$. So if $w_k$ has more than one negative end, then the building $w_k\cup w_{k-1}$ has nonzero genus. Of course this is impossible, since all of the curves in $\mathcal{M}(e_1,\ldots,e_{g+1})/\mathbb{R}$ are planar. So $w_k$ is a cylinder with positive end $e_{g+1}$ and negative end $h^1_{g+1}$, as desired.\
If instead the image of $\pi\circ w_{k-1}$ is contained in $(N_2^1)'$, then the same considerations lead us to conclude that $w_k$ is a cylinder with positive end $e_{g+1}$ and negative end $(h^1_{g+1})'$, and that $w_{k-1}$ is positively asymptotic to $e_1,\ldots,e_g,h^1_{g+1}$, with no negative ends. We thus define $v_{0,\pm}=w_{k-1}$ and $v_{1,\pm}=w_k$ in the case that $\pi\circ w_{k-1}$ is contained in $N_1^1$ and define $v'_{0,\pm}=w_{k-1}$ and $v'_{1,\pm}=w_{k}$ in the case that $\pi\circ w_{k-1}$ is contained in $(N_2^1)'$.
Now let $\mathcal{M}_{\widehat{W}}(e_1,\ldots,e_g,h^1_{g+1})$ be the index-1 moduli space of holomorphic curves in $\widehat{W}$ which are positively asymptotic to $e_1,\ldots,e_g,h^1_{g+1}$ and represent the same homology class as $v_{0,+}$ or $v_{0,-}$, the curves identified (up to translation) in Lemma \[lemma:boundary-curves\]. The following lemma will allow us to use this moduli space to interpolate between $v_{0,+}$ and $v_{0,-}$, producing what will serve as the middle part of our 1-parameter family.
\[lemma:middle-interval\] One component of the compactification $\overline{\mathcal{M}_{\widehat{W}}(e_1,\ldots,e_g,h^1_{g+1})}$ is a closed interval $I$ with $\partial I=\{v_{0,+},v_{0,-}\}$.
We denote $\mathcal{M}_{\widehat{W}}(e_1,\ldots,e_g,h^1_{g+1})$ by $\mathcal{M}_{\widehat{W}}$ and investigate the objects that could appear in the boundary of the compactification of $\mathcal{M}_{\widehat{W}}$. Because this is an index-1 family, the compactification will not contain any nodal curves, and the only possible boundary elements are holomorphic buildings in the symplectization end of $\widehat{W}$. Suppose we have such a building, and let $$w\colon S^2\setminus\{p_1,\ldots,p_k\}\to\mathbb{R}\times M$$ be its topmost level. As in the proof of Lemma \[lemma:boundary-curves\], the curves $\bar{u}_{\pm,i},\bar{u}'_{\pm,i},\tilde{u}_{\pm,i}$, and $\tilde{u}'_{\pm,i}$ are all disjoint from elements of $\mathcal{M}_{\widehat{W}}$ and hence, by the positivity of intersections, from $w$. So the image of the projection $\pi\circ w$ must be contained in one of the neighborhoods $N_1^j,N_2^j,(N_1^j)',(N_2^j)'$ identified above. We claim that this is only possible if $w$ is positively asymptotic to $e_1,\ldots,e_g,h^1_{g+1}$ and has no negative ends.\
We first show that $\pi\circ w$ cannot be contained in a neighborhood of the form $N_2^j$ or $(N_1^j)'$. To this end, suppose that $\pi\circ w$ is contained in $N_2^j$. Then $$w^+ \subset\{e_1,\ldots,e_g\}
\quad\text{and}\quad
w^- \subset\{e_1,\ldots,e_g,\overline{e}_j,\tilde{e}_j,\tilde{h}_j,e^{j+1}_{g+1}\}.$$ But the homology classes computed in equation \[eq:new-homology-classes\] tell us that curves chosen in this way can only satisfy $[w_k^+]=[w_k^-]$ if in fact $w_k^+=w_k^-$. Of course this violates the inequality $\mathcal{A}(w_k^+)>\mathcal{A}(w_k^-)$, and we see that $\pi\circ w$ cannot be contained in $N_2^j$ for any $j$. The same reasoning shows that $\pi\circ w$ also cannot be contained in a neighborhood of the form $(N_1^j)'$.\
Just as in the proof of Lemma \[lemma:boundary-curves\], the projection $\pi\circ w$ of the topmost level $w$ cannot be contained in $N^j_1$ or $(N_2^j)'$ if $j>1$. These leaves two possibilities — either $\pi\circ w$ is contained in $N_1^1$, or in $(N_2^1)'$ — which we now consider.\
Suppose that the image of $\pi\circ w$ is contained in $N_1^1$, meaning that $$w^+\subseteq \{e_1,\ldots,e_g,h^1_{g+1}\}
\quad\text{and}\quad
w^-\subseteq \{e_1,\ldots,e_{g+1},h^1_{g+1},\bar{e}_1,e^2_{g+1}\}.$$ Again we must have $[w^+]=[w^-]$. Because we could have $[w^+]=0$, it is possible that $w^-$ is empty, and we have a holomorphic building of height one. Suppose this is not the case. Because $[h^1_{g+1}]=[e^1_{g+1}]+[\overline{e}_1]=[e_{g+1}]$, one homological possibility is that as we move from $w^+$ to $w^-$ we replace the curve $h^1_{g+1}$ with $e_{g+1}$ or with $e^2_{g+1}$ and $\overline{e}_1$. That is, if $w^-\neq\emptyset$, then either $w^+=w^-$, $$w^+=\{h^1_{g+1}\}\cup\overline{w}
\quad\text{and}\quad
w^-=\{e_{g+1}\}\cup\overline{w}$$ for some $\overline{w}\subseteq\{e_1,\ldots,e_{g}\}$, or $$w^+=\{h^1_{g+1}\}\cup\overline{w}
\quad\text{and}\quad
w^-=\{e^2_{g+1},\overline{e}_1\}\cup\overline{w}.$$ However, all of these possibilities are prohibited by the action requirement $\mathcal{A}(w^-)<\mathcal{A}(w^+)$. The first possibility obviously violates this requirement, while the second and third do so because $\mathcal{A}(h^1_{g+1})<\mathcal{A}(e_{g+1})<\mathcal{A}(e^2_{g+1})+\mathcal{A}(\overline{e}_1)$. From all of this we conclude that $w^-=\emptyset$ and thus $w$ cannot be the topmost level of a building of height greater than one. The same reasoning shows that if $\pi\circ w$ is contained in $(N_2^1)'$ then $w^-=\emptyset$. So in any case, $w^-$ is empty, and $\pi\circ w$ is contained in either $N_1^1$ or $(N_2^1)'$. But if $\pi\circ w$ is contained in $(N_2^1)'$, then $$w^+\subseteq\{e_1,\ldots,e_{g}\},$$ so we cannot have $[w^+]=0$. So in fact the image of $\pi\circ w$ lies in $N_1^1$, and $w$ has no negative ends.\
So $w$ is a holomorphic curve in the symplectization end of $\widehat{W}$ positively asymptotic to $e_1,\ldots,e_{g},h^1_{g+1}$. In Lemma \[lemma:boundary-curves\] we showed that there are precisely two such curves — $v_{0,+}$ and $v_{0,-}$ — so $w$ must be one of these two. We conclude that $$\partial\overline{\mathcal{M}_{\widehat{W}}(e_1,\ldots,e_g,h^1_{g+1})} = \{v_{0,+},v_{0,-}\}.$$ So $\overline{\mathcal{M}_{\widehat{W}}(e_1,\ldots,e_g,h^1_{g+1})}$ contains the desired component $I$.
\[lemma:1-parameter-family\] There is a 1-parameter family $$\mathcal{S} = \{u_t\colon S^2\setminus\{p_1,\ldots,p_{g+1}\}\to \widehat{W}~|~du_t\circ j=J\circ du_t\}_{t\in\mathbb{R}}$$ of embedded holomorphic curves in $(\widehat{W},\widehat{\omega})$ such that
1. for $t\gg 0$, the images of $u_t$ and $u_{-t}$ are contained in the symplectization part of $\widehat{W}$;\[property:symplectization-part\]
2. for $t\gg 0$, the image of $\pi\circ u_{\pm t}$ is $R_{\pm}(\Sigma_g)$, where $\pi\colon[0,\infty)\times M\to M$ is the obvious projection;\[property:projections\]
3. the images of $u_{t_1}$ and $u_{t_2}$ are disjoint whenever $t_1\neq t_2$.\[property:disjoint-images\]
Consider the interval $I$ given by Lemma \[lemma:middle-interval\]. We take this interval to be the “middle part" of $\mathcal{S}$ and for $t\gg 0$ we take $u_{\pm t}$ to be $v_{0,\pm}$, translated by $t+c$ in the symplectization end $[0,\infty)\times M$, where $c$ is some constant. Property (\[property:symplectization-part\]) follows immediately. Because $v_{0,\pm}$ is positively asymptotic to $h^1_{g+1}$ and not $e_{g+1}$, we must isotope $\Sigma_g$ to ensure that $R_{\pm}(\Sigma_g)=\operatorname{im}(\pi\circ v_{0,\pm})$ and thus satisfy property (\[property:projections\]). Finally, notice that if $t_1\neq t_2$ are large then the images of $u_{t_1}$ and $u_{t_2}$ are disjoint; the positivity of intersections and the homotopy invariance of the intersection number tells us that in fact $u_{t_1}$ and $u_{t_2}$ are disjoint for any $t_1\neq t_2$.
\[lemma:embedded-handlebody\] The map $\iota\colon\mathbb{R}\times (S^2\setminus\{p_1,\ldots,p_{g+1}\})\to \widehat{W}$ defined by $$\iota(t,x) := u_t(x),$$ with $u_t$ as identified in Lemma \[lemma:1-parameter-family\], is an embedding of a genus-$g$ handlebody into $\widehat{W}$.
For an arbitrary $t\in\mathbb{R}$ the curve $u_{t}$ is an embedding and thus each curve $u_{t'}$, for $t'$ near $t$, can be thought of as a section of the normal bundle $N_{u_{t}}$. We can compute the first Chern number of this bundle according to $$c_1(N_{u_{t}}) = c_1(u_{t}^*T\widehat{W}) - \chi(S^2\setminus\{p_1,\ldots,p_{g+1}\}) = c_1(u_t^*T\widehat{W}) + g-1,$$ but first we must compute $c_1(u_{t}^*T\widehat{W})$. For this we appeal to [@wendl2008automatic Equation 1.1], which says that $$2c_1(u_{t}^*T\widehat{W}) = \operatorname{ind}(u_{t}) + \chi(S^2\setminus\{p_1,\ldots,p_{g+1}\}) - \mu_{CZ}(u_{t}),$$ where the last term is a signed count of the Conley-Zehnder indices of the orbits to which $u_{t}$ is asymptotic. Then $$2c_1(u_{t}^*T\widehat{W}) = 1 + (1-g) - g = 2-2g,$$ so $c_1(u_{t}) = 1-g$ and it follows that $c_1(N_{u_t})=0$. So sections of $N_{u_{t}}$ are zero-free, meaning that $\iota$ is an embedding.
The stage is now set for the construction of $(W',\omega')$, the symplectic manifold promised by Theorem \[main-theorem\]. This construction proceeds exactly as in [@menke2018jsj], with small changes to the statements of the lemmas found there. The strategy is to remove from $W$ the handlebody $H\subset\widehat{W}$ embedded by $\iota$ in Lemma \[lemma:embedded-handlebody\]. This is done in stages. First $W$ is enlarged to $W_R:=W\cup([0,R]\times M)$, with $R$ chosen large enough that the projection of $u_{\pm t}$ to $[0,R]\times M$ is $R_{\pm}(\Sigma_g)$ minus a small collar neighborhood whenever $t\gg 0$. From $W_R$ we remove $\tilde{N}(\Gamma_{\Sigma_g})$, a small tubular neighborhood of $\{R\}\times\Gamma_{\Sigma_g}$, leaving us with $W'_R:=W_R-\tilde{N}(\Gamma_{\Sigma_g})$. This allows us to decompose $\partial W'_R$ into its horizontal part $$\label{eq:horizontal-part}
\partial_hW'_R = \partial W'_R - \partial W_R \simeq \bigsqcup_{i=1}^{g+1} (S^1\times D^2)$$ and its vertical part $\partial_vW'_R=\partial W'_R-\partial_hW'_R$, not unlike the boundary of a Lefschetz fibration over a Weinstein domain. Note that the deletion of $\tilde{N}(\Gamma_{\Sigma_g})$ from $W_R$ removes small collar neighborhoods from $\{R\}\times R_{\pm}$, leaving us with $\{R\}\times R'_{\pm}$. We now begin modifying $H$ in preparation for its removal from $W'_R$.
\[lemma:liouville-embedding\] There exists an embedding $\Sigma_L\times[-T,T]\hookrightarrow W'_R$ so that
1. $\Sigma_L$ is a compact surface with genus 0 and $g+1$ boundary components;
2. $\Sigma_L\times\{\pm T\}=\{R\}\times R'_{\pm}$;
3. using the identification given in equation (\[eq:horizontal-part\]) we have\[item:meridians\] $$\partial\Sigma_L\times\{t\} = \bigsqcup_{i=1}^{g+1} (S^1\times\gamma(t)) \subset \partial_hW'_R$$ for $t\in[-T,T]$, where $\gamma(t)$ is the straight arc from $(-1,0)$ to $(1,0)$ in $D^2$.
We denote the embedded copy of $\Sigma_L\times[-T,T]$ by $H'\subset W'_R$ and endow it with the obvious coordinates $(x,t)$. The following two results are proven in [@menke2018jsj] and allow us to cut $W'_R$ along $H'$ to obtain a symplectic manifold $(W',\omega')$ that strongly fills its boundary.
Let $B=[-T,T]\times[-\epsilon,\epsilon]$ with coordinates $(t,w)$. After slight adjustments of $H'$ and $W'_R$, there exists a neighborhood $N(H')\simeq H'\times[-\epsilon,\epsilon]\subset W'_R$ and a 1-form $\lambda = \lambda_B+\lambda_{\Sigma_L}$ on $N(H')$ such that
1. $\Sigma_L\times\{\pm T\}\times[-\epsilon,\epsilon]\subset\partial_vW'_R$ and $(\partial\Sigma_L)\times B\subset\partial_hW'_R$;
2. $\lambda_{\Sigma_L}$ is the Liouville form for $R'_{\pm}$;
3. $\lambda_B=t~dw$;
4. $d\lambda$ is the symplectic form on $W'_R$;
5. $\lambda$ agrees with the Liouville form on $W'_R$ near $\partial W'_R$.
\[lemma:strong-filling\] There exists a modification $$\lambda'=\lambda+d(tw) = 2t~dw + w~dt + \lambda_{\Sigma_L},$$ whose Liouville vector field $Z'=2t\partial_t-w\partial_w+Z_{\Sigma_L}$ points into $N(H')$ along $w=\pm\epsilon$.
At last we define $W':=W'_R-N(H')$ and $\omega':=d\lambda'$ and from Lemma \[lemma:strong-filling\] we conclude that $(W',\omega')$ strongly fills its boundary. In case our original symplectic filling was exact we ask the same of $(W',\omega')$. Once again we may appeal to [@menke2018jsj], where the proof of the following lemma is genus-independent.
If $(W,\omega=d\beta)$ is an exact filling, then there exists a 1-parameter family of Liouville forms $\beta_\tau$, $\tau\in[0,1]$, on $W'_R$ such that $\beta_0=\beta$ and $\beta_1=\lambda'$ on $N(H')\cap\{-\epsilon/2\leq w\leq \epsilon/2\}$.
Let us give an informal summary of the relationship between $\partial W'$ and $M$. The first step in constructing $W'$ was to consider $W_R$, whose boundary is contactomorphic to $M$. From $W_R$ we deleted a neighborhood of the dividing set of $\Sigma_g$. This provided a decomposition of $\partial W'_R$ into its horizontal and vertical parts, but the overall effect on $\partial W_R$ was trivial. The last step in our construction — deleting $N(H')$ from $W_R'$ — made the most substantive changes to the boundary. We first identified $H'$, a handlebody in $W_R'$ which picked out for us two copies of $\Sigma_L$ in $\partial W'_R$. Namely, $H'$ distinguished the Liouville hypersurfaces $\Sigma_L\times\{\pm T\}=\{R\}\times R'_{\pm}$. Then $N(H')$ is a neighborhood of $H'$, part of whose boundary lies in $\partial W'_R$. The part of $\partial N(H')$ lying in the interior of $W'_R$ consists of two disjoint copies of $H'$, and the part lying in $\partial W'_R$ includes $\Sigma_L\times\{\pm T\}$. So deleting $N(H')$ from $W'_R$ cuts $\partial W'_R$ open along the Liouville hypersurfaces $\Sigma_L\times\{\pm T\}$ and glues in two handlebodies modeled on $H'$. This process is depicted in Figure \[fig:boundary-intuition\].\
All that remains is to use symplectic handle attachment to recover $W$ from $W'$. To this end we observe that the neighborhood $(N(H'),d\lambda')$ we have removed from $W'_R$ is precisely the abstract symplectic handle $(H_{\Sigma_L},\omega_{\lambda_{\Sigma_L}})$ constructed from the Liouville domain $(\Sigma_L,\lambda_{\Sigma_L})$. That is, we have obtained $W'$ from $W$ by removing a symplectic handle, and thus may recover $W$ by reattaching said handle as described in Section \[sec:background\].
[^1]: The curves $b_1,\ldots,b_g$ are not canonically oriented, but we fix their orientations according to equation \[eq:new-homology-classes\].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We classify the path-components of the space of circle-valued Morse functions on compact surfaces: two Morse functions ${f}, {g}: {M}\to {S^1}$ belong to same path-component of this space if and only if they are homotopic and have equal numbers of critical points at each index.'
author:
- Sergey Maksymenko
title: 'Connected components of the space of circle-valued Morse functions on surfaces'
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs.'
author:
- 'Peter Keevash[^1]'
title: Hypergraph matchings and designs
---
Introduction
============
Matching theory is a rich and rapidly developing subject that touches on many areas of Mathematics and its applications. Its roots are in the work of Steinitz [@St], Egerváry [@eg], Hall [@hall] and König [@konig] on conditions for matchings in bipartite graphs. After the rise of academic interest in efficient algorithms during the mid 20th century, three cornerstones of matching theory were Kuhn’s [@kuhn] ‘Hungarian’ algorithm for the Assignment Problem, Edmonds’ [@edmonds] algorithm for finding a maximum matching in general (not necessarily bipartite) graphs, and the Gale-Shapley [@GS] algorithm for Stable Marriages. For an introduction to matching theory in graphs we refer to Lovász and Plummer [@LP], and for algorithmic aspects to parts II and III of Schrijver [@schrijver].
There is also a very large literature on matchings in hypergraphs. This article will be mostly concerned with one general direction in this subject, namely to determine conditions under which the necessary ‘geometric’ conditions of ‘space’ and ‘divisibility’ are sufficient for the existence of a perfect matching. We will explain these terms and discuss some aspects of this question in the next two sections, but first, for the remainder of this introduction, we will provide some brief pointers to the literature in some other directions.
We do not expect a simple general characterisation of the perfect matching problem in hypergraphs, as by contrast with the graph case, it is known to be NP-complete even for $3$-graphs (i.e. when all edges have size $3$), indeed, this was one of Karp’s original 21 NP-complete problems [@Karp]. Thus for algorithmic questions related to hypergraph matching, we do not expect optimal solutions, and may instead consider Approximation Algorithms (see e.g. [@WS; @AFS; @LRS]).
Another natural direction is to seek nice sufficient conditions for perfect matchings. There is a large literature in Extremal Combinatorics on results under minimum degree assumptions, known as ‘Dirac-type’ theorems, after the classical result of Dirac [@dirac] that any graph on $n \ge 3$ vertices with minimum degree at least $n/2$ has a Hamiltonian cycle. It is easy to see that $n/2$ is also the minimum degree threshold for a graph on $n$ vertices (with $n$ even) to have a perfect matching, and this exemplifies the considerable similarities between the perfect matching and Hamiltonian problems (but there are also substantial differences). A landmark in the efforts to obtain hypergraph generalisations of Dirac’s theorem was the result of Rödl, Ruciński and Szemerédi [@RRS] that determined the codegree threshold for perfect matchings in uniform hypergraphs; this paper was significant for its proof method as well as the result, as it introduced the Absorbing Method (see section \[sec:ab\]), which is now a very important tool for proving the existence of spanning structures. There is such a large body of work in this direction that it needs several surveys to describe, and indeed these surveys already exist [@RR; @KOlarge; @KOicm; @yi; @Y]. The most fundamental open problem in this area is the Erdős Matching Conjecture [@ematch], namely that the maximum number of edges in an $r$-graph[^2] on $n$ vertices with no matching of size $t$ is either achieved by a clique of size $tr-1$ or the set of all edges hitting some fixed set of size $t-1$ (see [@FT Section 3] for discussion and a summary of progress).
The duality between matching and covers in hypergraphs is of fundamental important in Combinatorics (see [@fur]) and Combinatorial Optimisation (see [@C]). A defining problem for this direction of research within Combinatorics is ‘Ryser’s Conjecture’ (published independently by Henderson [@hen] and Lovász [@L]) that in any $r$-partite $r$-graph the ratio of the covering and matching numbers is at most $r-1$. For $r=2$ this is König’s Theorem. The only other known case is $r=3$, due to Aharoni [@A], using a hypergraph analogue of Hall’s theorem due to Aharoni and Haxell [@AH], which has a topological proof. There are now many applications of topology to hypergraph matching, and more generally ‘independent transversals’ (see the survey [@haxsurvey]). In the other direction, the hypergraph matching complex is now a fundamental object of Combinatorial Topology, with applications to Quillen complexes in Group Theory, Vassiliev knot invariants and Computational Geometry (see the survey [@wachs]).
From the probabilistic viewpoint, there are (at least) two natural questions:
\(i) does a random hypergraph have a perfect matching with high probability (whp)?
\(ii) what does a random matching from a given (hyper)graph look like?
The first question for the usual (binomial) random hypergraph was a longstanding open problem, perhaps first stated by Erdős [@e81] (who attributed it to Shamir), finally solved by Johansson, Kahn and Vu [@JKV]; roughly speaking, the threshold is ‘where it should be’, namely around the edge probability at which with high probability every vertex is in at least one edge. Another such result due to Cooper, Frieze, Molloy and Reed [@CFMR] is that random regular hypergraphs (of fixed degree and edge size) whp have perfect matchings.
The properties of random matchings in lattices have been extensively studied under the umbrella of the ‘dimer model’ (see [@kenyon]) in Statistical Physics. However, rather little is known regarding the typical structure of random matchings in general graphs, let alone hypergraphs. Substantial steps in this direction have been taken by results of Kahn [@Ka2] characterising when the size of a random matching has an approximate normal distribution, and Kahn and Kayll [@KaKa] establishing long-range decay of correlations of edges in random matchings in graphs; the final section of [@Ka2] contains many open problems, including conjectural extensions to simple hypergraphs.
Prequisite to the understanding of random matchings are the closely related questions of Sampling and Approximate Counting (as established in the Markov Chain Monte Carlo framework of Jerrum and Sinclair, see [@J]). An approximate counting result for hypergraph matchings with respect to balanced weight functions was obtained by Barvinok and Samorodnitsky [@BS]. Extremal problems also arise naturally in this context, for the number of matchings, and more generally for other models in Statistical Physics, such as the hardcore model for independent sets. Much of the recent progress here appears in the survey [@yufei], except for the very recent solution of (almost all cases of) the Upper Matching Conjecture of Friedland, Krop and Markström [@FKM] by Davies, Jenssen, Perkins and Roberts [@DJPR].
Suppose $2d \mid n$ and $1 \le k \le n/2$. Let $G$ be any $d$-regular graph on $n$ vertices. Then the number of matchings of size $k$ in $G$ is at most the number in the disjoint union of $n/2d$ copies of $K_{d,d}$. Proved for large $n$ and $k > o(n)$.
Space and divisibility
======================
In this section we discuss a result (joint work with Mycroft [@KM]) that characterises the obstructions to perfect matching in dense hypergraphs (under certain conditions to be discussed below). The obstructions are geometric in nature and are of two types: Space Barriers (metric obstructions) and Divisibility Barriers (arithmetic obstructions).
The simplest illustration of these two phenomena is seen by considering extremal examples for the simple observation mentioned earlier that a graph on $n$ vertices ($n$ even) with minimum degree at least $n/2$ has a perfect matching. One example of a graph with minimum degree $n/2-1$ and no perfect matching is obtained by fixing a set $S$ of $n/2-1$ vertices and taking all edges that intersect $S$. Then in any matching $M$, each edge of $M$ uses at least one vertex of $S$, so $|M| \le |S| < n/2$; there is no ‘space’ for a perfect matching. For another example, suppose $n=2$ mod $4$ and consider the graph that is the disjoint union of two cliques each of size $n/2$ (which is odd). As edges have size $2$, which is even, there is an arithmetic (parity) obstruction to a perfect matching.
There is an analogous parity obstruction to matching in general $r$-graphs, namely an $r$-graph $G$ with vertices partitioned as $(A,B)$, so that $|A|$ is odd and $|e \cap A|$ is even for each edge $e$ of $G$; this is one of the extremal examples for the codegree threshold of perfect matchings (see [@RRS]).
In general, space barriers are constructions for each $1 \le i \le r$, obtained by fixing a set $S$ of size less than $in/r$ and taking the $r$-graph of all edges $e$ with $|e \cap S| \ge i$. Then for any matching $M$ we have $|M| \le |S|/i < n/r$, so $M$ is not perfect.
General divisibility barriers are obtained by fixing a lattice (additive subgroup) $L$ in ${\mathbb{Z}}^d$ for some $d$, fixing a vertex set partitioned as $(V_1,\dots,V_d)$, with $(|V_1|,\dots,|V_d|) \notin L$, and taking the $r$-graph of all edges $e$ such that $(|e \cap V_1|,\dots,|e \cap V_d|) \in L$. For example, the parity obstruction corresponds to the lattice $\{(2x,y): x,y \in {\mathbb{Z}}\}$.
To state the result of [@KM] that is most conveniently applicable we introduce the setting of simplicial complexes and degree sequences. We consider a simplicial complex $J$ on $[n]=\{1,\dots,n\}$, write $J_i = \{e \in J: |e|=i\}$ and look for a perfect matching in the $r$-graph $J_r$. We define the degree sequence $({\delta}_0(J),\dots,{\delta}_{r-1}(J))$ so that each ${\delta}_i(J)$ is the least $m$ such that each $e \in J_i$ is contained in at least $m$ edges of $J_{i+1}$. We define the critical degree sequence ${\delta}^c = ({\delta}^c_0,\dots,{\delta}^c_{r-1})$ by ${\delta}^c_i=(1-i/r)n$. The space barrier constructions show that for each $i$ there is a complex with ${\delta}_i(J)$ slightly less than ${\delta}^c_i$ but no perfect matching. An informal statement of [@KM Theorem 2.9] is that if $J$ is an $r$-complex on $[n]$ (where $r \mid n$) with all ${\delta}_i(J) \ge {\delta}_i^c - o(n)$ such that $J_r$ has no perfect matching then $J$ is close (in edit distance) to a space barrier or a divisibility barrier.
One application of this result (also given in [@KM]) is to determine the exact codegree threshold for packing tetrahedra in $3$-graphs; it was surprising that it was possible to obtain such a result given that the simpler-sounding problems of determining the thresholds (edge or codegree) for the existence of just one tetrahedron are open, even asymptotically (the edge threshrold is a famous conjecture of Turán; for more on Turán problems for hypergraphs see the survey [@Kturan]). Other applications are a multipartite version of the Hajnal-Szemeredi theorem (see [@KM2]) and determining the ‘hardness threshold’ for perfect matchings in dense hypergraphs (see [@KKM; @han]).
We will describe the hardness threshold in more detail, as it illustrates some important features of space and divisibility, and the distinction between perfect matchings and almost perfect matchings. For graphs there is no significant difference in the thresholds for these problems, whereas for general $r$-graphs there is a remarkable contrast: the codegree threshold for perfect matchings [@RRS] is about $n/2$, whereas Han [@han2], proving a conjecture from [@RRS], showed that a minimum codegree of only $n/r$ guarantees a matching of size $n/r-1$, i.e. one less than perfect. The explanation for this contrast is that the divisibility barrier is no obstacle to almost perfect matching, whereas the space barrier is more robust, and can be continuously ‘tuned’ to exclude a matching of specified size.
To illustrate this, we consider a $3$-graph $G_0$ on $[n]$ where the edges are all triples that intersect some fixed set $S$ of size $(1/3 - c)n$, for some small $c>0$. Then the minimum codegree and maximum matching size in $G_0$ are both equal to $|S|$. Furthermore, if we consider $G = G_0 \cup G_1$ where all edges of $G_1$ lie within some $S'$ disjoint from $S$ with $|S'|=3cn$ then $G$ has a perfect matching if and only if $G_1$ has a perfect matching, which is NP-complete to decide for arbitrary $G_1$. Thus the robustness of the space barrier provides a reduction showing that the codegree threshold for the existence of an algorithm for the perfect matching is at least the threshold for an approximate perfect matching.
Now consider the decision problem for perfect matchings in $3$-graphs on $[n]$ (where $3 \mid n$) with minimum codegree at least ${\delta}n$. For ${\delta}<1/3$ the problem is NP-complete, and for ${\delta}>1/2$ it is trivial (there is a perfect matching by [@RRS]). For intermediate ${\delta}$ there is a polynomial-time algorithm, and this is in essence a structural stability result: the main ingredient of the algorithm is a result of [@KKM] that any such $3$-graph with no perfect matching is contained in a divisibility barrier. (For general $r$ the structural characterisation is more complicated.)
Fractional matchings
====================
The key idea of the Absorbing Method [@RRS] mentioned earlier is that the task of finding perfect matchings can often broken into two subproblems: (i) finding almost perfect matchings, (ii) absorbing uncovered vertices into an almost perfect matching until it becomes perfect. We have already seen that the almost perfect matching problem appears naturally as a relaxation of the perfect matching problem in which we eliminate divisibility obstacles but retain space obstacles. This turns out to fit into a more general framework of fractional matchings, in which the relaxed problem is a question of convex geometry, and space barriers correspond to separating hyperplanes.
The fractional (linear programming) relaxation of the perfect matching problem in a hypergraph is to assign non-negative weights to the edges so that for any vertex $v$, there is a total weight of $1$ on all edges incident to $v$. A perfect matching corresponds to a $\{0,1\}$-valued solution, so the existence of a fractional perfect matching is necessary for the existence of a perfect matching. We can adopt a similar point of view regarding divisibility conditions. Indeed, we can similarly define the integer relaxation of the perfect matching problem in which we now require the weights to be integers (not necessarily non-negative); then the existence of an integral perfect matching is necessary for the existence of a perfect matching.
The fractional matching problem appears naturally in Combinatorial Optimisation (see [@C; @schrijver]) because it brings in polyhedral methods and duality to bear on the matching problem. It has also been studied as a problem in its own right from the perspective of random thresholds (e.g. [@DK; @kriv]), and it appears naturally in combinatorial existence problems, as in dense hypergraphs almost perfect matchings and fractional matchings tend to appear at the same threshold. Indeed, for many open problems, such as the Erdős Matching Conjecture [@ematch] or the Nash-Williams Triangle Decomposition Conjecture [@NW], any progress on the fractional problem translates directly into progress on the original problem (see [@BKLO]).
This therefore makes the threshold problem for fractional matchings and decompositions a natural problem in its own right. For example, an asymtotic solution of the Nash-Williams Conjecture would follow from the following conjecture: any graph on $n$ vertices with minimum degree at least $3n/4$ has a fractional triangle decomposition, i.e. an assignment of non-negative weights to its triangles so that for any edge $e$ there is total weight $1$ on the triangles containing $e$. An extremal example $G$ for this question can be obtained by taking a balanced complete bipartite graph $H$ and adding a $(n/4-1)$-regular graph inside each part; indeed, this is a space barrier to a fractional triangle decomposition, as any triangle uses at least one edge not in $H$, but $|H| > 2|G {\setminus}H|$. The best known upper bound is $0.913n$ by Dross [@dross]. More generally, Barber, Kühn, Lo, Montgomery and Osthus [@BKLMO] give the current best known bounds on the thresholds for fractional clique decompositions (in graphs and hypergraphs), but these seem to be far from optimal.
There are (at least) two ways to think about the relationship between almost perfect matchings and fractional matchings. The first goes back to the ‘nibble’ (semi-random) method of Rödl [@R], introduced to solve the Erdős-Hanani [@EH] conjecture on approximate Steiner systems (see the next section), which has since had a great impact on Combinatorics (e.g. [@AKS; @BB; @Boh; @BFL; @BK; @BK2; @FGM; @FR; @Gr; @Ka; @KaLP; @Kim; @KR; @Kuz; @PS; @S; @Vu]). A special case of a theorem of Kahn [@KaLP] is that if there is a fractional perfect matching on the edges of an $r$-graph $G$ on $[n]$ such that for any pair of vertices $x,y$ the total weight on edges containing $\{x,y\}$ is $o(1)$ then $G$ has a matching covering all but $o(n)$ vertices. In this viewpoint, it is natural to interpret the weights of a fractional matching as probabilities, and an almost perfect matching as a random rounding; in fact, this random rounding is obtained iteratively, so there are some parallels with the development of iterative rounding algorithms (see [@LRS]).
Another way to establish the connection between almost perfect matchings and fractional matchings is via the theory of Regularity, developed by Szemerédi [@Sz] for graphs and extended to hypergraphs independently by Gowers [@Gow] and Nagle, Rödl, Schacht and Skokan [@NRS; @RSc1; @RSc2; @RSk]. (The connection was first established by Haxell and Rödl [@HR] for graphs and Rödl, Schacht, Siggers and Tokushige [@RSST] for hypergraphs.) To apply Regularity to obtain spanning structures (such as perfect matchings) requires an accompanying result known as a blowup lemma, after the original such result for graphs obtained by Komlós, Sárközy and Szemerédi [@KSS]; we proved the hypergraph version in [@Kblowup]. More recent developments (for graphs) along these lines include the Sparse Blowup Lemmas of Allen, Böttcher, Hàn, Kohayakawa and Person [@ABHKP] and a blowup-up lemma suitable for decompositions (as in the next section) obtained by Kim, Kühn, Osthus and Tyomkyn [@KKOT] (it would be interesting and valuable to obtain hypergraph versions of these results). The technical difficulties of the Hypergraph Regularity method are a considerable barrier to its widespread application, and preclude us giving here a precise statement of [@KM Theorem 9.1], which informally speaking characterises the perfect matching problem in dense hypergraphs with certain extendability conditions in terms of space and divisibility.
Designs and decompositions
==========================
A *Steiner system* with parameters $(n,q,r)$ is a $q$-graph $G$ on $[n]$ such that any $r$-set of vertices is contained in exactly one edge. For example, a Steiner Triple System on $n$ points has parameters $(n,3,2)$. The question of whether there is a Steiner system with given parameters is one of the oldest problems in combinatorics, dating back to work of Plücker (1835), Kirkman (1846) and Steiner (1853); see [@RobinW] for a historical account.
Note that a Steiner system with parameters $(n,q,r)$ is equivalent to a $K^r_q$-decomposition of $K^r_n$ (the complete $r$-graph on $[n]$). It is also equivalent to a perfect matching in the auxiliary $\tbinom{q}{r}$-graph on $\tbinom{[n]}{r}$ (the $r$-subsets of $[n]:=\{1,\dots,n\}$) with edge set $\{ \tbinom{Q}{r}: Q \in \tbinom{[n]}{q} \}$.
More generally, we say that a set $S$ of $q$-subsets of an $n$-set $X$ is a *design* with parameters $(n,q,r,{\lambda})$ if every $r$-subset of $X$ belongs to exactly ${\lambda}$ elements of $S$. (This is often called an ‘$r$-design’ in the literature.) There are some obvious necessary ‘divisibility conditions’ for the existence of such $S$, namely that $\tbinom{q-i}{r-i}$ divides ${\lambda}\tbinom{n-i}{r-i}$ for every $0 \le i \le r-1$ (fix any $i$-subset $I$ of $X$ and consider the sets in $S$ that contain $I$). It is not known who first advanced the ‘Existence Conjecture’ that the divisibility conditions are also sufficient, apart from a finite number of exceptional $n$ given fixed $q$, $r$ and ${\lambda}$.
The case $r=2$ has received particular attention due to its connections to statistics, under the name of ‘balanced incomplete block designs’. We refer the reader to [@CD] for a summary of the large literature and applications of this field. The Existence Conjecture for $r=2$ was a long-standing open problem, eventually resolved by Wilson [@W1; @W2; @W3] in a series of papers that revolutionised Design Theory. The next significant progress on the general conjecture was in the solution of the two relaxations (fractional and integer) discussed in the previous section (both of which are interesting in their own right and useful for the original problem). We have already mentioned Rödl’s solution of the Erdős-Hanani Conjecture on approximate Steiner systems. The integer relaxation was solved independently by Graver and Jurkat [@GJ] and Wilson [@W4], who showed that the divisibility conditions suffice for the existence of integral designs (this is used in [@W4] to show the existence for large ${\lambda}$ of integral designs with non-negative coefficients). Wilson [@W5] also characterised the existence of integral $H$-decompositions for any $r$-graph $H$.
The existence of designs with $r \ge 7$ and any ‘non-trivial’ ${\lambda}$ was open before the breakthrough result of Teirlinck [@T] confirming this. An improved bound on ${\lambda}$ and a probabilistic method (a local limit theorem for certain random walks in high dimensions) for constructing many other rigid combinatorial structures was recently given by Kuperberg, Lovett and Peled [@KLP]. Ferber, Hod, Krivelevich and Sudakov [@FHKS] gave a construction of ‘almost Steiner systems’, in which every $r$-subset is covered by either one or two $q$-subsets.
In [@Kexist] we proved the Existence Conjecture in general, via a new method of Randomised Algebraic Constructions. Moreover, in [@Kcount] we obtained the following estimate for the number $D(n,q,r,{\lambda})$ of designs with parameters $(n,q,r,{\lambda})$ satisfying the necessary divisibility conditions: writing $Q=\tbinom{q}{r}$ and $N=\tbinom{n-r}{q-r}$, we have $$D(n,q,r,{\lambda}) = {\lambda}!^{-\tbinom{n}{r}}
( ({\lambda}/e)^{Q-1} N + o(N))^{{\lambda}Q^{-1} \tbinom{n}{r}}.$$ Our counting result is complementary to that in [@KLP], as it applies (e.g.) to Steiner Systems, whereas theirs is only applicable to large multiplicities (but also allows the parameters $q$ and $r$ to grow with $n$, and gives an asymptotic formula when applicable).
The upper bound on the number of designs follows from the entropy method pioneered by Radhakrishnan [@rad]; more generally, Luria [@luria] has recently established a similar upper bound on the number of perfect matchings in any regular uniform hypergraph with small codegrees. The lower bound essentially matches the number of choices in the Random Greedy Hypergraph Matching process (see [@BB]) in the auxiliary $Q$-graph defined above, so the key to the proof is showing that this process can be stopped so that whp it is possible to complete the partial matching thus obtained to a perfect matching. In other words, instead of a design, which can be viewed as a $K^r_q$-decomposition of the $r$-multigraph ${\lambda}K^r_n$, we require a $K^r_q$-decomposition of some sparse submultigraph, that satisfies the necessary divisibility conditions, and has certain pseudorandomness properties (guaranteed whp by the random process).
The main result of [@Kexist] achieved this, and indeed (in the second version of the paper) we obtained a more general result in the same spirit as [@KM], namely that we can find a clique decomposition of any $r$-multigraph with a certain ‘extendability’ property that satisfies the divisibility conditions and has a ‘suitably robust’ fractional clique decomposition.
Glock, Kühn, Lo and Osthus [@GKLO; @GKLO2] have recently given a new proof of the existence of designs, as well as some generalisations, including the existence of $H$-decompositions for any hypergraph $H$ (a question from [@Kexist]), relaxing the quasirandomness condition from [@Kexist] (version 1) to an extendability condition in the same spirit as [@Kexist] (version 2), and a more effective bound than that in [@Kexist] on the minimum codegree decomposition threshold; the main difference in our approaches lies in the treatment of absorption (see the next section).
Absorption {#sec:ab}
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Over the next three sections we will sketch some approaches to what is often the most difficult part of a hypergraph matching or decomposition problem, namely converting an approximate solution into an exact solution. We start by illustrating the Absorbing Method in its original form, namely the determination in [@RRS] of the codegree threshold for perfect matchings in $r$-graphs; for simplicity we consider $r=3$.
We start by solving the almost perfect matching problem. Let $G$ be a $3$-graph on $[n]$ with $3 \mid n$ and minimum codegree ${\delta}(G)=n/3$, i.e. every pair of vertices is in at least $n/3$ edges. We show that $G$ has a matching of size $n/3-1$ (i.e. one less than perfect). To see this, consider a maximum size matching $M$, let $V_0 = V(G) {\setminus}V(M)$, and suppose $|V_0|>3$. Then $|V_0| \ge 6$, so we can fix disjoint pairs $a_1b_1$, $a_2b_2$, $a_3b_3$ in $V_0$. For each $i$ there are at least $n/3$ choices of $c$ such that $a_ib_ic \in E(G)$, and by maximality of $M$ any such $c$ lies in $V(M)$. We define the weight $w_e$ of each $e \in M$ as the number of edges of $G$ of the form $a_ib_ic$ with $c \in e$. Then $\sum_{e \in M} w_e \ge n$, and $|M|<n/3$, so there is $e \in M$ with $w_e \ge 4$. Then there must be distinct $c,c'$ in $e$ and distinct $i,i'$ in $[3]$ such that $a_ib_ic$ and $a_{i'}b_{i'}c'$ are edges. However, deleting $e$ and adding these edges contradicts maximality of $M$.
Now suppose ${\delta}(G)=n/2 + cn$, where $c>0$ and $n>n_0(c)$ is large. Our plan for finding a perfect matching is to first put aside an ‘absorber’ $A$, which will be a matching in $G$ with the property that for any triple $T$ in $V(G)$ there is some edge $e \in A$ such that $T \cup e$ can be expressed as the disjoint union of two edges in $G$ (then we say that $e$ absorbs $T$). Suppose that we can find such $A$, say with $|A| < n/20$. Deleting the vertices of $A$ leaves a $3$-graph $G'$ on $n'=n-|A|$ vertices with ${\delta}(G') \ge {\delta}(G)-3|A| > n'/3$. As shown above, $G'$ has a matching $M'$ with $|M'| = n'/3-1$. Let $T = V(G') {\setminus}V(M')$. By choice of $A$ there is $e \in A$ such that $T \cup e = e_1 \cup e_2$ for some disjoint edges $e_1,e_2$ in $G$. Then $M' \cup (A {\setminus}\{e\}) \cup \{e_1,e_2\}$ is a perfect matching in $G$.
It remains to find $A$. The key idea is that for any triple $T$ there are many edges in $G$ that absorb $T$, and so if $A$ is random then whp many of them will be present. We can bound the number of absorbers for any triple $T=xyz$ by choosing vertices sequentially. Say we want to choose an edge $e=x'y'z'$ so that $x'yz$ and $xy'z'$ are also edges. There are at least $n/2 + cn$ choices for $x'$ so that $x'yz$ is an edge. Then for each of the $n-4$ choices of $y' \in V(G) {\setminus}\{x,y,z,x'\}$ there are at least $2cn - 1$ choices for $z' \ne z$ so that $x'y'z'$ and $xy'z'$ are edges. Multiplying the choices we see that $T$ has at least $cn^3$ absorbers.
Now suppose that we construct $A$ by choosing each edge of $G$ independently with probability $c/(4n^2)$ and deleting any pair that intersect. Let $X$ be the number of deleted edges. There are fewer than $n^5$ pairs of edges that intersect, so ${\mathbb{E}}X < c^2 n/16$, so ${\mathbb{P}}(X < c^2 n/8) \ge 1/2$. Also, the number of chosen absorbers $N_T$ for any triple $T$ is binomial with mean at least $c^2 n/4$, so whp all $N_T > c^2 n/8$. Thus there is a choice of $A$ such that every $T$ has an absorber in $A$. This completes the proof of the approximate version of [@RRS], i.e. that minimum codegree $n/2 + cn$ guarantees a perfect matching.
The idea for the exact result is to consider an attempt to construct absorbers as above under the weaker assumption ${\delta}(G) \ge n/2 - o(n)$. It is not hard to see that absorbers exist unless $G$ is close to one of the extremal examples. The remainder of the proof (which we omit) is then a stability analysis to show that the extremal examples are locally optimal, and so optimal.
In the following two sections we will illustrate two approaches to absorption for designs and hypergraph decompositions, in the special case of triangle decompositions of graphs, which is considerably simler, and so allows us to briefly illustrate some (but not all) ideas needed for the general case. First we will conclude this section by indicating why the basic method described above does not suffice.
Suppose we seek a triangle decomposition of a graph $G$ on $[n]$ with $e(G) = {\Omega}(n^2)$ in which there is no space or divisibility obstruction: we assume that $G$ is ‘tridivisible’ (meaning that $3 \mid e(G)$ and all degrees are even) and ‘triangle-regular’ (meaning that there is a set $T$ of triangles in $G$ such that every edge is in $(1+o(1))tn$ triangles of $T$, where $t>0$ and $n>n_0(t)$). This is equivalent to a perfect matching in the auxiliary $3$-graph $H$ with $V(H)=E(G)$ and $E(H) = \{\{ab,bc,ca\}: abc \in T\}$. Note that $H$ is ‘sparse’: we have $e(H) = O(v(H)^{3/2})$. Triangle regularity implies that Pippenger’s generalisation (see [@PS]) of the Rödl nibble can be applied to give an almost perfect matching in $H$, so the outstanding question is whether there is an absorber.
Let us consider a potential random construction of an absorber $A$ in $H$. It will contain at most $O(n^2)$ triangles, so the probability of any triangle (assuming no heavy bias) will be $O(n^{-1})$. On the other hand, to absorb some fixed (tridivisible) $S {\subseteq}E(G)$, we need $A$ to contain a set $A_S$ of $a$ edge-disjoint triangles (for some constant $a$) such that $S \cup A_S$ has a triangle decomposition $B_S$, so we need ${\Omega}(n^a)$ such $A_S$ in $G$. To see that this is impossible, we imagine selecting the triangles of $A_S$ one at a time and keeping track of the number $E_S$ of edges that belong to a unique triangle of $S \cup A_S$. If a triangle uses a vertex that has not been used previously then it increases $E_S$, and otherwise it decreases $E_S$ by at most $3$. We can assume that no triangle is used in both $A_S$ and $B_S$, so we terminate with $E_S=0$. Thus there can be at most $3a/4$ steps in which $E_S$ increases, so there are only $O(n^{3a/4})$ such $A_S$ in $G$.
The two ideas discussed below to overcoming this obstacle can be briefly summarised as follows. In Randomised Algebraic Construction (introduced in [@Kexist]), instead of choosing independent random triangles for an absorber, they are randomly chosen according to a superimposed algebraic structure that has ‘built-in’ absorbers. In Iterative Absorption (used for designs and decompositions in [@GKLO; @GKLO2]), instead of a single absorption step, there is a sequence of absorptions, each of which replaces the current subgraph of uncovered edges by an ‘easier’ subgraph, until we obtain $S$ that is so simple that it can be absorbed by an ‘exclusive’ absorber put aside at the beginning of the proof for the eventuality that we end up with $S$. This is a powerful idea with many other applications (see the survey [@KOicm]).
Iterative Absorption
====================
Here we will sketch an application of iterative absorption to finding a triangle decomposition of a graph $G$ with no space or divisibility obstruction as in the previous subsection. (We also make certain ‘extendability’ assumptions that we will describe later when they are needed.) Our sketch will be loosely based on a mixture of the methods used in [@BKLO] and [@GKLO], thus illustrating some ideas of the general case but omitting most of the technicalities.
The plan for the decomposition is to push the graph down a ‘vortex’, which consists of a nested sequence $V(G) = V_0 {\supseteq}V_1 {\supseteq}\cdots {\supseteq}V_\tau$, where $|V_i|={\theta}|V_{i-1}|$ for each $i \in [\tau]$ with $n^{-1} \ll {\theta}\ll t$, and $|V_\tau|=O(1)$ (so $\tau$ is logarithmic in $n=v(G)$). Suppose $G$ has a set $T$ of triangles such that every edge is in $(1 \pm c)tn$ triangles of $T$, where $n^{-1} \ll {\theta}\ll c, t$. By choosing the $V_i$ randomly we can ensure that each edge of $G[V_i]$ is in $(1 \pm 2c)t|V_i|$ triangles of $T_i = \{ f \in T: f {\subseteq}V_i \}$. At step $i$ with $0 \le i \le t$ we will have covered all edges of $G$ not contained in $V_i$ by edge-disjoint triangles, and also some edges within $V_i$, in a suitably controlled manner, so that we still have good triangle regularity in $G[V_i]$.
At the end of the process, the uncovered subgraph $L$ will be contained in $V_\tau$, so there are only constantly many possibilities for $L$. Before starting the process, for each tridivisible subgraph $S$ of the complete graph on $V_\tau$ we put aside edge-disjoint ‘exclusive absorbers’ $A_S$, i.e. sets of edge-disjoint triangles in $G$ such that $S \cup A_S$ has a triangle decomposition $B_S$ (we omit here the details of this construction). Then $L$ will be equal to one of these $S$, so replacing $A_S$ by $B_S$ completes the triangle decomposition of $G$.
Let us now consider the process of pushing $G$ down the vortex; for simplicity of notation we describe the first step of covering all edges not within $V_1$. The plan is to cover almost all of these edges by a nibble, and then the remainder by a random greedy algorithm (which will also use some edges within $V_1$). At first sight this idea sounds suspicious, as one would think that the triangle regularity parameter $c$ must increase substantially at each step, and so the process could not be iterated logarithmically many times before the parameters blow up.
However, quite suprisingly, if we make the additional extendability assumption that every edge is in at least $c' n^3$ copies of $K_5$ (where $c'$ is large compared with $c$ and $t \ge c'$), then we can pass to a different set of triangles which dramatically ‘boost’ the regularity. The idea (see [@GKLO Lemma 6.3]) is that a relatively weak triangle regularity assumption implies the existence of a perfect fractional triangle decomposition, which can be interpreted as selection probabilities (in the same spirit as [@KaLP]) for a new set of triangles that is much more regular. A similar idea appears in the Rödl-Schacht proof of the hypergraph regularity lemma via regular approximation (see [@RSc1]). It may also be viewed as a ‘guided version’ of the self-correction that appears naturally in random greedy algorithms (see [@BK2; @FGM]).
Let us then consider $G^* = G {\setminus}G[V_1] {\setminus}H$, where $H$ contains each edge of $G$ crossing between $V_1$ and $V^* := V(G) {\setminus}V_1$ independently with some small probability $p \ll c,{\theta}$. (We reserve $H$ to help with the covering step.) Then whp every edge of $G^*$ is in $(1 \pm c) tn \pm |V_1| \pm 3pn$ triangles of $T$ within $G^*$. By boosting, we can find a set $T^*$ of triangles in $G^*$ such that every edge of $G^*$ is in $(1 \pm c_0) tn/2$ triangles of $T^*$, where $c_0 \ll p$. By the nibble, we can choose a set of edge-disjoint triangles in $T^*$ covering all of $G^*$ except for some ‘leave’ $L$ of maximum degree $c_1 n$, where we introduce new constants $c_0 \ll c_1 \ll c_2 \ll p$.
Now we cover $L$ by two random greedy algorithms, the first to cover all remaining edges in $V^*$ and the second to cover all remaining cross edges. The analysis of these algorithms is not as difficult as that of the nibble, as we have ‘plenty of space’, in that we only have to cover a sparse graph within a much denser graph, whereas the nibble seeks to cover almost all of a graph. In particular, the behaviour of these algorithms is well-approximated by a ‘binomial heuristic’ in which we imagine choosing random triangles to cover the uncovered edges without worrying about whether these triangles are edge-disjoint (so we make independent choices for each edge). In the actual algorithm we have to exclude any triangle that uses an edge covered by a previous step of the algorithm, but if we are covering a sparse graph one can show that whp at most half (say) of the choices are forbidden at each step, so any whp estimate in the binomial process will hold in the actual process up to a factor of two. (This idea gives a much simpler proof of the result of [@FHKS].)
For the first greedy algorithm we consider each remaining edge in $V^*$ in some arbitrary order, and when we consider $e$ we choose a triangle on $e$ whose two other edges are in $H$, and have not been previously covered. In general we would make this choice uniformly at random, although the triangle case is sufficiently simple that an arbitrary choice suffices; indeed, there are whp at least $p^2 {\theta}n/2$ such triangles in $H$, of which at most $2c_1 n$ are forbidden due to using a previously covered edge (by the maximum degree of $L$). Thus the algorithm can be completed with arbitrary choices.
The second greedy algorithm for covering the cross edges is more interesting (the analogous part of the proof for general designs is the most difficult part of [@GKLO]). Let $H'$ denote the subgraph of cross edges that are still uncovered. We consider each $x \in V^*$ sequentially and cover all edges of $H'$ incident to $x$ by the set of triangles obtained by adding $x$ to each edge of a perfect matching $M_x$ in $G[H'(x)]$, i.e.the restriction of $G$ to the $H'$-neighbourhood of $x$. We must choose $M_x$ edge-disjoint from $M_{x'}$ for all $x'$ preceding $x$, so an arbitrary choice will not work; indeed, whp the degree of each vertex $y$ in $G[H'(x)]$ is $(1 \pm c_2) p{\theta}tn$, but our upper bound on the degree of $y$ in $H'$ may be no better than $pn$, so previous choices of $M_{x'}$ could isolate $y$ in $G[H'(x)]$.
To circumvent this issue we choose random perfect matchings. A uniformly random choice would work, but it is easier to analyse the process where we fix many edge-disjoint matchings in $G[H'(x)]$ and then choose one uniformly at random to be $M_x$. We need some additional assumption to guarantee that $G[H'(x)]$ has even one perfect matching (the approximate regularity only guarantees an almost perfect matching).
One way to achieve this is to make the additional mild extendability assumption that every pair of vertices have at least $c' n$ common neighbours in $G[H'(x)]$, i.e. any adjacent pair of edges $xy, xy'$ in $G$ have at least $c' n$ choices of $z$ such that $xz$, $yz$ and $y'z$ are edges. It is then not hard to see that a random balanced bipartite subgraph of $G[H'(x)]$ whp satisfies Hall’s condition for a perfect matching. Moreover, we can repeatedly delete $p^{3/2} {\theta}c'n$ perfect matchings in $G[H'(x)]$, as this maintains all degrees $(1 \pm 2\sqrt{p}) p{\theta}tn$ and codegrees at least $p{\theta}c'n/2$.
The punchline is that for any edge $e$ in $G[H'(x)]$ there are whp at most $2p^2 n$ earlier choices of $x'$ with $e$ in $G[H'(x')]$, and the random choice of $M_{x'}$ covers $e$ with probability at most $(p^{3/2} {\theta}c'n)^{-1}$, so $e$ is covered with probability at most $2p^2 n (p^{3/2} {\theta}c'n)^{-1} < p^{1/3}$, say. Thus whp $G[H'(x)]$ still has sufficient degree and codegree properties to find the perfect matchings described above, and the algorithm can be completed. Moreover, any edge of $G[V_1]$ is covered with probability at most $p^{1/3}$, so whp we maintain good triangle regularity in $G[V_1]$ and can proceed down the vortex.
Randomised Algebraic Construction
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Here we sketch an alternative proof (via our method of Randomised Algebraic Construction from [@Kexist]) of the same result as in the previous subsection, i.e. finding a triangle decomposition of a graph $G$ with certain extendability properties and no space or divisibility obstruction. Our approach will be quite similar to that in [@Kcount], except that we will illustrate the ‘cascade’ approach to absorption which is more useful for general designs.
As discussed above, we circumvent the difficulties in the basic method for absorption by introducing an algebraic structure with built-in absorbers. Let $\pi:V(G) \to {\mathbb{F}}_{2^a} {\setminus}\{0\}$ be a uniformly random injection, where $2^{a-2} < n \le 2^{a-1}$. Our absorber (which in this context we call the ‘template’) is defined as the set $T$ of all triangles in $G$ such that $\pi(x)+\pi(y)+\pi(z)=0$. Clearly $T$ consists of edge-disjoint triangles. We let $G^* = \bigcup T$ be the underlying graph of the template and suppress $\pi$, imagining $V(G)$ as a subset of ${\mathbb{F}}_{2^a}$.
Standard concentration arguments show that whp $G {\setminus}G^*$ has the necessary properties to apply the nibble, so we can find a set $N$ of edge-disjoint triangles in $G {\setminus}G^*$ with leave $L := (G {\setminus}G^*) {\setminus}\bigcup N$ of maximum degree $c_1 n$ (we use a similar hierarchy of very small parameters $c_i$ as before). To absorb $L$, it is convenient to first ‘move the problem’ into the template: we apply a random greedy algorithm to cover $L$ by a set $M^c$ of edge-disjoint triangles, each of which has one edge in $L$ and the other two edges in $G^*$. Thus some subgraph $S$ of $G^*$, which we call the ‘spill’ has now been covered twice. The binomial heuristic discussed in the previous subsection applies to show that whp this algorithm is successful, and moreover $S$ is suitably bounded. (To be precise, we also ensure that each edge of $S$ belongs to a different triangle of $T$, and that the union $S^*$ of all such triangles is $c_2$-bounded.)
The remaining task of the proof is to modify the current set of triangles to eliminate the problem with the spill. The overall plan is to find a ‘hole’ in the template that exactly matches the spill. This will consist of two sets of edge-disjoint triangles, namely $M^o$ (outer set) and $M^i$ (inner set), such that $M^o {\subseteq}T$ and $\bigcup M^o$ is the disjoint union of $S$ and $\bigcup M^i$. Then replacing $M^o$ by $M^i$ will fix the problem: formally, our final triangle decomposition of $G$ is $M := N \cup M^c \cup (T {\setminus}M^o) \cup M^i$.
We break down the task of finding the hole into several steps. The first is a refined form of the integral decomposition theorem of [@GJ; @W4], i.e. that there is an assignment of integers to triangles so that total weight of triangles on any edge $e$ is $1$ if $e \in S$ or $0$ otherwise. Our final hole can be viewed as such an assignment, in which a triangle $f$ has weight $1$ if $f \in M^o$, $-1$ if $f \in M^i$, or $0$ otherwise. We intend to start from some assignment and repeatedly modify it by random greedy algorithms until it has the properties required for the hole. As discussed above, the success of such random greedy algorithms requires control on the maximum degree, so our refined version of [@GJ; @W4] is that we can choose the weights $w_T$ on triangles with $\sum_{T: v \in T} |w_T| < c_3 n$ for every vertex $v$. (The proof is fairly simple, but the analogous statement for general hypergraphs seems to be much harder to prove.) Note that in this step we allow the use of any triangle in $K_n$ (the complete graph on $V(G)$), without considering whether they belong to $G^*$: ‘illegal’ triangles will be eliminated later.
Let us now consider how to modify assignments of weights to triangles so as to obtain a hole. Our first step is to ignore the requirement $M^o {\subseteq}T$, which makes our task much easier, as $T$ is a special set of only $O(n^2)$ triangles. Thus we seek a signed decomposition of $S$ within $G^*$, i.e. an assignment from $\{-1,0,1\}$ to each triangle of $G^*$ so that the total weight on any $e$ is $1$ if $e \in S$ or $0$ otherwise, and every edge appears in at most one triangle of each sign.
To achieve this, we start from the simple observation that the graph of the octahedron has $8$ triangles, which can be split into two groups of $4$, each forming a triangle decomposition. For any copy of the octahedron in $K_n$ we can add $1$ to the triangles of one decomposition and subtract $1$ from the triangles of the other without affecting the total weight of triangles on any edge. We can use this construction to repeatedly eliminate ‘cancelling pairs’, consisting of two triangles on a common edge with opposite sign. (There is a preprocessing step to ensure that each triangle to be eliminated can be assigned to a unique such pair.) In particular, as edges not in $G^*$ have weight $0$, this will eliminate all illegal triangles. The boundedness condition facilitates a random greedy algorithm for choosing edge-disjoint octahedra for these eliminations, which constructs the desired signed decomposition of $S$.
Now we remember that we wanted the outer triangle decomposition $M^o$ to be contained in the template $T$. Finally, the algebraic structure will come into play, in absorbing the set $M^+$ of positive triangles in the signed decomposition. To see how this can be achieved, consider any positive triangle $xyz$, recall that vertices are labelled by elements of ${\mathbb{F}}_{2^a} {\setminus}\{0\}$, and suppose first for simplicity that $xyz$ is ‘octahedral’, meaning that $G^*$ contains the ‘associated octahedron’ of $xyz$, defined as the complete $3$-partite graph $O$ with parts $\{x,y+z\}$, $\{y,z+x\}$, $\{z,x+y\}$. Then $xyz$ is a triangle of $O$, and we note that $O$ has a triangle decomposition consisting entirely of template triangles, namely $\{x,y,x+y\}$, $\{y+z,y,z\}$, $\{x,z+x,z\}$ and $\{y+z,z+x,x+y\}$. Thus we can ‘flip’ $O$ (i.e. add and subtract the two triangle decompositions as before) to eliminate $xyz$ while only introducing positive triangles that are in $T$.
The approach taken in [@Kcount] was to ensure in the signed decomposition that every positive triangle is octahedral, with edge-disjoint associated octahedra, so that all positive triangles can be absorbed as indicated above without interfering with each other. For general designs, it is more convenient to define a wider class of triangles (in general hypergraph cliques) that can be absorbed by the following two step process, which we call a ‘cascade’. Suppose that we want to absorb some positive triangle $xyz$. We look for some octahedron $O$ with parts $\{x,x'\}$, $\{y,y'\}$, $\{z,z'\}$ such that each of the $4$ triangles of the decomposition not using $xyz$ is octahedral. We can flip the associated octahedra of these triangles so as to include them in the template, and now $O$ is decomposed by template triangles, so can play the role of an associated octahedron for $xyz$: we can flip it to absorb $xyz$. The advantage of this approach is that whp any non-template $xyz$ has many cascades, so no extra property of the signed decomposition is required to complete the proof. In general, there are still some conditions required for a clique have many cascades, but these are not difficult to ensure in the signed decomposition.
Concluding remarks
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There are many other questions of Design Theory that can be reformulated as asking whether a certain (sparse) hypergraph has a perfect matching. This suggests the (vague) meta-question of formulating and proving a general theorem on the existence of perfect matchings in sparse ‘design-like’ hypergraphs (for some ‘natural’ definition of ‘design-like’ that is sufficiently general to capture a variety of problems in Design Theory). One test for such a statement is that it should capture all variant forms of the basic existence question, such as general hypergraph decompositions (as in [@GKLO]) or resolvable designs (the general form of Kirkman’s original ‘schoolgirl problem’, solved for graphs by Ray-Chaudhuri and Wilson [@RW]).
In [@K2] we generalised the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes. This general framework includes the problem of decomposing hypergraphs with extra edge data, such as colours and orders, and so incorporates a wide range of variations on the basic design problem, notably Baranyai-type generalisations, such as resolvable hypergraph designs, large sets of hypergraph designs and decompositions of designs by designs. Our method also gives approximate counting results, which is new for many structures whose existence was previously known, such as high dimensional permutations or Sudoku squares. For an exposition of these results and further applications, see [@KLovasz70].
Could we be even more ambitious? To focus the ideas, one well-known longstanding open problem is Ryser’s Conjecture [@ryser] that every Latin square of odd order has a transversal. (A generalised form of this conjecture by Stein [@stein] was recently disproved by Pokrovskiy and Sudakov [@PoSu].) To see the connection with hypergraph matchings, we associate to any Latin square a tripartite $3$-graph in which the parts correspond to rows, columns and symbols, and each cell of the square corresponds to an edge consisting of its own row, column and symbol. A perfect matching in this $3$-graph is precisely a transversal of the Latin square. However, there is no obvious common structure to the various possible $3$-graphs that may arise in this way, which presents a challenge to the absorbing methods described in this article, and so to formulating a meta-theorem that might apply to Ryser’s Conjecture. The best known lower bound of $n-O(\log^2n)$ on a partial transversal (by Hatami and Shor [@HS]) has a rather different proof. Another generalisation of Ryser’s Conjecture by Aharoni and Berger [@AB] concerning rainbow matchings in properly coloured multigraphs has recently motivated the development of various other methods for such problems not discussed in this article (see e.g. [@grww; @KY; @Po]).
Recalling the theme of random matchings discussed in the introduction, it is unsurprising that it is hard to say much about random designs, but for certain applications one can extract enough from the proof in [@Kexist], e.g. to show that whp a random Steiner Triple System has a perfect matching (Kwan [@Kw]) or that one can superimpose a constant number of Steiner Systems to obtain a bounded codegree high-dimensional expander (Lubotzky, Luria and Rosenthal [@LLR]). Does the nascent connection between hypergraph matchings and high-dimensional expanders go deeper?
We conclude by recalling two longstanding open problems from the other end of the Design Theory spectrum, concerning $q$-graphs with $q$ of order $\sqrt{n}$ (the maximum possible), as opposed to the setting $n>n_0(q)$ considered in this article (or even the methods of [@KLP] which can allow $q$ to grow as a sufficiently small power of $n$).
[[**Hadamard’s Conjecture.**]{}]{} ([@had])\
There is an $n \times n$ orthogonal matrix $H$ with all entries $\pm n^{-1/2}$ iff $n$ is $1$, $2$ or divisible by $4$?
[[**Projective Plane Prime Power Conjecture.**]{}]{} (folklore)\
There is a Steiner system with parameters $(k^2+k+1,k+1,2)$ iff $k$ is a prime power?
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[^2]: An $r$-graph is a hypergraph in which every edge contains $r$ vertices.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We derive the bifurcation set for a not previously considered three-parametric Bogdanov-Takens unfolding, showing that it is possible express its vector field as two different perturbed cubic Hamiltonians. By using several first-order Melnikov functions, we obtain for the first time analytical approximations for the bifurcation curves corresponding to homoclinic and heteroclinic connections, which along with the curves associated to local bifurcations organize the parametric regions with different structures of periodic orbits.
As an application of these results, we study a family of 3D memristor oscillators, for which the characteristic function of the memristor is a cubic polynomial. We show that these systems have an infinity number of invariant manifolds, and by adding one parameter that stratifies the 3D dynamics of the family, it is shown that the dynamics in each stratum is topologically equivalent to a representant of the above unfolding. Also, based upon the bifurcation set obtained, we show the existence of closed surfaces in the 3D state space which are foliated by periodic orbits. Finally, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role played by the existence of invariant manifolds.
author:
- 'Andrés Amador$^{1}$ and Emilio Freire$^{2}$ and Enrique Ponce$^{3}$'
title: 'Bifurcation set for a disregarded Bogdanov-Takens unfolding. Application to 3D cubic memristor oscillators.'
---
[^1]
Introduction
============
In planar systems, the existence of some local bifurcations may reveal the presence of other bifurcations of global character [@dumortier1987; @dumortier1991] and the curves that determine these global phenomena are difficult to determine. This is, for instance, the case regarding the appearance of homoclinic or heteroclinic connections.
A homoclinic connection is an orbit of the system that joins a saddle equilibrium point to itself, and generally creates or destroys periodic orbits (see, for instance [@wiggins2003]). A heteroclinic connection joins two different equilibrium points of a system and the existence of this connection can determine changes in the basin of attraction of a positively invariant set.
Following [@freire2000], the techniques to study homoclinic orbits in planar vector fields were well developed during the 1920s in the works of Dulac. The fundamental idea is that the recurrent behavior near a connecting orbit should be studied in a fashion similar to that used in studying periodic orbits via a Poincaré return map. But there are some additional complications in the study of homoclinic orbits compared to that of periodic orbits which significantly complicate the analysis.
Usually, the bifurcation curves of homoclinic and heteroclinic connections are studied by numerical continuation techniques [@freire1999; @freire2000; @matcont2008; @matcont2012]. On the other hand, when a planar system can be written as a perturbed Hamiltonian system, we can calculate some Melnikov functions, introduced by Melnikov in [@melnikov63], and under certain hypotheses, the zeros of the associated Melnikov function determine the existence of periodic orbits, homoclinic loops or heteroclinic connections, see for instance [@dangelmayr1987; @guckenheimer1989; @perko1994].
As show later, we will resort to such Melnikov functions for setting information on such global bifurcations curves in a two-parametric plane for a specific family of differential systems.
The normal form of the Bogdanov-Takens (see [@wiggins2003]) bifurcation is given by $$\dot{x}=y,\quad\dot{y}=\mu_{1}+\mu_{2}x+x^{2}\pm xy.$$ Following the classification proposed in [@dumortier1991], the deformation of codimension three of the previous normal form is given by the unfolding $$%
\begin{split}
\dot{x} & =y,\\
\dot{y} & =\mu_{1}+\mu_{2}x+\alpha x^{3}+y(\mu_{3}+\mu_{4}x\pm x^{2}),
\end{split}
\label{unfolding}%$$ The presence of this type of systems has been reported in different applications, see [@freire2005; @kuznetsov2016; @kong2017]. On the study of bifurcation phenomena in these systems many contributions have been made. In [@freire1996b], the authors studied the global bifurcation diagram of the three-parameter family$$\begin{aligned}
\dot{x} & =y,\\
\dot{y} & =\mu_{1}+\mu_{2}x-x^{3}+y(\mu_{3}-3x^{2}),\end{aligned}$$ and fixing $\mu_{3}>0$, they obtained analytical approximations to the bifurcation curves of the homoclinic orbits, by using Melnikov functions. Later on, the above work was quoted in [@khibnik1998], where a numerical analysis of the same model was performed. In [@dumortier2001; @dumortier2003b; @dumortier2003], it is considered the system $$%
\begin{split}
\dot{x} & =y,\\
\dot{y} & =\mu_{1}+\mu_{2}x-x^{3}+y(\mu_{3}+\mu_{4}x-x^{2}),
\end{split}
\label{sys-aux}%$$ and the authors showed that it can be written as a perturbed Hamiltonian system, reporting the maximum number of limit cycles. Later, by taking the parameter $\mu_{4}=0$ in , the authors in [@chen2015; @chen2016; @Hebai2017; @Hebai2018] analyzed the system as a Liénard system, its local bifurcations were characterized, and a numerical study of the global bifurcations was done.
While all the above references dealt with the focus case, in this work we study the saddle case $$%
\begin{split}
\dot{x} & =y,\\
\dot{y} & =\mu_{1}+\mu_{2}x+x^{3}+y(\mu_{3}-3x^{2}),
\end{split}
\label{forma-canonica-1A}%$$ which up to the best of our knowledge, seems to be a disregarded case with rather interesting dynamic behavior.
In fact, our motivation comes from the analysis of certain 3D memristor oscillators [@amador2017; @ponce2017; @chua2008], where under specific hypotheses on the memristor characteristics, such system appears in a natural way after a dimensional reduction achieved thanks to the existence of a first integral.
The paper is organized in the following way. First, in section \[sec:2\] we review the information that can be gained by means of a local analysis of the system. Our main results appear in section \[sec:3\], where we apply Melnikov theory to approximate the homoclinic and heteroclinic curves on a convenient parameter plane. We obtain the global bifurcation set for system , by splitting such a plane in regions with different qualitative dynamical behavior. Next, in section \[sec:4\], We show how the above analysis is useful for deriving all the possible responses of certain 3D canonical memristor oscillators, when the flux-charge characteristics function is a specific cubic polynomial, generalizing some results given in [@amador2017]. As one of the possible dynamical behaviors, we focus or attention in showing, thanks to the previous analysis, the existence of a topological sphere in the 3D phase-space completely foliated by periodic orbits. Thus, we confirm previous numerical results reported in [@messias2010; @Korneev2017]. The necessity of incorporating rigorous techniques in the analysis of memristor oscillators is emphasized with the material of section \[sec:5\], where following a similar procedure for the dimensional reduction of section \[sec:4\], we can refute several recently published studies that report the existence of an infinite number of hidden attractors in a three-dimensional memristor-based autonomous Duffing oscillator. Some technical results are relegated to the appendix.
Local bifurcations {#sec:2}
==================
In this section, we study the local bifurcations that occur in system . First, we note that the system is invariant under the transformation $$(x,y,\mu_{1},\mu_{2},\mu_{3})\rightarrow(-x,-y,-\mu_{1},\mu_{2},\mu_{3}).$$ Therefore, it is sufficient to study the bifurcation diagram for $\mu_{1}>0.$ The equilibrium points of the system are of the form $(\overline{x}%
,\overline{y})=(\tilde{x},0)$, being $\tilde{x}$ a solution of the cubic $\mu_{1}+\mu_{2}x+x^{3}=0,$ and its Jacobian matrix of is given by$$J(x,y)=%
\begin{pmatrix}
0 & 1\\
\mu_{2}+3x^{2}-6yx & \mu_{3}-3x^{2}%
\end{pmatrix}
. \label{jacobian}%$$
Note that for $\mu_{3}\leq0$ the divergence of system does not change sign, thus from Bendixson’s criterion [@Kocak91], the system does not have periodic solutions.
First, we provide a technical result that provides a study of the number of equilibria in system and their topological nature.
\[equiCubi\] Consider system , the following statements hold.
- If $\mu_{2}\geq0$ or we have $\mu_{2}<0$ with $27\mu_{1}^{2}%
+4\mu_{2}^{3}>0,$ then the system has only one equilibrium point.
- If $\mu_{2}<0$ and $27\mu_{1}^{2}+4\mu_{2}^{3}=0$ we have two equilibrium points.
- If $\mu_{2}<0$ and $27\mu_{1}^{2}+4\mu_{2}^{3}<0,$ then the system has three equilibrium points $\mathbf{x}_{i}=(s_{i},0)$ with $i\in\{L,C,R\}$ such that$$s_{L}<-\left( -\mu_{2}/3\right) ^{1/2}<s_{C}<\left( -\mu_{2}/3\right)
^{1/2}<s_{R}, \label{cota-roots}%$$ and $s_{L}+s_{C}+s_{R}=0.$ Furthermore, $\mathbf{x}_{L}$ and $\mathbf{x}_{R}$ are saddles while $\mathbf{x}_{C}$ is an antisaddle (node or focus).
We study the roots of the polynomial $p(x)=\mu_{1}+\mu_{2}x+x^{3}.$ Since $p^{\prime}(x)=\mu_{2}+3x^{2},$ if $\mu_{2}\geq0$ we obtain $p^{\prime}%
(x)\geq0$ and so the polynomial has only one root. In the rest of the proof we assume $\mu_{2}<0$. The derivative $p^{\prime}(x)$ vanishes at the points $x_{\pm}=\pm(-\mu_{2}/3)^{1/2}$ being a maximum and minimum local respectively, also a direct computation gives $p(x_{\pm}%
)=\mu_{1}\mp2(-\mu_{2}/3)^{3/2}$. When $p(x_{-})<0$ or $p(x_{+})>0$ the graph of $p(x)$ only crosses once the $x$-axis and so these inequalities provides the condition $27\mu_{1}^{2}+4\mu_{2}^{3}>0,$ and the statement (a) follows. Assuming $p(x_{-})=0$ or $p(x_{+})=0,$ the statement (b) follows.
Finally, if $p(x_{+})<0<p(x_{-})$ then we have three roots as indicated in . Moreover, using the relation between roots and coefficients of polynomials, we get $$\mu_{1}=-s_{L}s_{C}s_{R},\quad\mu_{2}=s_{C}s_{L}+s_{C}s_{R}+s_{L}s_{R},\quad
s_{L}+s_{C}+s_{R}=0.$$ For the Jacobian matrix given in , we get $J(s_{i}%
,0)=-(\mu_{2}+3s_{i}^{2})=-p^{\prime}(s_{i}).$ As we know that $p^{\prime
}(s_{L})>0$, $p^{\prime}(s_{C})<0$ and $p^{\prime}(s_{R})>0,$ the conclusion follows and the proof is complete.
In the next result, we give a characterization of the local bifurcations of system on the parametric plane $(\mu_{2},\mu_{1}),$ assuming a fixed value of the parameter $\mu_{3}.$ Note that wee have put the $\mu_{1}$ axis in the plane $(\mu_{2},\mu_{1})$ vertically, being the $\mu
_{2}$ axis the horizontal one.
The following statements hold for system .
- Given $\mu_{3}\in\mathbb{R}$ the parameter values in the set $$\varphi_{sn}=\{(\mu_{2},\mu_{1}):27\mu_{1}^{2}+4\mu_{2}^{3}=0\},
\label{curveSN}%$$ correspond with saddle-node bifurcation points of equilibria. In particular, the system has a cusp bifurcation of equilibria at $\mu_{2}=\mu_{1}=0.$
- Given $\mu_{3}>0,$ the parameter values in the set$$\varphi_{H}=\{(\mu_{2},\mu_{1}):\mu_{1}=\pm\left( \mu_{3}/3\right) ^{3/2}%
\mp\left( \mu_{3}/3\right) ^{1/2}\mu_{2},\quad\text{ }\mu_{2}<-\mu_{3}\},
\label{curveH}%$$ represent Andronov-Hopf bifurcation points of codimension one for the central equilibrium point $\mathbf{x}_{C},$ see Lemma \[equiCubi\](c).
- The set defined in determines a symmetric pair of straight half lines emanating from two points corresponding to Bogdanov-Takens bifurcation points, namely$$BT_{\pm}\equiv\left( -\mu_{3},\pm2\left( \mu_{3}/3\right) ^{3/2}\right) .
\label{BT-points}%$$
Statement (a) is a direct consequence of the equations$$x^{3}+\mu_{2}x+\mu_{1}=0,\quad\mu_{2}+3x^{2}=0,$$ to be fulfilled for any non-hyperbolic equilibrium $(x,0)$ at a saddle-node bifurcation.
Let $\left( \tilde{x},0\right) $ an equilibrium point of system . Considering the Jacobian matrix $J$ given in , then $J(\tilde{x},0)$ has two purely imaginary eigenvalues when taking $\mu_{3}>0,$ the value $\tilde{x}$ satisfies $\tilde{x}=\pm
\sqrt{\mu_{3}/3}$ with $\mu_{2}<-\mu_{3}<0,$ because then $\mu_{2}+3\tilde
{x}^{2}<0.$ The last inequality is fulfilled only for the equilibrium point $\mathbf{x}_{C},$ see Lemma \[equiCubi\](c). Since $\left( \tilde
{x},0\right) $ is an equilibrium point we have$$\mu_{1}+\mu_{2}\left( \pm\sqrt{\mu_{3}/3}\right) +\left( \pm\sqrt{\mu
_{3}/3}\right) ^{3}=0,$$ and statement (b) follows. To show statement (c) is sufficient to consider the equations $\operatorname{trace}\left( J(\tilde{x},0)\right) =\det
(J(\tilde{x},0))=0.$
Global bifurcations {#sec:3}
===================
In this section we will complete the bifurcation analysis of system . We will write the system as a perturbed Hamiltonian to which the Melnikov theory can be applied. This can be done in different ways, as indicated in the next result. The possibility of resorting to one of the two next reparametrization forms will be helpful later.
System can be written as two different perturbed Hamiltonian systems, as follows.
- Taking $$\mu_{1}=\varepsilon^{4}\nu_{1},\quad\mu_{2}=-\varepsilon^{2}\nu_{2},\quad
\mu_{3}=\varepsilon^{2}\nu_{3}, \label{para-perturbedA}%$$ the system can be rewritten as $$%
\begin{split}
\dot{x} & =y,\\
\dot{y} & =-\nu_{2}x+x^{3}+\varepsilon\left( \nu_{1}+\nu_{3}y-3x^{2}%
y\right) ,
\end{split}
\label{perturbed-systemA}%$$ which for $\varepsilon=0$ corresponds to the Hamiltonian $$H_{1}(x,y)=\frac{y^{2}}{2}+\nu_{2}\frac{x^{2}}{2}-\frac{x^{4}}{4}.
\label{HamiltonianA}%$$
- Taking$$\mu_{1}=\varepsilon^{3}\nu_{1}\quad,\mu_{2}=-\varepsilon^{2}\nu_{2},\quad
\mu_{3}=\varepsilon^{2}\nu_{3}, \label{para-perturbed}%$$ the system can be rewritten as $$%
\begin{split}
\dot{x} & =y,\\
\dot{y} & =\nu_{1}-\nu_{2}x+x^{3}+\varepsilon(\nu_{3}y-3x^{2}y),
\end{split}
\label{perturbed-system}%$$
which for $\varepsilon=0$ corresponds to the Hamiltonian$$H_{2}(x,y)=\frac{y^{2}}{2}-\nu_{1}x+\nu_{2}\frac{x^{2}}{2}-\frac{x^{4}}{4}.
\label{Hamiltonian}%$$
The blow-up transformation $\ x_{1}=(1/\varepsilon)x,$ $y_{1}=(1/\varepsilon
^{2})y,$ and $\tilde{t}=\varepsilon t$, allows to rewrite system as $$x_{1}^{\prime}=y_{1},\quad y_{1}^{\prime}=x_{1}^{3}+\frac{\mu_{2}}%
{\varepsilon^{2}}x_{1}+\frac{\mu_{1}}{\varepsilon^{3}}+\frac{\mu_{3}%
}{\varepsilon}y_{1}-3\varepsilon x_{1}^{2}y_{1},$$ where the prime denotes derivatives with respect to the new time $\tilde{t}.$ Now, using and , after some elementary algebra we obtain systems and , respectively.
The phase portrait for the unperturbed Hamiltonian systems and are shown in Figure \[Hamiltonian-fig\]. Note that when $\nu_{1}=0$ we obtain $H_{1}%
(x,y)=H_{2}(x,y),$ and so in that case it is sufficient to study the properties of the Hamiltonian $H_{1}.$
Now, we will consider the heteroclinic connections of unperturbed Hamiltonian system . The Hamiltonian has a pair of heteroclinic connections $\Gamma_{\pm}(t)=(x\left( t\right) ,\pm y\left( t\right) ),$ parameterized by $$%
\begin{split}
x\left( t\right) & =\sqrt{\nu_{2}}\tanh\left( \sqrt{\nu_{2}/2}t\right)
,\\
y\left( t\right) & =\frac{\nu_{2}}{\sqrt{2}}\operatorname{sech}^{2}\left(
\sqrt{\nu_{2}/2}t\right) ,
\end{split}
\label{param-hetero}%$$ where $-\infty<t<\infty$ and $\nu_{2}>0.$ In the next result, we compute the Melnikov function along the heteroclinic connection $\Gamma_{+}$ for the unperturbed Hamiltonian system , and by using , we obtain the approximate bifurcation curves for heteroclinic connections for system .
![(a) Phase portrait of unperturbed Hamiltonian system with $\nu_{2}=0.2$. We show in green the two heteroclinic orbits, while the non-closing stable and unstable manifolds of the saddle points are shown in red. (b) Phase portrait of unperturbed Hamiltonian system with $\nu_{1}=0.3$ and $\nu_{2}%
=1$. We draw in green the homoclinic orbit, the stable and unstable manifolds of the saddle points are shown in red. []{data-label="Hamiltonian-fig"}](hamiltonianos.eps){width="15cm"}
If we consider perturbed Hamiltonian system and $\overline{\nu}=(\nu_{1},\nu_{2},\nu_{3})$ with $\nu_{2}>0$ and $\nu_{2}%
^{2}/4<\nu_{2}^{3}/27$ (see Lemma \[equiCubi\](c)) then the Melnikov function along of the heteroclinic connection $\Gamma_{\pm}$ is given by$$M_{ht}(\overline{\nu})=\frac{2}{15}\sqrt{\nu_{2}}\left( 15\nu_{1}+5\sqrt
{2}\nu_{2}\nu_{3}-3\sqrt{2}\nu_{2}^{2}\right) . \label{melnikovF1}%$$
The system can be written as $$(\dot{x},\dot{y})^{T}=f(x,y)+\varepsilon g(x,y),$$ where $f(x,y)=(y,-\nu_{2}x+x^{3})^{T}$ and $g(x,y)=(0,\nu_{1}+\nu_{3}%
y-3x^{2}y)^{T}.$ Thus, we have $f\wedge g=y\left( \nu_{1}+\nu_{3}%
y-3x^{2}y\right) $. Accordingly, the Melnikov function is defined by $$\begin{aligned}
M_{ht}(\overline{\nu}) & =\int_{-\infty}^{\infty}f(x(t),\pm y(t))\wedge
g(x(t),\pm y(t))dt=\\
& =\int_{-\infty}^{\infty}\pm y(t)\left[ \nu_{1}\pm(\nu_{3}-3x^{2}%
(t))y(t)\right] dt,\end{aligned}$$ where $x(t)$ and $y(t)$ are defined as in . After a direct computation we obtain .
By using the Melnikov theory and by fixing one parameter of system , we can give an approximation of the heteroclinic connection curves in the remaining parameters plane.
Consider system with $\mu_{3}>0$ sufficiently small and the parametric plane $(\mu_{2},\mu_{1})$. Then the system has a unique hyperbolic heteroclinic connection in a neighborhood of the curve $$\varphi_{ht}=\{(\mu_{2},\mu_{1})\in\mathbb{R}^{2}:\mu_{1}=\pm\frac{\sqrt{2}%
}{15}\mu_{2}\left( 3\mu_{2}+5\mu_{3}\right) ,\quad\mu_{2}\neq-5\mu_{3}/3\}.
\label{curve-ht}%$$
Fixing $\nu_{3}=1$ in the Melnikov function given in , and imposing the condition $M_{ht}(\nu_{1},\nu_{2})=0,$ we obtain $$\nu_{1}=\frac{\sqrt{2}}{15}\nu_{2}\left( 3\nu_{2}-5\right) .$$ From we get $\varepsilon=\sqrt{\mu_{3}}$, $\mu_{1}%
=\mu_{3}^{2}\nu_{1},$and $\mu_{2}=-\mu_{3}\nu_{2},$ so that$$\mu_{1}=\mu_{3}^{2}\nu_{1}=-\mu_{3}^{2}\frac{\sqrt{2}}{15}\frac{\mu_{2}}%
{\mu_{3}}\left( 3\left( -\frac{\mu_{2}}{\mu_{3}}\right) -5\right) ,$$ and the conclusion follows.
When $\mu_{1}=0$ and $\mu_{3}>0$ on the parameter plane $(\mu_{2},\mu_{1}),$ we obtain the point of double heteroclinic connections $$DHT\equiv\left( -5\mu_{3}/3,0\right) . \label{dhtpoint}%$$
We recall that Schecter’s points are co-dimension two points defined by the intersection of a saddle-node curve and a homoclinic or heteroclinic curve, for more details see [@schecter]. Taking the intersection points of the saddle-node bifurcation curve and the heteroclinic curves given in and respectively, we obtain a first-order approximation of Schecter’s points of the system. Since the system is symmetric with respect to the parameter $\mu_{1},$ the system has four Schecter’s points (see Figure \[bifurcation-sets\]), these points are $$%
\begin{split}
S_{1}^{\pm} & \equiv\rho_{1}\left( (5/27),\mp(5\sqrt{10}/729)\left(
\sqrt{18\mu_{3}+5}+\sqrt{5}\right) \right) ,\\
S_{2}^{\pm} & \equiv\rho_{2}\left( (5/27),\pm(5\sqrt{10}/729)\left(
\sqrt{18\mu_{3}+5}-\sqrt{5}\right) \right) ,
\end{split}
\label{schecter}%$$ where $$\rho_{1}=\left( 9\mu_{3}+5-\sqrt{5}\sqrt{18\mu_{3}+5}\right) ,\quad\rho
_{2}=\left( \sqrt{5}\sqrt{18\mu_{3}+5}-9\mu_{3}-5\right) .$$
Now, by using the homoclinic connection of Hamiltonian system , we compute the associated Melnikov function for system when $\nu_{3}=1$.
If we consider system and $\overline{\nu}=(\nu
_{1},\nu_{2},\nu_{3})$ with $\nu_{1}>0$, $\nu_{2}>0$ and $\nu_{3}=1$, then the Melnikov function associated to the homoclinic orbit with connection point $(0,s_{R})$, it is given by$$M(\overline{\nu})=\sqrt{2}\frac{\cosh^{2}(\theta)}{\cosh^{2}(\theta)+2}\left(
F_{1}(\theta)+\nu_{2}F_{2}(\theta)\right) , \label{melniF2B}%$$ where $$%
\begin{split}
F_{1}(\theta)= & 720\theta-320\sinh\theta+240\theta\cosh^{3}\theta
-320\cosh^{2}\theta\sinh\theta-\\
& -80\cosh^{4}\theta\sinh\theta+480\theta\cosh\theta,\\
F_{2}(\theta)= & 1440\theta\cosh\theta-768\sinh\theta-\cosh^{3}%
\theta-1344\cosh^{2}\theta\sinh\theta-\\
& -48\cosh^{4}\theta\sinh\theta
\end{split}
\label{melniF2B-2}%$$
and $0<\theta<\infty$, with $$\cosh\theta=\frac{2s}{\omega},\quad\omega^{2}=2(\nu_{2}-s_{R}^{2})>0,\quad
\nu_{1}=\nu_{2}s_{R}-s_{R}^{3},$$ being $s_{R}$ the biggest positive root of the equation $\nu_{1}-\nu
_{2}x+x^{3}=0$, see Figure \[Homoclina\].
We consider the unperturbed Hamiltonian system given in with $\nu_{2}>0$. From Lemma \[equiCubi\](c), the system has $3$ equilibrium points $\mathbf{x}_{i}=(s_{i},0)$, where $\mathbf{x}_{L}$ and $\mathbf{x}_{R}$ are saddle points and $\mathbf{x}_{C}$ is a focus or node and $$s_{L}<s_{C}<s_{R},\quad s_{L}+s_{C}+s_{R}=0,\quad s_{L}s_{C}s_{R}=-\nu_{1}.$$ We study only the case $\nu_{1}>0$, for the case $\nu_{1}<0$ is analogous.System can written as$$(\dot{x},\dot{y})^{T}=f(x,y)+\varepsilon g(x,y).$$ Now, assuming $\nu_{1}>0,$ by Green’s Theorem, the homoclinic Melnikov function of the system can rewritten as $$M_{h}(\overline{\nu})=\int\int_{D(\nu_{1},\nu_{2})}\left( \frac{-\partial
g(x,y)}{\partial y}\right) dA,$$ where $D$ is the region bounded by the homoclinic orbit which joins the equilibrium point $\left( s_{R},0\right) $ to itself. By fixing $\nu_{3}=1$ (that is $\mu_{3}>0$), and taking $\ p(x)=\nu_{1}-\nu_{2}x+x^{3},$ we get $p(s_{R})=\nu_{1}-\nu_{2}s_{R}+s_{R}^{3}=0,$ that is $$\nu_{1}=s_{R}(\nu_{2}-s_{R}^{2}), \label{auxilar-cubicA}%$$ and so $\nu_{2}-s_{R}^{2}>0.$ Taking the auxiliary function $$q(x)=\int_{0}^{x}p(x)dx=\nu_{1}x-\nu_{2}\frac{x^{2}}{2}+\frac{x^{4}}{4},$$ and using , the homoclinic loop is given by the points $\left( x,y_{s}^{\pm}(x)\right) $ where $\overline{x}\leq x\leq s_{R}$, $$y_{s}^{\pm}(x)=\pm\sqrt{2}\sqrt{q(x)-q(s_{R})},$$ and $y_{s}^{\pm}(\overline{x})=y_{s}^{\pm}(s_{R})=0,$ see Figure \[Homoclina\]. Now, the Melnikov function is thanks to the symmetry of the loop $$\begin{aligned}
M_{h}(\overline{\nu}) & =2\int_{\overline{x}}^{s_{R}}(3x^{2}-1)dx\int
_{0}^{y_{s}^{+}(x)}dy=\\
& =\sqrt{2}\int_{\overline{x}}^{s}(3x^{2}-1)(s_{R}-x)\sqrt{(x+s_{R}%
)^{2}-2(\nu_{2}-s_{R}^{2})}dx=\\
& =\sqrt{2}\int_{\overline{x}}^{s}(3x^{2}-1)(s_{R}-x)\sqrt{(x+s_{R}%
)^{2}-\omega^{2}}dx,\end{aligned}$$ where from $\omega^{2}=2(\nu_{2}-s_{R}^{2}),$ and we have used that $$q(x)-q(s_{R})=\frac{1}{4}(x-s_{R})^{2}\left[ (x+s_{R})^{2}-\omega^{2}\right]
.$$ Taking the change of variable $$x+s_{R}=\omega\cosh\theta,$$ and noting that $q(s_{R})-q(\overline{x})=0$ we see that $\overline{x}%
+s_{R}=\omega$, which corresponds to $\theta=0,$ while for $x=s_{R}$ the corresponding values of $\theta=\theta_{s_{R}}$ satisfy $\cosh\theta_{s_{R}%
}=2s_{R}/\omega,$ or also $\omega^{2}\cosh^{2}\theta_{s_{R}}=4s_{R}^{2},$ that is $(s_{R}^{2}+\nu_{2})\cosh^{2}\theta_{s_{R}}=2s_{R}^{2},$ and so we get $$s_{R}^{2}=\frac{\cosh^{2}\theta_{s_{R}}}{2+\cosh^{2}\theta_{s_{R}}}\nu_{2}.$$ Now we arrived to $$M_{h}(\overline{\nu})=\sqrt{2}\omega^{2}\int_{0}^{\theta_{s_{R}}}%
(1-3(\omega\cosh\theta-s_{R})^{2})(2s_{R}-\omega\cosh\theta)\sinh^{2}\theta
d\theta,$$ and after some computations, we obtain and , where $\theta_{s_{R}}$ has been simplified to $\theta$.
![ Homoclinic orbit which joins the saddle equilibrium point $(s_{R},0)$ to itself.[]{data-label="Homoclina"}](homoclina.eps){width="11cm"}
As a direct consequence of the above result, we give an analytical approximation of the bifurcation curves for homoclinic connections of system .
Consider system with $\mu_{3}>0$ sufficiently small and the parametric plane $(\mu_{2},\mu_{1}).$ Then the system has a unique homoclinic orbit in a neighborhood of the curve $$\varphi_{h}=\{(\mu_{2},\mu_{1})\in\mathbb{R}^{2}:\mu_{2}=-\mu_{3}\nu
_{2}(\theta),\quad\mu_{1}=\pm\mu_{3}^{3/2}\nu_{1}(\theta),\quad0<\theta
<\infty\}, \label{curve-h}%$$ where $$%
\begin{split}
\nu_{2}\left( \theta\right) & =\frac{10(\cosh2\theta+5)(9\sinh\theta
+\sinh3\theta-12\theta\cosh\theta)}{3(370\sinh\theta+115\sinh3\theta
+\sinh5\theta-60\theta(11\cosh\theta+\cosh3\theta))},\\
\nu_{1}(\theta) & =\nu_{2}\left( \theta\right) s-s^{3},\quad s^{2}%
=\frac{\cosh^{2}\theta}{2+\cosh^{2}\theta}\nu_{2}(\theta).
\end{split}
\label{parametric-curve}%$$ Moreover, for the points $(\mu_{2}(\theta),\mu_{1}(\theta))$ at the curve $\varphi_{h}$ we have.$$\lim_{\theta\rightarrow0^{+}}(\mu_{2},\mu_{1})=(-\mu_{3},\pm2/3\sqrt{\mu
_{3}^{3}/3}),\quad\lim_{\theta\rightarrow\infty}(\mu_{2},\mu_{1})=\left(
-5\mu_{3}/3,0\right) .$$
The Melnikov function given in - vanishes at the points $(\nu_{1}(\theta),\nu_{2}(\theta))$ defined in . Taking $\nu_{3}=1$ in we obtain $\mu_{1}=\mu_{3}^{3/2}\nu_{1}$ and $\mu_{2}=-\mu_{3}\nu_{2},$ and after some computations the conclusion follows.
Note that from the previous result we obtain the two points $$\lim_{\theta\rightarrow0^{+}}(\mu_{2},\mu_{1})\equiv BT,\quad\lim
_{\theta\rightarrow\infty}(\mu_{2},\mu_{1})\equiv DHT,$$ where the points $BT$ and $DHT$ are given in and respectively.
In Figure \[bifurcation-sets\], the complete bifurcation set of system is shown. Figures \[limit-cycle\] and \[no-limit-cycle\] give the different phase portrait in the labeled parameter regions, where in these figures we show the different configurations of the phase portrait of the system. In Figure \[limit-cycle\], since the homoclinic Melnikov function is positive, we can guarantee that there is no change on the relative position of the stable and unstable manifolds of each saddle points.
![The bifurcation diagram of system , taking $\mu_{3}>0$ sufficiently small.[]{data-label="bifurcation-sets"}](curva-canonico.eps){width="15cm"}
![Phase portrait of system in the parameter regions labeled with **1**,**2**, **3** and **4** in Figure \[bifurcation-sets\]. The thick lines are the boundary of the basin of attraction of a limit cycle, such boundary is formed by some stable and unstable manifolds of the saddle points. The green lines are the heteroclinic connections. []{data-label="limit-cycle"}](casesB.eps){width="14cm"}
![Phase portrait of system in the parameter regions labeled with **5** and **6** in Figure \[bifurcation-sets\]. The stable and unstable manifolds of the saddle points.[]{data-label="no-limit-cycle"}](cases2.eps){width="12cm"}
\[existence\]Note that considering the function $\nu_{2}$ defined in , and after some algebra, we obtain that finding the minimum of the function $\nu_{2}$ is equivalent to finding the zeros of the function $$h_{1}\left( x\right) =2x\left( 26\cosh2x+\cosh4x+33\right) -5\left(
10\sinh2x+\sinh4x\right) , \label{fun-h}%$$ where $h_{1}\left( 0\right) =0,$ $h_{1}\left( 1\right) <0$ and $h_{1}\left( 2\right) >0,$ see Figure \[estudio-homoclina\](c). Thus at $\theta^{\ast}\approx1.8630981$ the function $\nu_{2}$ has a minimum given by $\nu_{2}(\theta^{\ast})\approx2.454887$ (see Figure \[estudio-homoclina\](b)), so that$$-\frac{5}{2}\mu_{3}<-\nu_{2}(\theta^{\ast})\mu_{3}<-\frac{5}{3}\mu_{3}.$$ Now, if we consider system with $\mu_{3}>0$ sufficiently small, $\mu_{2}<-(5/2)\mu_{3}$ and $\mu_{1}$ such that $$|\mu_{1}|<\left( \frac{\mu_{3}}{3}\right) ^{3/2}-\left( \frac{\mu_{3}}%
{3}\right) ^{1/2}\mu_{2},$$ then the system has a stable limit cycle, see Figure \[estudio-homoclina\](a). This assertion is a direct consequence of and Poincaré-Bendixson Theorem (see for instance [@wiggins2003]), since the sign of the Melnikov function guarantees the existence of a compact positive invariant set with only one unstable equilibrium point in its interior.
![(a) The parametric curve defined in on the plane of parameters $(\mu_{2},\mu_{1})$. (b) The function $\nu_{2}(\theta)$ defined in . (c) The function $h(\theta)$ defined in .[]{data-label="estudio-homoclina"}](estudio-homoclina.eps){width="13cm"}
Just to illustrate the quality of the above analytical predictions for the homoclinic connection bifurcation curve, by using the shooting method (see for instance [@rodriguez1990]) and taking $\mu_{3}=0.1$, we show in Figure \[comparacion\], the numerical continuation curve for the homoclinic orbit of system , and in red the analytic approximation curve given by . As observed, there is a great similarity between the two approaches, and in general we can conclude that the above analytical predictions are really useful is getting a global view of the actual bifurcation set.
![System with $\mu_{3}=0.1$. In black (left panel), the numerical continuation curve in the parameter plane $(\mu_{2}%
,\mu_{1})$, and by using points of the curve, in black (right panel) the numerical computation of the stable and unstable manifold for a saddle point of the system . In red (left panel), the analytic approximation curve given by , and by using points of the curve, in red (right panel) the numerical computation of the stable and unstable manifolds for a saddle point of the system.[]{data-label="comparacion"}](comparacion3.eps){width="13cm"}
Application to 3D Canonical Memristor Oscillator {#sec:4}
================================================
As one of the possible applications of the above study, in this section we will show the existence of a topological sphere completely foliated by periodic orbits for a 3D canonical memristor oscillator, when the flux-charge characteristics of the memristor is a monotone cubic polynomial. The existence of this sphere was reported numerically in [@messias2010; @Korneev2017].
We start by considering the modeling of an elementary oscillator endowed with one flux-controlled memristor $M$, see Figure \[fig:mem3D\] and [@chua2008]. In the shown circuit the values of $L$ and $C$ for the impedance and capacitance are positive constants, while the resistor has a negative value $-R$. From Kirchoff’s laws we see that $$%
\begin{array}
[c]{rcl}%
i_{R}(\tau)-i_{L}(\tau) & = & 0,\\
i_{L}(\tau)-i_{C}(\tau)-i_{M}(\tau) & = & 0,\\
-v_{R}(\tau)+v_{L}(\tau)+v_{C}(\tau) & = & 0,\\
v_{C}(\tau)-v_{M}(\tau) & = & 0,
\end{array}$$ where $v,i$ stand for the voltage and current, respectively, across the corresponding element of the circuit as indicated by the subscript.
![The canonical memristor oscillator [@chua2008]. Note that the the only active element in the circuit is the resistor with a negative resistance $-R$.[]{data-label="fig:mem3D"}](RLCbasicfigure){width="9cm"}
In Section 3.2 of [@chua2008], this circuit is proposed as a third-order canonical memristor oscillator but the notation is slightly different as follows. They take $i_{1}=i_{C}$, $i_{3}=i_{L}=i_{R}$, $i=i_{M}$, $v_{1}=v_{C}=v_{M}$, $v_{3}=v_{L}$, $v_{4}=v_{R}$, $\varphi_{1}=\varphi_{C}$, $\varphi_{3}=\varphi_{L}$, $\varphi_{4}=\varphi_{R}$ and $\varphi=\varphi_{M}%
$. Thus, they write the two equations $$i_{1}=i_{3}-i,\quad v_{3}=v_{4}-v_{1},$$ and, after integrating respect to time, they arrive to $$q_{1}=q_{3}-q(\varphi),\quad\varphi_{3}=\varphi_{4}-\varphi_{1},\label{eq:2}%$$ where $q(\varphi)$ stands for the nonlinear flux-charge characteristics of the flux-controlled memristor. Solving now for $(q_{3},\varphi_{4})$ and taking into account that $\varphi_{1}=\varphi$ since $v_{1}=v_{M}$, it is immediate to obtain $$q_{3}=q_{1}+q(\varphi),\quad\varphi_{4}=\varphi+\varphi_{3},$$ so that authors conclude that a good choice for independent variables is the triple $(q_{1},\varphi_{3},\varphi)$, that is, the charge of capacitor $C$, the flux of the inductor $L$ and the flux of the memristor, respectively. Accordingly, by taking derivatives in , the following set of differential equations is proposed, $$%
\begin{array}
[c]{rcl}%
C\dot{v}_{1} & = & i_{3}-W(\varphi)v_{1},\\
L\dot{i}_{3} & = & Ri_{3}-v_{1},\\
\dot{\varphi} & = & v_{1},
\end{array}
\label{eq:3}%$$ where $\dot{q}_{1}=i_{1}=C\dot{v}_{1},$ $\dot{q}_{3}=i_{3},$ $\dot{\varphi
}_{4}=v_{4}=Ri_{3}$ and $$W(\varphi)=\frac{dq}{d\varphi}.$$ Finally, they rewrite the system as follows, $$%
\begin{array}
[c]{rcl}%
\dot{x} & = & \alpha\left( y-W(z)x\right) ,\\
\dot{y} & = & -\xi x+\beta y,\\
\dot{z} & = & x,
\end{array}
\label{chua2008}%$$ where $x=v_{1},\,y=i_{3},\,z=\varphi,$ and the parameters used are $\alpha=1/C,$ $\xi=1/L,$ and $\beta=R/L,$ so that, $\alpha,\xi,\beta>0$. An important observation is that the parameter $\alpha$ is not essential so that it can be removed with the change of variables and parameters $$%
\begin{split}
& \widetilde{x}=x,\quad\widetilde{y}=\alpha y,\quad\widetilde{z}%
=z,\quad\widetilde{\xi}=\alpha\xi,\\
& \widetilde{a}=\alpha a,\quad\widetilde{b}=\alpha b,\quad\widetilde
{W}=\alpha W,
\end{split}
\label{removing-alpha}%$$ to be assumed in the sequel, omitting also tildes to alleviate the notation. Therefore, we need to study the system $$%
\begin{array}
[c]{rcl}%
\dot{x} & = & -W(z)x+y,\\
\dot{y} & = & -\xi x+\beta y,\\
\dot{z} & = & x,
\end{array}
\label{ap1:1}%$$ where $W(z)=q^{\prime}(z)=3z^{2}+2az+b,$ and $$q(z)=z^{3}+az^{2}+bz,\label{cubica-ap-1}%$$ with $a^{2}-3b<0,$ which assumes that the memristor is passive ($q^{\prime
}(z)>0$).
System belongs to a more general class of systems whose reduction is possible thanks to the existence of a first integral, as shown in the Appendix. Like other models of memristor oscillators, system has some special feature. For instance, it has a continuum of equilibria on the z-axis. Furthermore, the Jacobian matrix at any of these points has a zero eigenvalue.
Taking the parameters $a_{11}=-1,\ a_{12}=1,\ a_{21}=-\xi$ and $a_{22}=\beta$ in Proposition \[theor:1\] of the appendix, we obtain that for all $h\in\mathbb{R}$, system has an invariant manifold $S_{h}$ defined by $$S_{h}=\{(x,y,z)\in\mathbb{R}^{3}:-\beta x+y-\beta z^{3}-a\beta z^{2}+\left(
\xi-b\beta\right) z=h\}. \label{invariant}%$$ Moreover, assuming $c=1$ in Corollary \[theor-cubic-q\] of the appendix, we obtain that on each invariant manifold $S_{h},$ the system is topologically equivalent to the Liénard system $$%
\begin{split}
\dot{x} & =y-x^{3}-x^{2}-(b-\beta)x,\\
\dot{y} & =\beta x^{3}+a\beta x^{2}+\left( b\beta-\xi\right) x+h.
\end{split}
\label{lienard:ap1}%$$ In Figure \[invariant-surface\], we show the invariant manifold corresponding to $h=0.3$ and the set of parameters $\xi
=100$, $a=b=1$, $\beta=5$, along with the phase space of the equivalent Liénard system .
![(a) The invariant manifold corresponding to the set of parameters $\xi=100$, $a=b=1$, $\beta=5$ and $h=0.3$ is shown. In black the infinite number of equilibrium points of the system and in blue a periodic orbit of the system contained in the invariant manifold. (b) The phase plane of the equivalent Liénard system corresponding to the set parameters given in (a) showing a limit cycle in blue, in red the function $g(X)$ and the function $F(X)$ in black. (c) A zoom of figure (b) is shown.[]{data-label="invariant-surface"}](cubica1.eps){width="13cm"}
From Proposition \[canonical-forms\](a), the system can be rewritten as $$\dot{x}=y,\quad\dot{y}=\mu_{1}+\mu_{2}x+\mu_{3}y+x^{3}-3x^{2}y,
\label{norma-form-ap1}%$$ where the new parameter are $$%
\begin{split}
\mu_{1} & =\frac{1}{27\beta^{5/2}}(27h+9a\xi+2a^{3}\beta-9ab\beta),\\
\mu_{2} & =\frac{1}{3\beta^{2}}\left( \beta(3b-a^{2})-3\xi\right)
,\quad\mu_{3}=\frac{1}{3\beta}\left( a^{2}-3b+3\beta\right) .
\end{split}
\label{parameters-ap1}%$$ From Remark \[existence\], for $\mu_{3}>0$ sufficiently small, we have that for all $\mu_{2}<-(5/2)\mu_{3}<0$ and $\mu_{1}$ such that $$|\mu_{1}|<\left( \frac{\mu_{3}}{3}\right) ^{3/2}-\left( \frac{\mu_{3}}%
{3}\right) ^{3/2}\mu_{2}, \label{cylinder-3}%$$ the system has a stable limit cycle. Therefore, we can give the following result in terms of the parameter $h$, which is associated with the invariant manifolds $S_{h}$ of 3D system . This result guarantees the existence of a topological sphere in the 3D phase-space completely foliated by periodic orbits.
\[prop-sphare\] Consider system with $\beta,\xi>0,$, the function $q$ defined as in , $a^{2}-3b<0$ and $$a^{2}-3b+3\beta>0 \label{cylinder-1}%$$ sufficiently small. Additionally, suppose that the following inequalities hold $$%
\begin{split}
0 & <3b-a^{2}<3\xi/\beta,\\
\beta(3b-a^{2})-3\xi & <(5/2)\left( 3b-a^{2}-3\beta\right) \beta<0.
\end{split}
\label{cylinder-2}%$$ Then for all $h\in\mathbb{R}$ with$$-\frac{A}{27}<h<\frac{B}{27},$$ where $$%
\begin{split}
A & =\left( 4a^{2}\beta+3\beta^{2}-12b\beta+9\xi\right) \sqrt
{a^{2}-3b+3\beta}+9a\xi+2a^{3}\beta-9ab\beta,\\
B & =\left( 4a^{2}\beta+3\beta^{2}-12b\beta+9\xi\right) \sqrt
{a^{2}-3b+3\beta}-9a\xi-2a^{3}\beta+9ab\beta,
\end{split}
\label{limits-AB}%$$ the system has a stable periodic orbit. Moreover, there exist a topological sphere $\Omega$ (see Figure \[sphereAp1\]) foliated by such periodic orbits.
From Remark \[existence\], and after substituting the values of $\mu_{1}%
,\mu_{2}$ and $\mu_{3}$ given in , we obtain the inequalities -. Now, from we obtain $|\mu_{1}|<(1/3)\left( \mu_{3}/3\right)
^{1/2}\left( \mu_{3}-3\mu_{2}\right) ,$ so that $\mu_{3}-3\mu_{2}>0,$ since from hypotheses we have $\mu_{2}<-(5/2)\mu_{3}<0.$ Now after some algebra we obtain $$|27h+9a\xi+2a^{3}\beta-9ab\beta|<\left( 4a^{2}\beta+3\beta^{2}-12b\beta
+9\xi\right) \sqrt{a^{2}-3b+3\beta}.$$ Taking into account the absolute value, and grouping terms, we obtain the values of $A$ and $B$ defined in . Finally, from Remark \[existence\] system has a stable periodic orbit on each $S_{h}$ defined in , so varying the parameter $h$, we obtain a sphere foliated by such periodic orbits.
![Using on each invariant manifold $S_{h}$ defined in , some slices of the surface $\Omega$ given by Proposition \[prop-sphare\] for system with parameters $a=1,b=4.8,\beta=5$ and $\xi=80$ are shown. For this set of parameters we get $\mu_{3}=0.106>0$, $\mu_{2}=-2.3<-(5/2)\mu_{3}$, $A=1180.1$ and $B=152.2$.[]{data-label="sphereAp1"}](surface-canonico.eps){width="12cm"}
False Hidden Attractors in Memristor-Based Autonomous Duffing Oscillators {#sec:5}
=========================================================================
An attractor is called a hidden attractor if its basin of attraction does not intersect any neighborhood of equilibria; otherwise, it is called a self-excited attractor, for more details see [@Kuznetsov2011; @Leonov2011b]. Recently in [@mentira2018; @mentira2018b; @mentira2018c] it was reported the existence of an infinite number of hidden attractors in a memristor-based autonomous Duffing oscillators, whose memristance function is a cubic polynomial. Here, by using a similar approach to the followed in the previous section, we will show that such hidden attractors are not possible, so that the numerical simulations included in [@mentira2018; @mentira2018b; @mentira2018c] are misleading.
The quoted memristor based autonomous Duffing oscillator is modeled by the dynamical system $$%
\begin{split}
\dot{x} & =y,\\
\dot{y} & =z,\\
\dot{z} & =-\alpha z-M(x)y,
\end{split}
\label{mentira-1}%$$ where the memristance function $M$ (possibly discontinuous) is defined as $$M(x)=\frac{d\phi(x)}{dx}\label{functionM}%$$ and $\phi$ is a continuous function. System has a continuum of equilibria, since any point of the $x$-axis is an equilibrium point. In the next result, we show that even system does not belong to the family of the appendix, the system also has the property of possessing an infinite number of invariant manifolds.
\[propo:mentira0\]Consider system with the function $M$ defined as in . For any $h\in\mathbb{R}$ the set $$S_{h}=\{(x,y,z)\in\mathbb{R}^{3}\colon H(x,y,z)=h\}\label{sh-cubica}%$$ is an invariant manifold for the system, where we have introduced the continuous function $$H(x,y,z)=\phi(x)+\alpha y+z.\label{invariante-cubica}%$$ Therefore, the system has an infinite number of invariant manifolds foliating the whole $\mathbb{R}^{3}$, and so the dynamics is essentially two-dimensional.
Taking $H$ as in , define for any solution $(x(\tau),y(\tau),z(\tau))$ of the auxiliary continuous function $$h(\tau)=H(x(\tau),y(\tau),z(\tau))$$ Now, a direct computation gives, excepting the points of possible non-differentiability, $$h^{\prime}(\tau)=\frac{d\phi(x)}{dx}\dot{x}+\alpha\dot{y}+\dot{z}=M(x)y+\alpha
z-\alpha z-M(x)y=0.$$ Then $h$ is piecewise constant along the orbits of , but as $h$ is continuous by definition, it should be globally constant. In short, the level sets of $H$ are invariant for the flow.
Now, by using the above result, we reduce the study of the dynamical behavior of the system, to the study of a planar system.
\[propo:mentira1\]Consider system with the function $M$ defined as in . Then on each invariant set $S_{h}$ defined in the system is topologically equivalent to the planar system $$%
\begin{split}
\dot{x} & =y,\\
\dot{y} & =-\phi(x)-\alpha y+h.
\end{split}
\label{reducido-mentira}%$$ Moreover, $(x\left( \tau\right) ,y\left( \tau\right) )\in\mathbb{R}^{2}$ is a solution of the above system if and only if $E_{h}\left( x\left(
\tau\right) ,y\left( \tau\right) \right) $ is a solution of system , where $$E_{h}\left( X\left( \tau\right) ,Y\left( \tau\right) \right) =%
\begin{pmatrix}
x\left( \tau\right) \\
y\left( \tau\right) \\
h-\phi(x(\tau))-\alpha y(\tau)
\end{pmatrix}$$
From Proposition \[propo:mentira0\] we can solve for $z$ in the equation $H(x,y,z)=h$, and write $$z=h-\phi(x)-\alpha y.$$ Replacing this expression into the first and second equation of we obtain system . Suppose that $(x\left( \tau\right) ,y\left( \tau\right) )\in\mathbb{R}^{2}$ is a solution of system . Taking $$z(\tau)=h-\alpha y(\tau)-\phi(x(\tau))$$ we obtain $$\begin{aligned}
\dot{z}(\tau) & =-\alpha\dot{y}(\tau)-\frac{d\phi(x(\tau))}{dx}\dot{x}%
(\tau)=-\alpha\left( h-\phi(x(\tau))-\alpha y(\tau)\right) -M(x(\tau
))y(\tau)=\\
& =-\alpha\left( z(\tau)\right) -M(x(\tau))y(\tau).\end{aligned}$$ and the proposition follows.
In the following result, we show that for $\alpha\neq0$, the system does not have periodic solutions.
\[propo:mentira\]Consider system . The following statements hold.
- For $\alpha=0$ the system is Hamiltonian.
- For $\alpha\neq0$ the system does not have periodic solutions.
The divergence of the system is $\Delta=-\alpha.$ Then, when $\alpha=0$ the system corresponds to the Hamiltonian$$H(x,y)=\frac{y^{2}}{2}+\phi^{\prime}(x).$$ For $\alpha\neq0$ the divergence of system does not change sign, thus from Bendixson’s criterion [@Kocak91], system does not have periodic solutions.
\[mentira\]Note that as a consequence of proposition \[propo:mentira1\] and \[propo:mentira\], the $3D$ system system cannot have periodic orbits for any continuous function $\phi$ and $\alpha\neq0.$ However, when $\alpha=0$ the system could have an infinite number of periodic orbits on each invariant set $S_{h}$ defined in .
Regarding [@mentira2018; @mentira2018b; @mentira2018c] authors consider system with the function $\phi(x)=\omega x+\beta x^{3}$ and the set of parameters $\alpha=0.0001,$ $\omega=0.35,$ $\beta=0.85.$ In both quoted references, authors reported the existence of an infinite number of stable periodic orbits coexisting with an infinite number of stable equilibria, by taking into account several numerical simulations, so concluding the existence of hidden attractors.
From Propositions \[propo:mentira0\] and \[propo:mentira1\], we obtain the invariant manifolds $$S_{h}=\{(x,y,z)\in\mathbb{R}^{3}\colon\omega x+\beta x^{3}+\alpha y+z=h\},$$ and the planar system ruling the dynamics on each $S_{h}$ given by $$\dot{x}=y,\quad\dot{y}=-\omega x-\beta x^{3}-\alpha y+h.$$
From Remark \[mentira\] we note that, the system cannot have periodic orbits, and so, the statement made in the quoted papers is clearly wrong, probably after giving too much credit to numerical simulations. This emphasizes the relevance of the approach followed in this work which allows to avoid misconceptions coming just from numerical simulations.
Conclusions
===========
Motivated by the dynamical analysis of 3D memristor oscillators whose nonlinear characteristics is a cubic polynomial, and after showing that their dynamics is essentially two-dimensional, the need to consider a disregarded unfolding of the Bogdanov-Takens singularity naturally arose. The corresponding bifurcation set, including both local and global bifurcations has been described. While local bifurcations can be easily detected, the characterization of global bifurcations parameters curves is much more involved; only by resorting to Melnikov’s theory it was possible to obtain such curves providing a complete description of the bifurcation set.
Regarding the considered 3D memristor oscillators, and by working within some parameters regions of the above bifurcation set, it has been possible to show rigorously the existence of multiple periodic orbits leading to a topological sphere.
When the same approach is applied to a different family of 3D memristor oscillators, it has been shown that the oscillations are not possible, contrarily to what had been recently claimed.
Acknowledgements {#acknowledgements .unnumbered}
================
The first author is supported by Pontificia Universidad Javeriana Cali-Colombia. E. Freire and E. Ponce are partially supported by MINECO/FEDER grant MTM2015-65608-P and by the *Consejería de Economía, Innovación, Ciencia y Empleo de la Junta de Andalucía* under grant P12-FQM-1658.
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We consider a family of three-dimensional systems, which is general enough to capture all the mathematical models of memristor oscillators given in . Such family was been studied in [@amador2017] and [@ponce2017], where the authors showed that the dynamics of such a family of three-dimensional systems is essentially ruled by a one parameter set of two-dimensional systems. We consider the system $$%
\begin{array}
[c]{rcl}%
\dot{x} & = & a_{11}W(z)x+a_{12}y,\\
\dot{y} & = & a_{21}x+a_{22}y,\\
\dot{z} & = & x,
\end{array}
\label{sisgeneral}%$$ where the constants $a_{11},a_{12},a_{21},a_{22}\in\mathbb{R}$ and the function $W$ allows to define a continuous function $$q(z)=\int_{0}^{z}W(s)ds.\label{wq}%$$
The next result guarantees that the dynamics of system is essentially two-dimensional, see [@amador2017] for a proof.
\[theor:1\]Consider system where the functions $W$ and $q$ are related as in . For any $h\in\mathbb{R}$, the set $$S_{h}=\{(x,y,z)\in\mathbb{R}^{3}:-a_{22}x+a_{12}y-a_{12}a_{21}z+a_{11}%
a_{22}q(z)=h\}\label{Sh-general}%$$ is an invariant manifold for the system. Therefore, the system has an infinite family of invariant manifolds foliating the whole $\mathbb{R}^{3}$, and so the dynamics is essentially two-dimensional.
In the following result we show that on each invariant set $S_{h}$ given in , and for any continuous function $q$ defined as in , the dynamics is topologically equivalent to a Liénard system. Furthermore, we give for any solution of the Liénard system with a given value of $h$, the corresponding solution of the 3D canonical model . This result is a generalization of Theorem 3 given in [@amador2017], where the function $q$ was considered to be a continuous piecewise linear function.
\[theor2-cubic\]Consider system with the function $q$ defined as in . If $a_{12}\neq0$, then on each invariant set $S_{h}$ given in , the dynamics is topologically equivalent to the Liénard system $$\dot{X}=Y-F(X),\quad\dot{Y}=-g(X)+h,\label{lienard-general}%$$ where $F$ and $g$ are given by$$F(X)=-a_{11}q(X)-a_{22}X,\quad g(X)=a_{11}a_{22}q(X)-a_{12}a_{21}%
X\label{funcionesFG-general}%$$ Moreover, $(X\left( \tau\right) ,Y\left( \tau\right) )\in\mathbb{R}^{2}$ is a solution of the Liénard system for a given $h\in\mathbb{R}$, if and only if $E_{h}\left( X\left( \tau\right) ,Y\left(
\tau\right) \right) $ $\in\mathbb{R}^{3}$ is a solution of system on $S_{h},$ where $$E_{h}\left( X\left( \tau\right) ,Y\left( \tau\right) \right) =%
\begin{pmatrix}
Y(\tau)-F(X(\tau))\\
\frac{1}{a_{12}}\left[ (a_{22}^{2}+a_{12}a_{21})Y\left( \tau\right)
-a_{22}Y\left( \tau\right) +h\right] \\
X\left( \tau\right)
\end{pmatrix}
.\label{condiciones-iniciales-general}%$$
First, with $a_{12}\neq0$ the change of variables $$\overline{x}=x,\quad\overline{y}=a_{22}x-a_{12}y,\quad\overline{z}%
=z\label{cambio-nuevo}%$$ transforms system into the system $$\begin{aligned}
\dot{\overline{x}} & =f_{1}\left( \overline{z}\right) \overline
{x}-\overline{y},\label{lienard1B}\\
\dot{\overline{y}} & =f_{2}\left( \overline{z}\right) \overline
{x},\nonumber\\
\dot{\overline{z}} & =\overline{x},\nonumber\end{aligned}$$ where the functions $f_{1}$ and $f_{2}$ are defined as$$f_{1}(\overline{z})=a_{11}W(\overline{z})+a_{22},\quad\quad f_{2}\left(
\overline{z}\right) =a_{22}a_{11}W(\overline{z})-a_{12}a_{21}%
.\label{f1f2continuas}%$$ From Proposition \[theor:1\], the invariant manifolds for system - can be written in the new variables as $$\widetilde{S}_{h}=\{(\overline{x},\overline{y},\overline{z})\in\mathbb{R}%
^{3}:-\overline{y}+g(\overline{z})=h\}.\label{sh2general}%$$ Now, replacing the condition given in in the first equation of and removing the unnecessary second equation, we obtain the system $$%
\begin{array}
[c]{l}%
\dot{\overline{x}}=f_{1}\left( \overline{z}\right) \overline{x}%
-g(\overline{z})+h,\\
\dot{\overline{z}}=\overline{x}.
\end{array}
\label{lienard2B}%$$ where the function $g$ is defined by $$g(u)=a_{11}a_{22}q(u)-a_{12}a_{21}u.\label{funcionesG-general}%$$ After the change of variables $$%
\begin{array}
[c]{l}%
X=\overline{z},\\
Y=-\tilde{F}(\overline{z})+\overline{x},
\end{array}
\label{cambio2B}%$$ where $F$ is $$\tilde{F}(z)=a_{11}q(z)+a_{22}z,\label{funcionesF-general}%$$ we obtain $$\dot{X}=\dot{\overline{z}}=\overline{x}=Y+\tilde{F}(X)=Y-(-\tilde{F}(X)),$$ so that $$\begin{aligned}
\dot{Y} & =-\tilde{F}^{\prime}(\overline{z})\dot{\overline{z}}+\dot
{\overline{x}}=-\left( a_{11}q^{\prime}(\overline{z})+a_{22}\right) +\left(
f_{1}\left( \overline{z}\right) \overline{x}-g(\overline{z})+h\right) =\\
& =-f_{1}\left( \overline{z}\right) \overline{x}+f_{1}\left( \overline
{z}\right) \overline{x}-g(\overline{z})+h=-g(\overline{z})+h,\end{aligned}$$ and taking $F(X)=-\tilde{F}(X)$ we obtain system -.
If $\left( X\left( \tau\right) ,Y\left( \tau\right) \right)
\in\mathbb{R}^{2}$ is a solution of system - for a given $h\in
\mathbb{R}$, we have from that $$%
\begin{pmatrix}
\overline{x}(\tau)\\
\overline{z}(\tau)
\end{pmatrix}
=%
\begin{pmatrix}
Y(\tau)-F(\overline{z}(\tau))\\
X(\tau)
\end{pmatrix}$$ is a solution of system . From , we obtain on $\widetilde{S}_{h}$ that $\overline{y}=g(\overline{z})-h$, with $g$ as in . Thus, $$%
\begin{pmatrix}
\overline{x}(\tau)\\
\overline{y}(\tau)\\
\overline{z}(\tau)
\end{pmatrix}
=%
\begin{pmatrix}
Y(\tau)-F(X(\tau))\\
g\left( X(\tau)\right) -h\\
X(\tau)
\end{pmatrix}
,$$ is a solution of system on $S_{h}$. Finally, from we obtain for system the solution $x(\tau)=\overline{x}(\tau)$, $$\begin{aligned}
y(\tau) & =\dfrac{1}{a_{12}}\left[ a_{22}\overline{x}(\tau)-\overline
{y}(\tau)\right] =\dfrac{1}{a_{12}}\left[ a_{22}Y(\tau)-a_{22}%
F(X(\tau))-g\left( X(\tau)\right) +h\right] \\
& =\dfrac{1}{a_{12}}\left[ a_{22}Y(\tau)+a_{22}\tilde{F}(X(\tau))-g\left(
X(\tau)\right) +h\right] ,\end{aligned}$$ and $z(\tau)=\overline{z}(\tau)$. The conclusion follows from the fact that for all $X$ we have $$a_{22}\tilde{F}(X)-g(X)=(a_{22}^{2}+a_{12}a_{21})X.$$
In order to apply the analysis performed to system , in what follows we consider the function $q$ defined by a cubic polynomial, that is, we assume $$W(z)=3cz^{2}+2az+b,\quad q(z)=cz^{3}+az^{2}+bz,\label{cubica}%$$ with $c\neq0$. As a direct consequence of Propositions \[theor:1\] and \[theor2-cubic\], we obtain the next result.
\[theor-cubic-q\]Consider system with the functions $q$ and $W$ defined as in . If $a_{12}\neq0$, then on each invariant set $S_{h}$ given by$$S_{h}=\{(x,y,z)\in\mathbb{R}^{3}:-a_{22}x+a_{12}y+a_{11}a_{22}cz^{3}%
+aa_{11}a_{22}z^{2}+(ba_{11}a_{22}-a_{12}a_{21})z=h\}$$ the dynamics is topologically equivalent to the Liénard system $$%
\begin{split}
\dot{x} & =y+ca_{11}x^{3}+aa_{11}x^{2}+\left( ba_{11}+a_{22}\right) x,\\
\dot{y} & =-a_{11}a_{22}cx^{3}-a_{11}a_{22}ax^{2}+(a_{12}a_{21}-a_{11}%
a_{22}b)x+h.
\end{split}
\label{lienard-curbica1}%$$ Moreover, $(x\left( \tau\right) ,y\left( \tau\right) )\in\mathbb{R}^{2}$ is a solution of the Liénard system for a given $h\in\mathbb{R}$, if and only if $E_{h}\left( x\left( \tau\right) ,y\left(
\tau\right) \right) $ $\in\mathbb{R}^{3}$ is a solution of system on $S_{h},$ where $$E_{h}\left( x\left( \tau\right) ,y\left( \tau\right) \right) =%
\begin{pmatrix}
y(\tau)+ca_{11}^{3}x(\tau)^{3}+aa_{11}^{2}x(\tau)^{2}+\left( ba_{11}%
+a_{22}\right) x(\tau)^{2}\\
\frac{1}{a_{12}}\left[ (a_{22}^{2}+a_{12}a_{21})y\left( \tau\right)
-a_{22}y\left( \tau\right) +h\right] \\
x\left( \tau\right)
\end{pmatrix}
.\label{solucion-ap}%$$
In the next Proposition, we show that system can be written into the form .
\[canonical-forms\] The following statements hold for system .
- If $a_{22}\neq0$ and $a_{11}a_{22}<0$ then the system can be written into the form$$\dot{x}=y,\quad\dot{y}=\mu_{1}+\mu_{2}x+cx^{3}+\mu_{3}y+3ca_{11}x^{2}y.
\label{forma-canonica-1}%$$ where the new parameters $\mu_{1},\mu_{2}$ and $\mu_{3}$ are given by $$\label{parameters-canonica-1}%
\begin{split}
\mu_{1} & =\frac{27ch+a_{11}a_{22}a(9cb-2a^{2})-9caa_{12}a_{21}}%
{27c^{2}\left( -a_{11}a_{22}\right) ^{5/2}},\\
\mu_{2} & =\frac{a_{11}a_{22}(a^{2}-3cb)+3ca_{12}a_{21}}{3c\left(
a_{11}a_{22}\right) ^{2}},\quad\mu_{3}=\frac{a_{11}(a^{2}-3cb)-3ca_{22}%
}{3ca_{11}a_{22}}.
\end{split}$$
- If $a_{22}=0$ then the system can be written into the form$$\dot{x}=y,\quad\dot{y}=\mu_{1}+\mu_{2}x+\mu_{3}y+3ca_{11}x^{2}y,
\label{forma-canonica-2}%$$ where the new parameters $\mu_{1},\mu_{2}$ and $\mu_{3}$ are defined by$$\mu_{1}=h-\frac{aa_{12}a_{21}}{3c},\quad\mu_{2}=a_{12}a_{21},\quad\mu
_{3}=ba_{11}-\frac{a^{2}a_{11}}{3c}. \label{parameters-canonica-2}%$$
First, the change of variables $$u=x+\frac{a}{3c},\quad v=y+\frac{2}{27}\frac{a^{3}}{c^{2}}a_{11}-\frac{1}%
{3}\frac{a}{c}a_{22}-\frac{1}{3}a\frac{b}{c}a_{11},$$ transforms system into $$%
\begin{split}
\dot{u} & =v+ca_{11}u^{3}+\lambda_{1}u,\\
\dot{v} & =-ca_{11}a_{22}u^{3}+\lambda_{2}u+\lambda_{3},
\end{split}
\label{sis-auxiliar-A}%$$ where the new parameters are $$%
\begin{split}
\lambda_{1} & =a_{22}+ba_{11}-\frac{1}{3}\frac{a^{2}}{c}a_{11},\quad
\lambda_{2}=a_{12}a_{21}-ba_{11}a_{22}+\frac{1}{3}\frac{a^{2}}{c}a_{11}%
a_{22},\\
\lambda_{3} & =h+\frac{1}{3}a\frac{b}{c}a_{11}a_{22}-\frac{1}{3}\frac{a}%
{c}a_{12}a_{21}-\frac{2}{27}\frac{a^{3}}{c^{2}}a_{11}a_{22}.
\end{split}
\label{parameter-auxiliar}%$$ If $a_{11}a_{22}<0,$ the change of variable $$x=\frac{1}{\left( -a_{11}a_{22}\right) ^{1/2}}u,\quad y=v,\quad\tau=\frac
{1}{-a_{11}a_{22}}t,$$ transforms system - into $$\begin{aligned}
\dot{x} & =\frac{1}{\left( -a_{11}a_{22}\right) ^{3/2}}y+ca_{11}x^{3}%
-\frac{\lambda_{1}}{a_{11}a_{22}}x,\\
\dot{y} & =\frac{\lambda_{3}}{-a_{11}a_{22}}+\frac{\lambda_{2}}{\left(
-a_{11}a_{22}\right) ^{1/2}}x+c\left( -a_{11}a_{22}\right) ^{3/2}x^{3}%\end{aligned}$$ and taking into account that $$\ddot{x}=\frac{1}{\left( -a_{11}a_{22}\right) ^{3/2}}\dot{y}+3ca_{11}%
x^{2}\dot{x}-\frac{\lambda_{1}}{a_{11}a_{22}}\dot{x},$$ and after some algebra, statement (a) follows. If $a_{22}=0,$ then from system , we obtain statement (b) after a direct computation.
[^1]: Facultad de Ingeniería y Ciencias, Departamento de Ciencias Naturales y Matemáticas, Pontificia Universidad Javeriana-Cali, Cali, Colombia.\
$^{2,3}$ Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería, Avda. de los Descubrimientos, 41092 Sevilla, Spain.\
$^{1}[email protected], $^{2}[email protected] $^{3}[email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The X-ray quasi-periodic oscillation (QPO) seen in RE J1034+396 is so far unique amongst AGN. Here we look at the another unique feature of RE J1034+396, namely its huge soft X-ray excess, to see if this is related in any way to the detection of the QPO. We show that all potential models considered for the soft energy excess can fit the 0.3–10 keV X-ray spectrum, but that the energy dependence of the rapid variability (which is dominated by the QPO) strongly supports a spectral decomposition where the soft excess is from low temperature Comptonization of the disc emission and remains mostly constant, while the rapid variability is produced by the power law tail changing in normalization. The presence of the QPO in the tail rather than in the disc is a common feature in black hole binaries, but low temperature Comptonization of the disc spectrum is not generally seen in these systems. The main exception to this is GRS 1915+105, the only black hole binary which routinely shows super-Eddington luminosities. We speculate that super-Eddington accretion rates lead to a change in disc structure, and that this also triggers the X-ray QPO.'
author:
- |
Matthew Middleton$^1$, Chris Done$^1$, Martin Ward$^1$, Marek Gierli[ń]{}ski$^1$ and Nick Schurch$^1$\
$^1$Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK\
title: 'RE J1034+396: The origin of the soft X-ray excess and QPO'
---
= -0.5cm
\[firstpage\]
accretion, accretion discs – galaxies: active – X-rays: galaxies
Introduction
============
The discovery of a significant quasi-periodic oscillation in the X-ray light curve of a Narrow Line Seyfert 1 AGN RE J1034+396 (Gierli[ń]{}ski et al. 2008) strengthens arguments stressing the similarities in the physics of the accretion flow between supermassive and stellar mass black holes. Previous evidence for a simple correspondence between AGN and the black hole binaries (BHB) included similarities in the broadband shape of the X-ray variability power spectra, with characteristic break timescales scaling with mass (e.g. M$^c$Hardy et al. 2006), but the characteristic QPOs often seen in BHB light curves remained undetected until now (Vaughan & Uttley 2005; Leighly 2005).
The QPOs in BHB can be split into two main groups, at high and low frequencies, respectively. While there is as yet no clear mechanism for producing either set of QPOs (or the correlated broad band variability) there are many pointers to their origin from the observations. Most clearly, their amplitude increases with increasing energy. The BHB spectra typically contain two components, a disc and tail, and this increasing amplitude is consistent with the QPO (and all the rest of the rapid variability) being associated with a variable tail while the disc remains constant. This shows that QPOs are produced by some mode of the hot coronal flow rather than a mode of the thin disc. (see e.g. the reviews by Van der Klis 2004; McClintock & Remillard 2006; Done, Gierli[ń]{}ski & Kubota 2007).
The rest of the QPO properties are more complex. The high frequency (HF) QPO may have a constant fundamental frequency (though it is seen at 3:2 harmonic ratios of this), which may relate to the mass of the black hole. By contrast, the frequency (and other properties) of the low frequency (LF) QPO change dramatically with mass accretion rate, correlating with the equally dramatic changes in the source spectra. Typically these show that at low mass accretion rates compared to Eddington, $L/L_{\rm Edd} \ll 1$, the LF QPO is at low frequencies, but is weak and rather broad. The corresponding energy spectra are dominated by a hard power law (photon index $\Gamma<2$) which rolls over at around 100 keV (low/hard spectral state). As the mass accretion rate increases, the low frequency QPO increases in strength and coherence as well as frequency, while the power law spectrum softens and the disc increases in strength relative to the power law. The QPO is at its highest frequencies, is strongest and most coherent where the spectrum has both a strong disc component and a strong soft tail of emission to higher energies ($\Gamma>2.5$). After this, the LF QPO frequency remains more or less constant as the tail declines and hardens to $\Gamma
\sim 2.2$, leaving the spectra dominated by the disc component (high/soft state), though it becomes harder to follow the QPO as increasing contribution from the stable disc (thermal dominant state) swamps the signal from the variable tail. (e.g. McClintock & Remillard 2006 and references therein)
The only major difference expected in scaling these models up to the supermassive black holes in AGN is the decrease in disc temperature down to the UV band. The X-ray spectra of AGN spectra should then be always dominated by the tail, irrespective of spectral state (e.g. Done & Gierli[ń]{}ski 2005). The predicted change in shape of the tail with spectral state/mass accretion rate gives an explanation for the variety of spectral properties seen in [*unobscured*]{} subtypes AGN such as Broad Line AGN, Narrow Line Seyfert 1’s (NLS1) and LINERs (e.g. Middleton et al. 2008). The generally hard X-ray spectra seen in LINERS could be explained as a low $L/L_{\rm Edd}$ flow (e.g. Yuan et al. 2008 but see Maoz 2007), while broad line AGN with a strong UV disc component and weak X-ray $\Gamma\sim 2$ tail would be analogous to the high/soft state in BHB. The NLS1 probably have the highest mass accretion rates, so have lower mass for a given luminosity (Boroson 2002), and their steeper X-ray spectra (Brandt, Mathur & Elvis 1997; Leighly 1999; Shemmer et al. 2006) make them natural counterparts for the very high state (Pounds, Done & Osborne 1995; Murashima et al. 2005; Middleton et al. 2007, M$^c$Hardy et al. 2007) where all QPOs (both high and low frequencies) are strongest. Intriguingly, RE J1034+396 is a NLS1, so its QPO detection is consistent with these scaling models.
All this supports models where the underlying physics of the accretion flow is very similar between BHB and AGN. However, the scaling clearly breaks down under more detailed study of the X-ray spectra from high mass accretion rate objects. These should have spectra dominated by the steep tail which is typical of the high and very high states, with no disc emission in the X-ray band. Yet [*all*]{} the high mass accretion rate AGN show an excess below 1 keV (Gierli[ń]{}ski & Done 2004; Brocksopp et al. 2006). This ’soft X-ray excess’ is completely inconsistent with the expected disc component, both as its temperature is higher than predicted from the mass and mass accretion rate of these AGN (e.g. Bechtold et al. 1987), and as its shape is much smoother than a sharply peaked disc spectrum (Czerny et al. 2003; Gierli[ń]{}ski & Done 2004). Either the soft X-ray excess is an additional component which breaks the scaling between AGN and BHB, or it is produced by some external distortion of the intrinsic emission. Obviously it is very important to distinguish between these alternatives and the objects with the strongest soft excesses offer the most stringent constraints.
RE J1034+396 has one of the largest soft excesses known, which appears to connect smoothly onto the enormous EUV peak of its spectral energy distribution (Middleton et al. 2007). The size and temperature of this peak is extreme even amongst NLS1’s (Puchnarewicz et al. 2002, Casebeer et al. 2006; Middleton et al. 2007), and an obvious question is whether this extreme soft excess is somehow linked to the detection of the QPO. Here we use XMM-Newton data on RE J1034+396 to test the various models for the soft X-ray excess, and speculate on its connection to the QPO.
The origin of the soft excess in high mass accretion rate AGN
=============================================================
There are multiple models for the origin of the soft X-ray excess seen ubiquitously in high mass accretion rate AGN. The most obvious possibility is that it is related somehow to the disc. One way to do this is if the mass accretion rate is super Eddington, so the disc spectrum is distorted by advection of radiation in the very optically thick flow. Such slim discs (Abramowicz et al. 1988) have spectra which are less peaked than a standard disc (Waterai et al. 2000) so give a better fit to the shape of the soft excess (Mineshige et al. 2000; Wang & Netzer 2003; Haba et al. 2008). Another way to modify the shape of the (standard or slim) disc emission is if it is Comptonized by low temperature electrons (Czerny & Elvis 1987; Kawaguchi 2003), perhaps produced by a hotter skin forming over the cooler disc (Czerny et al. 2003). However, the derived temperature for this skin is remarkably similar for all objects at 0.1–0.2 keV despite a large range in black hole mass (and hence disc temperature: Czerny 2003; Gierli[ń]{}ski & Done 2004, Crummy 2006). This argues against it being related to the disc, so it is unlikely to be a true continuum component.
Instead, a constant energy is most easily explained through atomic processes, in particular the abrupt increase in opacity in partially ionized material between $\sim$0.7–2 keV due to O[VII]{}/O[VIII]{} and Fe transitions. This results in an increase in transmitted flux below 0.7 keV, which could produce the soft excess either from absorbtion in optically thin material in the line of sight, or from reflection by optically thick material out of the line of sight. In both models the observed smoothness of the soft excess requires large velocity smearing (velocity dispersion $\ga$0.3 c) in order to hide the characteristic [*sharp*]{} atomic features, but with this addition then both reflection and absorption models fit the shape of the soft excess equally well (Fabian 2002, 2004; GD04; Crummy 2006; Chevallier 2006; Schurch & Done 2006; Middleton et al. 2007; Dewangan et al. 2007, D’Ammando et al. 2008). Such high velocities are naturally produced only close to the black hole, so both absorption and reflection models predict that the soft excess arises in regions of strong gravity. However, both models also require some extreme, and probably unphysical, parameters. In the reflection model, the inferred smearing can be so large that the required emissivity must be more strongly centrally peaked than expected from purely gravitational energy release, perhaps pointing to extraction of the spin energy itself. The amount of reflection required to produce the soft excess can also be extreme. Quasi- isotropic emission sets a limit to size of the soft excess of no more than a factor of 2–3 above the harder continuum emission (Sobolewska & Done 2007), yet the strongest observed soft excesses are a factor $4$ larger than this (a factor $8-10$ above the extrapolated continuum), requiring that the intrinsic illuminating spectrum is strongly suppressed ([*e.g.*]{} Fabian 2002). This issue becomes even more problematic when incorporating any pressure balance condition (such as hydrostatic equilibrium) as this strongly limits the section of the disc in which partially ionized material can exist, hence makes a much smaller soft excess (Done & Nayakshin 2007; Malzac, Dumont & Mouchet 2005).
Conversely, in the smeared absorption model, pressure balance rather naturally produces the required partially ionized zone (Chevallier et al. 2006). However, these models also imply extreme velocities, with a peak outflow velocity of $\sim$ 0.8c required in order to smooth away the characteristic absorption features from a smooth wind which covers the source (Schurch & Done 2007, 2008). This is much larger than the maximum velocity of $\sim$ 0.2-0.4c produced by models of a radiation driven disc wind (Fukue 1996), so the absorber must instead be associated with faster material in this description. The obvious physical component would be a jet/magnetic wind, yet the amount of material is probably far too large to be associated with either of these (Schurch & Done 2007; 2008).
Thus there are physical problems with all of these models: the Comptonized disc cannot explain the narrow range in inferred temperatures, smeared reflection cannot produce the strongest soft excesses seen if the disc is in hydrostatic equilibrium, and smeared absorption requires a fast wind/jet which is not readily associated with the properties of any known component.
An alternative possibility for the absorption model is that the characteristic absorption features are masked by dilution instead of smearing, perhaps due to the wind becoming clumpy so that it partially covers the source (Boller et al. 1996; Tanaka et al. 2004; Miller et al. 2007; 2008). These models lack diagnostic power as the potential complexity of the wind means that any number of partial absorbers can be added until the model fits the observed spectra. Nonetheless, this may correspond to the more messy reality of high Eddington fraction winds from UV luminous discs (Proga & Kallman 2004). Some of the clumps could even have high enough column to produced significant reflected emission as well (Malzac et al. 2005; Chevallier et al. 2006), giving a complex mix of reflection and partial absorption from a range of different columns, velocities and ionization states of the material.
It is important to distinguish between these very different potential origins of the soft excess as they make very different predictions about the environment and geometry of the accreting material close to the black hole. The extreme broad iron lines required in the pure reflection models are not required in the absorption models (either smeared: Sobolewska & Done 2007; Done 2007 or partial covering: Miller et al. 2008) as the broad feature redwards of the iron line is fit instead by continuum curvature. All the absorption models require the presence of material above the inner disc, probably in the form of a wind, whereas the reflection models instead suggest a clean line of sight to the inner disc. The large velocity shear in the smeared wind model requires that the material is strongly accelerated, so it potentially carries an enormous amount of kinetic energy (Chevallier et al. 2006; Schurch & Done 2006) with corresponding impact on AGN feedback/galaxy formation.
Spectral fitting alone cannot distinguish between these very different spectral models in the 0.3–10 keV bandpass (Crummey et al. 2007; Sobolewska & Done 2007; Middleton et al. 2007; Miller et al. 2007; 2008). Variability gives additional information, but the smeared reflection and smeared absorption models are known to be able to predict similar variability patterns (Ponti et al. 2005; Gierli[ń]{}ski & Done 2006). Here for the first time we fit [*all*]{} of these models to the data and calculate their predicted variability to see which description of the soft excess best matches the [*simultaneous*]{} constraints from both spectral and variability data.
Data Extraction
===============
[*XMM-Newton*]{} observed RE J1034+396 on 2007-05-31 and 2007-06-01 for about 93 ks (observation id. 0506440101, revolution no. 1369). We extracted source and background light curves and noticed background flares and data gaps in the final 7 ks. Therefore, we excluded this data segment from analysis and used 84.3 ks of clean data starting on 2007-05-31 20:10:12 UTC for further analysis.
Spectra
-------
We selected data from the PN (patterns 0–4) and MOS (patterns 0–12) in a region of radius 45 arcsec. The data are very similar in shape to a previous 16 ks XMM-Newton observation, but are much higher quality due to the longer exposure. Because of the extreme softness of this particular source both MOS and PN were piled up so we excised the central regions of the image out to 7.5 arcsec radius so as not to be affected by this. Background was taken from 6 source free regions of the same size.
We use only the MOS data in the 0.3–10 keV range for spectral fitting as there are well known discrepancies at soft energies between the PN and MOS spectra which makes simultaneous fitting of these two instruments very difficult.
We use [xspec]{} version 11.3.2, and fix the minimum galactic absorption at $1.31\times10^{20}$cm$^{-2}$ in all fits, but also allow a separate neutral absorption column to account for additional absorption in the host galaxy.
Energy dependence of the variability: rms spectra
-------------------------------------------------
A successful model must be able to describe [*both*]{} the spectrum [*and*]{} the variability. Fig. \[fig:lc\]a shows the full light curve for these data, with the clear QPO which is remarkably coherent in the latter part of the observation (25–85 ks: G08). There is also a large scale drop in flux at 40–55 ks. This looks very similar to an occultation event such as those recently recognised in other AGN (McKernan & Yaqoob 1998; Gallo et al. 2004; Risaliti et al. 2007; Turner et al. 2008). Fig. \[fig:lc\]b shows the light curve in the high energy bandpass, where this dip is [*not*]{} present. This shows that there is clearly energy dependent variability in this event which is not present in the rest of the light curve. Thus the variability is [*complex*]{} and made from more than one component.
[l]{}
-- --
-- --
We explore this further by calculating the root mean square fractional variability amplitude (hereafter we will use the term ‘rms’ for simplicity) as a function of energy (see Edelson et al. 2002; Markowitz, Edelson & Vaughan 2003 and Vaughan at el. 2003). We first calculate this for the total light curve with 100-s binning. Fig. \[fig:rms\]a shows this rms spectrum rising smoothly with energy. This behaviour is quite unlike the rms spectra seen from other NLS1’s although these can show a variety of shapes, including flat (e.g. Gallo et al. 2007; O’Neill et al. 2007), flat with a peak at 2 keV (Vaughan & Fabian 2004; Gallo et al. 2004; Gallo et al. 2007), and falling but with a peak at 2 keV (Ponti et al. 2006; Gallo et al. 2007; Larsson et al. 2008).
The rms is dominated by noise above $\sim 4$ keV as the count rate at high energies is very low due to the steep spectrum. In order to extend the rms to higher energies we extract light curves from the full source region, without excising the core to correct for pileup (red points in Fig. \[fig:rms\]b). This means that some fraction of the hard photons originate from much lower energies, but the steeply rising rms spectrum means that the variability of these pileup photons is rather small. Thus pileup adds an approximately constant offset to the hard spectrum, and so should dilute the variability seen at high energies. Instead, the rms spectrum continues to rise in a rather smooth fashion at high energies (red points in Fig. \[fig:rms\]b), showing that this is a good lower limit to the total variability at high energies.
The smooth increase of the rms as a function of energy seems initially most likely to be from a single component. However, there is clear evidence of different processes contributing to the variability from the different energy dependence seen in the dip event (Fig. \[fig:lc\]). We explore this by re-calculating the rms on a range of timescales by changing the bin time, $\Delta T$, of the light curves. The rms is the square root of the integrated power in the light curve from frequencies of 1/$T$ to 1/(2$\Delta T$) Hz, where $T$ = 84300 s is the length of the light curve. Thus the original binning of 100 s means that the rms at each energy is the integral of the power spectrum from 11.2 to 5000 $\mu$Hz. We compare this to the rms of the long timescale variability by increasing the binning to 3700 s (the QPO period), i.e. corresponding to the frequency range 11.2–135 $\mu$Hz. This is shown by the black data in Fig. \[fig:rms\]b, and has a very different shape to the total rms (red points), with a similar amount of variability at low energies, but a sharp break around 0.7 keV so that the variability at high energies is much lower. Subtracting this long timescale rms from the total rms (in quadrature) gives the rms of the rapid variability (including the QPO at 270 $\mu$Hz), i.e. the integral of the power spectrum from 135 to 5000 $\mu$Hz (green points in Fig. \[fig:rms\]b). This has a sharp rise with increasing energy at low energies. We can compare this directly to the rms of the QPO by calculating this from the folded light curve (blue points). This is indeed very similar in shape and normalization to the rms of rapid variability. This is expected as the QPO forms a large fraction (more than half) of the variability power in this frequency range. However, the shape at high energies is marginally different, with the rapid variability appearing to continue rising with energy while the QPO saturates at an rms of $\sim$0.2. Given the potential problems with pileup, and the size of the uncertainties, this difference is probably not significant.
Simultaneous constraints from spectra and variability
=====================================================
In the following sections we take each model for the soft excess and fit it to the 0.3–10 keV spectrum. The models are listed in Table \[tab:models\]. The fit results are shown in Table \[tab:fit\_results\]. Then we inspect this best-fitting model to see how varying these components might produce the observed energy dependence of the variability. We focus here on the rapid variability, which is dominated by the QPO, as it is clear that the longer timescale variability may contain contributions from multiple processes. We calculate the simulated rms spectra by randomly varying a spectral model parameter with the mean equal to the best-fitting value and a certain standard deviation (Gierli[ń]{}ski & Zdziarski 2005). We calculate the rms spectrum at high energy resolution, but then bin this to the same resolution as the rms for direct comparison.
Separate component for the soft excess: [disk]{}, [slim]{} and [comp]{} {#sec:comp}
-----------------------------------------------------------------------
We first test the models where the soft excess is a separate emission component. A standard accretion disc spectrum ([diskbb]{} in [xspec]{}) is much more strongly peaked than the data, giving a very poor fit, $\chi^2_\nu$ = 517/211, showing that this is an unacceptable description of the broad band curvature. Instead we test models of an advective (slim) disc, however the best estimates for mass give a mass accretion rate of $\sim 0.3 L_{\rm Edd}$, not formally high enough for advection to be important. Theoretical calculations show that the spectrum of a slim disc can be approximated by sum of blackbodies, similar to the standard disc models, but with $T \propto r^{-p}$, where $0.5<p<0.75$ (the higher limit is the standard disc profile, while the lower limit is a fully advection dominated disc: Waterai et al. 2000). Fig. \[fig:disc\_panel\]a shows that such models ([diskpbb]{}, available as an additional local model for [xspec]{}) are still not broad enough to match the shape of the soft excess ($\chi^2_\nu$ = 457/210). Instead, a low temperature Comptonization component ([comptt]{}) gives a reasonable fit to the data, except for the features below $\sim$0.6 keV (Fig. \[fig:disc\_panel\]b). Hereafter we refer to this model as [comp]{}.
[c]{}
It is plain from the residuals to this fit (lower panel of Fig 3b) that there are weak atomic features at low energies, especially at $\sim$0.6 keV. Adding a series of narrow Gaussians gives a reduction in $\chi^2$ of $\sim 30$ for significant emission features at $\sim 0.55$, $0.74$ and $0.87$ keV. These are consistent with He–like O Ly$\alpha$ and He- and H-like O radiative recombination continua i.e. indicating photoionized material. However, there are no corresponding narrow lines in the RGS data, so these features must be intrinsically broad. Their equivalent width is $\le 10$ eV, so these have negligible impact on the derived continuum which is the subject of this paper.
In BHB the power law tail is generically very variable, while the disc is remarkably constant (Churazov et al. 2000). If the soft excess is really related to the disc and is constant, then we expect the low energy variability to be suppressed, as observed, because of increasing dilution of the variable power law by the constant soft excess at low energies.
We quantify this effect by taking the same Comptonization plus power law model as fits the spectrum of the soft excess, and varying the normalization of the power law tail by 20 per cent. This gives a constant rms where the tail dominates (above 2 keV) and a dramatically increasing suppression of the variability below this, as the fractional contribution of the constant soft excess increases (Fig. \[fig:rms\_comptt\]a). This looks very similar to the drop observed at low energies in rms of the rapid variability (and QPO). This provides very strong evidence that a two component model is the correct spectral decomposition as both the spectrum and spectral variability can be easily described in this model. The soft excess is a separate component above the power-law tail at low energies, and the rapid variability (including the QPO) is a modulation of the tail.
The longer timescale variability can also be fit in this model. The data show a rise in rms with energy to $\sim$1 keV so this cannot be produced by simply by varying the norm of the Comptonized disc. Instead, varying the optical depth or temperature leads to spectral pivoting about the peak of the seed photons (at $\sim$3$kT_{\rm seed} \approx 0.15$ keV). Fig. \[fig:rms\_comptt\]b shows how changing the optical depth by 1.5 per cent can produce this characteristic rise, which is then diluted by the constant power law above $\sim 0.8$ keV. While this does indeed match the broad shape of the long timescale rms, we caution that there should also be some contribution to this from partial covering if the dip in the light curve is indeed an occultation event.
[c]{}
Smeared Reflection models: [ref1]{} and [ref2]{} {#sec:ref}
------------------------------------------------
We next explore how the reflection model for the soft excess can fit both the spectrum and spectral variability. We use the constant density reflection models of Ballantyne, Iwasawa & Fabian (2001) (available as a table model in [xspec]{}). These models (described in more detail in Ross & Fabian 2002) become inaccurate for $\Gamma>2.5$ (Done & Nayakshin 2007), so instead we use the spectra tabulated for $\Gamma=2.2$, and multiply the model by $E^{-(\Gamma - 2.2)}$, where $E$ is energy. This code also converts the normalization of the reflected emission to that of the illuminating power law so that the amount of reflection is parameterized by the solid angle of the illuminated material, $\Omega/2\pi$ and inclination (Done & Gierli[ń]{}ski 2006), which we fix at $30\deg$. We smear this using the convolution version of the [laor]{} code for the velocity structure of an extreme Kerr spacetime (Laor 1991).
[c]{}
Fig. \[fig:refl\]a shows that this is a very poor fit to the data ($\chi^2_\nu$ = 612/210), as a single ionization state reflector cannot simultaneously produce both the hard tail [*and*]{} the shape of the soft excess. This is a common feature in good signal-to-noise data: multiple reflectors are required, one to fit the soft excess, another to fit the iron line, dispelling a key attraction of the model (Crummy et al. 2006). Nonetheless, a double reflector can indeed fit the data well (Fig. \[fig:refl\]b), where the hard tail/iron line region requires a neutral reflector (modelled using the [thcomp]{} code: [Ż]{}ycki, Done & Smith 2000 as the [reflion]{} code does not extend down to the very low ionization states required). We hereafter refer to this model as [ref2]{}.
This is a very different spectral deconvolution to the one where the soft excess is a separate Comptonized component ([comp]{}). The spectrum at low energies is now modelled by the intrinsic power law, and instead there is a ‘hard excess’ at high energies which is formed by dramatically enhanced, strongly smeared, neutral reflection.
The only way to produce a steeply rising rms is to pivot the power law about its peak in seed photons (around 0.15 keV, as before). This gives a linearly rising rms as shown in Fig. \[fig:rms\_refl\]a, assuming both reflectors respond to this changing illumination (though we fix the ionization of each component for simplicity).
While this would fit the overall rms shown in Fig. \[fig:rms\]b, it plainly cannot match the two very different shaped components which underlie the variability. The rapid variability rms shape does [*not*]{} rise linearly with energy, there is a clear inflection point around $\sim$0.8 keV. This has an obvious origin in the separate Comptonized disc models discussed above, as this is the energy at which this component starts to dominate the spectrum and hence dilute the variability. However, it is possible to match this in the reflection model by assuming that the ionized reflector remains constant. Fig. \[fig:rms\_refl\]b shows how this dilutes the variability towards lower energies, giving a fairly good fit to the rapid variability rms.
Constant reflection components are a feature of these models for the soft excess in other AGN, and may arise from strong light bending effects (Miniutti & Fabian 2004). This simultaneously explains the lack of variability together with enhanced amplitude and extreme smearing. However, our spectral deconvolution and variability require that the constant reflector is the one which is [*not*]{} enhanced relative to the illuminating power law, making this model appear somewhat contrived in RE J1034+396.
[c]{}
Smeared Absorption Models: [swind]{} and [xscort]{} {#sec:swind}
---------------------------------------------------
We now fit the smeared absorption models. These assume the source is completely covered by a partially ionized wind from the disc, which has a large velocity shear to smear out the characteristic strong absorbtion lines. We first use the simple [swind]{} model (available on the local models page for [xspec]{}) of Done & Gierli[ń]{}ski (2004; 2006), where the absorption from a partially ionized column of material is convolved with a Gaussian velocity field. This is an excellent fit to the data (Fig. \[fig:abs\]a), but the best fit velocity dispersion goes to the upper limit of 0.5c allowed in the model. The gaussian form of the velocity dispersion means that this corresponds to a velocity shear greater than the speed of light, which is obviously unphysical (see Middleton et al. 2007, Schurch & Done 2007a, b).
[c]{}
[c]{}
Instead we fit more physical wind absorption models, calculated from the [xscort]{} code, which accelerate the wind from 0 to 0.3c, and use this internal velocity field self-consistently in the photoionization code. These models have only been calculated for a small number of parameters (Schurch & Done 2007a), so cannot be properly fit to the data. However, we have taken the model with parameters closest to those required by the soft excess spectral shape, and then corrected for the different spectral index, as in Section \[sec:ref\]. Fig. \[fig:abs\]b shows that while this gives a strong soft excess, the velocity shear is insufficient to smooth the strong atomic features in the 0.7–3 keV band into the observed smooth rise, so this gives a very poor fit to the data ($\chi^2_\nu$ = 2153/212). Thus the absorption does not arise in a line driven disc wind, but instead must be connected to much faster moving material in this model.
We explore the variability properties of the fast wind Fig. \[fig:abs\]a, since the more physical wind (Fig. \[fig:abs\]b) does not fit the data. Because the whole spectrum is again made from the intrinsically steep power law, then simply pivoting this at the peak flux of the seed photons gives a linearly rising rms (Fig. \[fig:rms\_abs\]a), similar to that of Fig. \[fig:rms\_refl\]a. Instead, allowing the ionization of the fast wind to change in response to this changing illumination means that the variability is enhanced over the range where the partially ionized material has most effect on the spectrum i.e. 0.7–2 keV (Gierli[ń]{}ski & Done 2006). Fig. \[fig:rms\_abs\]b shows how this does not produce enough enhancement of the variability to follow the sharp rise in rms at $\sim$0.7 keV and it also predicts that this enhancement does not affect the spectrum above $\sim$2 keV so the rms at high energies is also too small.
Thus while the smeared absorption model is a good fit to the spectrum, it cannot easily match the variability patterns seen, making it a less attractive solution.
Partial covering models: [ionpcf1]{} and [ionpcf2]{} {#sec:ionpcf}
----------------------------------------------------
If instead the absorption is clumpy then it can lead to clouds which partially cover the source. These can be ionized as they are illuminated by the central source, so we model this using partial covering by partially ionized material ([zxipcf]{}, available as an additional model for [xspec]{}). This fits the data fairly well (Fig. \[fig:pcf\]a) for a column with very low ionization covering 75 per cent of the source. We call this model [ionpcf1]{}. A second partially ionized, partial covering component is marginally significant ($\Delta\chi^2=12$ for 4 additional parameters), but this must be outflowing, with a blueshift of $0.12\pm 0.01c$. This is similar to the velocities sometimes seen in highly ionized K$\alpha$ absorption lines from iron in high mass accretion rate AGN (e.g. the compilation Reeves et al. 2008), and also close to the theoretical expectations of the maximum velocity of a UV line driven disc wind (Proga et al. 2004). Hereafter we call this model [ionpcf2]{}.
[c]{}
[c]{}
Similarly to Fig. \[fig:rms\_refl\]a and Fig. \[fig:rms\_abs\]a, pivoting the power law at the peak flux of seed photons gives a linearly rising rms (Fig. \[fig:rms\_ion\]a) which does not match the sharper rise in rms seen in the data. Again, dilution by a constant component at low energies is required, so Fig. \[fig:rms\_ion\]b shows the variability pattern produced by assuming that the ionized absorption component at low energies remains constant while the rest of the spectrum varies as before. Physically this could arise from light travel time delays if this is scattered from ionized material at some distance from the source. This produces some suppression of variability at low energies, but not as much as required to fit the rms of the rapid variability.
Name Description [xspec]{} syntax
------------- --------------------------------- ---------------------------------------------------------------
[disk]{} Standard disc [wabs(diskbb+powerlaw)]{}
[slim]{} Advective disc [wabs(diskpbb+powerlaw)]{}
[comp]{} Comptonized component [wabs(comptt+powerlaw)]{}
[swind]{} Smeared wind absorption [wabs\*swind(powerlaw)]{}
[xscort]{} Ionized absorption/emission [wabs(xscort)]{}
[ref1]{} Ionized reflection (single) [wabs(powerlaw+conline\*reflbal(powerlaw))]{}
[ref2]{} Ionized reflection (double) [wabs(powerlaw+conline(thcomp)+conline\*reflbal(powerlaw))]{}
[ionpcf1]{} Single ionized partial covering [zxipcf\*wabs(powerlaw)]{}
[ionpcf2]{} Double ionized partial covering [zxipcf\*zxipcf\*wabs(powerlaw)]{}
Model Parameter Value $\chi_\nu^{2}$/d.o.f
------------- --------------------------------- ------------------------ ----------------------
[comp]{} $\Gamma$ $2.28^{+0.13}_{-0.10}$ 289/210
$N_{\rm PL}$ ($\times10^{-4}$) $4.9^{+0.8}_{-0.3}$
$kT_e$ (keV) $0.26\pm0.03$
$N_{\rm comptt}$ $10.6^{+1.2}_{-0.8}$
[swind]{} $\Gamma$ $3.10^{+0.02}_{-0.05}$ 306/210
$N_{\rm PL}$ ($\times10^{-4}$) $23.1^{+0.9}_{-0.7}$
lg $\xi$ $2.82^{+0.09}_{-0.02}$
$\sigma$ $0.50^{+**}_{-0.05}$
[ref2]{} $\Gamma$ $3.61^{+0.08}_{-0.04}$ 268/206
$N_{\rm PL}$ ($\times10^{-4}$) $9.4^{+0.3}_{-2.4}$
$\Omega_1/2\pi$ $0.43^{+1.05}_{0.10}$
$R_{\rm in,1}$ $2.1^{+1.0}_{-0.8}$
lg $\xi$ $3.02^{+0.16}_{-0.07}$
$\Omega_2/2\pi$ $56^{+15}_{-18}$
$R_{\rm in,2}$ $3.2^{+0.4}_{-1}$
[ionpcf1]{} $\Gamma$ $3.69\pm0.05$ 287/210
$N_{\rm PL}$ ($\times10^{-4}$) $74\pm13$
lg $\xi$ $0.8^{+0.3}_{-0.5}$
$N_H$ ($10^{22}$ cm$^{-2}$) $14^{+2}_{-3}$
$f$ $0.84^{+0.02}_{-0.04}$
[ionpcf2]{} $\Gamma$ $3.68^{+0.09}_{-0.13}$ 233/206
$N_{\rm PL}$ ($\times10^{-4}$) $165^{+367}_{-69}$
lg $\xi_1$ $0.7^{+0.4}_{-0.9}$
$N_{H,1}$ ($10^{22}$ cm$^{-2}$) $12^{+2}_{-5}$
$f_1$ $0.80\pm 0.06$
lg $\xi_2$ $2.72^{+0.05}_{-0.5}$
$N_{H,2}$ ($10^{22}$ cm$^{-2}$) $153^{+13}_{-14}$
$f_2$ $0.63^{+0.07}_{-0.09}$
$v_2/c$ $0.12\pm 0.01$
: Best-fitting parameters of selected spectral models. Models described in the text that do not fit the spectrum are not shown here. Error bars are 90 per cent confidence, and \*\* denotes a parameter that reached its limit. $\Gamma$ is the photon spectral index, $N_{\rm PL}$ is power-law normalization at 1 keV, $\xi$ is the ionization (in the units of erg cm s$^{-1}$), $\sigma$ is the velocity dispersion in the units of $c$, $\Omega$ is the reflector solid angle, $R_{\rm in}$ is the inner disc radius (in the units of $GM/c^2$), $f$ is the covering factor.[]{data-label="tab:fit_results"}
Constraints from spectra and variability
========================================
Any viable model for the soft excess must simultaneously fit both the spectrum and spectral variability with the same model parameters. The spectra alone can be fit with a variety of continua. There is clear curvature but this is degenerate. The spectra can either be fit with low energy spectral curvature, together with a ($\Gamma\sim 2.3$) power law at high energies, or with a steep power law at low energies ($\Gamma\sim 3.6$) together with curvature at high energies from emerging reflected ([ref2]{}) or absorbed components ([ionpcf2]{}), or by a somewhat less steep power law ($\Gamma\sim 3.3$) with curvature around 0.7–2 keV from absorption in a rapidly accelerating wind ([swind]{}). However, the energy dependence of the rapid X-ray variability (which is dominated by the QPO in these data) breaks these degeneracies. The rms spectrum over the whole observation shows a smooth rise with energy, initially appearing to favour models where there is a single variable component (pivoting power law) which forms the spectrum. However, this rms is made up from two quite dissimilar variability patterns for the short and long timescales. Both these contain clear changes in the rms at $\sim
0.7$ keV, with the rapid variability amplitude strongly increasing with energy at this point while the longer timescale variability strongly decreases. These combine together in such a way as to produce an apparently featureless overall rms.
The rms of the rapid variability clearly supports the spectral decomposition where the soft excess is a separate, low temperature Comptonization component ([comp]{}). The variability drops just where the low temperature Compton component starts to dominate, so the rms shape is easily produced from models where the soft excess remains constant and simply dilutes the variability of a high energy power law tail. This is very similar to the sorts of spectral decompositions used in binary systems, where the QPO (and all other rapid variability) is associated with the tail, not with the disc.
By contrast, the alternative spectral models (where the soft excess is an artifact of some distortion on an intrinsic steep power law spectrum) have much more difficulty in matching the energy dependence of the rapid variability. Pivoting the steep intrinsic power law at the peak energy of the seed photons gives a linearly rising rms, so some additional assumptions are required to produce the observed faster drop in variability at low energies. Changing the ionization of the absorber in the smeared wind model ([swind]{}) gives enhanced variability in the 0.7–2 keV region which matches well to the shape of the rms at low energies. However, this then drops above 2 keV where ionization changes no longer affect the spectrum, which does not match the observed rms.
By contrast, both the double reflector ([ref2]{}) and double partial covering model ([ionpcf2]{}) have ionized components at low energies, so holding these constant while the rest of the spectrum responds to the pivoting power law dilutes the low energy variability. However, physically this seems somewhat contrived, and both these (but especially [ionpcf2]{}) have difficulties in suppressing enough variability at the lowest energies.
[c]{}
Thus the combined spectral and spectral variability constraints strongly favour the model where the soft excess is a true continuum component, described by a low temperature Comptonized disc component ([comp]{}). The long timescale variability can also be explained in this spectral decomposition by changing the temperature and/or optical depth of the Comptonization to give a predicted rms which matches the data very well. However, the light curve shows that there can also be discrete energy dependent events (the dip), which seems more likely to be from an occultation. If so, then there should also be some contribution from absorption in the spectrum, and this might also shape the longer timescale variability.
Constraints from the broad band spectral energy distribution
============================================================
This decomposition of the X-ray spectrum into a Comptonized disc with separate tail to high energies ([comp]{}) also looks sensible in the light of the overall spectral energy distribution. Both the UV/EUV/soft X-ray shape and the rms variability of the X-ray emission show that this peaks at $\sim 0.15$ keV (Puchnarewicz et al. 2000; Casebeer et al. 2006; Fig. \[fig:rms\]). Integrating the [comp]{} model down to this energy gives a luminosity of $\sim 7\times 10^{43}$ ergs s$^{-1}$. This is already more than 10 per cent of the bolometric luminosity of $\sim 5\times 10^{44}$ ergs s$^{-1}$. However, in the reflection and partial covering models, [ref2]{} and [ionpcf2]{}, the observed soft X-ray flux is only a small fraction of the total illuminating power law. The hidden emission required by both these models is a factor $\sim 10$ larger. This luminosity must be reprocessed, yet is more than the total bolometric luminosity of the source! Thus these two models fail on energetics as well as requiring somewhat contrived assumptions in order to match the rms variability and requiring an uncomfortably steep ($\Gamma\sim
3.6$) intrinsic continuum.
This all strongly supports the conclusion that in RE J1034+396 the soft excess is a Comptonized disc component, connecting smoothly onto the EUV peak of the spectral energy distribution and extending out to $\sim 1$ keV. This does not have an obvious counterpart in the typical black hole binary systems. These can show convincingly clean disc spectra (thermal dominant state, high/soft state, ultrasoft state), or higher temperature Comptonization (10–20 keV: very high state, intermediate state, steep power law state), but generally they do not show low temperature Comptonization (see e.g. Done, Gierli[ń]{}ski & Kubota 2007). However, most of these systems have $L/L_{\rm
Edd}<1$, so they may not be a good guide to the spectra of super Eddington flows. The disc structure should change at such high mass accretion rates, with strong winds potentially increasing the amount of material in the corona, leading to mass loading of the acceleration mechanism and decreasing energy per particle. Whatever the origin, a similar process of low temperature disc Comptonization probably happens in the unique galactic black hole binary GRS 1915+105. This is the only black hole binary in our Galaxy which consistently shows super Eddington luminosities (Done, Wardzi[ń]{}ski & Gierli[ń]{}ski 2004), and it can show spectra which are dominated by a huge, low temperature Comptonized disc component, with a weak power law tail to higher energies. Fig. \[fig:1915\] shows how one of these spectra from GRS 1915+105 (the low $\omega$ state in fig. 7 of Zdziarski et al. 2005) fits almost exactly onto the XMM-Newton spectrum of RE J1034+396 with a shift in energy scale by a factor 20. This gives a mass estimate for RE J1034+396 of $\sim 2\times 10^6 $M$_\odot$ from scaling the disc temperature as $M^{-1/4}$, which seems quite reasonable (Puchnarewicz et al. 2002).
Understanding the origin of the soft excess in RE J1034+396 may not necessarily solve the problem of the soft excess in general. The spectral energy distribution of this object is plainly rather different from that in most other high mass accretion rate AGN in that the soft excess contains a large fraction of the bolometric luminosity of the source (Middleton et al. 2007). Spectral distortion models (reflection or absorption) are clearly more likely to explain a feature carrying only a small fraction of the bolometric luminosity. Instead, the continuous EUV/soft X-ray excess in RE J1034+396 clearly favours a common origin for the disc and soft excess, unlike that for most other quasars, where the UV and soft X–rays do not appear to smoothly connect to each other in individual objects (Haro-Corzo et al. 2007).
Conclusions
===========
The combined constraints from the spectrum and variability show the soft excess in RE J1034+396 is most likely a smooth extension of the accretion disc peak in the EUV probably arising from low temperature Comptonization of the disc. This remains more or less constant on short timescales, diluting the QPO and rapid variability seen in the power law tail at the low energies where the soft excess dominates. As in the black hole binary systems, the QPO is a feature of the tail [*not*]{} of the disc.
The conclusion that the soft excess is a low temperature Comptonization of the disc emission may not necessarily be more widely applicable to other NLS1. The spectrum of RE J1034+396 is unique, so extrapolating results from this object may not be justified. In particular, if this object has a very high (super Eddington?) mass accretion rate then it could enter a new accretion state (perhaps analogous to the unique galactic binary GRS 1915+105 and the Ultra Luminous X-ray sources: Roberts 2008) which may also be the trigger for its so far unique QPO. A different origin for the soft excess in RE J1034+396 has the advantage that it would not require some unknown physical mechanism to restrict the temperature of the continuum in all objects to a narrow range (Gierli[ń]{}ski & Done 2004), though does require a coincidence that this temperature in RE J1034+396 is so close to that ubiquitously seen for the soft excess. Nonetheless, it is clearly possible that in selecting the most extreme soft excess object, we have also selected the one where the the soft excess has a different origin.
\[lastpage\]
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Films of cerium-doped LaMnO$_3$, which has been intensively discussed as an electron-doped counterpart to hole-doped mixed-valence lanthanum manganites during the past decade, were analyzed by x-ray photoemission spectroscopy with respect to their manganese valence under photoexcitation. The comparative analysis of the Mn 3s exchange splitting of La$_{0.7}$Ce$_{0.3}$MnO$_3$ (LCeMO) films in the dark and under illumination clearly shows that both oxygen reduction and illumination are able to decrease the Mn valence towards a mixed 2$+$/3$+$ state, independently of the film thickness and the degree of CeO$_2$ segregation. Charge injection from the photoconductive SrTiO$_3$ substrate into the Mn e$_g$ band with carrier lifetimes in the range of tens of seconds and intrinsic generation of electron-hole pairs within the films are discussed as two possible sources of the Mn valence shift and the subsequent electron doping.'
author:
- Andreas Thiessen
- Elke Beyreuther
- Stefan Grafström
- 'Lukas M. Eng'
- Kathrin Dörr
- Robert Werner
- Reinhold Kleiner
- Dieter Koelle
title: 'Mn$^{2+}$/Mn$^{3+}$ state of La$_{0.7}$Ce$_{0.3}$MnO$_3$ by oxygen reduction and photodoping'
---
Introduction
============
Mixed-valence manganites have been in the focus of extended research activities for decades due to their intriguingly manifold magnetic and electronic phases, high spin polarization, or the occurrence of the colossal magnetoresistance (CMR) effect in a number of such compounds [@ram97; @coe99; @hag03; @dor06]. It is of fundamental academic interest to fully understand microscopically the strong coupling of different degrees of freedom, the resulting subtle phase equilibria, the sensitivity of the latter to external stimuli, as well as other peculiarities such as electronic phase separation or completely new physical properties in thin film heterostructures. Furthermore, some compounds, such as La$_{0.7}$Sr$_{0.3}$MnO$_3$ (LSMO), show high spin polarization at room temperature, which makes them interesting for applications in upcoming oxide-electronic devices.
In the past, most frequently hole-doped rare-earth manganites such as La$_{1-x}$Ca$_{x}$MnO$_3$ (LCMO), the above-mentioned La$_{1-x}$Sr$_{x}$MnO$_3$ (LSMO), Pr$_{1-x}$Ca$_{x}$MnO$_3$ (PCMO), or Pr$_{1-x}$Sr$_{x}$MnO$_3$ (PSMO) – all with 0$<$x$<$1, were investigated. There, in the parent compounds (LaMnO$_3$ or PrMnO$_3$, respectively) part of the trivalent rare earth ions (La$^{3+}$ or Pr$^{3+}$) are replaced by divalent cations such as Sr$^{2+}$ or Ca$^{2+}$. To preserve charge neutrality part of the originally trivalent Mn ions are forced into a tetravalent state, finally leading to a mixed Mn$^{3+/4+}$ state, which corresponds to hole doping. The mixed Mn$^{3+/4+}$ valence is crucial for the specific electronic transport mechanisms (e.g. the double-exchange scenario) of the manganites and the strong coupling of magnetic order and electric transport.
With regard to possible all-oxide or even all-manganite devices, the question whether LaMnO$_3$ accepts a La$^{3+}$ substitution by tetravalent cations such as Ce$^{4+}$, Te$^{4+}$, Sn$^{4+}$ was raised around a decade ago. Nominally such a substitution would result in a mixed Mn$^{2+/3+}$ state and electron doping. However, for the most intensively investigated case of Ce doping, it was found that single-phase materials can be grown only in the form of thin films in a limited range of doping concentrations [@ray03], because of the unfavorable ionic sizes of Ce$^{4+}$ and Mn$^{2+}$. While the most previous investigations deal with fundamental preparation [@mit01c; @yan04b], transport [@man97; @ray99], magnetism [@ray99; @geb99; @lee01], electronic structure [@min01; @jol02; @mit03a; @han04a], or valence [@phi99; @kan01b] issues – always motivated by the question whether an electron-doping is really present or not – a few studies went a step further and focused on possible heterostructures including LCeMO films [@mit01a; @mit03b; @cho05]. A more detailed review of the debate was given in . In general, one can summarize, that La$_{1-x}$Ce$_{x}$MnO$_3$ (LCeMO) films tend to CeO$_2$ phase segregation and oxygen enrichment, the latter leading to an effective hole doping of as-grown films.
On the other hand, as discussed in our former XPS investigation [@bey06], oxygen reduction by heating in an ultrahigh-vacuum environment is a suitable way to drive the Mn valence towards $2+/3+$, which is, however, connected with a decisive resistance increase and the loss of an important functional property, namely the manganite-typical metal-insulator transition (MIT) [@bey09]. As a further intriguing aspect, those oxygen-reduced, insulating, electron-doped LCeMO thin films exhibited a large photoconductivity and the recovery of the MIT, in contrast to the non-photosensitive as-grown hole-doped films.
Besides a number of microscopic explanations of intrinsic photoinduced effects in manganites (as summarized in ), the injection of photogenerated charge carriers from the substrate [@bey09; @bey10; @kat00a; @kat01] seems to play a major and possibly technologically interesting role in thin-film-substrate heterostructures. In two studies of Katsu et al. [@kat00a; @kat01] La$_{0.7}$Sr$_{0.3}$MnO$_3$ films on SrTiO$_3$ were illuminated with a broad-band white-light source and exhibited a negative photoresistivity (PR), according to the definition $PR=(R_{dark}-R_{illum})/R_{illum}$.
The negative PR was interpreted as the injection of optically generated electrons from the SrTiO$_3$ substrate into the hole-doped film followed by the recombination of both carrier types in the film leading to a resistance increase. We picked up this stream of thinking in our two previous works, in which we observed a positive PR in La$_{0.7}$Ce$_{0.3}$MnO$_{3-\delta}$ [@bey09] and La$_{0.7}$Ca$_{0.3}$MnO$_{3-\delta}$ [@bey10] films on SrTiO$_3$. By comparatively evaluating the wavelength dependence of the PR and the surface photovoltage we concluded that the photogenerated carriers must stem from interband transitions in the substrate or be excitated from interface states.
In the present work, we aim at deepening our understanding of the light-induced charge carrier generation in a tetravalent-doped thin-film manganite. We performed a comparative investigation of the Mn valence of LCeMO films of different thicknesses on SrTiO$_3$ under broadband white-light illumination, by evaluating the Mn 3s exchange splitting in the x-ray photoemission spectrum in order to clarify the doping type under photoexcitation.
Experimental
============
Samples
-------
label thickness (nm) p(O$_2$) (mbar) XRD
------- ---------------- ----------------- ----------------- --
A 10 0.53 single phase
B 30 0.25 single phase
C 100 0.25 single phase
D 100 0.03 CeO$_2$ cluster
: \[tab\_1\]Samples of this study: LCeMO-film parameters; the full details to sample A can be found in , the details concerning samples B, C, D in . Note, that transmission electron microscopy had shown nanoscopic cerium oxide clusters in B and C, which were not visible in the XRD results.
Four different LCeMO films, in the following labelled A, B, C, D, and summarized in table \[tab\_1\], were grown by pulsed laser deposition on SrTiO$_3$ (100) single crystal substrates. The substrates were 10$\times$5$\times$0.5 mm$^3$ in size. Sample A was already the subject of our previous investigations – the corresponding references [@bey06; @bey09] contain the preparation details. The growth and structural analysis of samples B, C, D are given in . Samples B and C were grown under identical oxygen partial pressure of p(O$_2$)$=$0.25 mbar but are different in thickness, while samples C and D have the same thickness but D was grown under a much lower oxygen pressure, which led to microscopic segregation of CeO$_2$ clusters. Thus, the comparison of B and C provides information on the thickness dependence of the observed effects, while contrasting C and D is relevant to quantify the influence of the phase segregation.
To separate the influence of oxygen stoichiometry from any of the above parameters, each sample was studied in two different states of oxygen reduction, which were prepared by heating in a low-pressure oxygen atmosphere: A *slightly reduced* state was achieved by annealing the samples at 480$\symbol{23}$C in 10$^{-6}$ mbar oxygen partial pressure for 1 h, while a *highly reduced* state was prepared by heating at 700$\symbol{23}$C in 10$^{-8}$ mbar oxygen partial pressure for 2 h.
XPS measurements
----------------
The photoemission equipment for the XPS measurements uses a Mg anode and has been specified elsewhere [@bey06], where also details on the LCeMO overview spectra, surface cleaning, and background correction are elaborated. All spectra were recorded at room temperature.
For studying the effect of photoexcitation on the Mn core signals the respective sample was simultaneously illuminated by the broadband output of a Hg arc lamp providing a maximum intensity of 18.6 W mm$^{-2}$ (integrated over the whole spectrum). Between 390 nm and 250 nm the integrated intensity measures 9.6 W mm$^{-2}$. This interval is the spectral range relevant for band-to-band excitation of electrons in the SrTiO$_3$ substrate. The white light was focused by a fused-silica lens and transmitted through an UV-transparent viewport into the UHV apparatus.
In contrast to our former XPS work [@bey06], which was limited to one 10-nm-thick, gradually oxygen-reduced LCeMO film, now the Mn 3s exchange splitting energy $\Delta E_{3s}$ is measured for a whole set of LCeMO films, cf. table \[tab\_1\], in two different states of oxygen reduction.
As pointed out earlier, the Mn valence $V_{Mn}$ can be calculated from the measured value of $\Delta E_{3s}$ via the empirical equation:
$$\label{eq_valence}
V_{Mn}=9.67 - 1.27 \mbox{(eV)}^{-1} \cdot \Delta E_{3s} \quad ,$$
which is based on XPS results of a number of Mn compounds with well-known Mn valences [@zha84; @gal02].
 Mn 3s doublet and neighboring La 4d signals: Deconvolution into individual peaks in order to find a realistic value of the Mn 3s exchange splitting energy. (b) Increase of the Mn 3s exchange splitting energy under illumination in an oxygen-reduced LCeMO film.](figure1.pdf){width="45.00000%"}
As shown in figure \[fig1\](a), the intensity of the Mn 3s lines is comparably low and, more problematic, they are located in the direct neighborhood of the La 4d lines and part of their satellites. For large Mn 3s exchange splittings the smaller peak of the Mn 3s doublet even sits on the shoulder of the La 4d K$\alpha_3$ satellite. Consequently, the quantitative determination of $\Delta E_{3s}$ and thus $V_{Mn}$ needs a careful deconvolution of the spectrum into the individual peaks of the La 4d and Mn 3s structures. The La 4d structure consists of four peaks and the corresponding K$\alpha_3$ and K$\alpha_4$ satellites. Together with the two peaks of the Mn 3s level this gives 14 peaks, each described by three parameters (position, height, and width). Thus, we obtain a regression function with 42 variables, which are fortunately not completely independent.
The La 4d signal is made up of four peaks with a fixed energetic separation and an intensity ratio between La 4d$_{5/2}$ and La 4d$_{3/2}$ of 0.69, which results in eight independent fit parameters, see [@how78]. Since the K$\alpha_3$ and K$\alpha_4$ satellites are energetically shifted images of the original La 4d signal, they theoretically cannot have free parameters. However, due to the noise of the measured data, it was necessary to treat the intensity and width of La 4d$_{5/2}$ K$\alpha_3$ and the width of La 4d$_{3/2}$ K$\alpha_3$ as free parameters initially – in order to make the regression algorithm more flexible. In practice, first the regression analysis [@fityk] (based on the Levenberg-Marquardt algorithm) was performed for the whole La 4d structure, then the regression function was extended by the two Mn 3s peaks and the regression analysis was rerun. Figure \[fig1\](a) represents a typical result of the fitting procedure.
After the respective oxygen reduction procedure the sample was moved from the preparation chamber to the photoemission chamber – all in the same UHV apparatus– and an overview x-ray photoemission spectrum as well as a detailed spectrum of the region of a possible C 1s peak (as a test for possible organic contaminations) were taken. To study the influence of illumination on the Mn valence and to conclude on the nature of the photogenerated carriers, spectra of the La4d/Mn 3s region in the dark and afterwards under illumination were recorded. Figure \[fig1\](b) depicts two exemplary spectra of this region with and without light; the increase of the exchange splitting, which corresponds to a decrease of the Mn valence towards 2$+$/3$+$ according to (\[eq\_valence\]), is clearly visible.
Due to the weakness of the Mn 3s signal, noise influences the fitting results. Depending on the concrete case, each measurement was repeated between 5 and 13 times. For each of those measurements the sample was freshly prepared. Note that heating in oxygen-rich atmosphere approximately recovers the as-prepared state, as shown formerly [@bey06]. In some selected cases, a second dark measurement was performed after the measurement under light excitation. There was no remarkable difference observed between the two dark values of the valence shift. Thus we can exclude that the valence shifts are caused by further outdiffusion of oxygen into the UHV surroundings. Finally, the mean values of $V_{Mn}^{dark}$, $V_{Mn}^{illuminated}$, and $\Delta V_{Mn}=V_{Mn}^{dark}-V_{Mn}^{illuminated}$ were statistically calculated based on 95% confidence intervals. Figure \[fig2\] and table \[tab\_2\] show the mean values and the confidential intervals (errors).
In the dark, as clearly visible in figure \[fig2\], the slight reduction procedure is not able to drive the Mn valence below $+3$, while the stronger reduction leads to Mn valences below $+3$ and thus indeed to an electron doping. This is in qualitative accordance with our previous study [@bey06]. Within the error bars, it is hard to see systematic differences between the four individual samples. Besides the fact that the lowest Mn valence is achieved in the thinnest film, no serious systematic dependence on the thickness or the degree of phase segragation is visible.
Under illumination, both the slightly and the highly reduced films exhibit a *further* decrease of the Mn valence towards the mixed 2$+$/3$+$ state, which is equivalent to a photoinduced increase of the electron density in the Mn-$e_g$ orbital.
![\[fig2\]Overview of the main results: Mn valence of the four LCeMO films in two states of oxygen reduction, in the dark and under illumination, respectively. For better orientation, the nominal value of 2.7 is depicted as dashed line.](figure2.pdf){width="45.00000%"}
sample reduction $\Delta V_{Mn}$ $\Delta N_{e_g}$ ($\times$10$^{15}$) $\tau$ (s)
---------- ----------- ------------------ -------------------------------------- ----------------
A slight -0.32 $\pm$ 0.09 2.8 $\pm$ 0.8 2.9 $\pm$ 0.8
(10 nm) high -0.16 $\pm$ 0.08 1.4 $\pm$ 0.7 1.5 $\pm$ 0.7
B slight -0.15 $\pm$ 0.11 3.9 $\pm$ 2.9 4.1 $\pm$ 3.0
(30 nm) high -0.14 $\pm$ 0.04 3.6 $\pm$ 1.1 3.8 $\pm$ 1.2
C slight -0.15 $\pm$ 0.05 13.8 $\pm$ 4.2 13.8 $\pm$ 4.2
(100 nm) high -0.15 $\pm$ 0.08 13.8 $\pm$ 7.4 13.8 $\pm$ 7.4
D slight -0.11 $\pm$ 0.06 9.5 $\pm$ 5.3 9.5 $\pm$ 5.3
(100 nm) high -0.11 $\pm$ 0.03 9.5 $\pm$ 2.5 9.5 $\pm$ 2.5
: \[tab\_2\] XPS results of the light-induced Mn-valence shift $\Delta V_{Mn}$, the amount of the corresponding injected charge carriers $\Delta N_{e_g}$ according to eq. (\[eq\_N\_eg\]) and the associated carrier lifetime $\tau$. The latter are calculated using eq. (\[eq\_time\]) and assuming carrier injection from the substrate to be the dominant mechanism.
Importantly, control experiments with only-cleaned films in the dark and under illumination showed a mixed 3$+$/4$+$ valence *and no change* under illumination.
Discussion
==========
In general, at least two scenarios have to be discussed as possible explanations for the observed Mn valence decrease under illumination. First, the photoinduced generation of electron-hole pairs within the LCeMO films and a delayed recombination due to hole-trapping at oxygen vacancies, as already observed for other manganite compounds [@gil00], would be conceivable. Second, as pointed out earlier, for the 10-nm-thick LCeMO film, carrier injection from the substrate was suggested to be the main origin for the previously observed photoconductivity [@bey09]. Here, all films show a similar shift of the Mn valence towards the electron-doped state under illumination. Thus it seems likely that the injection mechanism plays a role also for the thicker films. Assuming a semiconductor band model, which has been successfully employed to explain a number of photoexcitation effects in manganites in the past, the samples are heterostructures of a wide-gap semiconductor (SrTiO$_3$, E$_g$=3.2 eV) and a narrow-gap semiconductor (LCeMO, E$_g$ around 1.0 eV), showing a band bending at the interface leading to a built-in field. Electron-hole pairs which are generated by photoexcitation in the substrate are separated by this field and the electrons can be injected into the LCeMO film.
In the following, the results for $V_{Mn}^{dark}$, $V_{Mn}^{illuminated}$, and $\Delta V_{Mn}$ are used to calculate the carrier densities $n_{e_g}$ in the Mn-$e_g$ orbitals of the LCeMO films according to:
$$\label{eq_n_eg}
n_{e_g}=\frac{3-V_{Mn}}{a^3} \quad .$$
The lattice constant $a$ measure 0.388 nm for samples A–C and 0.389 nm for sample D [@wer09]. The total number of charge carriers injected from the substrate into the film, $\Delta N_{e_g}$, can be calculated as:
$$\label{eq_N_eg}
\Delta N_{e_g}=\frac{\Delta V_{Mn}}{a^3}\cdot V \quad ,$$
with $V$ being the volume of the film. The results for $\Delta N_{e_g}$ are listed in table \[tab\_2\]. Note that equations (\[eq\_n\_eg\]) and (\[eq\_N\_eg\]) are valid for the charge-injection as well as for the intrinsic-carrier-generation scenario.
Assuming $\Delta N_{e_g}$ to be the equilibrium number of photogenerated electrons in the film, the absorbed photon flux $\Phi$ and the life time $\tau$ are connected with $\Delta N_{e_g}$ via:
$$\label{eq_time}
\Delta N_{e_g}=\Phi \cdot \tau \quad .$$
Using the charge-injection scenario, we can calculate $\tau$ from this relation. However, we have to be aware of the fact that the formula gives only a rough estimate of the carrier life times, since eq. (\[eq\_time\]) is only valid for a quantum efficiency for the electron-hole pair generation in the substrate of 100% and a simple exponential relaxation process. However, with the known flux of photons with energies above the SrTiO$_3$ bandgap of $\Phi=$9.45$\times$10$^{14}$ s$^{-1}$ and under the assumption that all photons from the light source are absorbed in the SrTiO$_3$ bulk, life times between 1 and 10 s are calculated, see table \[tab\_2\]. Obviously, the life time increases with the film thickness. Within the mechanism postulated above this can be understood as follows: the spatial separation of the electron-hole pairs by the internal field prevents the electrons from recombining. For thicker films the mean distance of the electrons from the interface is larger and consequently the life time is longer. Similarly long life times (22 s) were also observed by Gao et al. in electron-doped La$_{0.8}$Te$_{0.2}$MnO$_3$ films on SrTiO$_3$ by analyzing the relaxation of the photoresistivity [@gao09]. For the intrinsic-carrier-generation scenario, a serious value for $\Phi$ cannot easily be determined, since the LCeMO absorption coefficients for the differently oxygen-reduced states are unknown. Thus, we are reluctant to estimate life times for this scenario.
Furthermore, we have to comment on why the illumination-induced Mn valence shift is exclusively seen in oxygen-reduced samples. In principle, this observation is compatible with both scenarios. For the intrinsic carrier generation scenario, the carrier (hole) trapping at oxygen vacancies is indeed essential. If there were no centers for pinning the holes, a fast recombination of the photogenerated electron-hole pairs could take place and would prevent the films from any electron doping – exactly as observed in the as-prepared case. However, the charge-injection scenario would be possible in the as-prepared (oxygen-rich) samples as well. At the current state of knowledge one may only speculate that the probably dramatically changed band alignment at the interface – note that SrTiO$_3$ changes from p-type in the ideal stoichiometric case to n-type under reduction – leads to very different recombination conditions.
Finally, possible systematic errors of the Mn valences due to the surface sensitivity of XPS measurements have to be considered. Taking into account that only the uppermost 2–3 nm of the films are probed and that former XAS results [@wer09] revealed a higher Mn valence deeper inside the films than at their surfaces, the mean Mn valence can be higher than the values of table \[tab\_2\]. The finding that $\Delta V_{Mn}$ is independent of the film thickness might be, at least partially, an artifact produced by this systematic error.
Summary and Outlook
===================
The Mn valence of 10-, 30-, and 100-nm-thick La$_{0.7}$Ce$_{0.3}$MnO$_3$ films was determined from the Mn 3s exchange splitting in the x-ray photoemission spectrum in the dark and under white-light illumination. Independent of the film thickness and the degree of CeO$_2$ segregation, both oxygen reduction and illumination turned out to be effective ways to drive the Mn valence towards 2$+$/3$+$ and thus make the LCeMO films electron-doped. Nevertheless, the two routes are not equivalent: While *oxygen reduction* alone drives the system towards a Mn valence below $+$3, *photoexcitation* lowers the Mn valence only in films with a certain *initial* degree of oxygen reduction.
Discussing (i) a scenario postulating charge injection from the photoconductive SrTiO$_3$ substrate into the Mn e$_g$ band and (ii) a scenario assuming intrinsic photostimulated carrier generation and subsequent hole-trapping at oxygen vacancies within the LCeMO films as possible (and maybe coexisting) origins for the decreased Mn valence in LCeMO, we estimated the number of photogenerated injected carriers and their lifetimes in the films. To completely clarify which of the scenarios is the dominating one, a similar investigation of LCeMO on a non-photoconductive substrate as well as a more extended study of the spectral dependence of the photoconductivity than in ref. would be illuminative.
The question whether electron-doped manganites can be established by tetravalent-ion doping can be answered as follows: It is not primarily the tetravalent doping ions but post-deposition oxygen reduction and – optionally – illumination which lead to an electron-doped state. The fact that reduction and illumination can induce an electron-doped state in divalent-ion-doped manganites (whose growth is commonly easier) as well [@bey10] questions the need of tetravalent-ion substitution in manganites.
This work was financially supported by the German Research Foundation (DFG, project no. BE 3804/2-1 and EN 434/31-1).
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abstract: 'In [@zh], R. B. Zhang found a way to link certain formal deformations of the Lie algebra $\oo(2l+1)$ and the Lie superalgebra $\osp(1,2l)$. The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebras [*à la*]{} Drinfeld-Jimbo and to show how this construction can explain the main theorem of [@gl2]: the annihilator of a Verma module over the Lie superalgebra $\osp(1,2l)$ is generated by its intersection with the centralizer of the even part of the enveloping algbra.'
address: ' Dept. of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel, [email: lanzmann@@wisdom.weizmann.ac.il]{} '
author:
- Emmanuel Lanzmann
title: 'The Zhang transformation and $\cU_q\bigl(\osp(1,2l)\bigr)$-Verma modules annihilators'
---
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[^1]
Introdution
===========
A well known theorem of Duflo claims that the annihilator of a Verma module over a complex semi-simple Lie algebra is generated by its intersection with the centre of the enveloping algbra. In [@gl2] we show that in order for this theorem to hold in the case of the Lie superalgebra $\osp(1,2l)$ one has to replace the centre by the centralizer of the even part of the enveloping algbra. The purpose of this article is to show how quantum groups can illucidate this phenomemon.
Let $\fk$, $\fg$ be respectively the complex simple Lie algebra $\oo(2l+1)$ and the complex superalgebra $\osp(1,2l)$. From many point of views, the algebras $\fg$ and $\fk$ are very similar objects. For instance, identifying properly Cartan subalgebras of $\fk$ and $\fg$, the root systems $\Delta_{\fk}$, $\Delta_{\fg}$ are contained one into the other, and the set of irreductible roots of $\Delta_{\fg}$ is $\Delta_{\fk}$. Moreover, given a simple finite dimensional $\fg$-module, the corresponding simple $\fk$-module of the same highest weight is also finite dimensional and has the same formal character (and even the same crystal graph). Nevertheless, there is no obvious direct way to link the algebras $\fg$, $\fk$. To bridge the gap, one has to go through the quantum level: in his article published in 1992, R. B. Zhang (see [@zh], 3) found a recipe to pass from a certain formal deformation of $\cU(\fk)$ to a formal deformation of $\cU(\fg)$.
In this article we present a reformulation of the Zhang transformation in the more algebraic context of the quantizations [*à la*]{} Drinfeld-Jimbo. The idea is to start with the Drinfeld-Jimbo quantum enveloping algebra $\cU:=\cU_{\sqrt q}(\oo(2l+1))$ and to extend the torus by the finite group $\Gamma:=\Delta_{\fg}/ 2\Delta_{\fg}$. In other words, we introduce the semi-direct product $\wU:=\cU\rtimes k\Gamma$ where $\Gamma$ acts on $\cU$ in an obvious manner. Twisting the generators of $\cU$ by elements of $\Gamma$, we build a subalgebra $\bU$ of $\wU$ isomorphic to the quantum enveloping algebra $\cU_{-q}(\fg)$, and such that $\wU\simeq \bU\rtimes k\Gamma$. This construction provides an involution of the vector space $\wU$ mapping $\cU$ onto $\bU$. We call it the Zhang transformation.
A first obvious consequence of this construction is that the algebras $\cU$ and $\bU$ have the “same” finite dimensional modules. To be more precise, the quantum simple spinorial $\cU$-modules, viewed as $\bU$-modules (via $\wU$), are not deformation of $\fg$-modules (even up to tensorization by one dimensional modules). Moreover, roughly speaking, given a simple finite dimensional $\cU$-modules which is not of spinorial type, the specialization $q\rightarrow 1$ provides a simple $\fk$-module and the specialization $q\rightarrow -1$ a simple $\fg$-module. These classical $\fk$ and $\fg$-modules have therefore the same formal character. We believe that the characters of their Demazure modules are also equal, and this construction might be an interesting approach to this problem.
Another consequence, which is the crucial observation for our purpose, is that the Zhang transformation maps the centre of $\cZ(\cU)$ to $\cA(\bU)$, the commutant of the even part of the $\bU$. As in the classical case, $\cA(\bU)$ turns out to be the direct sum of the centre $\cZ(\bU)$ of $\bU$ and of the anticentre $\cZp(\bU)$, the subspace of elements which commute with even elements and anticommute with odd elements. The subspace $\cZp(\bU)$ is a cyclic module over the centre $\cZ(\bU)$. We construct a generator of this module which is a quantization of the element $T$ introduced in 4.4.1 [@gl2]. The set $\cZp(\bU)$ has another interpretation: this is the set of invariant elements under a certain twisted superadjoint action. This twisted action is the quantum analogue of the “non-standard” action considered in the classical case by Arnaudon, Bauer, Frappat (see [@abf], 2). More generally, we show that the locally finite part $\F(\cU)$ of $\cU$ for the adjoint action is mapped to the direct sum $\bF(\bU)\oplus \bFp(\bU)$ of the locally finite parts of $\bU$ for the superadjoint action and for its twisted version. This allows us to deduce from the work of Joseph and Leztzer (see [@jl]) that the annihilator of a $\bU$-Verma module in $\bF(\bU)\oplus \bFp(\bU)$ is generated by its intersection with $\cA(\bU)$ (theorem \[qthmduflo\]). We also prove a quantum analogue of theorem 7.1 [@gl].
The article is organized as follows. In section \[qsecwU\] we present the algebra $\wU$, the main object of this article, and we define the Zhang transformation. In section \[qtwisad\] we study the locally finite parts of $\wU$ for different actions and analyse how these actions are transformed by the Zhang transformation. We define in \[secalgstrucbU\] the subalgebra $\bU$. We show that $\bU$ is isomorphic to $\cU_{-q}(\fg)$ and we deduce from section \[qtwisad\] algebraic structure theorems for $\bU$. Section \[qsecdufloth\] is devoted to a proof of the annihilation theorem for $\bF(\bU)\oplus \bFp(\bU)$. In section \[qseccom\] we show that $\cA(\bU)$ coincides with the centralizer of the even part of $\bU$.
[*Acknowledgement*]{}. I am grateful to M. Gorelik for reading earlier versions of this paper and making numerous important remarks. I would like to thank T. Joseph for his support and his comments. I wish to express my gratitude to P. Cartier, M. Duflo, T. Levasseur and G. Perets for helpful discussions.
Background
==========
Notations {#notation4}
---------
Let ${\frak k}$ be the complex simple Lie algebra $\oo(2l+1)$, $l\geq 1$. Fix a Cartan subalgebra ${\frak h}$ of $\fk$ and denote by $\Delta_{\fk}$ the root system of $\fk$. We fix a basis of simple roots $\pi$ of $\Delta_{\fk}$. Denote by $W$ the Weyl group of $\Delta_{\fk}$, and set $\rho:=\displaystyle\mathop{\sum}_{\alpha\in
\Delta_{\fk}^+}\alpha$. Denote by $(-,-)$ the non-degenerate bilinear form on ${\frak h}^*$ coming from the restriction of the Killing form of ${\frak g}_0$ to ${\frak h}$. For any $\lambda,\mu\in
{\frak h}^*,\
(\mu,\mu)\not=0$ one defines $\langle\lambda,\mu\rangle:=2(\lambda,\mu)/ (\mu,\mu).$ One has the following useful realization of $\Delta_{\fk}$. Identify ${\frak h}^*$ with ${\Bbb C}^l$ and consider $(-,-)$ as an inner product on ${\Bbb C}^l$. Then there exists an orthonormal basis $\{\beta_1,\ldots,\beta_l\}$ such that $$\pi=\{\beta_1-\beta_2,\ldots,\beta_{l-1}-\beta_l,\beta_l\},\ \
\Delta_{\fk}=
\{\pm\beta_i\pm\beta_j, 1\leq i<j\leq l,\ \pm\beta_i,\ 1\leq i\leq l\}$$ and the Weyl group $W$ is just the group of the signed permutations of the $\beta_i$. We set $\alpha_i:=\beta_{i}-\beta_{i+1}$ with the convention $\beta_{l+1}=:0$. Let also $w_i$, $1\leq i\leq l$, be the elements $w_i:=\beta_1+\ldots +\beta_i$. With these notations, the set of weights $P_{\fk}(\pi)$ of $\Delta_{\fk}$ is $$P_{\fk}(\pi):=\Z w_1\oplus\ldots \Z w_{l-1}\oplus \Z (w_l/ 2).$$ We consider also the set of dominant weights $
P_{\fk}^+(\pi)$.
###
Let $\fg$ be the complex Lie superalgebra $\osp(1,2l)$. Denote by ${\frak g}_0$ the even part of $\fg$ and by ${\frak g}_1$ its odd part. We identify $\fh$ with a Cartan subalgebra of $\fg$ in such a way that $\pi$ is also a basis of simple roots of the root system $\Delta_{\fg}$ of $\fg$ with respect to $\fh$. The sets of even and odd roots of $\fg$ equal respectively $\{\pm\beta_i\pm\beta_j, 1\leq i<j\leq l,\ \pm 2\beta_i,
1\leq i\leq l\}$ and $\{\pm \beta_i,\ 1\leq i\leq l\}$. The set of weights $P_{\fg}(\pi)$ of $\Delta_{\fg}$ is $$P_{\fg}(\pi):=\Z w_1\oplus\ldots \Z w_{l-1}\oplus \Z w_l
=\Z \alpha_1\oplus\ldots \Z \alpha_{l-1}\oplus \Z \alpha_l.$$ In a standard manner we define the set of dominant weights $P_{\fg}^+(\pi)$.
### {#qcoef}
Let $q$ be an indeterminate and $k=\C(\sqrt{q})$. Let $\nu\in k$, be such that $\nu\not= 0,\pm 1$. For all $n\in \N$, we set $[n]_{\nu}:=\displaystyle{\nu^n-\nu^{-n}\over
\nu-\nu^{-1}}$ and $[n]_{\nu}!:=[n]_{\nu}\times[n-1]_{\nu}
\times\ldots\times [1]_{\nu}$ with the convention $[0]_{\nu}=1$. If $1\leq m\leq n\in \N$ we set $\binome{\nu}nm:=
\displaystyle{[n]_{\nu}!\over [m]_{\nu}![n-m]_{\nu}!}$.
The algebra $\cU$ {#qsecdefU}
-----------------
Let $\cU$ be the algebra over the field $k$ generated by the elements $E_i$, $F_i$, $1\leq i\leq l$, $K_{\mu}$, $\mu\in P_{\fg}(\pi)$ under the relations $$\label{defwueq1}
K_0=1,\ K_{\lambda}K_{\mu}=K_{\lambda+\mu}$$ $$\label{defwueq3}
K_{\lambda}E_jK_{\lambda}^{-1}=q^{(\lambda,\alpha_j)}E_j,\
K_{\lambda}F_jK_{\lambda}^{-1}=
q^{-(\lambda,\alpha_j)}F_j$$ $$\label{reluhef}\label{defwueq4}
E_iF_j-F_jE_j=\delta_{ij}\displaystyle{ K_{\alpha_i}-K_{\alpha_i}^{-1}
\over q-q^{-1}}$$ $$\label{reluhsere}
\displaystyle\sum_{k=0}^{1-\langle\alpha_j,\alpha_i\rangle}{(-1)}^k
\binome {q_i}{1-\langle\alpha_j,\alpha_i\rangle}k
E_{i}^{1-\langle\alpha_j,\alpha_i\rangle-k}
E_{j}E_{i}^{k}=0$$ $$\label{reluhserf}
\displaystyle\sum_{k=0}^{1-\langle\alpha_j,\alpha_i\rangle}{(-1)}^k
\binome {q_i}{1-\langle\alpha_j,\alpha_i\rangle}k
F_{i}^{1-\langle\alpha_j,\alpha_i\rangle-k}
F_{j}F_{i}^{k}=0$$ where $q_i:={q}^{(\alpha_i,\alpha_i)\over 2}$ and $\langle\alpha_j,\alpha_i\rangle:= 2(\alpha_j,\alpha_i)/
(\alpha_i,\alpha_i)$. Relations (\[reluhsere\]), (\[reluhserf\]) are called the quantum Serre relations.
### {#qsubsecU}
Replacing $E_l$ by $\left({\displaystyle{q-q^{-1}}\over \sqrt{q}-\sqrt{q}^{-1}}\right)
E_l$ in the above equations, we see that $\cU$ is just the Drinfeld-Jimbo quantum enveloping algebra $\cU_{\sqrt q}(\fk)$.
### {#qnottorus}
Let $\cU^+$ (resp. $\cU^-$) be the subalgebra of $\cU$ generated by the $E_i$ (resp. $F_i$). We also denote by $\cU^o$ the subalgebra generated by the $K_{\lambda}$. If $T$ stands for the multiplicative group $\{K_{\mu},\ \mu\in P_{\fg}(\pi)\}$, then the group algebra of $T$ identifies with $\cU^o$. One has the triangular decomposition $\wU\simeq \cU^-\otimes \cU^o \otimes \cU^+$, this isomorphism of vector spaces being given by multiplication.
### {#qHopf}
The algebra $\cU$ is a $k$-Hopf algebra with counit $\varepsilon$, coproduct $\Delta$ and antipode $S$ defined by $$\varepsilon(K_{\lambda})=1,\
\varepsilon(E_i)=\varepsilon(F_i)=0$$ $$\begin{array}{c}\Delta(K_{\lambda})=K_{\lambda}\otimes K_{\lambda},\\
\Delta(E_i)=E_i\otimes 1+K_{\alpha_i}\otimes E_i,\
\Delta(F_i)=F_i\otimes K_{\alpha_i}^{-1}+1\otimes F_i\\
\end{array}$$ $$S(K_{\lambda})=K_{\lambda}^{-1},\
S(E_i)=-K_{\alpha_i}^{-1}E_i,\
S(F_i)=-F_iK_{\alpha_i}$$
### Representations {#repU}
Let $\widehat{T}$ be the group of characters of $T$ with values in $k$. Given a character $\Lambda\in
\widehat{T}$, we say that $\Lambda$ is linear if there exists $\lambda\in P_{\fk}(\pi)$ such that $\Lambda(K_{\mu})=q^{(\lambda,\mu)}\ \forall \mu\in
P_{\fg}(\pi)$. In that case we write $\Lambda:=q^{\lambda}$.
Let $M$ be a $T$-module. An element $m\in M$ is said to be an element of weight $\Lambda\in \widehat{T}$ if $K_{\mu}m=\Lambda(K_{\mu})m\ \forall \mu\in P_{\fg}(\pi)$. We also call $m$ a $T$-weight element. We denote by $M_{\Lambda}$ the subspace of elements of weight $\Lambda$.
For any $\Lambda\in \widehat{T}$, we denote by $M(\Lambda)$ the $\cU$-Verma module of highest weight $\Lambda$, and by $V(\Lambda)$ its unique simple quotient. We recall now basic properties of the representation theory of $\cU$ (see for instance [@jq], 4.4.9). If $\Lambda=q^{\lambda}$ is linear, and if there exists $\alpha\in \Delta^+_{\fk}$ such that $\langle \lambda+\rho,\alpha\rangle\in \N^+$ then $M(q^{s_{\alpha}.\lambda})$ is a submodule of $M(q^{\lambda})$. The simple module $V(\Lambda)$ is finite dimensional if and only if $\Lambda=\phi q^{\lambda}$ where $\lambda\in P_{\fk}^+(\pi)$ and $\phi\in \widehat{T}$ is such that $\phi(K_{\mu})=\pm 1$ for all $\mu\in P_{\fg}(\pi)$. Any finite dimensional $\cU$-module is completely reducible.
### {#weightU}
The group $T$ acts on $\cU$ by inner automorphisms. Thus we can speak of weight elements in $\cU$. By a slight abuse of notation we shall say that $u\in \cU$ is of weight $\lambda\in P_{\fk}(\pi)$ if it is actually of weight $q^{\lambda}$.
The algebra $\wU$ {#qsecwU}
=================
In this section we introduce the main object of this article.
Definition and basic properties
-------------------------------
### Definition {#qdefwU}
Let $\Gamma$ be the multiplicative group $\{\xi_{\mu}, \mu\in P_{\fg}(\pi)\}\simeq P_{\fg}(\pi)\slash 2P_{\fg}(\pi)$ and $k\Gamma$ its group algebra. The group $\Gamma$ acts on $\cU$ in the following natural way: $$\xi_{\mu}. u ={(-1)}^{(\mu,\lambda)}u,\ \ \forall u\in\cU \mbox{ of
weight $\lambda$},\ \forall \lambda,\mu\in P_{\fg}(\pi).$$ Using this action of $\Gamma$ we introduce the algebra $$\wU:=\cU\rtimes k \Gamma.$$ Throughout this article, we shall use the shortened notation $u\xi_{\mu}$ for an element $u\otimes \xi_{\mu}\in \wU$.
### The Hopf structure {#HopfwU}
Recall that the algebra $k\Gamma$ is a Hopf algebra for the counit $\varepsilon$, coproduct $\Delta$ and antipode $S$ defined by $$\varepsilon(\xi_{\mu})=1,\
\Delta(\xi_{\mu})=\xi_{\mu}\otimes \xi_{\mu},\ S(\xi_{\mu})=\xi_{\mu}.$$ Since $\Gamma$ acts on $\cU$ by Hopf algebra automorphisms, $\wU=\cU\rtimes k \Gamma$ is a Hopf algebra for the obvious coalgebra structure (resp. antipode) on tensor product.
### A $\Z_2$-gradation {#qgrad}
The algebra $\wU$ is also endowed with the following $\Z_2$-gradation. Introduce $\xi:=\xi_{w_l}$, and define $$\ \forall \overline{i}\in\Z_2,\ \ {\wU}_{|_i}=\{x\in \wU,\
\xi x\xi={(-1)}^{i}x\}.$$ An element $x\in {\wU}_{|_0}\cup {\wU}_{|_1} $ is called $\Z_2$-homogenous, and we write $|x|=i$ if $x\in {\wU}_{|_i}$, $x\not=0$.
### The Harish-Chandra projection {#defhcq}
Recall the triangular decomposition given in \[qnottorus\]. Introduce the subalgebra $\wU^{o}:=\cU^o\otimes
k\Gamma$. One has the triangular decomposition $$\wU\simeq \cU^-\otimes \wU^o\otimes \cU^+.$$ Let $\cU^{++}$ (resp. $\cU^{--}$) be the augmentation ideal of $\cU^+$ (resp. $\cU^-$). The triangular decomposition of $\wU$ implies $\wU=(\cU^{--}\wU+\wU\cU^{++})\oplus
\wU^0$ which allows to define a Harish-Chandra projection $\Upsilon: \wU\longrightarrow \wU^o$ with respect to this decomposition.
Representations of $\wU$
------------------------
### Generalities {#qrepgen}
Let $\widehat{\Gamma}$ be the group of characters of $\Gamma$, identified with the set of group morphisms $P_{\fg}(\pi)\slash
2P_{\fg}(\pi)\longrightarrow \{1,-1\}$. Any $\lambda\in
P_{\fg}(\pi)$ defines a character $(-1)^{\lambda}\in \widehat{\Gamma}$ by the formula: $(-1)^{\lambda}(\mu):={(-1)}^{(\lambda,\mu)}.$ Observe that $\Gamma$ embeds in $\widehat{T}$ as the set $\{\Lambda\in \widehat{T}\mbox{ s.t. }\ \forall \mu\in P_{\fg}(\pi),\
\Lambda(K_{\mu})=\pm 1\}$.
Recall \[repU\]. Let $M$ be a $T\times \Gamma$-module. We say that an element $m\in M$ is a $T\times \Gamma$-weight element of weight $(\Lambda,\theta)
\in\widehat{T}\times \widehat{\Gamma}$ if $\xi_{\mu'}K_{\mu}m=\Lambda(K_{\mu})\theta(\mu')m\
\forall \mu,\mu'\in P_{\fg}(\pi)$.
Take $\Lambda\in\widehat{T}$ and $\theta\in\widehat{\Gamma}$. There is an obvious way to endow $M(\Lambda)$ with a structure of a $\wU$-module. Define for any $x\in M(\Lambda)$ of weight $q^{-\nu}\Lambda$, $\nu\in P_{\fg}(\pi)$, $$\xi_{\mu}x:={(-1)}^{(\mu,\nu)}\theta(\mu)x\ \ \forall \mu\in P_{\fg}(\pi).$$ We call this $\wU$-module a $\wU$-Verma module and we denote it by $M(\Lambda,\theta)$. By definition, $M(\Lambda,\theta)$ and $M(\Lambda)$ have the same submodules. Let $\Lambda=q^{\lambda}$ be linear. Assume that there exists $\alpha\in \Delta^+_{\fk}$ such that $\langle \lambda+\rho,\alpha\rangle\in \N^+$ and define $\theta':={(-1)}^{s_{\alpha}.\lambda-\lambda}\theta$. Then $M(q^{s_{\alpha}.\lambda},\theta')$ is a $\wU$-submodule of $M(q^{\lambda},\theta)$. The $\wU$-module $M(\Lambda,\theta)$ has a unique simple quotient, $V(\Lambda,\theta)$. As a $\cU$-module, $V(\Lambda,\theta)\simeq V(\Lambda)$.
### {#propcomplredwU}
All finite dimensional $\wU$-modules are completely reducible. Moreover, any simple finite dimensional $\wU$-module $M$ is isomorphic to a $V(q^{\lambda}\phi,\theta)$, $\lambda\in P^+_{\fk}(\pi)$, $\phi,\theta\in \widehat{\Gamma}$.
Let $M^{{\cU}^+}$ be the subspace of $M$ of invariant vectors by $\cU^+$. This subspace is stable by the action of the commutative algebra $\wU^o$. Since $\Gamma$ is finite and $M$ is a $\cU$-module of finite dimension, $\wU^o$ acts diagonally on $M$ and hence on $M^{{\cU}^+}$. Let $\{v_1,\ldots,v_r\}$ be basis of $M^{{\cU}^+}$ composed of $T\times \Gamma$-weight vectors. The representation theory of $\cU$ (see [@jan] chap. 5) asserts that on the one hand $M_i:=\cU v_i=\wU v_i$ is a simple $\cU$-module, and so a $\wU$-module of the form $V(q^{\lambda_i}\phi_i,\theta_i)$ with $\lambda_i\in P^+_{\fk}(\pi)$, $\theta,\phi_i\in \widehat{\Gamma}$, and on the other hand that $M=\oplus M_i$.
### The $\Z_2$-gradations {#qrepgengradM}
Both $M(\Lambda,\theta)$ and $V(\Lambda,\theta)$ inherits a $\Z_2$-gradation. Let $v_{\Lambda}$ be the highest weight vector of $M(\Lambda,\theta)$. They are two natural $\Z_2$-gradations on $M(\Lambda,\theta)$. Fix ${j}\in \Z_2$ and define ${M(\Lambda,\theta)}_{|_{ i}}=
\wU_{|_{ {i}+{j}}}v_{\lambda}\ \forall
{i}\in \Z_2$. The gradations on $V(\Lambda,\theta)$ are defined similarly using the highest weight vector of $V(\Lambda,\theta)$.
Three gradations and the Zhang transformation {#qsecUUs}
---------------------------------------------
### The gradation by the weights {#notweight}
The considerations of \[weightU\] extend to $\wU$. We shall denote by $\nu(x)$ the weight of a weight element $x\in\wU$ and by $\wU_{\nu}$ the subspace of elements of weight $\nu$. The algebra $\wU$ is graded by its weight subspace: $$\wU:=\bigoplus_{\nu\in P_{\fg}(\pi)} \wU_{\nu}.$$
### The $\mu$-gradation and the Zhang transformation {#defzhtransfor}
Define the $P_{\fg}(\pi)/2P_{\fg}(\pi)$-gradation $$\wU:=\bigoplus_{\mu\in P_{\fg}(\pi)/2P_{\fg}(\pi)}{}^{\mu}\wU$$ for which $\xi_{\lambda}\in {}^{0}\wU$, $K_{\lambda}\in {}^{\lambda}\wU$, $E_i\in {}^{\beta_{i+1}}\wU$, and $F_i\in{}^{\beta_{i}}\wU$ for all $\lambda\in P_{\fg}(\pi)$, $1\leq i\leq l$. If $x\in {}^{\mu}\wU$, we set $\mu(x):=\mu$. We call Zhang transformation the involution of vector space $\Psi:\wU\longrightarrow \wU$ such that $$\Psi(x):=\xi_{\mu} x\ \ \forall
x\in {}^{\mu}\cU,\ \forall \mu\in P_{\fg}(\pi)/2P_{\fg}(\pi).$$ We shall show in \[qalgstUs\] that $\Psi(\cU)$ is a subalgebra isomorphic to $\cU_{-q}(\fg)$ (see [@mz] for the definition of this algebra). For any homogenous elements $a,b$ for the respective gradations $(\wU_{\nu})$ and $({}^{\mu}\wU)$, $$\label{qcomPsi}
\Psi(ab)={(-1)}^{(\nu(a),\mu(b))}\Psi(a)\Psi(b).$$
### The $\delta$-gradation {#qdefgraddelta}
We introduce another $P_{\fg}(\pi)/2 P_{\fg}(\pi)$-gradation on $\wU$ (compare this gradation with the filtration defined in 5.3.1 [@jq]) $$\wU=\bigoplus_{\delta\in P_{\fg}(\pi)/2 P_{\fg}(\pi) } \wU^{\delta}$$ for which $\xi_{\mu},E_i\in \wU^0$, $K_{\mu}\in \wU^{\mu}$, $F_i\in \wU^{\alpha_i}$ for all $\mu\in P_{\fg}(\pi)$, $1\leq i\leq l$. A glance at the defining relations of $\wU$ ensures that this does define a gradation on $\wU$. If $x\in \wU^{\delta}$, we shall use the notation $\delta(x):=\delta$.
### Compatibilities between the gradations
The gradation $(\wU^{\delta})$ is invariant under the action of $T$ (see \[notweight\]). Thus the $\wU^{\delta}$ are direct sums of there weight subspaces. If $\delta,\nu\in P_{\fg}(\pi)$, we set $\wU_{\nu}^{\delta}:=\wU^{\delta}
\cap \wU_{\nu}$. One has the bigradation on $\wU$ $$\label{eqbigrad}
\wU=\bigoplus_{\nu\in P_{\fg}(\pi)\atop
\delta\in P_{\fg}(\pi)/2 P_{\fg}(\pi)}\wU_{\nu}^{\delta}.$$ The relation between the gradations $(\wU_{\nu})$, $({}^{\mu}
\wU)$, $(\wU^{\delta})$ reads as follows. The gradation $({}^{\mu}
\wU)$ is also $T$-invariant, and hence induces a bigradation $\wU=\oplus \bigl({}^{\mu}\wU\cap \wU_{\nu})$. Then, this bigradation coincides with the bigradation (\[eqbigrad\]). To be more precise, one has $$\label{qcompagrad}
\wU_{\nu}^{\delta}= {}^{\delta+ \eta(\nu)}\wU\cap \wU_{\nu}$$ where $\eta: P_{\fg}(\pi)\rightarrow P_{\fg}(\pi)/2P_{\fg}(\pi)$ is the map defined by $$\eta( \sum n_i\alpha_i):=\sum n_i\beta_{i+1}.$$ Indeed, it is enough to check (\[qcompagrad\]) on the generators. One has $\delta(E_i)+\eta(\nu(E_i))=\beta_{i+1}=\mu(E_i)$, $\delta(F_i)+\eta(\nu(F_i))=\alpha_i+\beta_{i+1}=\beta_i=\mu(F_i)$, $\delta(K_{\mu})+\eta(\nu(K_{\mu}))=\mu=\mu(K_{\mu})$, $\delta(\xi_{\mu})=\nu(\xi_{\mu})=\mu(\xi_{\mu})=0$.
### {#qparity}
Recall the $\Z_2$-gradation defined in \[qgrad\]. If $\nu\in P_{\fg}(\pi)$, we set $|\nu|:=(\nu,w_l)\pmod 2$. Observe that for all $\nu,\nu'\in P_{\fg}(\pi)$, the following identity holds in $\Z_2$: $$\label{qeqparity}
(\nu,\eta(\nu'))+(\eta(\nu),\nu')+(\nu,\nu')=|\nu||\nu'|$$ Since both sides of (\[qeqparity\]) are bilinear in $\nu,\nu'$, the identity (\[qeqparity\]) reduces to the case where $\nu=\alpha_i$, $\nu'=\alpha_j$. In that case, the left hand side of (\[qeqparity\]) is equal (in $\Z_2$) to $$(\alpha_i,\beta_{j+1})+(\beta_{i+1},\alpha_j)+(\alpha_i,\alpha_j)
= (\beta_i,\beta_j)+(\beta_{i+1},\beta_{j+1}) =\left\{\vcenter{
\hbox{$0$ if $(i,j)\not= (l,l)$}
\hbox{$1$ if $i=j=l$}}\right.= |\alpha_i||\alpha_j|$$
### {#qcomHopfbigrad}
Recall the definition of $\wU_{\nu}^{\delta}$ (see \[qdefgraddelta\]). One has $$\label{qcomcopbigrad}
\begin{array}{l} \Delta (\wU^{\delta}_{\nu})\subset
\displaystyle\bigoplus_{\nu_1+\nu_2=\nu}
\wU^{\delta+\nu_2}_{\nu_1}\otimes \wU^{\delta}_{\nu_2}\\
S(\wU^{\delta}_{\nu})\subset \wU^{\delta+\nu}_{\nu}
\end{array}$$ Indeed, according to \[qHopf\], \[HopfwU\] these inclusions are satisfied for the generators of $\wU$.
A Hopf superalgebra structure on $\wU$ {#sHopfwU}
--------------------------------------
The algebra $\Psi(\cU)$ is not a Hopf subalgebra of $\wU$ for the coproduct and antipode defined in \[qHopf\], \[HopfwU\]. In this subsection, we endow $\wU$ with a structure of Hopf superalgebra for which $\Psi(\cU)$ is a Hopf subalgebra.
Recall definitions of $\Psi$ (see \[defzhtransfor\]) and $\wU_{\nu}^{\delta}$ (see \[qdefgraddelta\]). Define for all homogenous elements $a\in \wU$ for the bigradation $\wU_{\nu}^{\delta}$: $$\label{qeqdefsHopf}
\begin{array}{lcl}
\overline{\Delta} \Psi(a)&:=& {(-1)}^{(\nu(a_1),\nu(a_2)+\eta(\nu(a_2))}
\Psi(a_1)\otimes \Psi(a_2)\\
\overline{S}\Psi(a)&:=& {(-1)}^{(\nu(a),\delta(a))}\Psi(S(a))
\end{array}$$ with Sweedler notation $\Delta(a)=a_1\otimes a_2$.
### {#qdefsgen}
Intoduce $e_i:=\Psi(E_i)$, $f_i=\Psi(F_i)$, $k_{\mu}:=\Psi(K_{\mu})$. On the generators the definitions (\[qeqdefsHopf\]) give: $$\begin{array}{c}\overline{\Delta}(\xi_{\mu})=\xi_{\mu}\otimes \xi_{\mu},
\ \overline{\Delta}(k_{\lambda})=
k_{\lambda}\otimes k_{\lambda},\\
\overline{\Delta}(e_i)=e_i\otimes 1+k_{\alpha_i}\otimes e_i,\
\overline{\Delta}(f_i)=f_i\otimes k_{\alpha_i}^{-1}+1\otimes f_i\\
\end{array}$$ $$\overline{S}(\xi_{\mu})=\xi_{\mu},\
\overline{S}(k_{\lambda})=k_{\lambda}^{-1},\
\overline{S}(e_i)=-k_{\alpha_i}^{-1}e_i,\
\overline{S}(f_i)=-f_ik_{\alpha_i}$$
### {#section-1}
$\wU$ endowed with $(\overline{\Delta},
\overline{S}, \varepsilon)$ is a Hopf superalgebra.
Retain the definition of the $\Z_2$-gradation on $\wU$ (see \[qgrad\]). Let us prove that for any $a,b\in\wU$ homogenous for the bigradation $(\wU_{\nu}^{\delta})$ $$\label{qscopcompmult}
\overline{\Delta}\bigl(\Psi(a)\Psi(b)\bigr)={(-1)}^{|a_2||b_1|}
\overline{\Delta}\Psi(a)\overline{\Delta} \Psi(b)$$ with Sweedler notation $\Delta(a)=a_1\otimes a_2$, $\Delta(b)=b_1\otimes b_2$. Recall (\[qcompagrad\]), (\[qcomcopbigrad\]) which assert in particular that $a,b,a_1,a_2,b_1,b_2$ are graded for the three gradations $(\wU_{\nu})$, $({}^{\mu}
\wU)$, $(\wU^{\delta})$. By definition of $\overline{\Delta}$ and by (\[qcomPsi\]) one has $$\begin{array}{lcl}
\overline{\Delta}\bigl(\Psi(a)\Psi(b)\bigr)&=&
{(-1)}^{(\nu(a),\mu(b)}\overline{\Delta}\Psi(ab)\\
&=&
{(-1)}^{(\nu(a),\mu(b))+(\nu(a_1b_1),a_2b_2+\eta\nu(a_2b_2))}
\Psi(a_1b_1)\otimes \Psi(a_2b_2)\\
&=& {(-1)}^s \Psi(a_1)\Psi(b_1)\otimes \Psi(a_2)\Psi(b_2)
\end{array}$$ where $s\in \Z_2$ and $$s:=\bigl(\nu(a),\mu(b)\bigr)+
\bigl(\nu(a_1b_1),\nu(a_2b_2)+\eta\nu(a_2b_2)\bigr) +
\bigl(\nu(a_1),\mu(b_1)\bigr)+\bigl(\nu(a_2),\mu(b_2)\bigr).$$ According to (\[qcompagrad\]) and (\[qcomcopbigrad\]), one has $\mu(b)=\delta(b)+\eta\nu(b)$, $\mu(b_1)=\delta(b)+\nu(b_2)+\eta\nu(b_1)$, $\mu(b_2)=\delta(b)+
\eta\nu (b_2)$, and hence $$\begin{array}{lcl}
s&=&\bigl(\nu(a_1a_2),\eta\nu (b_1b_2\bigr))+\bigl(\nu(a_1b_1),
\nu(a_2b_2)+ \eta\nu(a_2b_2)\bigr)+\bigl(\nu(a_1),\nu(b_2)\bigr)\\
&&\hskip 6.5truecm+\bigl(\nu(a_1),\eta\nu(b_1)\bigr)
+\bigl(\nu(a_2)+\eta\nu(b_2)\bigr).
\end{array}$$ Expending all scalar products in the above expression of $s$, we find $$\begin{array}{lcl}s&=&
\bigl(\nu(a_1),\nu(a_2)+\eta\nu(a_2)\bigr)+
\bigl(\nu(b_1),\nu(b_2)+\eta\nu(b_2)\bigr)\\
&&\hskip 3truecm
+\bigl(\nu(a_2),\nu(b_1)\bigr)+\bigl(\nu(b_1),\eta\nu(a_2)\bigr)+
\bigl(\nu(a_2),\eta\nu(b_1)\bigr).
\end{array}$$ Using (\[qeqparity\]), $s$ can be rewritten as $$s= \bigl(\nu(a_1),\nu(a_2)+\eta\nu(a_2)\bigr)+\bigl(\nu(b_1),\nu(b_2)+
\eta\nu(b_2)\bigr)
+|a_2||b_1|$$ and therefore $$\begin{array}{lcl}
\overline{\Delta}\bigl(\Psi(a)\Psi(b)\bigr)&=&
{(-1)}^{|a_2||b_1|}\Bigl({(-1)}^{(\nu(a_1),\nu(a_2)+\eta\nu(a_2))}
\Psi(a_1)\otimes \Psi(a_2)\Bigr)\\
&&\hskip 3.5truecm \times \Bigl({(-1)}^{(\nu(b_1),\nu(b_2)+
\eta\nu(b_2))} \Psi(b_1)\otimes \Psi(b_2)\Bigr)\\
&=&{(-1)}^{|a_2||b_1|} \overline{\Delta}\Psi(a)\overline{\Delta} \Psi(b)
\end{array}$$ which proves (\[qscopcompmult\]).
Let $m:\wU\otimes \wU\longrightarrow \wU$ be the multiplication map. We show next that for any element $a\in \wU$ homogenous for the bigradation $\wU_{\nu}^{\delta}$, $$\label{qsantipodcompmult}
m(1\otimes \overline{S})\overline{\Delta}\Psi(a)=
m(\overline{S}\otimes 1)\overline{\Delta}\Psi(a)=\varepsilon\Psi(a)$$ By definition of $\overline{\Delta}$, and $\overline{S}$ $$\begin{array}{lcl}
m(1\otimes \overline{S})\overline{\Delta}\Psi(a)&=&
{(-1)}^{(\nu(a_1),\nu(a_2)+\eta\nu(a_2))}\Psi(a_1)\otimes \overline{S}
\Psi(a_2)\\
&=&{(-1)}^{(\nu(a_1),\nu(a_2)+\eta\nu(a_2))+(\nu(a_2),\delta(a_2))}
\Psi(a_1)\otimes \Psi(Sa_2).
\end{array}$$ According to (\[qcomcopbigrad\]), $S(a_2)\in \wU^{\nu(a_2)+
\delta(a_2)}_{\nu(a_1)}$ and $\delta(a_2)=\delta(a)$. Then formula (\[qcomPsi\]) gives $\Psi(a_1)\Psi(Sa_2)={(-1)}^{(\nu(a_1),\delta(a)+\nu(a_2)+
\eta\nu(a_2))}\Psi(a_1Sa_2)$, and we obtain finally $$m(1\otimes \overline{S})\overline{\Delta}\Psi(a)={(-1)}^{(\nu(a),
\delta(a))}\Psi(\varepsilon(a))= {(-1)}^{(\nu(a),
\delta(a))}\varepsilon \Psi(a)=\varepsilon \Psi(a)$$ since $\varepsilon \Psi(a)=0$ if $\nu(a)\not =0$. The other equality of (\[qsantipodcompmult\]) can be established in the same way.
It remains to check that $$\begin{array}{c}
(\varepsilon \otimes 1)\overline{\Delta}\Psi(a)=(1\otimes \varepsilon)
\overline{\Delta}\Psi(a)=\Psi(a)\\
(\overline{\Delta}\otimes 1)\overline{\Delta}\Psi(a)=
(1\otimes\overline{\Delta})\overline{\Delta}\Psi(a).
\end{array}$$ These identities are straightforward from the definition of $\overline{\Delta}$.
Twisted adjoint actions and locally finite parts {#qtwisad}
================================================
The object of this section is to compute the locally finite parts of $\wU$ for certain twisted adjoint actions.
A general construction {#qactadgenconstr}
----------------------
Let $X=X_{|_0}\oplus X_{|_1}$ be a Hopf superalgebra. We recall that the Hopf structure of $X$ provides an adjoint action defined by the formula $\ad a(x)={(-1)}^{|x||a_2|}a_1xS(a_2)$, using the Sweedler notation: $\Delta(a)=a_1\otimes a_2$. There is an elementary way to construct new actions by twisting the adjoint action by an algebra morphism (similar twisted actions have been considered by Joseph in [@jprim]). One proceeds as follows. Let $\psi:X\longrightarrow X$ be an algebra morphism. For any $\Z_2$ homogenous element $a,x\in X$, set: $$\ad_{\psi}a(x)={(-1)}^{|a_2||x|}a_1xS(\psi(a_2)).$$ This formula defines an action since for any homogenous elements $a,b,x$ $$\begin{array}{lcl}(\ad_{\psi}a)(\ad_{\psi}b)(x)&=&
{(-1)}^{|b_2||x|+|a_2|(|b_1|+|x|+|b_2|)}
a_1b_1xS(\psi(b_2))S(\psi(a_2))\\
&=&{(-1)}^{|x|(|a_2|+|b_2|)+|b_1||a_2|}a_1b_1xS(\psi(a_2)\psi(b_2))\\
&=&{(-1)}^{|x|(|a_2|+|b_2|)+|b_1||a_2|}a_1b_1xS(\psi(a_2b_2))\\
&=&{(-1)}^{|x||(ab)_2|}(ab)_1xS(\psi((ab)_2))\\
&=&\ad_{\psi}(ab)(x)
\end{array}$$
{#qdeftwistadact}
Let $\mu\in
P_{\fg}(\pi)\slash 2P_{\fg}(\pi)$, and $\psi_{\mu}$ be the inner automorphism of $\wU$ defined by $\psi_{\mu}(a)=\xi_{\mu} a \xi_{\mu}$. In what follows we shall consider the twisted adjoint actions obtained by applying the construction \[qactadgenconstr\] to the cases of
- the genuine Hopf algebra $\wU$ (for the Hopf structure given in \[qHopf\]) and the morphisms $\psi_{\mu}$.
- the Hopf superalgebra $\wU$ (for the Hopf superstructure given in \[sHopfwU\]) and the morphisms $\psi_{\mu}$.
In order to avoid any confusion, we shall write $\ad$ the adjoint action of $\wU$, and $\bad$ the super adjoint action. The twisted actions are denoted respectively by $\ad_{\mu}:=\ad_{\psi_{\mu}}$, $\bad_{\mu}:=\bad_{\psi_{\mu}}$. Of course, $\ad_{0}=\ad$ and $\bad_{0}=\bad$. In the case $\mu=w_l$, we shall often prefer to write $\badp$ instead of $\bad_{w_l}$. The twisted adjoint action $\badp$ is the quantum version of the “non-standard” adjoint action considered by Arnaudon, Bauer, Frappat in [@abf], 2. Recall \[qdefsgen\]. By definition, $\badp a=\bad a$ for any generator $a\in \{k_{\lambda},\ \xi_{\lambda} \lambda\in P_{\fg}(\pi);\
e_i, f_i, 1\leq i<l\}$ and $$\label{eqadsdef}\begin{array}{c}
\bad(e_l)x=e_lx-{(-1)}^{|x|}k_lxk_l^{-1}e_l,\ \
\bad(f_l)x=f_lxk_l-{(-1)}^{|x|}xf_lk_l\\
\badp(e_l)x=e_lx+{(-1)}^{|x|}k_lxk_l^{-1}e_l,\ \
\badp(f_l)x=f_lxk_l+{(-1)}^{|x|}xf_lk_l
\end{array}$$
### {#qlocfindef}
If $N$ is any $\ad_{\lambda}$-stable (resp. $\bad_{\lambda}$-stable) subspace of $\wU$, we denote by $\F_{\lambda}(N)$ (resp. $\bF_{\lambda}(N)$) its locally finite part for the action $\ad_{\lambda}$ (resp. $\bad_{\lambda}$). If $\mu=0$ we shall write respectively $\F(N)$, $\bF(N)$ instead of $\F_0(N)$, $\bF_0(N)$. Also, we shall often prefer to use the notation $\bFp(N)$ instead of $\bF_{w_l}(N)$.
### {#qlemtwadact}
Let $\lambda\in P_{\fg}(\pi)$. By definition of $\ad_{\lambda}$, for all $x\in \wU$, and for all weight element $a\in \wU$, $(\ad_{\lambda} a)(\xi_{\lambda}x)=
a_1\xi_{\lambda}xS(\xi_{\lambda}a_2\xi_{\lambda})=
\xi_{\lambda} (\xi_{\lambda}a_1\xi_{\lambda}) x
S(\xi_{\lambda}a_2\xi_{\lambda})={(-1)}^{(\lambda,\nu(a))}
\xi_{\lambda}(\ad a)x.$ The same holds replacing $\ad$ by $\bad$. It follows that $$\F_{\lambda}(\wU)=\xi_{\lambda} \F(\wU)\mbox{ and }
\bF_{\lambda}(\wU)=\xi_{\lambda} \bF(\wU).$$
### {#section-2}
\[adactns0\] Let $\lambda$ be in $P_{\fg}(\pi)\slash
2P_{\fg}(\pi)$. Then
1. $\F_{\lambda}(\cU)=0$ if $\lambda\not =0$
2. $\F_{\lambda}(\wU)=\xi_{\lambda}\F(\cU)$
Assume that $\lambda\not=0$, and let $V\subset \cU$ be a simple $\ad_{\lambda}\cU$-module. Take $a$ an element of lowest weight of $V$. Since $\lambda\not=0$, there exists $\alpha_i\in \pi$ such that $(\lambda,\alpha_i)=1+2\Z$. Hence $0=\ad_{\mu} F_ia=(F_ia+aF_i)K_{\alpha_i}$. Proposition 1.7, [@dk], forces $a=0$. This establishes the assertion (i).
Let $a\in \F(\wU)$. Write $x=\sum_{\mu\in
P_{\fg}(\pi)\slash 2P_{\fg}(\pi)} \xi_{\mu}a_{\mu}$, $x_{\mu}\in
\cU$. According to \[qlemtwadact\] $\ad (\wU)x=\ad(\cU)x=
\bigoplus_{\mu\in
P_{\fg}(\pi)\slash 2P_{\fg}(\pi)} \xi_{\mu}\ad_{\mu}(\cU)(x_{\mu}).$ Thus $\ad_{\mu}(\cU)(x_{\mu})\in \F_{\mu}(\cU)$ and hence $x_{\mu}=0$ if $\mu\not=0$ by (i). This proves $\F(\wU)=\F(\cU)$ and (ii) follows from \[qlemtwadact\].
{#section-3}
Retain the definitions of \[qdefgraddelta\]. If follows from (\[qcomcopbigrad\]) and (\[qeqdefsHopf\]) that the gradation $(\wU^{\delta})$ possesses a very striking property: it is invariant by the actions $\ad_{\lambda}, \bad_{\lambda}$. Recall the definition of $\Psi$ (see \[defzhtransfor\]).
\[qlemtransfadbad\] Fix $\lambda,\ \delta\in {P_{\fg}(\pi)/2P_{\fg}(\pi)}$. Let $a,x\in \wU$ be homogenous elements for the bigradation $(\wU^{\delta}_{\nu})$. Then $$\label{formadadb}
\Psi\bigl(\ad_{\lambda}a(x)\bigr)=\pm
\bad_{\lambda+\delta}\Psi(a)\bigl(\Psi(x)\bigr).$$
Let $a,x\in \wU$ be as in the Lemma. By definition, $\ad_{\lambda}a x={(-1)}^{(\nu(a_2),\lambda)}
a_1 x S(a_2)$, where $\Delta(a)=a_1\otimes a_2$ in Sweedler notation. Recall (\[qcompagrad\]), (\[qcomcopbigrad\]) which assert in particular that $a,x,a_1,a_2$ are graded for the gradations $(\wU_{\nu})$, $({}^{\mu}
\wU)$, $(\wU^{\delta})$. Using (\[qcomPsi\]) one obtains $$\begin{aligned}
\Psi(\ad_{\lambda}a x)&={(-1)}^{(\nu(a_2),\lambda)+(\nu(a_1),\mu(x))+
(\nu(x),\mu (Sa_2))+(\nu(a_1),\mu (Sa_2))}
\Psi(a_1)\Psi(x)\Psi(Sa_2)\nonumber\\
&= {(-1)}^s \Psi(a_1)\Psi(x)\overline{S}\Psi(a_2)\label{qeqadcom1}\end{aligned}$$ where $s\in \Z_2$, and $$s=\underbrace{\bigl(\nu(a_2),\lambda\bigr)+\bigl(\nu(a_1),\mu(x)\bigr)+
\bigl(\nu(x),\mu (Sa_2)\bigr)}_{s_2}+
\underbrace{\bigl(\nu(a_1),\mu (Sa_2)\bigr)+\bigl(\nu(a_2),\delta(a_2)
\bigr)}_{s_2}.$$ According to (\[qcomcopbigrad\]) $\mu(x)=\delta(x)+\eta\nu(x)$, $\delta(a_2)=\delta(a)$, $\mu(S(a_2))=\delta(a)+\nu(a_2)+\eta\nu(a_2)$, so $$s_1=\bigl(\nu(a),\delta(a)\bigr)+\bigl(\nu(a_1),\nu(a_2)+\eta\nu(a_2)
\bigr)$$ and $$\begin{array}{lcl}
s_2&=& \bigl(\nu(a_2),\lambda\bigr)+\bigl(\nu(a_2)+\nu(a),\delta(x)+
\eta\nu(x)\bigr)
+\bigl(\nu(x),\nu(a_2)+\eta\nu(a_2)+\delta(a)\bigr)\\
&=& \bigl(\nu(a),\mu(x)\bigr)+\bigl(\nu(x),\delta(a)\bigr)+\bigl(\nu(a_2),
\lambda+\delta(x)\bigr)\\
&& \hfill+ \bigl(\nu(a_2),\eta\nu(x)\bigr)+
\bigl(\nu(x),\eta\nu(a_2)\bigr)+\bigl(\nu(x),\nu(a_2)\bigr)\\
&=& \bigl(\nu(a),\mu(x)\bigr)+\bigl(\nu(x),\delta(a)\bigr)+\bigl(\nu(a_2),
\lambda+\delta(x)\bigr)+
|a_2||x|\ \ \mbox{ by (\ref{qeqparity})}.
\end{array}$$ Consequently $$s=s_1+s_2=t + |a_2||x| +
(\nu(a_2),\lambda+\delta(x))+\bigl(\nu(a_1),\nu(a_2)+\eta\nu(a_2)\bigr)$$ where $t=(\nu(a),\delta(a))+ (\nu(a),\mu(x))+(\nu(x),\delta(a))$ depends only on the “degrees” (for the different gradations) of $a$ and $x$. Substituting the above expression of $s$ in (\[qeqadcom1\]), and using definitions of $\overline{\Delta}$, $\overline{S}$ we derive that $$\label{qeqfinecompad}
\Psi(\ad_{\lambda}a x)={(-1)}^t \bad_{\lambda+\delta(x)}\Psi(a)\Psi(x)$$ as required.
{#qalgstrucrec}
We recall the results of Joseph and Letzter (see [@jl2]): $$\F(\cU)=\displaystyle\bigoplus_{\lambda\in P^+_{\fk}(\pi)} (\ad\cU)
K_{-2\lambda}$$ and each $(\ad\cU) K_{-2\lambda}$ contains a unique (up to a non-zero scalar) central element denoted by $z_{2\lambda}$. The centre $\cZ(\cU)$ of $\cU$ is the polynomial algebra $$\label{tonycentre}
\cZ(\cU)=\C[z_{2w_1},\ldots, z_{2w_{l-1}}, z_{w_l}].$$ We shall need the following submodules of $\F(\cU)$: $$N_0:=\displaystyle\bigoplus_{\lambda\in P^+_{\fg}(\pi)}
(\ad\cU)K_{-2\lambda},\ \ \ N_1:=
\displaystyle\bigoplus_{\lambda\in P^+_{\fk}(\pi)
\backslash P^+_{\fg}(\pi)} (\ad\cU) K_{-2\lambda}.$$ If $\lambda\in P^+_{\fg}(\pi)$, then $\delta(K_{-2\lambda})=2\lambda=0$ and so $N_0\subset \cU^0$. If ${\lambda\in P^+_{\fk}(\pi)
\backslash P^+_{\fg}(\pi)}$, i.e. $\lambda=\lambda'+{w_l\over 2}$, $\lambda'\in P^+_{\fg}(\pi)$, then $\delta(K_{-2\lambda})=w_l$ and so $N_1\subset \cU^{w_l}$. Consequently $$\label{qN0N1}
\begin{array} {l} \F(\cU)\cap \cU^0=N_0\\
\F(\cU)\cap \cU^{w_l}=N_1.
\end{array}$$ It follows from that $\bF(\wU)\cap \wU^{\mu}=\Psi\bigl(\F_{\mu}(\wU)\cap \wU^{\mu}\bigr)$. By we know that $\F_{\mu}(\wU)=\xi_{\mu}\F(\cU)$. Hence, using (\[qN0N1\]) and recalling \[qlemtwadact\], we obtain $\forall \lambda\in P_{\fg}(\pi)/2 P_{\fg}(\pi)$, $$\label{qeqbFfctF2}
\begin{array}{lcl}
\bF_{\lambda}(\wU)&=&\bigoplus_{\mu} \xi_{\lambda+\mu}\Psi\bigl(\F(\cU)
\cap \cU^{\mu}\bigr)\\
&=&\xi_{\lambda}\bigl( \Psi(N_0)\oplus \xi \Psi(N_1)\bigr).
\end{array}$$
Algebraic structures of $\cU_{q}(\fg)$ {#secalgstrucbU}
======================================
Recall the definition of $\Psi$ (see \[defzhtransfor\]). We define $\bU:=\Psi(\cU)$. By definition of $\Psi$, $$\wU\simeq \bU\rtimes k\Gamma.$$ With the notations of \[qdefsgen\], $\bU$ is the subalgebra of $\wU$ generated by the $e_i$, $f_i$, $k_{\lambda}$. The algebra $\bU$ is graded for all the different gradations we defined on $\wU$. By definition of $\overline{\Delta}, \overline{S}$, the subalgebra $\bU$ is a Hopf subalgebra of $(\wU,\overline{\Delta}, \overline{S},
\varepsilon)$.
We shall show in \[qalgstUs\] that $\bU\simeq \cU_{-q}(\fg)$.
Representations of $\bU$
------------------------
We shall now give the classification of the finite dimensional $\bU$-modules.
### Generalities {#fdUsmod.1}
Let us prove that every simple $\bU$-module is the restriction of a simple $\wU$-module and that every finite dimensional $\bU$-modules is completely reducible.
Let $V$ be a simple finite dimensional $\bU$-module. Assume for the moment that we work over the algebraic closure $\overline{k}$ of $k$ (we extend the scalars of all our objects). Take any non-zero weight vector of $V$ (that is a common eigenvector for the $k_{\lambda}$). The simplicity forces $V=\bU v$. Choose any character $\theta\in \Gamma$. Then the the following formula defines an action of $\wU$ on $V$ : $\xi_{\mu}.{a_{\nu}v}=\theta(\mu){(-1)}^{(\mu,\nu)}a_{\nu}v$, for any $\nu\in P_{\fg}(\pi)$ and any $a_{\nu}\in \bU$ of weight $\nu$. Indeed, the vector $v$ being a weight vector, the annihilator $\Ann_{\bU} v$ is the sum of its weight subspaces, and hence the previous formula makes sense. As a $\wU$-module, $V$ is necessarily simple, and thus is a $V(\phi q^{\lambda},\theta)$ by lemma \[propcomplredwU\] (which obviously also holds over $\overline{k}$). This shows in particular that all the eighenvalues of the $k_{\lambda}$ actually lie in our ground field $k$. Therefore, we could have chosen $v$ to be in the $k$-vector space $V$, and so all that precedes actually holds over $k$. We have proved that $V$ is the restriction of a simple $\wU$-module.
Remark that we have just showed that the group $\{k_{\mu},\ \mu\in P_{\fg}(\pi)\}$ acts diagonally on a simple $\bU$-module. Hence the restriction of a simple finite dimensional $\wU$-module to $\bU$ is also simple.
Consider now $M$, a finite dimensional $\bU$-module. The induced module $\Ind_{\bU}^{\wU}M$ is a finite dimensional $\wU$-module, and therefore completely reducible by lemma \[propcomplredwU\]. Thus, $\Ind_{\bU}^{\wU}M$ is also completely reducible as a $\bU$-module (see the previous remark). But as a $\bU$-module, $M$ lies in $\Ind_{\bU}^{\wU}M$. Hence $M$ is completely reducible.
### {#section-4}
It follows from what preceedes, that for any fixed $\theta\in\widehat{\Gamma}$, the set $\{V(q^{\lambda}\phi,\theta),\
(\lambda,\phi)\in P_{\fk}^+(\pi)\times\widehat{\Gamma}\}$ is a complete set of non-isomorphic finite dimensional simple modules for both $\cU$ and $\bU$.
### {#section-5}
Recall that we shall prove in \[qalgstUs\] that $\bU\simeq \cU_{-q}(\fg)$. The classification of the finite dimensional modules over the “quantum” enveloping algebra of $\fg$ has been obtained by R. B. Zhang (see [@zh]) in the context of formal deformations and by Zou (see [@zo]) for the Drinfeld-Jimbo quantization $\cU_q(\fg)$, through the standard approach.
### Crystals
We admit for a moment that $\bU\simeq\cU_{-q}(\fg)$. Fix $\theta\in \widehat{\Gamma}$ and let $V(\lambda)$, $\lambda\in P_{\fk}^+(\pi)$, be the simple finite dimensional $\wU$-module $V(q^{\lambda},\theta)$. [*A priori*]{}, one can associate to $V(\lambda)$ two crystals. One is given by the work of Kashiwara (see [@kash]), considering $V(\lambda)$ as a $\cU$-module. We denoted it by $B(\lambda)$. The other one, ${\cal B}(\lambda)$, follows from the work of Musson and Zou (see [@mz]), viewing this time $V(\lambda)$ as a $\bU$-module. Both sets $\{B(\lambda),\ \lambda\in P_{\fk}^+(\pi)\}$, $\{{\cal B}(\lambda),\ \lambda\in P_{\fk}^+(\pi)\}$ are closed family of highest weight normal crystals, in the sense of [@jq] 6.4.21. Hence there are equal (up to isomorphisms) by proposition 6.4.21, [@jq]. Another way to see that $B(\lambda)\simeq {\cal B}(\lambda)$ is to remark (keeping the notations of [@kash] and [@mz]) that ${\cal L}(\lambda)=L(\lambda)$ and that the crystalline operators of Musson and Zou act on a given weight subspace of $L(\lambda)$ as the crystalline operators of Kashiwara up to signs (depending on the weight of the subspace and on the “color” of the operators).
{#section-6}
Recall the definition of $\cU_q(\fg)$ given in [@mz]. Let $\bZ(\bU)$ (resp. $\bZp(\bU)$) be the supercentre (resp. the anticentre) of $\bU$, that is the subspace of invariants elements of $\bU$ with respect to $\bad$ (resp $\badp$). One has $\bZp(\bU):=\{a\in \bU,\ ax={(-1)}^{|x|}xa\ \forall \ \Z_2
\mbox{-homogenous } x\in \bU\}$. We also introduce the algebra $\cA(\bU):=\bZ(\bU)+\bZp(\bU)$.
We deduce from section \[qtwisad\] the (compare (i) with 3.3 [@zh])
\[qalgstUs\]
1. The subalgebra $\bU$ is isomorphic to $\cU_{-q}(\fg)$.
2. One has $\Psi\bigl(\F(\cU)\bigr)=\bF(\bU)\oplus \bFp(\bU)$ with $$\begin{array}{l}
\bF(\bU)=\Psi(N_0)= \displaystyle\bigoplus_{\lambda\in P^+_{\fg}(\pi)}
\bad_{}\bU k_{-2\lambda}\end{array}$$$$\begin{array}{l}
\bFp(\bU)=\Psi(N_1)=\displaystyle\bigoplus_{\lambda\in P^+_{\fk}(\pi)
\backslash P^+_{\fg}(\pi)}\bad_{}\bU
k_{-2\lambda}\end{array}$$
3. Recall that $\xi:=\xi_{\beta_l}$ and (\[tonycentre\]). One has $\cA(\bU)=\bZ(\bU)\oplus
\bZp(\bU)=\Psi(\cZ(\cU))$ and $$\begin{array}{l}
\bZ(\bU)=\C[z_{2w_1},\ldots,z_{2w_{l-1}},z_{w_l}^2]\\
\bZp(\bU)=(\xi z_{w_l})\bZ(\bU)
\end{array}$$
We start by proving (i). The relations (\[defwueq3\]), (\[defwueq4\]), (\[reluhsere\]), (\[reluhserf\]) can be respectively rewritten as $\ad K_{\mu}E_i=q^{(\mu,\alpha_i)}E_i$, $\ad K_{\mu}F_i=q^{-(\mu,\alpha_i)}F_i$, $\ad F_iE_j =\delta_{ij}
(1-K_{\alpha_i}^2)/ (q-q^{-1})$, $\ad E_i^{1-\langle \alpha_j,
\alpha_i\rangle} E_j=0$, $\ad F_i^{1-\langle \alpha_j,
\alpha_i\rangle} F_j=0$. Take the image of these relations by $\Psi$. According to Lemma \[qlemtransfadbad\] (and more precisely to formula (\[qeqfinecompad\])) we obtain $\bad k_{\mu}e_i={(-q)}^{(\mu,\alpha_i)}e_i$, $\bad k_{\mu}f_i={(-q)}^{-(\mu,\alpha_i)}f_i$, $\bad f_ie_j =-\delta_{ij}{(-1)}^{\delta_{il}}
(1-k_{\alpha_i}^2)/ (q-q^{-1})$, $\ad e_i^{1-\langle \alpha_j,
\alpha_i\rangle} e_j=0$, $\ad f_i^{1-\langle \alpha_j,
\alpha_i\rangle} f_j=0$. It is easy to see that these relations are exactly the relations defining $\cU_{-q}(\fg)$.
The assertion (ii) results from (\[qeqbFfctF2\]). And (ii) implies that $\Psi(\cZ(\cU)\cap N_0)=\bZ(\bU)$ and $\Psi(\cZ(\cU)\cap N_1)=\bZp(\bU)$. On the other hand, elements of the centre $\cZ(\cU)$ are of weight zero. Therefore combining (\[qN0N1\]) with (\[qcompagrad\]), one has $\Psi(\cZ(\cU)\cap N_0)=\cZ(\cU)\cap N_0$ and $\Psi(\cZ(\cU)\cap N_1)=\xi (\cZ(\cU)\cap N_1)$, which ends the proof.
### {#section-7}
The element $\xi z_{w_l}$ is a quantization of the element $T$ introduced in 4.4.1 [@gl2]. See also formula (\[qformhcT\]) and remark \[qpropqT\]. Notice also that this element coincides with the sCasimir element constructed in [@ab] for the algebra $\cU_q(\osp(1,2))$
### {#section-8}
Let $\bFp(\cU(\fg))$ be the locally finite part of $\cU(\fg)$ for the “non-standard” action of Arnaudon, Bauer, Frappat (see [@abf], 2). Then $\bFp(\cU(\fg)))=\bF(\cU(\fg))$ since $\fg_1$ is finite dimensional. The situation in the quantum case is therefore radically different on this point.
The separation theorem for $\bF(\bU)\oplus \bFp(\bU)$ {#qsecsepthbU}
-----------------------------------------------------
In [@jl3], Joseph and Letzter established a separation theorem for the algebra $\F(\cU)$. They proved the existence of $\ad$-submodules $\cH(\cU)(\lambda)$ of $(\ad\cU) K_{-2\lambda}$ such that if $\cH(\cU):=\oplus_{\lambda\in
P^+_{\fk}(\pi)}\cH(\cU)(\lambda)$, then the multiplication $\cH(\cU)\otimes \cZ(\cU)\longrightarrow \cU$ is an isomorphism of $\ad \cU$-modules. Introduce $$\bH(\bU):= \Psi\bigl(N_0\cap \cH(\cU)\bigr)
\mbox{ and }\bHp(\bU):=\Psi\bigl(N_1\cap \cH(\cU)\bigr)$$ Let $h_i\in \cH(\cU)$ be weight elements, and $z_i\in \cZ(\cU)\cap N_{j_i}$ $j_i\in \{0,1\}$. It follows from (\[qN0N1\]) and (\[qcompagrad\]) that $\sum \Psi(h_i)\Psi(z_i)=\Psi(\sum \pm h_iz_i)$. This is enough to deduce the separation theorem for $\bF(\bU)\oplus \bFp(\bU)$:
### {#qsepthUs}
The multiplication $$(\bH(\bU)\oplus \bHp(\bU))\otimes
\cA(\bU)\longrightarrow \bF(\bU)\oplus \bFp(\bU)$$ is an isomorphism.
The annihilation theorem {#qsecdufloth}
========================
The goal of this section (theorem \[qthmduflo\]) is to establish that the annihilator of any $\bU$-Verma module in $\bF(\bU)\oplus\bFp(\bU)$ is generated by its intersection with $\cA(\bU)$.
{#qgenopVmod}
By definition (see \[qrepgen\]), a $\wU$-Verma module is a $\cU$-Verma module and a character of $\Gamma$ which describes the action of $\Gamma$ on the highest weight vector. Of course, the same holds replacing $\cU$ by $\bU$. Hence, throughout this subsection, we shall not make any distinctions between the $\cU$, $\bU$ and $\wU$-Verma modules.
We fix now once for all a $\wU$-Verma module $M:=M(\Lambda,\theta)$ and we define for all $\mu\in P_{\fg}(\pi)$, $\Lambda(k_{\mu}):=
\Lambda(K_{\mu})\theta(\xi_{\mu})$.
{#qsecdeflocaendVm}
The Verma module $M$ being $\Z_2$-graded (see \[qrepgengradM\]), $\End(M)$ inherits the natural gradation: $${\End(M)}_{|_{{j}}}=
\{f\in \End(M), \forall i\in\Z_2\ f(M_{|_{ {i}}})\subset
M_{|_{{i}+{j}}}\}$$
Consider the adjoint action of $\wU$ on $\End(M)$ defined by $(\ad a f)(x)=a_1
(f({S}(a_2)x))$ for all $a\in \wU,\ f\in \End(M)$ and all $x\in M$. Let $\F(M,M)$ be the locally finite part of $\End(M)$ for the adjoint action $\ad$. The subspace $\F(M,M)$ is $\Z_2$-graded for the above gradation. The restriction of $\wU\longrightarrow \End(M)$ induces a morphism of $\ad\wU$-modules: $\F(\wU)\longrightarrow \F(M,M)$. Its image coincides with the image of $\F(\cU)\longrightarrow \F(M,M)$.
We recall (see Lemma 8.3. in [@jq]) that $\F(M,M)$ is a domain.
{#section-9}
\[qinteropVm\] Let $f\in \F(M,M)$ and $i\in \Z_2$ be be such that $f(M_{|_i})=0$. Then $f=0$.
Let $f$ be as in the lemma. By definition of the $\Z_2$-gradation on $\F(M,M)$ we may assume that $f$ is $\Z_2$-homogenous, and hence that $f^2$ is even. Take any non-zero $p\in {\F(M,M)}_{|_1}$ (obviously such $p$ exists; for instance $\ad E_l K_{-w_l}=(1-q^{-1})E_lK_{-w_l}\in \F(\cU)$ has a non-trivial image in ${\F(M,M)}_{|_1}$). Then $f^2pf^2=0$ which implies $f=0$ since $\F(M,M)$ is a domain.
{#section-10}
Recall that $\Lambda(k_{\mu}):=\Lambda(K_{\mu})\theta(\xi_{\mu})$.
\[qlemqduflo\] For any $\bU$-Verma module $M$, the following equivalence holds $$\Ann_{\cA(\bU)}M=\cA(\bU)\Ann_{\cZ(\bU)}M\Longleftrightarrow
\forall \ 1\leq i\leq l,\ \Lambda(k_{\beta_i})\not=
\pm iq^{-(\rho,\beta_i)}.$$
By proposition \[qalgstUs\], one has $\cA(\bU)=(k\oplus k(\xi z_{w_l}))\otimes \cZ(\bU)$. The centre $\cZ(\bU)$ acts by scalars on $M$. Hence the equality $\Ann_{\cA(\bU)}M=\cA(\bU)\Ann_{\cZ(\bU)}M$ is equivalent to $\Ann_{k\oplus k(\xi z_{w_l})}M=0$. The element $\xi z_{w_l}$ acts on the $\Z_2$-graded components of $M$ by the two opposite scalars $\pm\Lambda(\Upsilon(z_{w_l}))$, $\Lambda$ being linearly extended to $\cU^o$. Thus, $\Ann_{k\oplus k(\xi z_{w_l})}M=0$ is equivalent to $\Lambda(\Upsilon(z_{w_l}))\not=0$. Retain notation of \[defhcq\]. By [@jq] 7.1.19, $$\label{qformhcz}
\Upsilon (z_{w_l})=\displaystyle \sum_{\mu\in P_{\fk}(\pi)}
\dim {V(q^{{w_l\over 2}})}_{\mu} q^{-2(\rho, \mu)}K_{-2\mu}$$ where $V(q^{{w_l\over 2}})$ is simple $\cU$-module of highest weight ${w_l\over 2}$. Since ${w_l\over 2}$ is a minuscule weight, $\dim {V(q^{{w_l\over 2}})}_{\mu}=1$ if $\mu\in W ({w_l\over 2})=
\{ {1\over 2}\sum_{i=1\ldots l}
\varepsilon_i\beta_i,\ \varepsilon_i=\pm 1\}$ and $0$ otherwise. So (\[qformhcz\]) can be rewritten $$\label{qformhcT}
\Upsilon (z_{w_l})=\displaystyle \prod_{i=1,\ldots l}
(q^{-(\rho,\beta_i)}K_{-\beta_i}+q^{(\rho,\beta_i)}K_{\beta_i}).$$ The assertion follows.
### {#section-11}
\[qpropqT\] If $\widehat{T}_d:=\{\Lambda\in\widehat{T},\ \exists 1\leq i\leq t
\mbox { such that } \Lambda(K_{\beta_i})=\pm iq^{-(\rho,\beta_i)}\}$ then the formula (\[qformhcT\]) implies that $$\xi z_{w_l}\in \bigcap_{\Lambda\in \widehat{T}_d,\ \theta\in
\widehat{\Gamma}}
\Ann_{\bU}M (\Lambda,\theta).$$ This is the quantum version of a property satisfied by the element $T$ constructed in [@gl2] (see [@gl2] 4.4.1 and 6.1.3).
{#section-12}
In [@jl], Joseph and Letzter prove the annihilation theorem of Duflo for $\F(\cU)$. We deduce from this result the (compare with Theorem 7.1 [@gl], and Theorem 6.2 [@gl2])
\[qthmduflo\] Let $M$ be a $\bU$-Verma module. Then
1. For any $i\in \Z_2$, $\Ann_{\bF(\bU)\oplus \bFp(\bU)}M_{|_i}=(\bF(\bU)\oplus \bFp(\bU))
\Ann_{\cA(\bU)}M_{|_i}.$
2. $\Ann_{\bF(\bU)\oplus \bFp(\bU)}M=(\bF(\bU)\oplus \bFp(\bU))
\Ann_{\cA(\bU)}M.$
3. $\Ann_{\bF(\bU)\oplus
\bFp(\bU)}M=(\bF(\bU)\oplus \bFp(\bU))
\Ann_{\cZ(\bU)}M \Longleftrightarrow
\forall \ 1\leq i\leq l,\ \Lambda(k_{\beta_i})\not=\pm iq^{-(\rho,\beta_i)}$
We start by (i). Recall (iii). As the centre $\cZ(\bU)$ acts by scalars on $M$, the algebra $\cA(\bU)$ acts by scalars on the homogenous components $M_{|_{{i}}}$. It follows from Proposition \[qsepthUs\] that (i) is equivalent to the statement $\forall i\in\Z_2$, $\Ann_{\bH(\bU)\oplus \bHp(\bU)}M_{|_i}=0$. Let $\Psi(h),\Psi(h')\in \bH(\bU),\bHp(\bU)$ and $i\in \Z_2$ be such that $\Psi(h)+\Psi(h')\in\Ann_{\bH(\bU)\oplus \bHp(\bU)}M_{|_i}$. Since $M_{|_i}$ is $T$-invariant, we can assume that $\Psi(h),\Psi(h')$ (and hence $h,h'$) are elements of the same weight $\nu$. Combining (\[qN0N1\]) and (\[qcompagrad\]) one has $\Psi(h)=\xi_{\eta(\nu)} h$ and $\Psi(h')=\xi_{\eta(\nu)+w_l} h'$. Hence $h +\xi h'\in \Ann_{\cH(\cU)} M_{|_i}$. The element $\xi$ acts by $\pm id$ on $M_{|_i}$, so we can assume that $h + h'\in \Ann_{\cH(\cU)} M_{|_i}$. From Lemma \[qinteropVm\] we derive that $h+h'\in \Ann M$. Therefore $h'=-h$ using 4.2, [@jl]. But $h\in N_0$, $h'\in N_1$ and $N_0\cap N_1=0$, which forces finally $h=h'=0$. This finishes the proof of (i).
For (ii), one has the equalities $$\begin{array}{lcl} \Ann_{\bF(\bU)\oplus
\bFp(\bU)}M&=& \bigcap_i \Ann_{\bF(\bU)\oplus
\bFp(\bU)}M_{|_i}\\
&=& \bigcap_i (\bF(\bU)\oplus \bFp(\bU))
\Ann_{\cA(\bU)}M_{|_i}\ \mbox{ by (i)}\\
&=& \bigcap_i (\bH(\bU)\oplus \bHp(\bU))\otimes
\Ann_{\cA(\bU)}M_{|_i}\ \mbox { by~\Prop{qsepthUs}} \\
&=&(\bH(\bU)\oplus \bHp(\bU))\otimes \bigcap_i \Ann_{\cA(\bU)}M_{|_i}\\
&=&(\bH(\bU)\oplus \bHp(\bU))\otimes \Ann_{\cA(\bU)} M\\
&=& (\bF(\bU)\oplus \bFp(\bU))
\Ann_{\cA(\bU)}M
\end{array}$$ And (iii) is a consequence of (ii), and of Lemma \[qlemqduflo\].
### {#section-13}
We believe that (i) should also hold in the classical case.
$\cA(\bU)$ is the commutant of ${\bU}_{|_0}$ {#qseccom}
=============================================
In this section we shall prove that $\cA(\bU)$ is the commutant of the even part of $\bU$, that is $\cA(\bU)=\cC(\bU_{|_0})$. Let $\cA(\wU)$ be the subalgebra $\cA(\wU):=\cZ(\cU)\oplus \xi \cZ(\cU)$. Since $\cC({\wU}_{|_0})\cap \bU=\cC(\bU_{|_0})$ and $\cA(\wU)\cap \bU=\cA(\bU)$ (recall (iii)) it is enough to prove the equality $\cA(\wU)=\cC({\wU}_{|_0})$. For this, we shall proceed by quantizing the mechanics of 4 [@gl2].
{#interannusimq}
For every subset $\Omega$ of $P_{\fk}(\pi)$ dense for the Zariski topology, one has $$\bigcap_{\lambda\in\Omega\atop \theta\in\widehat{\Gamma}}
\Ann_{\wU} V(q^{\lambda},\theta)=0.$$
Fix $\lambda\in\Omega$. One has $$\label{eqannusm}
\bigcap_{\theta\in\widehat{\Gamma}}\Ann_{\wU} V(q^{\lambda},\theta)
=(k\Gamma)\Ann_{\cU} V(q^{\lambda}).$$ where $ V(q^{\lambda})$ stands for the $\cU$-simple module of highest weight $q^{\lambda}$.
Indeed, let $e_{\chi}\in k\Gamma,\ \chi\in\Gamma$, be the projector corresponding to $\chi$, that is the projector such that $ge_{\chi}=\chi(g)e_{\chi}$, $\forall g\in \Gamma$. Let $x\in\wU$ and write $x=\sum_{\chi\in \widehat{\Gamma}}x_{\chi}e_{\chi}$, $x_{\chi}\in \cU$. As a $\cU$-module, $V(q^{\lambda},\theta)$ canonically identifies with the $\cU$-module $V(q^{\lambda})$ (see \[qrepgen\]). Under this identification, $x$ acts on the subspace of $T$-weight $q^{\lambda-\nu}$ of $V(q^{\lambda},\theta)$ as $x_{{(-1)}^{\nu}\theta}$ on the subspace of the same weight of $V(q^{\lambda})$. Hence, $x\in
\bigcap_{\theta\in\widehat{\Gamma}}\Ann_{\wU}
V(q^{\lambda},\theta)$ implies that $x_{{(-1)}^{\nu}\theta}$ vanishes on $V(q^{\lambda})_{q^{\lambda-\nu}}$ for all $\nu$ and $\theta$. This gives (\[eqannusm\]).
{#section-14}
As in 4.1 [@gl2], the previous lemma implies
\[qcommUactscal\] The algebra $\cC({\wU}_{|_0})$ coincides with the subalgebra of elements of $\wU$ acting by scalars on the homogenous components of simple highest weight modules.
{#qpfcentnotchar}
Retain the notation of \[defhcq\]. Take $x\in \wU^o$ and write $x:=\sum a_{\mu,\mu'}\xi_{\mu}K_{\mu'}$, $a_{\mu,\mu'}\in k$. For any $(\lambda,\theta)\in P_{\fk}(\pi)\times
\widehat{\Gamma}$, we set $x(\lambda,\theta):=\sum a_{\mu,\mu'}
\theta(\mu)q^{(\lambda,\mu')}$.
With these conventions, $a\in\cC(\wU_{|_0})$ acts by the scalar $\Upsilon(a)(\lambda,\theta)$ on the homogenous component of $V(q^{\lambda},\theta)$ containing the highest weight vector.
{#section-15}
\[lemhccentwu1\] The restriction of $\Upsilon$ to $\cC({\wU}_{|_0})$ is injective.
For all $(\lambda,\theta)\in P^+_{\fk}(\pi)\times
\widehat{\Gamma}$ denote by $v_{\lambda}$ the highest weight vector of $V(q^{\lambda},\theta)$. Let $a$ be in $\cA(\wU)$. We recall that $a$ acts on $\wU_{|_0}v_{\lambda}$ by the scalar $\Upsilon(a)(\lambda,\theta)$. On the other hand, if $\lambda\in \Omega:=\{\lambda\in
P^+_{\fk}(\pi),\ \langle
s_{\beta_l}.(\lambda)+\rho,\beta_l\rangle\in 2\N+1\}$ and $\theta':= {(-1)}^{s_{\beta_l}.\lambda-\lambda}\theta$ we claim that $a$ acts on $\wU_{|_{1}}v_{\lambda}$ by the scalar $\Upsilon(a)(s_{\beta_l}.\lambda,\theta')$. Indeed, assume that $\lambda\in \Omega$ and $\theta'= {(-1)}^{s_{\beta_l}.\lambda-\lambda}\theta$. Then (see \[qrepgen\]) $M(q^{\lambda},\theta)\subset M(q^{s_{\beta_l}.\lambda},\theta')$. Moreover, if $u_{\lambda}$, $u_{s_{\beta_l}.\lambda}$ are the respective highest weight vectors of these Verma modules, one has $\wU_{|_1}u_{\lambda}
\subset \wU_{|_0}u_{s_{\beta_l}.\lambda}$ and the claim follows. Hence $\Upsilon(a)=0$ implies $a\in\Ann_{\wU}V(q^{\lambda},\theta)$ for all $(\lambda,\theta)\in \Omega\times \widehat{\Gamma}$. The density of $\Omega$ allows us to use \[interannusimq\] and then to conclude.
{#qpfcentnotU0ev}
Set $$\cU^o_{ev}:=\sum_{\mu\in P_{\fk}(\pi)}kK_{2\mu}\subset \cU^o.$$ The Weyl group $W$ acts on $\cU^o$ and on $\cU^o_{ev}$ in the following way: $$w.K_{\mu}:=q^{(\mu,w^{-1}\rho-\rho)}K_{w\mu}.$$
\[lemhccentwu2\] $\Upsilon\bigl(\cC(\wU_{|_0})\bigr)\subset
(\cU^o_{ev})^W\oplus
\xi (\cU^o_{ev})^W$
Firstly, we shall check that $$\label{etp1q}
\Upsilon\bigl(\cC(\wU_{|_0})\bigr)\subset (\cU^o)^W\oplus \xi (\cU^o)^W.$$ We start by fixing some notations. Recall \[qnottorus\]. For any $1\leq i\leq l$, we define $\Gamma_i:=\{ \xi_{\mu},\ \mu\in \bigoplus_{j\not =i} (\Z\slash 2\Z)
w_j\}$, and the subalgebra $\wU_i^o:=(k\Gamma_i)\wU^o$. By definition, $\wU^o=\wU_i^o\oplus \xi_{w_i}\wU_i^o$.
Fix $a\in \cC(\wU_{|_0})$. For each $i=1,\ldots,l$, write $$\label{eqecrq}
\Upsilon(a)=P_i+\xi_{w_i} Q_i$$ with $P_i,Q_i\in \wU^o_i$. We fix $i<l$ and show that $Q_i=0$.
Let $(\lambda,\theta)$ be in $P_{\fk}(\pi)\times\widehat{\Gamma}$ such that $\langle\lambda,\alpha_i\rangle
\in \N$. Consider $\theta'$ defined by $\theta':={(-1)}^{\langle \lambda+\rho,\alpha_i\rangle\alpha_i}\theta$. In other words, $\theta'(w_j)=\theta(w_j)\ \forall j\not=i\mbox{ and }
\theta'(w_i)={(-1)}^{\langle \lambda+\rho,\alpha_i\rangle}\theta(w_i)$. According to \[qrepgen\], $M(q^{s_{\alpha_i}.\lambda},\theta')$ is a submodule of $M(q^{\lambda},\theta)$. Moreover, as $i<l$, $\wU_{|_0}v_{s_{\alpha_i}.\lambda}\subset
\wU_{|_0}v_{\lambda}$ where $v_{\lambda}, v_{s_{\alpha_i}.\lambda}$ stand for the vectors of highest weight of $M(q^{\lambda},\theta)$ and $M(q^{s_{\alpha_i}.\lambda},\theta')$. It follows that $$\label{forupsilemq}
\Upsilon(a)(s_{\alpha_i}.\lambda,{(-1)}^{\langle \lambda+\rho,\alpha_i
\rangle\alpha_i}\theta)=\Upsilon(a)(\lambda,\theta)$$ if $\langle \lambda,\alpha_i \rangle \in \N$. We shall now check that formula (\[forupsilemq\]) extends to all $\lambda\in P_{\fk}(\pi)$. Indeed if $ \langle \lambda,\alpha_i \rangle =-1$ then $ \langle \lambda+\rho,\alpha_i \rangle =0$, $s_{\alpha_i}.\lambda=\lambda$ and (\[forupsilemq\]) is obvious. If $\langle \lambda,\alpha_i\rangle=-p-2,\ p\geq 0$, then $s_{\alpha_i}.\lambda=\lambda+(p+1)\alpha_i$ and $\langle s_{\alpha_i}.\lambda,
\alpha_i\rangle=p$. We can then apply (\[forupsilemq\]) to $s_{\alpha_i}\lambda$, which establishes (\[forupsilemq\]) for $\lambda$.
Using notation (\[eqecrq\]), (\[forupsilemq\]) can be rewritten as follows $$P_i(s_{\alpha_i}.\lambda,\theta_i) +\theta(w_i)
{(-1)}^{\langle \lambda+\rho,\alpha_i\rangle} Q_i(s_{\alpha_i}.\lambda,
\theta_i)=P_i(\lambda,\theta_i)+\theta(w_i)Q_i(\lambda,\theta_i)$$ where $\theta_i$ is the restriction of $\theta$ to $\Gamma_i$. Taking successively $\theta(w_i)=\pm$ in the last equation, we obtain $$\begin{aligned}
P_i(s_{\alpha_i}.\lambda,\theta_i)&=&P_i(\lambda,\theta_i)\\
\label{eqecrq2}
Q_i(s_{\alpha_i}.\lambda,\theta_i)&=&{(-1)}^{\langle
\lambda+\rho,\alpha_i\rangle}
Q_i(\lambda,\theta_i)\end{aligned}$$ Write $Q_i=\displaystyle\sum_{(\mu,\gamma)\in P_{\fg}(\pi)\times \Gamma_i}
a_{\mu,\gamma}K_{\mu}\xi_{\gamma}$. Then (\[eqecrq2\]) implies that for all $(\lambda,\theta)\in P_{\fk}(\pi)\times
\widehat{\Gamma_i}$, $$\sum_{(\mu,\gamma)\in P_{\fg}(\pi)\times \Gamma_i}
a_{\mu,\gamma}{(-1)}^{\langle \lambda,\alpha_i\rangle}
q^{(\mu,s_{\alpha_i}.\lambda)}\theta(\gamma)
+\sum_{(\mu,\gamma)\in P_{\fg}(\pi)\times \Gamma_i}
a_{\mu,\gamma}
q^{(\mu,\lambda)}\theta(\gamma)=0.$$ Since the characters $P_{\fk}(\pi)\times \widehat{\Gamma_i}\longrightarrow
k$, $(\lambda,\theta)\mapsto
{(-1)}^{\langle \lambda,\alpha_i\rangle}
q^{(\mu,s_{\alpha_i}.\lambda)}\theta(\gamma)$ and $(\lambda,\theta)\mapsto
\ q^{(\mu,\lambda)}\theta(\gamma)$ are pairwise distinct, the lemma of linear independence of the characters of Dedekind forces $Q_i=0$.
Finally, $i$ running from $1$ to $l$, we have proved that $$\Upsilon(a)=P+\xi Q$$ where $P,Q\in \cU^o$ are invariant under the action of the subgroup of $W$ generated by the $s_i, i<l$.
If $\lambda$ is such that $\langle \lambda+\rho,\beta_l\rangle\in 2\N+2$, then $M(q^{s_{\beta_l}.\lambda},\theta)\subset M(q^{\lambda},\theta)$ with $\wU_{|_0}v_{s_{\beta_l}.\lambda}\subset \wU_{|_0}v_{\lambda}$, and one shows, proceeding as above, that $$\label{idPQ}
P(s_{\beta_l}.\lambda)=P(\lambda),\ \ Q(s_{\beta_l}.\lambda)=Q(\lambda)$$ for all $\lambda\in P_{\fk}(\pi)$ such that $\langle \lambda+\rho,\beta_l\rangle\in 2\Z$. We shall check that (\[idPQ\]) actually holds for all $\lambda\in P_{\fk}(\pi)$. Let us treat the case of $P$ for instance. The identity (\[idPQ\]) can be rewritten as $(s_{\beta_l}.P-P)(\displaystyle{w_l\over
2}+\lambda')=0$ for all $\lambda'\in P_{\fg}(\pi)$. Write $P=\sum_{\mu\in P_{\fg}(\pi)}
a_{\mu}K_{\mu}$. Then $\sum_{\mu\in P_{\fg}(\pi)} q^{{1\over
2}(\mu,w_l)}(a_{s_{\beta_l}\mu}
q^{(\mu,\rho-s_{\beta_l}\rho)}-a_{\mu})q^{(\mu,\lambda)}=0$ for all $\lambda\in P_{\fg}(\pi)$. The linear independence of the characters $P_{\fg}(\pi)\rightarrow k$, $
\lambda \mapsto q^{(\mu,\lambda)}$ forces the equalities $a_{s_{\beta_l}\mu}
q^{(\mu,\rho-s_{\beta_l}\rho)}=a_{\mu}$ and therefore $s_{\beta_l}.P=P$.
Finally, $P,Q$ are $W.$-invariant and we have proved (\[etp1q\]).
It remains to show that $P,Q$ are actually elements of $\cU^o_{ev}$. For this, we should reproduce the reasoning above, analyzing now the action of $a$ on the $M(q^{\lambda}\phi,\theta)$ where $\phi\in \Gamma$.
Another way to do it is to imitate [@jan] 6.6, that is to consider for each $\phi\in \widehat{\Gamma}$, the automorphism $\sigma_{\phi}$ which keeps $\cC(\wU_{|_0})$ invariant, and sends $K_{\alpha_i},E_i,F_i,\xi_i$ respectively to $\phi(\alpha_i)K_{\alpha_i},\phi(\alpha_i)E_i,F_i,\xi_i$.
{#section-16}
$\cC(\wU_{|_0})=\cA(\wU)$.
By [@jq] 7.17, $\Upsilon(\cZ(\cU))=(\cU^o_{ev})^W$. Since $\cA(\wU)\subset\cC(\wU_{|_0})$, we deduce from lemma \[lemhccentwu2\] that $\Upsilon\bigl(\cA(\wU)\bigr)=
\Upsilon(\cC(\wU_{|_0}))$. And lemma \[lemhccentwu1\] ends the proof.
[MMMMM]{}
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[^1]: The author was partially supported by the EC TMR network Algebraic Lie Representations Grant No. ERB FMRX-CT97-0100 and Minerva grant 8337
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present an analysis of archival Chandra observations of the mixed-morphology remnant 3C400.2. We analysed spectra of different parts of the remnant to observe if the plasma properties provide hints on the origin of the mixed-morphology class. These remnants often show overionization, which is a sign of rapid cooling of the thermal plasma, and super-solar abundances of elements which is a sign of ejecta emission. Our analysis shows that the thermal emission of 3C400.2 can be well explained by a two component non-equilibrium ionization model, of which one component is underionized, has a high temperature ($kT\approx 3.9$ keV) and super-solar abundances, while the other component has a much lower temperature ($kT\approx0.14$ keV), solar abundances and shows signs of overionization. The temperature structure, abundance values and density contrast between the different model components suggest that the hot component comes from ejecta plasma, while the cooler component has an interstellar matter origin. This seems to be the first instance of an overionized plasma found in the outer regions of a supernova remnant, whereas the ejecta component of the inner region is underionized. In addition, the non-ionization equilibrium plasma component associated with the ejecta is confined to the central, brighter parts of the remnant, whereas the cooler component is present mostly in the outer regions. Therefore our data can most naturally be explained by an evolutionary scenario in which the outer parts of the remnant are cooling rapidly due to having swept up high density ISM, while the inner parts are very hot and cooling inefficiently due to low density of the plasma. This is also known as the relic X-ray scenario.'
author:
- |
Sjors Broersen,$^1$ Jacco Vink$^{1,2}$\
$^1$ Astronomical Institute ‘Anton Pannekoek’, University of Amsterdam, Postbus 94249, 1090 GE Amsterdam, The Netherlands\
$^2$ GRAPPA, University of Amsterdam, Postbus 94249, 1090 GE Amsterdam, The Netherlands
title: ' A Chandra X-ray study of the mixed-morphology supernova remnant 3C400.2 '
---
ISM: supernova remnants – supernovae: general – supernovae: individual: 3C400.2
Introduction
============
An important and not well-understood class of supernova remnants (SNRs) are the so-called thermal composite or mixed-morphology remnants (MMRs, @rhopetre1998 [@lazendicslane2006; @vink2012]). These remnants are characterised by thermal X-ray emission that is centrally peaked whereas the radio emission has the familiar shell-type morphology. They have several other interesting characteristics: they are often associated with GeV gamma-ray emission [e.g. @giulianietal2010; @uchiyamaetal2012], there is evidence for enhanced metal abundances in some of them [@lazendicslane2006] although they are usually mature remnants, and they often show spectral features associated with strong cooling in the form of radiative recombination continua (RRCs) or strong He$-\alpha$/Ly$-\alpha$ X-ray line ratios of alpha-elements such as Si, S and Ar [@kawasakietal2005].
The centrally peaked X-ray emission of MMRs poses a problem for SNR evolution models. They typically have an age on the order of 20.000 years and are therefore expected to have a shell-like density structure with a hot, tenuous plasma in the centre, based on a Sedov evolutionary scenario. A flat interstellar matter (ISM) structure is therefore unsuited to create the centre-filled X-ray morphology that is observed, if the temperature is relatively uniform across the remnant. @rhopetre1998 mention two possible scenarios for the formation of the emission structure typical for MMRs, the *relic X-ray emission* and the *evaporating cloudlet* scenario [@whitelong1991]. The former scenario is the most simple scenario, in which the outer layers of the remnant have become radiative and have cooled strongly so that they hardly emit in the X-ray band, whilst the centre consists of hot plasma that has not yet cooled below $10^{6}$ K. This scenario requires a high surrounding ISM density. @whitelong1991 give a self-similar solution for a supernova that exploded in an environment where it is surrounded by a number of small, dense clouds. Due to the large filling fraction and small size of the clouds they do not alter the dynamics of the forward shock, and they survive the passage of the forward shock due to their large density. The clouds then slowly evaporate due to heating by thermal conduction, which increases the density and decreases the temperature in the centre of the remnant. Of these two scenarios, the relic X-ray emission scenario was preferred by @harrusetal1997 to explain the morphology of W44. One year later, @coxetal1999 [@sheltonetal1999] used a similar scenario as @harrusetal1997 to model the characteristics of the MMR W44, but added a density gradient for the ISM and thermal conduction. Thermal conduction in the model by @coxetal1999 smoothes the temperature gradient from the centre to the outer layers, thereby reducing the pressure in the centre. A lower central pressure reduces the need to expand, which allows for a higher density to remain in the centre of the remnant. A problem with the scenarios using thermal conduction is that it is not a priori clear whether it can be important in SNRs, since it is strongly suppressed by magnetic fields [@spitzer1981; @tao1995]. Clear evidence for overionization of thermal SNR plasmas is almost exclusively found in MMRs. Thermal plasmas in supernova remnants are often found in an underionized state, and reach ionization equilibrium on a density dependent timescale $t \simeq 10^{12.5} / n_e$ s [@smithhughes2010]. Due to their ages and their frequent association with high density regions, MMRs are expected to have a plasma that is in ionization equilibrium. The fact that there is ample evidence for overionization, means the cooling rate of the plasma was faster than the recombination rate. How the rapid cooling proceeds is still unclear. Efforts have been made to localise regions in MMRs which are cooling rapidly, as an association with a high density region may mean that thermal conduction is important. For example: @micelietal2010 [@lopezetal2013] find that the amount of cooling increases away from a molecular cloud for W49B, suggesting adiabatic cooling as the dominant cooling mechanism. @uchidaetal2012 suggest the cooling timescale of the plasma cannot be explained by thermal conduction, and that adiabatic expansion is probably the dominant cause for cooling there. In addition, @broersenetal2011 have shown that even at small expansion rates, simple adiabatic cooling combined with cooling through X-ray radiation can lead to a cooling rate of the plasma larger than the recombination rate. So far observational evidence for strong thermal conduction is lacking.
The ionization state of a thermal plasma can be determined by using both the electron temperature $T_{\rm e}$ and the so-called ionization temperature $T_{\rm z}$ [@masai1997]. The ionization temperature is the electron temperature that one would deduce from the ionization state of X-ray emitting elements alone, which may be different from the thermodynamic temperature in case the plasma is not in ionization equilibrium. When $T_{\rm z} < T_{\rm e}$, the plasma is underionized, when $T_{\rm z} = T_{\rm e}$ the plasma is in collisional ionization equilibrium (CIE) and when $T_{\rm z} > T_{\rm e}$ the plasma is overionized. There are different ways in which $T_{\rm z}$ can be determined. @kawasakietal2002 determine $T_{\rm z}$ by comparing the observed Ly$-\alpha$ / He$-\alpha$ ratio of a certain element to the ratios of CIE plasmas of different temperatures. The temperature of the CIE plasma that produces the observed ratio is then $T_{\rm z}$. A different method was used by @ohnishietal2011, who determined $T_{\rm z}$ by using the CIE model in [spex]{} to fit the plasma. This model has an additional parameter $\xi$, compared to the [xspec]{} CIE model. This parameter can be used to simulate a non-equilibrium plasma so that $T_{\rm z} = \xi\times T_{\rm e}$. Finally there is the method used by @broersenetal2011 and @uchidaetal2012, who fit a plasma using the [spex]{} NEI model with initial temperature $kT_1 > kT_2$. $kT_1$ has a similar function as the ionization temperature, but it has a slightly different physical meaning as it describes the initial ionization state. The NEI model then follows the ionization state as a function of ionization age as usual, assuming a rather sudden drop of electron temperature. Note that although using the He$-\alpha$ / Ly$-\alpha$ line ratios of alpha elements can show that a plasma is overionized, it has quite a large systematic error in determining the $T_{\rm z}$ [@lopezetal2013]. This is the result of the fact that there are emission lines of other elements which contaminate the He$-\alpha$ /Ly$-\alpha$ ratio. The strengths of the contaminating lines are $T_{\rm e}$ dependent, so that the systematic error is $T_{\rm e}$ dependent as well. This can be taken into account by calculating the CIE line ratios including the contaminating lines.
The difference in ionization temperature between the line ratio and NEI model methods is displayed in Fig. \[fig:Tz\_net\]. This figure shows the evolution of the ionization temperature as a function of ionization age, for an NEI plasma that is cooling abruptly from 4 keV to 0.2 keV. It is clear from the figure that different elements show different ionization temperatures as the plasma evolves, so that a plasma cannot be characterised by a single ionization temperature. This also explains why different elements in SNR plasmas often show different ionization temperatures, as found by @lopezetal2013. On the other hand the NEI model is not a perfect model for a cooling plasma, as it assumes the electron temperature instantly drops to a certain value. In reality the electron temperature will drop more gradual, so that the plasma evolution will be slightly different as a result of different recombination and ionization rates.
![The ionization temperature of a plasma versus the ionization age. The NEI plasma starts at an electron and ionization temperature of 4.0 keV, and it evolves after the electron temperature instantly drops to 0.2 keV. The figure illustrates that the NEI and line-ratio/ionization-temperature methods for characterizing overionized plasma do not provide identical ionization states. Moreover, the differences depend on the element: at certain ionization ages, different elements may have different ionization temperatures. \[fig:Tz\_net\]](temp_ionization_age_cooling_abun.pdf){width="\columnwidth"}
Here we present the first analysis of a Chandra X-ray observation of 3C400.2 (also known as G53.6 – 2.2), which is an important member of the MMR class. This remnant has centrally peaked X-ray emission, as shown by Einstein IPC and ROSAT observations [@longetal1991; @sakenetal1995]. The radio morphology can be described as two overlapping circular shells of diameters 14’ and 22’ [@dubneretal1994]. This has led to the speculation that 3C400.2 might be two supernova remnants in contact with each other, which would make it a rare event. @yoshitaetal2001 conclude however, using ASCA observations, that the remnant is the result of a single supernova explosion, based on the similar plasma properties found in the two shells. Hydrodynamical simulations, including thermal conduction, show that the morphology of the remnant can be explained by a supernova exploding in a cavity, where the larger shell is a part of the remnant expanding into a lower density region than the smaller shell [@schneiteretal2006]. This is consistent with HI observations performed by @giacanietal1998, who find a denser region to the northwest of the remnant where the smaller shell is located. In the optical, the remnant is characterised by a shell-like structure with a smaller radius (about 8’) than the radio shell [@winkleretal1993; @ambrociocruzetal2006]. The optical emission is located in regions of low X-ray emissivity. Optical emission suggest that those parts of the remnant contain radiative shocks, and are therefore cooling efficiently. Distance estimates to 3C400.2 range from 2.3 – 6.9 kpc [@rosado1983; @milne1979; @dubneretal1994; @giacanietal1998]. However, the distance estimates based on radio data are obtained with the uncertain $\Sigma-D$ relationship, and the kinematic estimate of @rosado1983 is based on interferograms of a small part of the remnant. We therefore consider the most recent distance estimate of $2.3\pm0.8$ kpc [@giacanietal1998], which is based on HI measurements, as the more reliable one and we will use a round value of 2.5 kpc for the distance throughout this paper.
MMRs and in particular overionization of plasmas have gathered increasing attention over the past few years. In this work, we therefore aim to characterise the plasma properties of 3C400.2 in order to increase our understanding both of overionized plasmas, and the possible evolutionary scenarios of MMRs. We start with the data reduction and spectral analysis of different regions of the remnant, which is followed by a discussion on the measured plasma properties. We end with the conclusions.
![Rosat PSPC image in an inverted grey scale, with NRAO VLA Sky Survey contours overlaid, The Chandra ACIS-I field of view is indicated in red. []{data-label="fig:rosat"}](ROSAT_chandra_fov.pdf){width="\columnwidth"}
Data Analysis and results
=========================
Data reduction
--------------
In this paper we report on the analysis of a $34$ ks archival Chandra observation (ObsID 2807) taken on August 11, 2002 with Chandra observing in imaging mode with the ACIS-I CCD array. The total number of counts in the 0.3-7.0 keV band in the full spectrum is $\sim58$k. We extracted spectra using the default tasks in the Chandra analysis software [ciao]{} version 4.5, using the task *specextract* to create spectra and weighted responses (RMF and ARF). We removed point sources from the spectral extraction regions. The spectra where grouped so that each bin contained a minimum of 15 counts. The remnant covers the whole area of the ACIS-I chips (see \[fig:rosat\]), and there was no region available for background extraction. We therefore used the standard ACIS-I background files to create background spectra. We used [spex]{} version 2.03 [@SPEX] for the spectral modelling. The NEI model in [spex]{} is an extended version of the MEKAL model used in [xspec]{} [@xspec], with the added advantage that the initial temperature of the NEI model can be varied to mimic an overionized plasma.
Spectral analysis {#sec:spectral_analysis}
-----------------
The Chandra ACIS-I field of view covers the part of the remnant that is mostly associated with the smaller radio shell. It includes the brightest part of the remnant in X-rays, as can be deduced from the ROSAT image in Fig. \[fig:rosat\]. With our spectral analysis, we aim to find differences in plasma properties between different parts of the remnant, and to investigate if this remnant conforms to the general properties of MMRs mentioned in the introduction, with regards to overionization and abundances.
We fitted the spectra using absorbed non-equilibrium ionization (NEI) models, of which the parameters are the ionization age $\tau = n_et$, the electron temperature $kT_2$, the elemental abundances and the normalisation $n_{\rm e}n_{\rm H}V$. A CIE plasma is identical to an NEI plasma when $\tau \geq 10^{12.5}$ cm$^{-3}$ s. In addition to the above listed parameters, the [spex]{}NEI model has the initial temperature of the plasma, $kT_1$, as an optional parameter. As mentioned in the introduction putting $kT_1 > kT_2$ makes the NEI model mimic an overionized plasma, where the ionization state of the plasma is determined by $kT_1$ and $\tau$, while the continuum shape is determined by the electron temperature $kT_2 = kT_{\rm e}$. The method to check for overionization using $Ly-\alpha$ and $He-\alpha$ line strengths is not feasible for 3C400.2, since elements with isolated emission lines such as Si and S show no significant $Ly-\alpha$ lines in this remnant.
We aimed to first find a satisfactory fit for the region covering almost the full area of the ACIS-I chips (see Fig. \[fig:chandra\]), where we initially tried to fit the spectrum with an absorbed, single, underionized NEI model with fixed abundances, freeing abundances only when it led to a significant improvement in C-stat / degrees of freedom [@cash]. The $N_{\rm H}$ was always allowed to vary. A single absorbed NEI model was not sufficient to obtain an acceptable C-stat / d.o.f. in any of the extracted spectra, however.
![Chandra image in the 0.3 - 7.0 keV band with NVSS radio contours. The extraction regions of the spectra are labelled in yellow. []{data-label="fig:chandra"}](chandra_final.pdf){width="95mm"}
The next step was to try to fit the spectrum with a double NEI model. Although the C-stat / d.o.f. improved significantly with respect to a fit with a single NEI model, the fit was still not acceptable. The only way the fit of the double NEI models became acceptable, was by allowing the $kT_1$ of the cooler NEI component to vary, so that $kT_1 > kT_2$, and therefore the NEI plasma is overionized. We show in detail why the overionized model fits the data better than an underionized model in section \[sec:overionization\].
Fig. \[fig:full\] shows the spectrum of the full region. The best-fit model for this region consists of a $kT=3.86$ keV plasma with super-solar abundances plotted as a green dashed line, combined with a $kT_2=0.14$ keV, overionized NEI component which is plotted as a blue dot-dashed line. The parameters of the model are listed in Tab. \[tab:full\_param\]. It is clear from the figure that the high $kT$ model accounts for the bulk of the Fe-L (0.7-1.2 keV), (1.85 keV) and (2.46 keV) line emission, while the low $kT$ model accounts for the (0.56 and 0.65 keV) line emission, and continuum emission in the form of , and RRCs. The super-solar abundances of Si, S and Fe in the hot NEI component suggest an ejecta origin for the plasma. The abundances of the high $kT$ component as well as the emission measure have quite large errors. This is caused by the fact that the continuum shape at energies larger than 2.5 keV is not well defined by the data. An ill-defined continuum strength causes large formal errors in the abundances, since the strength of the continuum and the height of the abundances are anti-correlated. Nonetheless the abundances are significantly super-solar, which means that 3C400.2 belongs to the group of MMRs with super-solar abundances. The ill-defined continuum at higher energies may be the result of the uncertainty introduced by the blank-sky background subtraction. Overionization is found only in the lower $kT$ plasma, and not in the high $kT$ plasma. Making the initial temperature a free parameter in the hot NEI component did not improve the fit. There is therefore no evidence for overionization in the ejecta component, which is further corroborated by the absence of strong $Ly-\alpha$ lines of Si and S. The abundances of the low $kT$ NEI component are sub-solar in the case of NE and O and solar for all other elements, which suggest a swept-up ISM origin for this plasma.
Overall the fit is acceptable at a C-stat / d.o.f. of 257 / 227, although there are some significant residuals. The most notable residual feature is found at the position of the line at 1.85 keV. The model fits this line partly with the RRC from the cool NEI component, and partly with Si emission from the hot component. It seems that the ionization state of Si in the hot component is somewhat too high. This may be caused by the fact that this spectrum was taken from a large region of the remnant, in which in different parts the Si may be in different ionization states. The parameters obtained from this spectral fit will be used in the discussion to determine the overall physical parameters of the remnant.
Our best-fit model differs significantly from results obtained previously. The most recent X-ray observations of 3C400.2 were performed by @yoshitaetal2001, using the GIS instrument onboard ASCA X-ray telescope. The $N_{\rm H}$ that we find is quite similar to theirs, but they find an acceptable fit for this region using a single NEI model with $kT=0.8$ keV and $log (\tau)= 10.7-11.2$ cm$^{-3}$ s. We have tried to fit our data with a single NEI model like this, but this gives a C-stat / d.o.f. = 606.23 / 229, even when allowing multiple abundances to vary. The differences in best-fit parameters most likely stem from the fact that the Chandra ACIS instrument has a higher spectral resolution than the GIS instrument, allowing for better constraints on plasma parameters.
### Overionization {#sec:overionization}
![A close up of the full spectrum, showing the effects of overionization in 3C400.2. The red solid line shows the best fit model without using cooling, while the blue line shows the best fit cooling model for this region. The bottom panel shows the residuals of the non-cooling model with plotted in blue the difference between the cooling and non-cooling model divided by the error. This shows that the cooling model fits the spectral features around 0.87 keV and 1.4 keV much better than the non-cooling model.[]{data-label="fig:mix"}](full_mix.pdf){width="\columnwidth"}
Although overionization is a common feature of MMRs, it is not immediately clear that the plasma is overionized in 3C400.2, due to the lack of strong, isolated RRC features. Overionization in this remnant has a more subtle presence, which we illustrate in Fig. \[fig:mix\]. This figure shows a close-up of the full spectrum shown in Fig. \[fig:full\] in the energy range 0.5 - 1.7 keV, where the red line represents the best-fit model without overionization, and the dashed blue line the best-fit overionized model. The bottom panel of the figure shows the residuals of the non-overionized model, with plotted as a blue line the two models subtracted and divided by the error. It is clear from the figure that the overionized model fits the data much better, especially in the 0.7-1.0 and the 1.3-1.7 keV region. [ In the 0.5-1.7 keV range, the best-fit single absorbed NEI model has a C-stat / d.o.f. = 334/72, a double absorbed NEI model has C-stat / d.o.f. = 119/64, and the overionized model with the parameters shown in table \[tab:full\_param\] has a C-stat / d.o.f. = 92/70. The overionized model therefore fits the data significantly better than any combination of a single or double absorbed underionized NEI or CIE model. We have performed such a fitting routine where we first attempted a single underionized NEI model, then a double underionized NEI model, then allowing the initial temperature of the lower $kT$ NEI model to vary for every spectrum shown, allowing in every case the abundances to vary only if the fit improved significantly. The $N_{\rm H}$ was always allowed to vary. The C-stat / d.o.f. for the different fitting attempts are listed in table \[tab:spectral\_parameters\]. ]{}
In the 0.7-1.0 keV energy range, the strongest emission feature is present at 0.87 keV which is not well-fit by the non-overionized model. There are two main spectral emission options which could account for the emission feature at 0.87 keV: emission, which has mainly emission lines at 0.77 and 0.87 keV, and the radiative recombination continuum (RRC) of . Fe-L emission in general is often not well predicted by plasma models, as there are many different emission lines of which the strengths are not entirely known or understood [e.g. @bernittetal2012]. Of the different ionization states of Fe which produce Fe-L emission, the Fe-L lines produced by are often the most prominent in the hot plasmas found in SNRs, but they have a weak presence in 3C400.2. Therefore, for a plasma to show strong emission unaccompanied by emission, the ionization state of the Fe plasma needs to be high, with most of the Fe in ionization states. We have tried to fit this feature solely with a highly ionized Fe plasma, but due to other Fe emission lines this always resulted in an unacceptable fit, as the higher ionization states of Fe produce too much emission around 1.2 keV. The non-overionized NEI model attempts to fit the feature at 0.87 keV using a combination of the $He-\alpha$ line and emission. The Ne line has a centroid of 0.92 keV, however, which results in a bad fit. The overionized model fits this feature much better due to the addition of the RRC. In the 1.3 - 1.7 keV range, the non-overionized model shows large residuals, where it attempts to fit continuum emission using lines of , which has emission lines with centroids of 1.35 and 1.47 keV respectively. The overionized model fits this region much better, using a RRC at 1.36 keV, which improves the fit with respect to the non-overionized model.
The above example shows that although the presence of overionization in 3C400.2 is indeed subtle, the overionized models provide a significantly better fit to the data due to the presence of RRCs.
Spectra
-------
In this section we apply a cooling model to several regions of the remnant, which show significant differences in their plasma properties. The best-fit models of the different regions are listed in Tab. \[tab:spectral\_parameters\], where we also listed the C-stat / d.o.f. of the single NEI and double, non-cooling NEI models that we attempted to fit to the data.
Component Parameter Unit value
----------- ----------------------- ----------------------- ------------------------
Ejecta $N_{\rm H}$ $10^{21}$ cm$^{-2}$ $6.08^{+0.15}_{-0.12}$
$n_{\rm e}n_{\rm H}V$ $10^{55}$ cm$^{-3}$ $2.31^{+0.83}_{-0.67}$
$kT_{\rm e}$ $3.86^{+0.30}_{-0.28}$
$\tau$ $10^{10}$ cm$^{-3}$ $2.02^{+0.05}_{-0.06}$
Si $3.11^{+1.23}_{-0.84}$
S $6.09^{+2.73}_{-1.80}$
Fe $16.6^{+7.2}_{-4.8}$
Luminosity $10^{31}$erg s$^{-1}$ $68$
ISM $n_{\rm e}n_{\rm H}V$ $10^{58}$ cm$^{-3}$ $1.17^{+0.21}_{-15}$
$kT_{\rm 1}$ $0.42^{+0.03}_{-0.02}$
$kT_{\rm e}$ $0.14^{+0.01}_{-0.01}$
$\tau$ $10^{10}$ cm$^{-3}$ s $26.4^{+7.8}_{-6.03}$
Ne $0.40^{+0.04}_{-0.04}$
Luminosity $10^{31}$erg s$^{-1}$ $4.5$
C-stat / d.o.f. 257.84 / 228
: The best-fit model parameters of the spectrum covering the whole area of the ACIS-I chips. We used the solar abundances from @abundances. Note that the hot ejecta plasma is *under*ionized, while the ISM component is *over*ionized. The errors represent the 1$\sigma$ confidence interval.
\[tab:full\_param\]
### Region 1: southern part of the remnant
This spectral extraction region coincides with the part of the remnant that is weak in X-ray emission, as is apparent from Fig. \[fig:chandra\], but is the brightest in terms of optical emission [@winkleretal1993]. This means that these parts of the remnant either show radiative shocks, where the shock velocity is lower than 200 km s$^{-1}$ for which the post-shock plasma cools efficiently, or the plasma has otherwise cooled to below $10^{6}$ K. This region has two models which fit nearly equally well. The first model (C-stat / d.o.f. = 81 / 87), of which the parameters are shown in Tab. \[tab:region1\_alt\_param\], has similar properties to the model for the full region. It has an ISM component with $kT_{\rm e} < kT_1$ and an ejecta component with $kT_{\rm e} = 3.1$ keV. Compared to the full model, however, the ejecta component contains no overabundance of Si. The feature in the spectrum around 1.8 keV is fit by the Mg IX RRC of the cooling component. The $\tau = 3.20^{+0.89}_{-0.68}\times10^{10}$ cm$^{-3}$ s combined with the high temperature produces a Si line with a centroid at $\sim 1.82$ keV, which is too high to fit the feature. In addition, the Fe abundance is very poorly constrained: when left as a free parameter the best fit value raises to $\sim250$ times solar abundance, suggesting a plasma consisting purely of Fe. This is not realistic for 3C400.2, especially when considering the best fit values of the other regions, and therefore we fixed the Fe abundance at 15, close to the value obtained for the full region. The spectrum with the best-fit model is shown in Fig. \[fig:reg1\_alt\_spectrum\].
Component Parameter Unit value
----------- ----------------------- ----------------------- ------------------------
Ejecta $N_{\rm H}$ $10^{21}$ cm$^{-2}$ $5.61^{+0.47}_{-0.45}$
$n_{\rm e}n_{\rm H}V$ $10^{54}$ cm$^{-3}$ $2.59^{+0.28}_{-0.26}$
$kT_{\rm e}$ $3.20^{+0.89}_{-0.68}$
$\tau$ $10^{10}$ cm$^{-3}$ s $2.12^{+0.08}_{-0.08}$
Fe $015$ (fixed)
Luminosity $10^{31}$erg s$^{-1}$ $4.2$
ISM $n_{\rm e}n_{\rm H}V$ $10^{56}$ cm$^{-3}$ $7.49^{+2.13}_{-1.41}$
$kT_{\rm 1}$ keV $0.77^{+0.27}_{-0.15}$
$kT_{\rm e}$ keV $0.19^{+0.01}_{0.01}$
$\tau$ $10^{10}$ cm$^{-3}$ $86.5^{+24.9}_{-21.8}$
Luminosity $10^{31}$erg s$^{-1}$ $1.6$
C-stat / d.o.f. 81 / 87
: Parameters of the alternative model for region 1. The errors represent the 1$\sigma$ confidence interval. []{data-label="tab:region1_alt_param"}
![The spectrum of region 1, fitted with the model of which the parameters are listed in Tab. \[tab:region1\_alt\_param\]. The red solid line shows the complete model. The blue long dashed line shows the low $kT$ NEI component while the green short-dashed line shows the high $kT$ component. []{data-label="fig:reg1_alt_spectrum"}](region1_alternative.pdf){width="84mm"}
The second model for this region (C-stat / d.o.f. = 78.2 / 84) consists of two cooling NEI components of which the parameters are listed in Tab. \[tab:spectral\_parameters\]. The coolest component has $kT_{\rm e} = 0.06$ keV, the lowest electron temperature of all regions in the remnant. The spectral features around 0.8-0.9 keV (see Fig. \[fig:spectra\]) are in this model fit by a combination of RRCs at 0.74 and 0.87 keV. In addition it shows a RRC at 1.2 keV. There is virtually no line emission present in this component. The hotter component with $kT_{\rm e}=0.53$ keV shows the emission line coupled with an RRC, but otherwise again very little line emission. The abundances of both components are mostly solar, with only O being overabundant in the hotter component. Contrary to the full spectrum and spectra of other extraction regions, there is no super solar abundance of Si, S or Fe present in the hot component for this region. The Fe abundance is sub-solar in the hotter component, while in the cooler component the temperatures are such that Fe has an ionization state $<$, which does not show significant emission in the X-ray band. The fact that there is practically no line emission from elements with higher mass than Mg is not unexpected in a plasma that is cooling rapidly, since the recombination and ionization rates of an element depend strongly on its charge.
The two models are statically almost equally probable. The first model fits the 0.8-0.9 keV features slightly better, while the second model fits the data better in the 0.4-0.8 keV region. Based on the abundances, especially the absence of Si and the high over abundance of Fe, we deem the double cooling NEI model slightly more realistic. The strong cooling fits well with the spatial coincidence with the optical emitting region in the remnant.
### Region 2: brightest part of the remnant
This region was extracted from the brightest part of the remnant in X-rays (see Fig. \[fig:chandra\]). The spectrum is shown in Fig. \[fig:spectra\], top right. The best fit model again contains a hot ($kT_{\rm e}=3.59$ keV) underionized ($\tau=1.87\times10^4$ cm$^{-3}$ s) NEI component with super-solar abundances, plotted as a green dashed line, and a cooler ($kT_{\rm e}=0.63$ keV) NEI component that is overionized, plotted as a blue dot-dashed line. As expected, the best fit model is very similar to the model of the full region, since the full spectrum is dominated by emission from the brightest part of the remnant. The plasma parameters of the hot component are identical to the full region within the errors, but the $N_{\rm H}$ is significantly higher at $7.62^{+0.26}_{-0.24}\times10^{21}$ cm$^{-2}$, compared to $6.08^{+0.15}_{-0.12}\times10^{21}$ cm$^{-2}$. In addition, the initial temperature $kT_{\rm 1}=0.63$ keV for the cooler model is somewhat higher in this region than for the central region ($kT_1$=0.42), while the electron temperatures are identical within the errors. In general the electron temperatures for the cool component are very similar throughout the remnant, while the initial temperature shows significant variation.
[llllll]{}
------------------------------------------------------------------------
& &\
Parameter & Unit & 1 & 2 & 3 & 4\
\
$N_{\rm H}$&$10^{21}$ cm$^{-2}$ & $4.76_{-0.88}^{+0.45}$ & $7.62_{-0.24}^{+0.26}$ & $9.72_{-1.32}^{+0.53}$ & $5.49_{-0.23}^{+0.23}$\
$n_en_{\rm H}V$& $10^{58}$ cm$^{-3}$& $0.03_{-0.01}^{+0.01}$ & $1.79_{-0.40}^{+0.42}(\times10^{-3})$ & $8.80_{-2.70}^{+0.59}(\times10^{-3})$ & $0.275_{-0.24}^{+0.42}(\times10^{-3})$\
$kT_{\rm 1}$ & keV &$0.53_{-0.04}^{+0.04}$ &$-$ &$-$ & $-$\
$kT_{\rm e}$ & keV & $0.25_{-0.02}^{+0.04}$ & $3.59_{-0.39}^{+0.44}$ & $0.71_{-0.13}^{+0.06}$ & $3.23_{-0.72}^{+0.67}$\
$\tau$ & $10^{10}$ cm$^{-3}$ s & $ 1.87_{-1.87}^{+2.77}$ & $2.03_{-0.06}^{+0.06}$ & $19.58_{-1.08}^{+2.55}$ & $1.94_{-0.14}^{+0.17}$\
O & & $2.67_{-0.61}^{+0.91}$ &$-$ &$-$ &$-$\
Ne & &$<0.24$ &$-$ &$-$ & $-$\
Si& & $-$ & $2.75_{-0.57}^{+0.81}$ & $1.34_{-0.18}^{+0.27}$ & $9.53_{-5.75}^{+25.05}$\
S& & $-$ & $5.05_{-1.09}^{+1.62}$ & $2.98_{-0.56}^{+1.22}$ & $12.2_{-7.9}^{+60.8}$\
Fe& & $0.62_{-0.24}^{+0.20}$ & $12.8_{-2.8}^{+6.6}$ & $0.47_{-0.47}^{+0.28}$ & $31.8_{-4.5}^{+78.2}$\
Luminosity & erg s$^{-1}$ & $2.7\times10^{31}$ &$4.1\times10^{32}$ & $9.53\times10^{31}$ & $1.4\times10^{32}$\
$n_en_{\rm H}V$& $10^{58}$ cm$^{-3}$& $0.16_{-0.06}^{+0.15}$ & $0.31_{-0.06}^{+0.07}$ & $2.08_{-1.24}^{+0.13}$ & $0.29_{-0.05}^{+0.07}$\
$kT_{\rm 1}$ & keV & $0.21_{-0.02}^{+0.08}$ & $0.63_{-0.07}^{+0.11}$ & $0.15_{-0.01}^{+0.01}$ & $0.32_{-0.04}^{+0.03}$\
$kT_{\rm e}$&keV & $0.06_{-0.01}^{+0.01}$ & $0.11_{-0.01}^{+0.01}$ & $0.10_{-0.01}^{+0.01}$ & $0.14_{-0.01}^{+0.02}$\
$\tau$& cm$^{-3}$ s & $9.40_{-9.40}^{+70.0}(\times10^{10})$ & $5.49_{-0.87}^{+0.89}(\times10^{11})$ & $2.18_{-2.18}^{+0.52}(\times10^{8})$ & $4.63_{-4.63}^{+7.62}(\times10^{10}) $\
O & & $-$ & $3.08_{-0.49}^{+0.53}$&$-$ &$-$\
Ne& & $-$ & $-$ & $2.07_{-0.51}^{+0.25}$ & $0.59_{-0.15}^{+0.35}$\
Luminosity & erg s$^{-1}$ & $5.3\times10^{26}$&$1.6\times10^{31}$ &$ 7.2\times10^{28}$ & $7.5\times10^{30}$\
C-stat / d.o.f. & & 78.2 / 84 & 118.40 /103 & 93.68 / 108 & 90.57 / 90\
C-stat / d.o.f. single NEI & &140.5 / 87 & 226 / 106 & 135 / 110 & 253 / 93\
C-stat / d.o.f. double NEI & &115 / 86 & 163 / 102 & 120 / 107 & 149 / 88\
$
\begin{array}{cc}
\includegraphics[trim=0 0 0 0,clip=true,width=0.5\textwidth,angle=0]{region1_spectrum.pdf} &
\includegraphics[trim=0 0 0 0,clip=true,width=0.5\textwidth,angle=0]{region2_spectrum.pdf} \\
\includegraphics[trim=0 0 0 0,clip=true,width=0.5\textwidth,angle=0]{region3_spectrum.pdf} &
\includegraphics[trim=0 0 0 0,clip=true,width=0.5\textwidth,angle=0]{region4_spectrum.pdf} \\
\end{array}$
### Region 3: The northwestern part of the remnant
This region was taken from the utmost NW part of the Chandra FOV, which overlaps with the radio shell. The spectrum is shown in Fig. \[fig:spectra\] bottom left. The hydrogen column of the best-fit model of this region is $9.7\times10^{21}$ cm$^{-2}$. Although equal within the errors to the $N_{\rm H}$ of region 2, it is significantly higher than the rest of the remnant. An increasing $N_{\rm H}$ towards the NW of the remnant is consistent with the notion of the remnant expanding into a dense ISM cloud in the NW, where it is expected that the emission from the outer parts of the remnant travel through a larger amount of material than the emission from the inner parts. Significant variation in foreground absorption cannot be ruled out, however.
The best-fit model again has a somewhat hotter, underionized NEI component with super-solar abundances combined with a cooler, overionized NEI component. However, the electron temperature of the hotter component is significantly smaller at $kT_e = 0.71$ keV, than those of the full spectrum and region 2 ($\approx 3.75$ keV), while the $\tau$ is significantly larger at $\sim2\times10^{11}$ cm$^{-3}$ s, compared to $\tau=2\times10^4$ cm$^{-3}$ s. And while the abundances are significantly super solar for Si and S, the Fe abundance is lower. It seems, therefore, that the Fe emission is confined to the central parts of the remnant.
### Region 4: The centre of the remnant
Region 4 was taken from the SW part of the Chandra FOV, which is situated roughly in the centre of the total remnant, as can be deduced from the ROSAT image. The spectrum of this region has again similar parameters as the spectra from the full remnant and region 2. It is plotted in Fig. \[fig:spectra\] bottom right, where the hot NEI component is plotted in a green dashed line, while the blue dot-dashed line represents the cooler NEI component. The $N_{\rm H}=5.49\times10^{21}$ cm$^{-2}$ of the best-fit model for this region confirms the trend of increasing $N_{\rm H}$ towards the NW part of the remnant. The plasma parameters of the hot component are equal within the errors to the temperatures of the full spectrum and the spectrum of region 2. Overall the model for this region confirms the notion of hot ejecta being confined to the central, brighter part of the remnant.
Discussion and Conclusion
=========================
From our spectral modelling we obtain the following. The overall spectrum is well-fitted by a two component NEI model plasma, of which one is *under*ionized with a high $0.71<kT<3.86$ keV and super-solar abundances of Si, S and Fe, while the other NEI component has a lower $0.06<kT_{\rm e}<0.15$ keV, is *over*ionized and has approximately solar abundances. The central parts of the remnant show significantly higher abundances than the outer parts, which is apparent in region 1, where no super-solar abundances are found. Region 1 coincides with the optically emitting region, and shows the lowest electron temperature. Although different parts of the remnant show slightly different plasma properties in terms of initial temperature $kT_{\rm 1}$ and electron temperature $T_{\rm e}$, the parameters are consistent with a hot ejecta plasma confined to the central part of the remnant, which is surrounded by a rapidly cooling, overionized swept up ISM plasma.
Shocked mass
------------
Using the parameters listed in Tab. \[tab:full\_param\], we can estimate of the amount of X-ray emitting mass, both in ejecta and ISM, present in the SNR. The emission measure of the cool component is $1.2\pm0.2\times10^{58}$ cm$^{-3}$. We use a distance of 2.5 kpc and a spherical volume of the emitting region. The spectral extraction region has a radius of 6.67 arcmin (4.85 pc), which corresponds to $V_{\rm total} = 1.4\times10^{58}d_{2.5}^{3}$ cm$^{3}$. If we assume that the hot and the cool component are two separate plasmas, which both occupy part of the total emitting volume and which are in pressure equilibrium, we can get a unique solution for the density and shocked mass of the different components. The reason is that if $P_{\rm hot}$ = $P_{\rm cool}$ then also $n_{hot}kT_{hot} = n_{cool}kT_{cool}$, where $n$ is the number density. For $n_{\rm e}$ = 1.2 n$_{\rm H}$, the number density $n_{\rm H}$ of a component is given by $(EM / 1.2 / V )^{1/2}$, so that:
$$\left(\frac{EM_{\rm cool}}{1.2 V_{\rm total}(x)}\right)^{1/2} kT_{\rm cool} = \left(\frac{EM_{\rm hot}}{1.2 V_{\rm total}(1-x)}\right)^{1/2} kT_{\rm hot},$$
where EM = $n_{\rm e}n_{\rm H}V$, $V_{\rm hot} = (1-x)V_{\rm total}$ and $V_{\rm cool}=xV_{\rm total}$. The above equation is equal for x = 0.4, so that $n_{\rm cool}=~1.3(d_{2.5})^{-1/2}$ cm$^{-3}$ and $n_{\rm hot}~=~0.05(d_{2.5})^{-1/2}$ cm$^{-3}$. The respective masses are $M_{\rm cool} = 8.7(d_{2.5})^{5/2}$ M$_{\odot}$ and $M_{\rm hot}~=~0.46(d_{2.5})^{5/2}$ M$_{\odot}$.
The total emitting volume is uncertain, since the line of sight depth of the remnant is unknown and therefore the emitting volume might be a factor of two greater. Note that the total mass in the remnant is larger than we calculate here, since the Chandra FOV covers about half of the total area of 3C400.2, and we only estimate the mass of the plasma with $T>10^6$ K.
The total mass of ejecta and ISM is quite similar to the mass found by @yoshitaetal2001, who found a mass of $6.7\pm 1.2$ M$_{\odot}d_{2.5}^{5/2}$ for the whole remnant. This is surprisingly small for a mature remnant like 3C400.2, even if the remnant would be located at twice the assumed distance. However, it should be noted that this is only the X-ray emitting mass, and most of the mass may in fact be ‘hiding’ in the plasma cooled below $10^6$ K. The hot component, with super-solar abundances, only makes up a small fraction of the mass, and the total ejecta mass seems very low if it is the ejecta component of a massive star. However, here it should be noted that only the inner regions of a massive star have enhanced metallicity. The total metallicity is a function of stellar mass, with stars around 13 M$_\odot$ producing only 0.3 M$_\odot$ of oxygen [@vink2012]. Therefore a low mass of ejecta-rich material is consistent with a relatively low mass for the exploding massive star. This does, however, also suggest that the cooler plasma, which we designated ISM, may partially or completely consist of the hydrogen-rich envelope of the star. An explanation for why, in particular, ejecta material remains hot needs to be addressed by detailed hydrodynamical simulations, which should incorporate the effects of the stellar wind of the progenitor.
The low mass of the metal-rich component is also consistent with a Type Ia origin for the remnant. In that case the cooler component could be solely shocked ISM. The enhanced iron abundances may indeed hint at a Type Ia origin, although the total mass of Fe at an abundance of 15 times solar is still much lower than the H mass. In addition, the association with the HI regions in the NW part of the remnant makes a core-collapse origin more likely. To settle on the origin of the remnant, it would be important to reconfirm the distance estimate of 2.5 kpc by @giacanietal1998, as a larger distance estimate would favour a core collapse origin. It would also be helpful to identify a stellar remnant in 3C400.2. The Chandra image does not show any evidence for a bright point source that could be the cooling neutron star.
Evolutionary scenario
---------------------
The above densities suggest a hot, metal enriched, tenuous plasma surrounded by a dense, cooler plasma which is cooling rapidly. This is consistent with the shell-like density structure expected from a Sedov evolutionary scenario. This is not the only MMR remnant in which such a temperature and density gradient is observed as @kawasakietal2002 also find a two temperature best-fit model and overionization in IC 443. They find that the central ejecta region has a temperature of 1 keV, compared to 0.2 keV in the outer layers. However, overionization is present only in the central ejecta rather than in the outer layers, while we find overionization in the outer layers and not in the ejecta.
In the introduction we mentioned three different evolutionary scenarios that might explain the centrally peaked X-ray emission: the evaporating cloudlet scenario [@whitelong1991], the @coxetal1999 scenario with high surrounding density including thermal conduction, and the relic X-ray emission scenario. Our observations show that the plasma is best explained by a low density, hot interior surrounded by a high density, lower temperature plasma. The total X-ray emitting mass is relatively low, for a mature remnant. But this is likely an indication that most of the remnant mass is not emitting in X-rays. In addition, overionization is only present in the cool plasma, suggesting that it is cooling more efficiently than the hot plasma. Our results are perhaps most naturally explained by the simplest evolutionary scenario of the three: the relic X-ray emission scenario [@rhopetre1998]. As a result of the high surrounding ISM density, the outer layers have cooled below temperatures capable of emitting in X-rays and the interior is still hot but has a low density and is therefore not cooling efficiently. We cannot rule out the presence of thermal conduction, but we do not find evidence for overionization as a result of rapid cooling for the hot centrally confined plasma. Moreover, the X-ray emitting plasma inside the remnant is clearly not isothermal, as indicated by the model of @coxetal1999. As a final note, thermal conduction has mainly been introduced to models explaining the evolutionary scenarios of MMRs based on the then current observations of generally lower spectral and spatial resolution, which showed little to no temperature gradient in the remnants. However, more modeling is needed to understand whether local density alone determines if a remnant will evolve into a MMR, or whether some other conditions, such as pre-supernova evolution or ejecta structure, are important as well.
The overionization of thermal plasmas can quite naturally occur in MMRs. The high initial ISM density allows the plasma to reach CIE on a timescale smaller than the age of the remnant, after which a combination of adiabatic and radiative cooling can make the cooling rate of the plasma higher than the recombination rate, as shown in @broersenetal2011. The higher surrounding ISM density of MMRs might then be the determining factor for the occurrence of overionization compared to non-MMRs, as already noted in @vinkreview. Indeed, all remnants cool adiabatically and by radiation, but not all remnants expand in a high enough ISM density to reach CIE and then overshoot to overionization.
Summary
=======
We have analysed an archival Chandra observation of the mixed morphology remnant 3C400.2. Our results can be summarised as follows:
- The plasma of the mixed-morphology SNR 3C400.2 is best fitted by a combination of a hot, underionized plasma with low density, and a cooler, overionized plasma with high density. To our knowledge, this is the first evidence for a combination of an overionized outer shell surrounding an ejecta-rich, underionized inner region in an MMR.
- The hot plasma shows significant overabundances of Fe, Si and S, suggesting an ejecta origin, with Fe enhanced in the central part.
- Overionization is significantly present in all parts of the remnant covered by the Chandra field of view.
- The X-ray emitting masses of the plasma components are $M_{\rm cool} = 8.7(d_{2.5})^{5/2}$ M$_{\odot}$ and $M_{\rm hot} = 0.46(d_{2.5})^{5/2}$ M$_{\odot}$.
- This low overall mass suggests that most of the X-ray emitting mass is from mix of metal-rich and hydrogen-rich (envelope) ejecta from a not too massive core collapse supernova, or the remnant has a Type Ia origin.
- The observations are best explained by a scenario in which the centrally peaked X-ray emission is caused by a hot, metal enriched, tenuous plasma. Due to the high surrounding ISM density the outer parts of the remnant have cooled efficiently towards a temperature below which they do not radiate in observable X-ray emission.
- The overionization can be naturally explained by efficient cooling due to a high ISM density in combination with adiabatic expansion.
Acknowledgements
================
The authors would like to thank the anonymous referee for helpful comments and suggestions. The scientific results reported in this article are based on data obtained from the Chandra Data Archive. We also made use of the ROSAT and NVSS archives.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Time-resolved polarisation measurements of pulsars offer an unique insight into the geometry of their emission regions. Such measurements provide observational constraints on the different models proposed for the pulsar emission mechanisms. Optical polarisation data of the Crab Nebula was obtained from the HST archive. The dataset consists of a series of observations of the nebula taken with the HST/ACS. We produced polarisation vector maps of the inner nebula and measured, for the first time, the degree of linear polarisation (P.D.) and the position angle (P.A.) of the pulsar’s integrated pulse beam, and of its nearby synchrotron knot. This yielded $\rm P.D.=5.2\pm0.3\%$ and $\rm P.A.=105.1\pm1.6\degr$ for the pulsar, and $\rm P.D.=59.0\pm1.9\%$ and $\rm P.A.=124.7\pm1.0\degr$ for the synchrotron knot. This is the first high-spatial resolution multi-epoch study of the polarisation of the inner nebula and pulsar. None of the main features in the nebula show evidence of significant polarisation evolution in the period covered by these observations. The results for the pulsar are consistent with those obtained by @Aga using the high-time resolution photo-polarimeter OPTIMA, once the DC component has been subtracted. Our results clearly prove that the knot is the main source of the DC component.'
date: Released 2012 Xxxxx XX
title: Optical Polarimetry of the Inner Crab Nebula and Pulsar
---
\[firstpage\]
Neutron Stars, polarisation, Crab pulsar, synchrotron radation.
Introduction
============
Strong polarisation is expected when the pulsar optical emission is generated by synchrotron radiation. @Shklovsky suggested that the continuous optical radiation from the Crab Nebula was due to synchrotron radiation. This was later confirmed by @Dombrovsky and @Vashakidze who found that the optical radiation was polarised. Incoherent synchrotron emission follows a simple relationship between its polarisation profile and underlying geometry. Hence, optical polarisation measurements of pulsars provide an unique insight into the geometry of their emission regions, and therefore observational constraints on the theoretical models of the emission mechanisms. From an understanding of the emission geometry, one can limit the competing models for pulsar emission, and hence understand how pulsars work - a problem which has eluded astronomers for almost 50 years.
Polarimeters are sensitive in the optical, but the majority of pulsars are very faint at these wavelengths with V $\ge$ 25 [@Shearer]. Polarimetry in the very high-energy domain, X-ray and gamma-ray, using instruments on board space telescopes, is of limited sensitivity. So far, detailed results have only been reported for the Crab pulsar [@Weisskopf; @Dean; @Forot]. Although the number of pulsars detected in the optical is growing, only five pulsars have had their optical polarisation measured; Crab [@Wampler; @Kristian; @Smith; @Aga], Vela [@Wagner; @Mignani], PSR B0540-69 [@Middleditch; @Chanan; @Wagner; @Mignani10] PSR B0656+14 [@Kern], and PSR B1509-58 [@Wagner]. Nonetheless, the optical currently remains invaluable for polarimetry in the energy domain above radio photon energies. The Crab pulsar, being the brightest optical pulsar with V $\approx 16.8$ [@Nasuti], has had several measurements of its optical polarisation, including both phase-averaged and phase-resolved studies.
The first phase-resolved observations of the optical linear polarisation of the Crab pulsar were those of @Wampler, @Cocke, and @Kristian. Those studies showed that the polarisation position angle changes through each peak in the pulsar lightcurve, and that the degree of polarisation falls and rises within each peak, reaching its minimum value shortly after the pulse peak. These observations were limited to the main and inter pulse phase regions only, because at the time it was thought that the pulsar radiated its optical emission through the pulse peaks only. However, a number of phase-resolved imaging observations of the pulsar [@Peterson; @Jones; @Percival; @Golden] showed that the optical emission actually persists throughout the pulsar’s entire rotation cycle, at the level of $\sim1\%$ of the maximum main-pulse intensity.
With this in mind, observations of the optical linear polarisation of the Crab pulsar were made by @Jones and @Smith. These results confirmed the previous observations, and were the first studies to reveal the polarisation profile of the pulsar during the bridge and off-pulse phase regions. They also found that the off-pulse region is highly polarised. The degree of polarisation was 70% and $47\pm10\%$ for @Jones and @Smith, respectively. @Aga report the most detailed phase-resolved observations of the optical linear polarisation of the Crab pulsar. Their results are consistent with previous observations albeit with better definition and statistics, and can be explained in the context of the two-pole caustic model [@Dyks], the outer-gap model [@Romani; @Takata], and the striped-wind model [@Petri].
Detailed observations of the inner nebula, in the optical and X-rays, have revealed a torus structure, that is bisected by oppositely-directed jets, knots, and a series of highly variable synchrotron wisps [@Scargle; @Hester; @Weisskopf2000]. A bright knot of synchrotron emission is located 065 SE of the pulsar. It is highly polarised and is slightly variable in both its location and its brightness [@Hester; @Hester02; @Hester08]. @Komissarov proposed that it is radiation from an oblique termination shock in the pulsar wind nebula. In this model, the Earth line-of-sight is tangent to the flow at the position of the knot, hence the intensity is Doppler boosted for synchrotron emission in the mildly relativistic post-shock flow. The results of their relativistic MHD simulations show that the knot is highly variable and can dominate the gamma-ray synchrotron emission. It has been suggested that this knot is responsible for the highly polarised off-pulse emission seen in time-resolved observations in the optical [@Aga; @Aga13] and gamma-rays [@Forot]. The knot has no counterpart to the NW of the pulsar, but there is a faint second knot located 38 SE of the pulsar, the so called outer knot.
The wisps consist of Wisp 1, a Thin wisp located 18 NW of the pulsar, and the Counter wisp located 83 SE of the pulsar. Wisp 1, located 73 NW of the pulsar, breaks into three separate and distinct components; 1-A, 1-B, and 1-C [@Scargle]. However, one must note that this does not represent the constant configuration of the wisps. The wisps are interpreted as magnetic flux tubes that undergo unstable synchrotron cooling [@Hester98]. A number of follow-up observations of the inner nebula have confirmed the presence of the wisps [@Hester; @Bietenholz]. The NW wisps are more prominent than the SE wisps due to Doppler beaming of the flow. @Schweizer studied the behaviour of the wisps NW of the pulsar in both the optical and X-rays. They observed that the wisps form and move off from the region associated with the termination shock of the pulsar wind, roughly once per year. Moreover, they found that the precise locations of the NW wisps in the optical and X-rays are similar but not exactly coincident, with X-ray wisps located closer to the pulsar. This would suggest that the optical and X-ray wisps are not produced by the same particle distribution. In terms of MHD models, they found that the optical wisps are more strongly Doppler-boosted than the X-ray wisps. For a more detailed review of the Crab Nebula see @Hester08.\
The first optical linear polarisation maps of the Crab Nebula were produced by @Oort, @Hiltner, and @Woltjer. X-ray observations of the linear polarisation of the nebula, in the range 2.6–5.2 keV, yield polarisation of 19% at a position angle of 152–156within a radius of 3of the pulsar [@Weisskopf]. These results are in agreement with the optical measurements of the polarisation, which give polarisation of 19% at a position angle 162for the central nebular region within a radius of $\approx$ 05 [@Oort]. @Dean measured the polarisation of the Crab Nebula and pulsar in the off-pulse phase using the INTEGRAL/SPI telescope, and showed that the polarisation E-vector (124$\pm11\degr$) is aligned with the spin-axis of the neutron star (@Kaplan08; 110$\pm2\pm9\degr$, where the first uncertainty is the measurement uncertainty and the second is from the reference frame uncertainty). This result was later confirmed by @Forot, in the off-pulse phase region using the INTEGRAL/IBIS telescope (120.6$\pm8.5\degr$), and has also been seen in optical observations [@Smith; @Aga]. The SPI and IBIS measurements both encompass the entire nebula and pulsar, and so are dominated by nebular emission. As with the optical observations [@Smith; @Aga], they found the off-pulse region to be highly polarised.\
The purpose of this work is two fold. Firstly, we want to check the polarisation of the pulsar, knot, and wisps for variability. It is difficult to determine the polarisation for objects embedded in a strong nebular background. So, in order to determine the Crab pulsar’s polarisation profile, we need to know the level of background polarisation. Therefore, the second purpose of this work is to accurately map the polarisation of the inner Crab Nebula. This will then act as a guideline for future time-resolved polarisation measurements of the Crab pulsar using the Galway Astronomical Stokes Polarimeter (GASP). This is an ultra-high-speed, full Stokes, astronomical imaging polarimeter based on the Division of Amplitude Polarimeter (DOAP). It has been designed to resolve extremely rapid variations in objects such as optical pulsars and magnetic cataclysmic variables [@Kyne].
Observations and Analysis
=========================
Date Exposure (s) Filter Polariser Roll-Angle (PA\_V3) () Pulsar Position on Chip (x,y)
------------- --------------- -------- ----------- ------------------------ -------------------------------
2003 Aug 08 2$\times$1200 F606W POL0V 87.6 1320.30 1045.02
2$\times$1200 POL60V
2$\times$1200 POL120V
2005 Sep 06 2$\times$1150 F550M CLEAR2L 87.2 1316.23 1034.54
2$\times$1150 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
2005 Sep 15 2$\times$1150 F550M CLEAR2L 87.4 1315.58 1042.01
2$\times$1150 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
2005 Sep 25 2$\times$1150 F550M CLEAR2L 87.6 1315.92 1041.17
2$\times$1150 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
2005 Oct 02 2$\times$1150 F550M CLEAR2L 87.8 1315.97 1040.11
2$\times$1150 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
2005 Oct 12 2$\times$975 F550M CLEAR2L 88.0 1316.44 1039.62
2$\times$975 F606W POL0V
2$\times$1000 POL60V
2$\times$1000 POL120V
2005 Oct 22 2$\times$1150 F550M CLEAR2L 88.3 1316.66 1038.13
2$\times$1150 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
2005 Oct 30 2$\times$1150 F550M CLEAR2L 88.6 1316.40 1036.31
2$\times$1150 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
2005 Nov 08 2$\times$1150 F550M CLEAR2L 89.0 1316.23 1034.54
2$\times$1140 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
2005 Nov 16 2$\times$1150 F550M CLEAR2L 89.6 1316.57 1031.40
2$\times$1150 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
2005 Nov 25 2$\times$1150 F550M CLEAR2L 90.6 1315.74 1025.14
2$\times$1150 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
2005 Dec 05 2$\times$1150 F550M CLEAR2L 120.0 1279.52 871.13
2$\times$1150 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
2005 Dec 14 2$\times$1150 F550M CLEAR2L 125.0 1267.44 851.24
2$\times$1150 F606W POL0V
2$\times$1150 POL60V
2$\times$1150 POL120V
{width="100mm"}
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{width="90mm"}
The raw Hubble Space Telescope/Advanced Camera for Surveys (HST/ACS) polarisation science frames of the Crab Nebula were obtained from the Mikulski Archive for Space Telescopes (MAST). The dataset consists of 13 observations of the nebula taken in three different polarisers (0, 60, and 120) between 2003 August and 2005 December (Proposal ID: 9787) (see Table 1). The Wide Field Camera (WFC) detector, called ACS/WFC, employs a mosaic of two $4096\times2048$ Scientific Imaging Technologies (SITe) CCDs, with a pixel-scale of $\sim0.05$ arcsecond/pixel, covering a nominal FOV $\sim202\arcsec\times202\arcsec$ [@Pavlovsky]. For these observations, with the polarisers in place, the FOV was $\approx102\arcsec\times102\arcsec$. The filter used was F606W ($\lambda=590.70$ nm, $\Delta\lambda=250.00$ nm). The raw images, which had already been flat-fielded, were geometrically aligned, combined and averaged with cosmic-ray removal using IRAF (see Figure 1). We used a total of five field stars and the IRAF task *ccmap* and the *2MASS* catalogue to fit the astrometry. The pulsar was found at $\rm \alpha=05^{\rm h} 34 ^{\rm m} 31\fs930\pm0\fs001$, $\rm \delta=+22\degr00\arcmin51\farcs990\pm0\farcs110$, whilst the synchrotron knot, located 065 SE of the pulsar [@Hester] is found at $\rm \alpha=05^{\rm h}34^{\rm m}31\fs980\pm0\fs001$, $\rm \delta=+22\degr00\arcmin51\farcs630\pm0\farcs110$ (the errors denote the rms of the astrometric fit). For each set of observations, the images taken in the different polarisers were analysed by the IMPOL[^1] software [@Walsh], which produces polarisation maps (see Figures 2 and 3).
In order to determine the polarimetry, aperture photometry was first performed on the pulsar and synchrotron knot in each image using the IRAF task *phot*. The pulsar is saturated in each frame per epoch. @Gilliland describes the well behaved response of the ACS, and shows that electrons are conserved after saturation. The response of the ACS CCDs remains linear up to and beyond the point of saturation provided one uses a GAIN value that samples the full well depth. For ACS this is a GAIN equal to 2 $\rm e^{-1}/ADU$, which is the GAIN setting used for these observations. Over a range of almost 4 magnitudes, photometry remains linear to $<1$%. One can perform aperture photometry of isolated point sources by summing over all the pixels that were bled into [@Pavlovsky]. We tested this method by performing aperture photometry on the pulsar and Trimble 28. We used images taken at the same epoch as the polarimetric observations (2005 September to Decemeber) in the F550M filter ($\lambda=558.15$ nm, $\Delta\lambda=54.70$ nm) but with no polariser in place. We computed the visual magnitudes of both targets and found that the values are consistent with those of @Sandberg, once the different pass bands are taken into account.
We used an aperture of radius 025 to measure the flux from the pulsar. The sky counts were measured using an annulus of width $\approx$ 01, located 015 beyond the central aperture. We added to this flux the flux from the pixels that were bled into. An aperture of radius 015 was used to measure the flux from the synchrotron knot. The sky counts were measured in a region close to the pulsar and knot.
The Stokes vectors were then calculated using the following formulae:
$$I = \frac{2}{3} \ [r(0) + r(60) + r(120)]\\$$
$$Q = \frac{2}{3} \ [2r(0) - r(60) - r(120)]\\$$
$$\hspace{-0.85cm} U = \frac{2}{\sqrt{3}} \ [r(60) - r(120)] \\$$
\
\
where r(0), r(60), and r(120) are the calibrated count rates in the 0, 60, and 120 degrees polarised images respectively [@Pavlovsky].
Computing the degree of linear polarisation of a target
-------------------------------------------------------
The degree of linear polarisation (P.D.) is calculated using the Stokes vectors, and factors which correct for cross-polarisation leakage in the polarising filters. This correction is useful for the POLUV filters; values for the the parallel and perpendicular transmission coefficients (T$_{\rm par}$ and T$_{\rm perp}$) can be found in Figure 5.4 of the ACS Instrument Handbook [@Pavlovsky]. These corrections together with the calibration of the source count rates removes the instrumental polarisation of the WFC ($\sim2\%$) (see Eqn. 4).
Computing the polarisation position angle on the sky of the polarisation E-vector
---------------------------------------------------------------------------------
The position angle (P.A.) is calculated using the Stokes vectors, the roll angle of the HST spacecraft (PA$\_$V3 in the data header files), and $\chi$, which contains information about the camera geometry that is derived from the design specifications; for the WFC $\chi$= -38.2 degrees (see Eqn. 5).\
\
$$\rm P.D. = \frac{\sqrt{Q^{2} + U^{2}}}{I} \ \frac{T_{\rm par} + T_{\rm perp}}{{T_{\rm par} - T_{\rm perp}}} \times 100 \\$$
$$\hspace{-0.4cm}\rm P.A. = \frac{1}{2} \ tan^{-1}\left(\frac{U}{Q} \right) + PA\_V3 + \chi \\$$
\
An important property of polarisation that needs to be considered during analysis is that of bias. This is due to instrumental errors which tend to increase the observed polarisation of a target from its true polarisation. The effect is negligible when $\eta = \rm p \times S/N$ is high ($> 10$), where p is the fractional polarisation of the target, and S/N is the signal-to-noise per image. See for example Fig. 4 of @Sparks. Since the targets are in the high $\eta$ regime, the debiasing correction is small and therefore we omit it. We note that for stars 3 and 4, which have low $\eta$ values, that there will be a systematic over estimate of the polarisation, see @Simmons and @Sparks. However, as these have a polarisation consistent with zero no further analysis was performed on them to remove bias.
@Naghizadeh-Khouei investigated the statistical behaviour of the position angle of linear polarisation using both numerical integrations and data simulations. They found that the distribution of the angle is essentially Gaussian for $\eta > 6$. Hence, we used the formulae of @Serkowski58 [@Serkowski62] for our error analysis. We propagated the errors in the count rates to obtain errors for the Stokes vectors I, Q, and U. Lastly, the errors in the Stokes vectors were propagated through the equations for the errors in the degree of polarisation (Eqn. 6) and position angle (Eqn. 7). As negative polarisation is impossible, we used asymmetric errorbars for stars 3 and 4. Below are the formulae used for calculating the errors in the degree of polarisation and position angle:
$$\rm \frac{\sigma_{P.D.}}{P.D.} = \sqrt{\frac{Q^{2}\sigma_Q^{2} + U^{2}\sigma_U^{2}}{Q^{2} + U^{2}} + \left(\frac{\sigma_I}{I} \right)^{2}} \\$$
$$\hspace{-2.0cm}\rm \sigma_{P.A.} = 28.65\degr \frac{\sigma_{P.D.}}{P.D.}\\$$
\
where $\sigma_{\rm I}$, $\sigma_{\rm Q}$, and $\sigma_{\rm U}$ are the errors in Stokes vectors I, Q, and U, respectively. These errors take into account those introduced by instrumention and systematics.
The polarisation of the synchrotron wisps was also studied (see Tables 2 and 3). To measure the total flux of each wisp, we summed the flux from a series of apertures (r $\approx$ 03) placed along the extent of each wisp. We adopt the standard nomenclature as discussed by [@Scargle], who noted their temporal variability and strong polarisation (see Figure 4). We have accounted for their temporal motion in our analysis. For the sky background subtraction we use the same region of the nebula as used for the knot. Since the contribution of zodiacal and scattered light to the background is low, we therefore ignore the effect of the background polarisation in our analysis.
As a guide to our analysis, a number of fore/background stars in the nebula (see Figure 3) were also analysed to confirm the methodology which we used, and to cross-check for any systematics. Stars 3 and 4 are not saturated in each frame per epoch, but stars 1, 2, and Trimble 28 are saturated. Therefore we employed the same photometric method as used for the pulsar. We used an aperture of radius 035 to measure the flux from each star. The sky counts were measured using an annulus of width $\approx$ 01, located 015 beyond the central aperture. We have found that all of the stars are consistent, within the errors, with unpolarised sources.
We have omitted the results of the analysis of the 2003 August dataset. The sky background is higly variable in one of the raw images. This then causes errors when one calculates the polarisation of the wisps and plots the polarisation maps for this epoch. We also note that for the 2005 December dataset that the roll-angle of the spacecraft was significantly different to other observations. For this dataset the diffraction spike from the pulsar crosses the knot. Furthermore, we note that for the December 14th dataset the full Moon was 9 degrees away from the pulsar. This might have introduced spurious background levels which would have impacted upon the polarisation of the faint extended sources such as the wisps.
We have investigated the effects of photometric losses due to charge transfer efficiency (CTE) in the CCDs of the WFC. The effect reduces the apparent brightness of sources, and it requires a photometric correction to restore the measured integrated counts to their true value. The ACS team claim that there is no evidence of photometric losses due to CTE for WFC data taken after 2004. Nonetheless, we applied the correction for CTE (see Eqn. 8) to our photometry and found that it does not change the results of the polarimetry. Below is the formula for the correction for CTE loss. This value is then added to the measured flux.\
$$\hspace{-0.2cm} YCTE = 10^{A} \times Sky^{B} \times Flux^{C} \times \frac{Y}{2048} \times \frac{(MJD-52333)}{365} \\$$\
where MJD is the modified julian date of the observation, and reflects the linear degradation of the CCD with time. The parameters A, B, and C are found in Table 6.1 of the ACS Instrument Handbook [@Pavlovsky].
In order to determine the performance of the ACS as a polarimeter, the ACS team have modeled the complete instrumental effects and the calibration together. This is done so as to quantify the impacts of the remaining uncalibrated systematic errors. They claim that the fractional polarisations will be uncertain at the one-part-in-ten level (e.g. a 20% polarisation has an uncertainty of 2%) for strongly polarised targets; and at about the 1% level for weakly polarised targets. The position angles will have an uncertainty of about 3. This is in addition to uncertainties which arise from photon statistics [@Pavlovsky]. They then checked this calibration against polarised standard stars ( 5% polarised) and found it to be reliable within the quoted errors [@Cracraft].
Photometry and morphology of the knot in unpolarised light
----------------------------------------------------------
We also retrieved from the MAST archive a series of 12 ACS/WFC datasets, collected through the F550M filter ($\lambda=558.15$ nm, $\Delta\lambda=54.70$ nm) at the same epoch as the polarimetric observations (from 2005, September 6 to 2005, December 14). Each observation consists of a sequence of two images collected in a single orbit, to allow for cosmic-ray rejection. We retrieved pipeline-calibrated, [*drizzled*]{}[^2] images from the archive. Total exposure times range from 1950 s to 2300 s per epoch (see Table 1). We superimposed the images on the first-epoch one by using the coordinates of 30 non saturated field sources as a reference grid. The rms accuracy was better than 0.07 pixels per coordinate. We performed multi-epoch photometry of the knot with the [Sextractor]{} software [@Bertin]. We used an implementation of the Kron method [@Kron], which measures the flux of an object within an optimised elliptical aperture, evaluated using the second moments of the object’s brightness distribution. The parameters of the Kron ellipse (center, semiaxes, and orientation) also yield a measure of the object coordinates and morphology, which is useful for the case of the knot, which is possibly variable in both position and shape as a function of time. The measured count rate was converted to flux using the standard ACS photometric calibration tabulated in the image headers. Correction for CTE losses proved to have a negligible effect.
Since the knot is a diffuse source located very close to a much brighter and saturated point source (the Crab pulsar), particular care was devoted to estimate systematic errors possibly affecting the flux measurements. To this aim, we have performed simulations with the ESO/Midas software[^3], adding to the ACS images a synthetic knot. A 2-dimensional Gaussian function was used to generate the artificial source, setting one of the symmetry axes aligned to the pulsar spin-axis. The synthetic knot was positioned to the NW of the pulsar, along the direction of the pulsar spin-axis, at an angular distance comparable to that of the true knot. By varying the flux and the position of the artificial source we evaluated the uncertainty on the flux measurement to be $\sim2.5\%$.
We also measured the fluxes of a sample of non-saturated stars in the field as a further assessment of the stability of photometry in the variable background of the Crab nebula (see Figure 3).
Results
=======
Included here are the measurements of the degree of polarisation and position angle of each target per epoch (see Tables 2–5 inclusive). We have plotted the degree of polarisation and position angles for each target as a function of time (see Figures 5 and 6). Using a $\chi^{2}$ goodness-of-fit, we found no significant variation (at the 95% confidence level) in the polarisation of the pulsar, knot, and wisps over the 3 month period of these observations. As a final comparison, we present the mean values. These are the values obtained from using the weighted mean and error of the degree of polarisation and position angle (see Table 6). Stars 3 and 4 have asymmetric error bars. Hence, we use the method of @Barlow for calculating the weighted mean for asymmetric error bars. As seen from Figure 5 and 6 and Table 6, the 2005 December dataset shows evidence of a possible variation of the knot polarisation at the $2 \sigma$ level. This variation is due neither to a known systematic effect nor to the contribution of the diffraction spikes from the pulsar (see Sectn. 2.2), which only affect the knot’s flux by $\la 2\%$. Future polarimetry observations on a longer time span will help us to address the possible knot variability. Similarly, we note that the polarisation of Wisps 1-A and 1-B also shows a possible variation at the $2 \sigma$ level in the 2005 December dataset. As discussed in Sectn. 2.2, this may be partially ascribed to the enhanced Moon contribution to the background. This possible variation may be also ascribed to the unresolved contribution of the bright torus in the nebula, to which Wisps 1-A and B have moved closest in the 2005 December observations.
The polarisation maps (Figure 2 and 7) show the variation of the polarisation throughout the inner nebula and particularly in the vicinity of the pulsar itself. Each vector has magnitude equal to the degree of polarisation, and its orientation is the position angle at that point. Such maps allow one to visualise the direction of the magnetic field lines within the nebula. One can distinctly see the overall structure of the inner nebula, the degree of polarisation of the knots and the synchrotron emission. In particular, the filaments are unpolarised and the structures that are visible in polarised light (Figure 3) do not map exactly the continuum (Figure 1).
The F550M band images were used to study with high accuracy possible displacements of the knot. The light curve of the knot is shown in Figure 8, compared to the one of the reference stars. The knot is seen to brighten by $\sim40\%$ on a 60-day time scale, then to fade to its initial flux level. Reference stars, conversely, do not display any significant variability. Focusing on the knot, we note that changes in flux are accompanied by shifts in position, the knot centroid being closer to the pulsar when the feature is brighter (the displacement between maximum and minimum flux is $0\farcs075\pm0.025$). There is no statistically significant evidence of a change in the FWHM of the knot as a function of time. We checked our photometric results by repeating the analysis with simple aperture photometry, using an aperture of 015 positioned on the knot centroid (as measured in each epoch). Such an exercise yielded consistent results ($\sim40\%$ brightening in two months), confirming the significant variability in flux.
Date Crab Knot Wisp 1-A Wisp 1-B Wisp 1-C Thin Wisp Counter Wisp
------------- ------------- -------------- -------------- -------------- -------------- -------------- --------------
2005 Sep 06 4.9$\pm$1.0 59.4$\pm$7.3 41.7$\pm$3.8 32.1$\pm$4.5 36.1$\pm$4.0 31.0$\pm$4.3
2005 Sep 15 5.0$\pm$1.0 60.0$\pm$7.0 45.0$\pm$4.0 34.5$\pm$4.6 37.5$\pm$4.4 35.6$\pm$4.6
2005 Sep 25 5.4$\pm$1.0 62.1$\pm$7.1 44.2$\pm$4.3 40.7$\pm$4.7 38.5$\pm$4.7 40.1$\pm$5.1
2005 Oct 02 4.9$\pm$1.0 62.4$\pm$6.9 41.3$\pm$4.5 39.7$\pm$4.7 37.2$\pm$4.5 39.2$\pm$5.1
2005 Oct 12 4.6$\pm$1.0 60.3$\pm$6.6 43.4$\pm$4.5 36.8$\pm$4.8 32.6$\pm$4.7 37.5$\pm$5.2
2005 Oct 22 4.3$\pm$1.0 61.1$\pm$6.6 43.6$\pm$4.7 46.2$\pm$4.5 37.5$\pm$5.0 36.9$\pm$4.7 45.1$\pm$5.4
2005 Oct 30 4.9$\pm$1.0 61.4$\pm$6.4 44.1$\pm$4.6 44.9$\pm$4.6 35.2$\pm$4.8 34.8$\pm$4.9 42.7$\pm$5.6
2005 Nov 08 5.1$\pm$1.0 60.9$\pm$6.2 41.6$\pm$4.3 45.6$\pm$4.5 39.3$\pm$4.6 35.0$\pm$5.0 41.0$\pm$5.6
2005 Nov 16 5.6$\pm$1.0 59.9$\pm$6.4 41.5$\pm$4.4 46.0$\pm$4.8 42.4$\pm$4.7 38.1$\pm$5.3 47.9$\pm$6.3
2005 Nov 25 5.8$\pm$1.0 59.9$\pm$6.9 41.4$\pm$4.4 40.1$\pm$4.8 38.9$\pm$4.5 38.6$\pm$5.5 41.7$\pm$5.9
2005 Dec 05 5.9$\pm$1.0 42.8$\pm$6.2 32.1$\pm$3.7 40.9$\pm$4.3 38.9$\pm$4.5 38.1$\pm$4.7 43.2$\pm$5.1
2005 Dec 14 5.1$\pm$1.0 43.9$\pm$6.6 21.2$\pm$3.7 25.2$\pm$3.7 40.5$\pm$4.5 35.4$\pm$5.1 39.1$\pm$5.5
Date Crab Knot Wisp 1-A Wisp 1-B Wisp 1-C Thin Wisp Counter Wisp
------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- -- -- --
2005 Sep 06 103.5$\pm$5.9 123.9$\pm$3.5 124.8$\pm$2.6 129.1$\pm$4.0 128.0$\pm$3.1 131.5$\pm$3.9
2005 Sep 15 103.2$\pm$5.9 124.7$\pm$3.4 123.7$\pm$2.5 124.7$\pm$3.8 125.4$\pm$3.3 128.1$\pm$3.7
2005 Sep 25 106.6$\pm$5.4 125.0$\pm$3.3 125.7$\pm$2.8 125.6$\pm$3.3 125.7$\pm$3.5 128.6$\pm$3.7
2005 Oct 02 103.5$\pm$5.8 125.1$\pm$3.2 126.5$\pm$3.1 125.6$\pm$3.4 126.3$\pm$3.5 128.6$\pm$3.7
2005 Oct 12 102.6$\pm$6.4 124.9$\pm$3.2 129.5$\pm$3.0 129.8$\pm$3.7 127.7$\pm$4.2 131.0$\pm$3.9
2005 Oct 22 109.7$\pm$6.7 125.7$\pm$3.1 124.8$\pm$3.1 127.2$\pm$2.8 125.5$\pm$3.8 124.2$\pm$3.6 126.7$\pm$3.5
2005 Oct 30 104.6$\pm$5.8 125.7$\pm$3.0 125.0$\pm$3.0 130.3$\pm$3.0 127.8$\pm$3.9 126.8$\pm$4.0 129.6$\pm$3.8
2005 Nov 08 106.6$\pm$5.7 125.4$\pm$2.9 123.4$\pm$3.0 130.6$\pm$2.8 129.3$\pm$3.3 127.1$\pm$4.1 130.7$\pm$3.9
2005 Nov 16 108.0$\pm$5.1 124.7$\pm$3.1 124.5$\pm$3.0 130.0$\pm$3.0 129.6$\pm$3.2 128.0$\pm$4.0 131.5$\pm$3.8
2005 Nov 25 102.2$\pm$5.1 125.4$\pm$3.3 125.5$\pm$3.0 127.4$\pm$3.5 130.2$\pm$3.3 125.9$\pm$4.1 129.5$\pm$4.1
2005 Dec 05 107.5$\pm$4.9 121.6$\pm$4.2 122.0$\pm$3.3 125.7$\pm$3.0 130.5$\pm$3.2 124.4$\pm$3.5 127.3$\pm$4.4
2005 Dec 14 102.5$\pm$5.7 119.9$\pm$4.3 132.6$\pm$5.1 131.4$\pm$4.2 134.4$\pm$3.1 139.2$\pm$4.2 143.2$\pm$4.0
Date Trimble 28 Star 1 Star 2 Star 3 Star 4
------------- ------------- --------------------- --------------------- --------------------- ---------------------
2005 Sep 06 1.0$\pm$0.7 1.5$\pm$1.3 0.9$\pm$0.9 1.6$^{+2.8}_{-1.6}$ 3.4$^{+3.7}_{-3.4}$
2005 Sep 15 1.1$\pm$0.6 2.4$\pm$1.3 1.2$\pm$0.9 2.6$^{+2.8}_{-2.6}$ 3.4$^{+3.7}_{-3.4}$
2005 Sep 25 1.7$\pm$0.7 2.4$\pm$1.4 1.0$\pm$0.9 2.7$^{+2.8}_{-2.7}$ 3.3$^{+3.7}_{-3.3}$
2005 Oct 02 1.1$\pm$0.6 3.1$\pm$1.3 1.2$\pm$0.9 1.7$^{+2.8}_{-1.7}$ 2.8$^{+3.6}_{-2.8}$
2005 Oct 12 0.9$\pm$0.7 2.0$\pm$1.3 0.9$\pm$0.9 2.0$^{+2.8}_{-2.0}$ 2.2$^{+3.7}_{-2.2}$
2005 Oct 22 2.1$\pm$0.7 2.5$\pm$1.3 1.5$\pm$0.9 2.9$\pm$2.8 4.3$\pm$3.7
2005 Oct 30 0.7$\pm$0.6 1.5$\pm$1.3 1.4$\pm$0.9 2.0$^{+2.8}_{-2.0}$ 2.7$^{+3.7}_{-2.7}$
2005 Nov 08 1.2$\pm$0.7 2.3$\pm$1.4 0.7$^{+0.9}_{-0.7}$ 3.3$\pm$2.8 3.7$\pm$3.7
2005 Nov 16 1.6$\pm$0.7 1.2$^{+1.3}_{-1.2}$ 1.6$\pm$0.9 2.1$^{+2.8}_{-2.1}$ 1.8$^{+3.7}_{-1.8}$
2005 Nov 25 1.5$\pm$0.7 2.0$\pm$1.4 1.5$\pm$0.9 2.5$^{+2.9}_{-2.5}$ 3.5$^{+3.8}_{-3.5}$
2005 Dec 05 1.8$\pm$0.7 2.2$\pm$1.4 1.2$\pm$0.9 2.2$^{+2.9}_{-2.2}$ 3.0$^{+3.8}_{-3.0}$
2005 Dec 14 1.9$\pm$0.7 2.5$\pm$1.4 1.5$\pm$0.9 1.3$^{+2.9}_{-1.3}$ 1.5$^{+3.7}_{-1.5}$
Date Trimble 28 Star 1 Star 2 Star 3 Star 4
------------- ---------------- ---------------- ---------------- ---------------- ----------------
2005 Sep 06 116.3$\pm$18.5 149.5$\pm$25.8 177.0$\pm$28.1 139.3$\pm$49.5 146.6$\pm$30.7
2005 Sep 15 170.8$\pm$17.6 145.3$\pm$15.9 174.4$\pm$22.2 138.7$\pm$31.4 136.2$\pm$30.7
2005 Sep 25 159.4$\pm$11.1 148.2$\pm$12.6 124.2$\pm$25.2 145.7$\pm$30.5 144.4$\pm$32.0
2005 Oct 02 178.5$\pm$16.3 151.1$\pm$19.4 1.0$\pm$20.5 125.9$\pm$47.0 148.1$\pm$36.9
2005 Oct 12 74.8$\pm$20.5 151.1$\pm$18.6 97.6$\pm$27.1 148.9$\pm$39.7 145.8$\pm$47.1
2005 Oct 22 152.4$\pm$8.0 147.4$\pm$15.4 138.8$\pm$17.1 148.8$\pm$28.3 148.6$\pm$24.5
2005 Oct 30 140.0$\pm$24.9 153.8$\pm$26.3 136.7$\pm$17.9 150.8$\pm$41.0 142.2$\pm$38.3
2005 Nov 08 155.6$\pm$15.1 151.3$\pm$16.5 124.2$\pm$36.8 149.0$\pm$24.5 139.5$\pm$29.0
2005 Nov 16 156.8$\pm$12.0 144.5$\pm$33.2 116.3$\pm$15.7 140.2$\pm$38.7 144.4$\pm$60.1
2005 Nov 25 142.6$\pm$12.2 153.8$\pm$19.9 136.6$\pm$17.0 135.3$\pm$33.0 149.6$\pm$30.9
2005 Dec 05 135.5$\pm$10.4 132.4$\pm$17.3 144.2$\pm$22.2 146.4$\pm$37.4 146.1$\pm$36.2
2005 Dec 14 144.1$\pm$10.1 148.7$\pm$15.8 164.9$\pm$17.0 150.4$\pm$64.5 149.1$\pm$69.2
--------------------------------------------------------------------------------------------------------
Polarisation Degree (%) Position Angle ($^{\circ}$)
------------------------------------------------ ------------------------- -----------------------------
**[Pulsar]{} &5.2$\pm$0.3 &105.1$\pm$1.6\
Synchrotron Knot &59.0$\pm$1.9 &124.7$\pm$1.0\
Wisp 1-A &39.8$\pm$1.6 &124.7$\pm$1.2\
Wisp 1-B &43.0$\pm$1.3 &127.4$\pm$0.9\
Wisp 1-C &38.5$\pm$1.3 &128.8$\pm$1.0\
Thin Wisp &36.7$\pm$1.4 &127.1$\pm$1.0\
Counter Wisp &40.6$\pm$1.5 &130.3$\pm$1.1\
Trimble 28 &1.6$\pm$0.2 &147.5$\pm$3.7\
Star 1 &2.3$\pm$0.4 &147.7$\pm$5.2\
Star 2 &1.2$\pm$0.3 &128.3$\pm$5.9\
Star 3 &1.6$\pm$0.7 &144.0$\pm$10.2\
Star 4 &2.2$\pm$1.0 &144.8$\pm$9.9\
**
--------------------------------------------------------------------------------------------------------
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Discussion
==========
We have studied the phase-averaged polarisation properties of the Crab pulsar and its nearby synchrotron knot using archival HST/ACS data. We note that the dataset analysed in this paper has previously been used by @Hester08 to examine the morphology and structure of the polarised components of the inner nebula. However, we have produced polarisation vector maps of the inner nebula and measured, for the first time, the degree of linear polarisation and the position angle of the pulsar’s integrated pulse beam, and of its nearby synchrotron knot. Furthermore, this work marks the first high-spatial resolution multi-epoch study of the variability of the polarisation of the inner nebula and pulsar.
The results for the Crab pulsar are $\rm P.D.=5.2\pm0.3\%$, and $\rm P.A.=105.1\pm1.6\degr$ (see Table 6). These values are in good agreement with those of @Aga using the high-time resolution photo-polarimeter OPTIMA[^4] [@Kanbach], once DC substracted. They measure phase-averaged values of $\rm P.D.=9.8\pm0.1\%$, and $\rm P.A.=109.5\pm0.2\degr$, which is not DC subtracted and includes the emission from the inner knot due to the OPTIMA aperture. They measure values of $\rm P.D.=5.4\%$, and $\rm P.A.=96.4\degr$ after DC subtraction, and it is this later measurement that agrees with our own. The optical polarisation of the Crab pulsar has also been measured by @Wampler ($\rm P.D.=6.5\pm0.9\%$, $\rm P.A.=107.0\pm6.0\degr$), and @Kristian ($\rm P.D.=6.8\pm0.5\%$, $\rm P.A.=98.0\pm3.0\degr$).
We note that the polarisation of the inner knot ($59.0\pm1.9\%$) is a factor of two larger than the off-pulse polarisation of 33% obtained from OPTIMA observations [@Aga] and consistent with the older measurements of [@Jones] (70%) and [@Smith] ($47\pm10\%$). This discrepancy is partially due to the uncertainty of determining the phase interval bracketing the minimum of the Crab’s lightcurve, hence the contribution of the DC component [see Fig. 5 of @Aga]. It could also be partially due to uncertainties in the estimate of the sky background in the OPTIMA data and/or the contribution from the sky and pulsar off-pulse flux. @Golden give the unpulsed pulsar flux to be 0.02 mJy compared to 0.03 mJy from the knot (this work). We estimate the contribution from the sky for OPTIMA data to be equivalent to 0.04 mJy based on the pupil size of 235 and a 21 magnitude/arcsec$^{2}$ sky background. This would be sufficient to explain the difference. Two-dimensional phase-resolved polarisation observations will allow us to better quantify the knot contribution to the DC component.
X-ray observations of the nebula taken by Chandra [@Weisskopf2000], reveal a torus with bipolar jets emanating outwards from SE and NW of the pulsar. @Ng06 found that the axis of symmetry of the jet is rougly aligned with the pulsar’s proper motion vector. The Crab torus, bisecting the synchrotron wisps, can be traced back to the knot of synchrotron emission seen $\approx0\farcs65$ SE of the pulsar. Our measurement of the polarisation PA of the synchrotron knot, $\rm PA=124.7\pm1.0\degr$, agrees with the Crab torus $\rm PA=126.31\pm0.03\degr$ [@Ng04]. We also found evidence for an apparent alignment between the pulsar polarisation PA (105.1$\pm1.6\degr$) and proper motion vector (@Kaplan08; 110$\pm2\pm9\degr$) (see Figure 9). @Mignani have found the same scenario for the Vela pulsar. Those authors found an apparent alignment between the polarisation position angle of the pulsar, the axis of symmetry of the X-ray arcs and jets (Chandra; @Pavlov, @Helfand), and the pulsar’s proper motion vector. This suggests that the kick given to neutron stars at birth is directed along the rotation axis [@Lai]. The alternative view is that the apparent alignment is an effect of projection onto the sky plane, and that there is no physical jet along the axis of rotation [@RD]. More concrete measurements of the optical polarisation of pulsars will yield the needed observational restraints on these hypotheses.\
As mentioned previously, the polarisation of the wisps was also studied. Our photometry accounts for the outward motion of the wisps. From analysis of the wisps in each epoch, we find that the wisps show variation in both location and brightness on time scales of a few weeks. We found that all of these wisps have similar values of degree of polarisation ($\sim 40\%$) and position angles equal to that of the synchrotron knot ($\sim 125\degr$). Hence, as with the synchrotron knot, they are aligned with the spin-axis of the pulsar. Also, Wisps 1-A is not visible in the frames from 2005 Spetember to 2005 October 12 inclusive, and may be merged with Wisp 1-B during this period. Examining the polarisation vectors maps, one can see that the position angles of the wisps are different to those of the rest of the nebula, where the position angles are aligned NS (Figures 2 and 7). Figure 10 is a histogram of the distribution of the polarisation position angles of the inner nebula. The position angles were extracted from the values in the polarisation map. From this histogram we see that the polarisation position angle of the pulsar environment ($\sim 125\degr$) is away from the peak of the nebula distribution ($\sim 165\degr$). This means that the polarisation properties of the structures close to the pulsar are different from those of the rest of the inner nebula.\
As discussed earlier, using a $\chi^{2}$ goodness-of-fit, we found no significant variation (at the 95% confidence level) in the polarisation of the pulsar, knot, and wisps over a 3 month period. The knot is variable in flux but fairly constant in polarisation. This variation in flux may be explained in terms of an increased plasma density in the vicinity of the knot. Whereas the wisps have constant flux and constant polarisation over this period of time. This would suggest that the magnetic fields within the nebula are uniform over time. However, more detailed follow-up observations will be needed to determine if there is any longer term variation.
Conclusions
===========
We have studied the phase-averaged polarisation properties of the Crab pulsar and its nearby synchrotron knot using archival HST/ACS data. This marks the first high-spatial resolution multi-epoch study of the polarisation of the inner nebula and pulsar. We found an apparent alignment between the polarisation position angle of the pulsar and the pulsar’s proper motion vector. We confirm that the inner knot is responsible for the highly polarised off-pulse emission seen in observations in the optical. We found that the inner knot is variable in position, and brightness over the period of these observations. These are the first quantified measurements of such a variation. We note that we found evidence of a possible variation of the knot polarisation (at $2 \sigma$) which is due neither to a known systematic effect nor to the spike contribution. Future observations will help to address this point. We have also measured the polarisation of the wisps in the inner nebula, and found no significant variation in their polarisation over this 3 month period of observations.\
Polarisation measurements give an unique insight into the geometry of the pulsar emission regions. More multi-wavelength polarisation observations of pulsars, both phase-averaged and phase-resolved, with instruments such as HST/ACS and GASP (optical), and INTEGRAL/IBIS (gamma-ray), will help to provide the much needed data to constrain the theoretical models.\
For example, @McDonald have developed an inverse mapping approach for determining the emission height of the optical photons from pulsars. It uses the optical Stokes parameters to determine the most likely geometry for emission, including: magnetic field inclination angle ($\alpha$), the observers line of sight angle ($\chi$), and emission height.
Acknowledgments {#acknowledgments .unnumbered}
===============
All of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. We thank Jeremy Walsh, ESO, for the use of his polarimetry software IMPOL to produce the polarisation maps. PM is grateful for his funding from the Irish Research Council (IRC). RPM thanks the European Commission Seventh Framework Programme (FP7/2007-2013) for their support under grant agreement n.267251. ASł acknowledges support from the Foundation for Polish Science grant FNP HOM/2009/11B, as well as from the FP7 Marie Curie European Reintegration Grant (PERG05-GA-2009-249168). This work was in part supported under the FP7 Opticon European Network for High Time Resolution Astrophysics (HTRA) project.
[99]{}
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\[lastpage\]
[^1]: http://www.stecf.org/software/IRAFtools/stecf-iraf/impol
[^2]: single, calibrated ACS images were combined using the [multidrizzle]{} software, which also produces a mosaic image of the two ACS chips and applies a correction for the geometric distortions of the camera.
[^3]: http://www.eso.org/sci/software/esomidas/
[^4]: http://www.mpe.mpg.de/OPTIMA
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report Karl G. Jansky Very Large Array (VLA) and Atacama Large Millimeter Array (ALMA) spectroscopy in the redshifted molecular oxygen (O$_2$) 56.265 GHz and 424.763 GHz transitions from the $z=0.88582$ gravitational lens towards PKS1830$-$21. The ALMA non-detection of O$_2$ 424.763 GHz absorption yields the $3\sigma$ upper limit $N({\rm O}_2) \leq 5.8 \times 10^{17}$ cm$^{-2}$ on the O$_2$ column density, assuming that the O$_2$ level populations are thermalized at the gas kinetic temperature of 80 K. The VLA spectrum shows absorption by the CH$_3$CHO 56.185 GHz and 56.265 GHz lines, with the latter strongly blended with the O$_2$ 56.265 GHz line. Since the two CH$_3$CHO lines have the same equilibrium strength, we used the known CH$_3$CHO 56.185 GHz line profile to subtract out the CH$_3$CHO 56.265 GHz feature from the VLA spectrum, and then carried out a search for O$_2$ 56.265 GHz absorption in the residual spectrum. The non-detection of redshifted O$_2$ 56.265 GHz absorption in the CH$_3$CHO-subtracted VLA spectrum yields $N({\rm O}_2) \leq 2.3 \times 10^{17}$ cm$^{-2}$. Our $3\sigma$ limits on the O$_2$ abundance relative to H$_2$ are then $X({\rm O}_2) \leq 9.1 \times 10^{-6}$ (VLA) and $X({\rm O}_2) \leq 2.3 \times 10^{-5}$ (ALMA). These are $5-15$ times lower than the best previous constraint on the O$_2$ abundance in an external galaxy. The low O$_2$ abundance in the $z= 0.88582$ absorber may arise due to its high neutral carbon abundance and the fact that its molecular clouds appear to be diffuse or translucent clouds with low number density and high kinetic temperature.'
author:
- 'Nissim Kanekar, David S. Meier'
title: 'A new constraint on the molecular oxygen abundance at $z \sim 0.886$'
---
Introduction {#sec:intro}
============
Molecular oxygen ([O$_2$]{}) has long been identified as a critical species for the understanding of cooling and energy balance in molecular clouds, and of interstellar chemistry [e.g. @goldsmith78; @goldsmith11]. In standard models of chemistry, the [O$_2$]{} abundance relative to that of molecular hydrogen ${\rm H}_2$ is expected to rise to $X({\rm O}_2) \equiv
N({\rm O}_2)/N({\rm H}_2) \sim 10^{-5}$, comparable to the carbon monoxide abundance, at times beyond $\sim 3 \times 10^5$ years [e.g. @herbst73; @marechal97]. Remarkably, despite numerous searches with the [*Submillimeter Wave Astronomy Satellite*]{}, and the [*Odin*]{} and [*Herschel*]{} satellites, [O$_2$]{} has been detected in only two directions in the Galaxy, towards $\rho$OphA [@larsson07; @liseau12] and Orion H$_2$ Peak 1 [@goldsmith11; @chen14], with abundances $X({\rm O}_2) \approx 5 \times
10^{-8}$ [$\rho$OphA; @larsson07; @liseau12] and $\approx 10^{-6}$ . The majority of searches have yielded low [O$_2$]{} abundances in both diffuse and dark clouds, $X({\rm O}_2) < 10^{-7}$ [e.g. @pagani03; @yildiz13], two orders of magnitude lower than expected. Although many attempts have been made to explain the paucity of [O$_2$]{} [e.g. @bergin00; @charnley01; @quan08; @hollenbach09; @whittet10], the low [O$_2$]{} abundances in molecular clouds remain a serious problem for models of chemistry.
For cosmologically-distant galaxies, the [O$_2$]{} lines are redshifted outside the telluric bands and can be observed with ground-based telescopes. Unlike satellite-based [O$_2$]{} [*emission*]{} searches, where the large telescope beam means that the derived ${\ensuremath{{\rm X(O_2)}}}$ is an average over multiple molecular clouds, searches for [O$_2$]{} in [*absorption*]{} towards compact radio sources provide estimates of ${\ensuremath{{\rm X(O_2)}}}$ in individual clouds along the sightline. Such observations are especially interesting for high-$z$ systems as they allow studies of interstellar chemistry in much younger galaxies.
The two best targets for a search for redshifted [O$_2$]{} in absorption are the spiral gravitational lenses at $z \sim 0.685$ and $z \sim 0.886$ towards B0218+357 and PKS1830$-$21, respectively, which show absorption in a variety of molecular species [e.g. @wiklind95; @wiklind96; @wiklind98; @combes97; @chengalur99b; @kanekar03c; @henkel05; @muller14]. Molecular absorption studies of these galaxies have been used to determine physical conditions in the absorbing clouds [e.g. @henkel08; @menten08], to estimate the temperature of the microwave background [e.g. @muller13], and even to constrain changes in the fundamental constants of physics [e.g. @kanekar11; @bagdonaite13; @kanekar15].
Searches for [O$_2$]{} absorption have been carried out at $z = 0.685$ towards B0218+357 in the [O$_2$]{}368 GHz and 424 GHz transitions [@combes95] and the [O$_2$]{}56 GHz and 119 GHz transitions [@combes97b]. These yielded the upper limit ${\rm N(O_2)} < 2.9 \times 10^{18}$ [cm$^{-2}$]{} on the [O$_2$]{} column density, where we have updated the results of @combes97b for an [O$_2$]{} excitation temperature equal to the inferred gas kinetic temperature [55 K; @henkel05]. The [H$_2$]{} column density of the $z \sim 0.685$ absorber is $\approx 2 \times 10^{22}$ [cm$^{-2}$]{} [@gerin97; @kanekar02]; this yields $X({\rm O}_2) \leq 1.5 \times 10^{-4}$, three orders of magnitude poorer than the limits from Galactic studies [e.g. @pagani03].
We have used the Karl G. Jansky Very Large Array (VLA) and the Atacama Large Millimeter Array (ALMA) to search for redshifted [O$_2$]{} absorption in the $z = 0.886$ spiral lens towards PKS1830$-$21. In this [*Letter*]{}, we report results from our observations, which yield stringent constraints on the [O$_2$]{} abundance in this galaxy.
 
Observations, data analysis and results {#sec:data}
=======================================
   
VLA observations
----------------
The Ka-band receivers of the VLA were used in July 2010 to carry out a search for [O$_2$]{} $1_{2}\rightarrow 1_{1}$ 56.2648 GHz absorption at $z = 0.88582$ towards [PKS1830$-$21]{} (proposal AK725). The observations used the WIDAR correlator as the backend, with a single 128 MHz band, sub-divided into 256 channels, centred at the redshifted [O$_2$]{} line frequency of 29.835 GHz, and two circular polarizations. Observations of 3C286 and the bright sources 3C273 and J2253+1608 were used to calibrate the flux density scale and the system bandpass, respectively. The total on-source time was 2 hours, with 19 working antennas in the VLA C-configuration
The VLA data were analysed in “classic” [aips]{} using standard procedures. Note that [PKS1830$-$21]{} is unresolved by our 19-antenna VLA C-array at Ka-band. After initial calibration, the tasks [uvsub]{} and [uvlin]{} were used to subtract the image of [PKS1830$-$21]{} from the calibrated visibilities, and then to subtract out any residual continuum by fitting a linear baseline to line-free channels. The residual visibilities were then imaged and the final spectrum covering the redshifted [O$_2$]{} 56.265 GHz transition obtained by taking a cut through the spectral cube at the location of [PKS1830$-$21]{}.
The final VLA spectrum is shown in the left panel of Fig. \[fig:vla\], with optical depth against the S-W image component of [PKS1830$-$21]{} plotted versus heliocentric frequency, in GHz. The root-mean-square (RMS) noise on the spectrum is $\approx 1.1 \times 10^{-3}$ per 5 [km s$^{-1}$]{} channel, in optical depth units [assuming that the S-W component contains $\approx 38$% of the total flux density of [PKS1830$-$21]{} at these frequencies; e.g. @muller11]. A strong absorption feature, with an integrated optical depth of $\approx (0.146 \pm 0.014)$ [km s$^{-1}$]{}, is clearly visible at the expected frequency of the redshifted [O$_2$]{} 56.265 GHz line (indicated by the dashed vertical line). However, it was realized that there is a CH$_3$CHO transition ($3_{-1,3}
\rightarrow 2_{-1,2}$ E) at a rest frequency of 56.2652 GHz that would be strongly blended with the [O$_2$]{} 56.265 GHz line, and that might cause the observed absorption. Further, a second absorption feature is visible at $\approx 29.793$ GHz, which could be redshifted CH$_3$CHO absorption, in the $3_{1,3} \rightarrow
2_{1,2}$ A++ transition. If the two features indeed arise from CH$_3$CHO, it would be difficult to draw conclusions about the [O$_2$]{} abundance (although see below). We hence carried out an ALMA search for redshifted [O$_2$]{} 424 GHz absorption, to test whether the VLA absorption feature indeed arises from the [O$_2$]{} 56.265 GHz line.
ALMA observations
-----------------
The Band-6 receivers of ALMA were used in March 2014 to search for redshifted [O$_2$]{} $1_{2}\rightarrow 3_{2}$ 424.7631 GHz absorption at $z=0.88582$ towards [PKS1830$-$21]{}. The observations used four 1.875 GHz intermediate frequency (IF) bands, each sub-divided into 3840 channels, and with 2 polarizations. The four IF bands were centred at 225.780 GHz (covering the redshifted [O$_2$]{} 424 GHz line frequency), 228.530 GHz, 241.033 GHz and 243.733 GHz. Observations of Titan, J1733$-$1304, J1923$-$2104, and a few calibrators were used to calibrate the flux density scale and the system bandpass and gain. The total on-source time was $\approx 2$ hours, with 25 ALMA antennas.
The ALMA data were analysed in two stages, first using the [casa]{} pipeline to carry out the initial calibration procedure, and then self-calibrating the data of [PKS1830$-$21]{} in [aips]{}. The flux density scale was calibrated using the short-baseline data on Titan, and this was then extended to longer baselines by bootstrapping the data of J1923$-$2104. The data of J1733$-$1304 and J1923$-$2104 were, respectively, used to calibrate the system bandpass and initial gain. After applying the initial calibration in [casa]{}, a standard self-calibration procedure was used in [aips]{}, with a few rounds of phase-only self-calibration followed by a single round of amplitude-and-phase self-calibration. The final image has a synthesized beam of $\approx 1.0'' \times 0.8''$ (with the two strong image components of [PKS1830$-$21]{} marginally resolved), and an RMS noise of $\approx 0.14$ mJy/Beam. The task [jmfit]{} was used to measure the flux densities of the N-E and S-W image components, via a 2-Gaussian fit to the final image; this yielded flux densities of $549.98 \pm 0.43$ mJy (N-E) and $342.63 \pm 0.43$ mJy (S-W). The continuum image of [PKS1830$-$21]{} was then subtracted from the calibrated visibilities of each IF band using the task [uvsub]{}, and the residual visibilities of each band were then imaged to produce a spectral cube, after shifting the data to the heliocentric frame. The spectrum for each IF band was then produced via a cut through the cube at the location of the S-W image component. The final spectra have an RMS noise of $\approx 1.0-1.3$ mJy at the re-sampled velocity resolution of $\approx 1.3$ km/s.
The final Hanning-smoothed and re-sampled spectra from the four ALMA IF bands (after subtracting a second-order baseline) against the S-W component are shown in the four panels of Fig. \[fig:alma\], with optical depth plotted versus heliocentric frequency, in GHz. All spectra are shown after smoothing to, and re-sampling at, a velocity resolution of $\approx 6.5$ [km s$^{-1}$]{} the resolution at which the search for redshifted [O$_2$]{} 424 GHz absorption was carried out. No evidence for [O$_2$]{} 424 GHz absorption can be discerned in the spectrum in the top left panel of Fig. \[fig:alma\]. The final RMS noise on the spectrum is $\approx
1.8 \times 10^{-3}$ per 6.5 [km s$^{-1}$]{} channel, in optical depth units.
In passing, we note that four absorption features were clearly detected in the ALMA spectra; three of these correspond to the CO (4–3) and two CN (4–3) transitions (see Fig. \[fig:alma\]). However, we have been unable to identify the fourth transition, at $\approx 226.033$ GHz, i.e. at rest-frame frequencies of 426.257 GHz (at $z=0.88582$, the absorber being studied here), 792.698 GHz [at $z=2.507$, the redshift of [PKS1830$-$21]{}; @lidman99] or 269.567 GHz [at $z=0.1926$, the redshift of another known absorber towards [PKS1830$-$21]{}; @lovell96]. The line width is $\approx 5$ [km s$^{-1}$]{}, similar to that of other high-frequency transitions from the $z = 0.88582$ absorber. It appears that this is not a known low-energy transition of a species expected to be abundant in the ISM.
Discussion {#sec:alpha}
==========
![Low energy level diagram for [O$_2$]{}, adapted from @marechal97, with levels labeled by their (N,J) quantum numbers. The observed transitions and transitions of comparable energy are displayed with solid and dashed arrows, respectively. Transition are labeled by their rest frequency [in GHz, from Splatalogue; e.g. @pickett98; @remijan07; @drouin10] and radiative/collisional rate coefficients \[as (${\rm log}[A_{ul}],{\rm log}[C_{ul}]$), with values for $C_{ul}$ from the RADEX database; [@schoier05; @lique10]\]. \[fig:otwo\]](fig3.eps)
The first question is whether the absorption feature seen at $\approx 29.836$ GHz in the VLA spectrum of [PKS1830$-$21]{} arises from [O$_2$]{} or from CH$_3$CHO (or, indeed, some other transition). The lower energy level of the [O$_2$]{} 56 GHz and [O$_2$]{} 424 GHz transitions is the same (the $1_{2}$ state; see Fig. \[fig:otwo\]), permitting a direct comparison between the expected optical depths in the two lines. Of course, the ratio of the line strengths depends on the respective excitation temperatures. In the case of the $z = 0.88582$ absorber, the number density $n_{\rm H_2}$ and kinetic temperature $T_k$ of the molecular gas have been estimated to be $n_{\rm H_2} \sim 1700-2600$ cm$^{-3}$ and $T_k \approx 80$ K [@henkel08; @henkel09]. For number densities $\gtrsim10^3$ cm$^{-3}$ and $T_k \geq 30$ K, the [O$_2$]{} line populations are expected to be thermalized [e.g. @goldsmith00], i.e. $T_x \approx T_k$. For $T_x \approx 80$ K, the 424 GHz line is expected to be slightly stronger than the 56 GHz line, $\tau_{424} \approx 1.1 \times \tau_{56}$. Our ALMA $3\sigma$ limit on the integrated [O$_2$]{} 424 GHz optical depth is $\approx 0.037$ [km s$^{-1}$]{}, a factor of 5 lower than the integrated optical depth ($\approx 0.146 \pm 0.014$ [km s$^{-1}$]{}) of the 29.836 GHz absorption feature in the VLA spectrum. We can thus conclusively rule out the possibility that the VLA absorption feature arises from the [O$_2$]{} 56 GHz transition. The feature is most likely to arise from the CH$_3$CHO $3_{-1,3} \rightarrow 2_{-1,2}$E transition.
The ALMA upper limit to the [O$_2$]{} 424 GHz optical depth can be used to place a limit on the total [O$_2$]{} column density. For $T_x = 80$ K, this gives ${\ensuremath{{\rm N(O_2)}}}\leq 5.8
\times 10^{17}$ [cm$^{-2}$]{}, at $3\sigma$ significance.
Interestingly, the two CH$_3$CHO transitions ($3_{-1,3} \rightarrow 2_{-1,2}$ E and $3_{1,3} \rightarrow 2_{1,2}$ A$++$) seen in the VLA spectrum at, respectively, 29.836 GHz and 29.793 GHz, have the same line strengths. One can hence subtract one from the other to search for any additional absorption arising from the [O$_2$]{} 56 GHz line. This was done by using two-point interpolation to resample the CH$_3$CHO $3_{1,3} \rightarrow 2_{1,2}$ A$++$ line profile at the measured velocities of the CH$_3$CHO $3_{-1,3} \rightarrow 2_{-1,2}$ E line, and then subtracting out the resampled line profile from the latter spectrum. This procedure is unlikely to yield any systematic effects, as both CH$_3$CHO line profiles are well-sampled, with at least 3 independent spectral points detected at $> 5\sigma$ significance. No absorption is detected in the residual VLA spectrum, yielding an integrated [O$_2$]{} 56 GHz optical depth of $< 0.0131$ [km s$^{-1}$]{}, again at $3\sigma$ significance, against the S-W component of [PKS1830$-$21]{}. Again using $T_x = 80$ K, this yields ${\ensuremath{{\rm N(O_2)}}}\leq 2.3 \times 10^{17}$ [cm$^{-2}$]{}, a factor of $\approx 2.5$ more stringent than the ALMA upper limit.
The [H$_2$]{} column density of the $z \sim 0.886$ lens has been estimated to be $\approx 2.5 \times 10^{22}$ [cm$^{-2}$]{} [@gerin97; @wiklind98]. These are broadly consistent with estimates of the total hydrogen column density towards both lensed images from Chandra and ROSAT X-ray spectroscopy, N(H) $= (1.8-3.5) \times 10^{22}$ [cm$^{-2}$]{} [@mathur97; @dai06]. Note that, while @muller08 argue that the [H$_2$]{} column density may be an order of magnitude larger than the above values to account for the detection of species such as HC$^{17}$O$^+$ and HC$^{15}$N in absorption, such a high value of ${\ensuremath{{\rm N(H_2)}}}$ appears to be ruled out by the X-ray data. Using a value of ${\ensuremath{{\rm N(H_2)}}}= 2.5 \times 10^{22}$ [cm$^{-2}$]{} yields [O$_2$]{} abundances of $X({\rm O}_2) \leq 9.1 \times 10^{-6}$ and $X({\rm O}_2) \leq 2.3 \times 10^{-5}$ from the VLA (CH$_3$CHO-subtracted) [O$_2$]{} 56 GHz and the ALMA [O$_2$]{} 424 GHz non-detections, respectively. Of course, if the higher [H$_2$]{} column density estimate of @muller11 is correct, then our constraints on the [O$_2$]{} abundance would be more stringent by an order of magnitude, i.e. $X({\rm O}_2) \leq 9.1 \times 10^{-7}$.
Prior to this work, the strongest constraint on the [O$_2$]{} abundance outside the Milky Way was $X({\rm O}_2) \leq 1.5 \times 10^{-4}$ in the $z = 0.685$ absorber towards B0218+357 [see Section 1; @combes97b]. Our VLA upper limit on the [O$_2$]{} abundance in the $z = 0.886$ absorber towards [PKS1830$-$21]{} is a factor of $\approx 15$ lower than this, and comparable to the measured [O$_2$]{} abundance towards the Orion H$_2$ Peak 1 [@goldsmith11]. However, our limit is nearly two orders of magnitude weaker than the constraints on, or measurements of, [O$_2$]{} abundances in the Milky Way [e.g. @pagani03; @larsson07; @liseau12]. Unfortunately, the high gas kinetic temperatures in the two gravitational lenses, $\approx 55$ K in the $z= 0.685$ absorber and $\approx 80$ K in the $z= 0.886$ absorber, imply that it will not be easy to improve upon our present constraint and achieve an [O$_2$]{} abundance sensitivity comparable to those in the Milky Way.
While our [O$_2$]{} abundance constraints for the $z=0.88582$ absorber are less stringent than those in the Galaxy, these are by far the most sensitive constraints in an external galaxy. Further, the Galactic estimates stem from emission studies with differing angular resolution in the [O$_2$]{} and CO lines. The derived abundances are hence an average over multiple molecular clouds with different excitation conditions; this can imply large uncertainties in $X({\rm O}_2)$, of upto two orders of magnitude [e.g. @liseau10]. The resolution of the present interferometric absorption study is determined by the size of the background radio continuum at the observing frequency. For [PKS1830$-$21]{}, the emission from the S-W image at high frequencies ($14.5-43$ GHz) arises in a compact source of size $< 0.5$ mas [@jin03; @sato13], i.e. transverse size $< 4$ pc at $z = 0.88582$. The [O$_2$]{} abundance estimates are hence likely to be reliable here, as both the [O$_2$]{} and the [H$_2$]{} column densities are inferred from absorption studies probing the same pencil beam towards the S-W image.
Finally, it is clear that we rule out [O$_2$]{} abundances of $\approx 10^{-5}$ at $3\sigma$ significance in the $z = 0.886$ lens towards [PKS1830$-$21]{}. As noted earlier, the [O$_2$]{} abundance is expected to reach about this level, comparable to the CO abundance, in standard models of molecular chemistry within $\approx 3 \times 10^5$ years [e.g. @herbst73; @marechal97]. The low [O$_2$]{} abundance thus appears a conundrum, even for the $z = 0.886$ absorber. A possible explanation lies in the high derived abundance of neutral carbon in this system by @bottinelli09, who obtain N(C)/N([H$_2$]{}) $\approx 10^{-4}$, somewhat larger than the CO abundance. In typical molecular clouds, [O$_2$]{} is destroyed by the reactions ${\rm C} + {\rm O}_2 \rightarrow {\rm CO} + {\rm O}$, ${\rm C}^+ + {\rm O}_2 \rightarrow {\rm CO} + {\rm O}^+$, and ${\rm C}^+ + {\rm O}_2 \rightarrow {\rm CO}^+ + {\rm O}$. The high carbon abundance in the $z = 0.886$ absorber is thus unfavourable for the survival of [O$_2$]{}, and can account for its low abundance. @bottinelli09 also note that the high carbon abundance relative to CO suggests that the absorbing gas arises in translucent clouds, or clouds in an early phase of the transition from diffuse to dense gas, with low densities and mild-UV fields. This is consistent with the high gas kinetic temperature, ($\approx 80$ K), and relatively low densities ($\approx 1700 - 2600$ cm$^{-3}$) obtained by @henkel09, implying that the absorber at $z = 0.88582$ does not arise in a classical dark cloud.
In summary, we have used the VLA and ALMA to obtain tight constraints on the [O$_2$]{} abundance (relative to [H$_2$]{}), ${\ensuremath{{\rm X(O_2)}}}\leq 9.1 \times 10^{-6}$, in the $z= 0.88582$ spiral gravitational lens towards [PKS1830$-$21]{}. This is a factor of $\gtrsim 15$ more stringent than the best previous constraint on the [O$_2$]{} abundance in an external galaxy. We argue that the low [O$_2$]{} abundance in the $z \sim 0.886$ lens may arise due to its high neutral carbon abundance (resulting in the efficient destruction of [O$_2$]{}), and the fact that the absorbing clouds are probably not dark clouds, but instead diffuse or translucent clouds, with relatively low number density and high gas kinetic temperature.
This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2012.1.00581.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This paper also makes use of VLA data (proposal 10A-110). The National Radio Astronomy Observatory is operated by Associated Universities, Inc, under cooperative agreement with the NSF. NK acknowledges support from the Department of Science and Technology via a Swarnajayanti Fellowship. DSM acknowledges partial support by the National Science Foundation through grant AST-1009620.
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Zhi-Bin Zhang, Si-Wei Kong, Yong-Feng Huang, Di Li, Long-Biao Li'
title: 'Detecting Radio Afterglows of Gamma-Ray Bursts with FAST $^*$ '
---
Introduction {#sect:intro}
============
With prompt developments of space and ground telescopes, studies of Gamma-Ray Bursts (GRBs) have come into an era of full wavelengths (e.g. Gehrels & Razzaque 2013). Particularly, with recent space missions such as Swift satellite, the rapid response together with accurate localization enables more detailed follow-up observations by the ground facilities at longer times and lower frequencies. It has led to deeper understanding of the physical origins of GRBs. In addition, statistical analysis becomes possible and it is very helpful to compare the properties of GRBs and their afterglows (Sakamoto et al. 2011). Prompt $\gamma$-rays and their follow-up X-ray and Optical afterglows are found to be correlated with each other, more or less, for both short and long-duration bursts (Gehrels et al. 2008; Nysewander et al. 2009; Kann et al. 2011). It is interesting that Chandra & Frail (2012) found the detectability of radio afterglows to be correlated with neither the $\gamma$-ray fluences nor the X-ray fluxes, but only with optical brightness in statistics. However, observational constraints of radio afterglows are relatively insufficient, although some authors have recently compiled larger datasets (Postigo, 2012; Chandra & Frail 2012; Ghirlanda et al. 2013; Staley et al. 2013) since the first radio afterglow of GRB 970508 was discovered (Frail et al. 1997). We need to synthesize the GRBs and their afterglows at multiple energy bands separately and then combine them to explore their comprehensive physics.
In the framework of the fireball internal-external shock model, GRBs are produced when the kinetic energy of an ultra-relativistic flow is dissipated by internal collisions, while the afterglows are emitted when the flow is slowed down by external shocks with the surrounding matter of the burst. The fireball model has given numerous successful predictions on GRB afterglows, such as the afterglow itself, jet break in the light curve of afterglows, the optical flash and the afterglow shallow-decay phenomena, etc. (see Lu et al. 2004, Piran 2004, Zhang 2007 and **Gao et al. 2013** for a review). Huang et al. (1999, 2000b) proposed a set of simplified dynamical equations that is consistent with the self-similar solution of Blandford & McKee (1976) in the ultra-relativistic phase, and also consort with the Sedov solution (Sedov 1969) in the non-relativistic phase. Therefore, these equations can conveniently describe GRB afterglows at all post-burst times. For instance, the beaming effects (Rhoads 1997, 1999; Huang, Dai & Lu 2000c), the rebrightening at multiple-wavelengths(Huang et al. 2004; Xu & Huang 2010; Kong, Wong, Huang & Cheng 2010; Yu & Huang 2013) and the multi-band afterglow modeling (Huang, Dai & Lu 2000c; Huang, Dai & Lu 2002; Huang & Cheng 2003; Wang, Huang & Kong 2009; Kong, Huang, Cheng & Lu 2009) can be easily dealt with by these equations.
Square Kilometer Array (SKA) will be the largest and most sensitive radio telescope group in the world. The SKA project is designed to be constructed via two phases and will receive radio signals at a frequency range from 70 MHz to 10 GHz. As the largest worldwide single-dish radio telescope, the Five-hundred-meter Aperture Spherical radio Telescope (FAST, Nan et al. 2011; Li, Nan & Pan 2013) is a Chinese megascience project that is being built in Guizhou province of southwestern China with an expected first light in Sep. of 2016. FAST continuously covers radio frequencies between 70 MHz and 3 GHz. A possible extension to 8GHz is being considered in the 2nd phase of FAST. FAST will be equipped with a variety of instruments and has been designed for different scientific purposes including the radio afterglows of GRBs. According to the current data sets presented by Chandra & Frail (2012), the detection rate of radio afterglows is 30 %, in which more than half of the radio flux measurements are made at 8.5 GHz. However, nearly 10 percent of the detected radio afterglows are from bright long GRBs (Ghirland et al. 2013; Salvaterra et al. 2012). The reason is that the low-frequency observations could be more affected by the bias of receivers.
In this work, we apply the fireball model to calculate a variety of numerical afterglow curves with changing redshifts for different cases representing failed, low luminosity, high luminosity and standard GRBs, respectively. These theoretical light curves based on diverse physical considerations are directly compared with FAST’s sensitivity in order to diagnose the detectability by FAST. Our radio light curves within the FAST’s window are derived for both low (70 MHz-0.5 GHz) and medium/high frequencies (0.5-3 GHz). The structure of our paper is as follows. Firstly, we provide an overview on observations of GRB radio afterglows in Section 2. Theoretical dynamical model of afterglow and sensitivity of FAST are introduced in Section 3. Numerical results are shown in Section 4 and we end with discussions and brief conclusions in Section 5.
Overview of radio afterglow observations
========================================
Recently, Chandra & Frail (2012) presented a large sample of GRB radio observations for a 14-year period since 1997. Despite 304 radio afterglows consisting of 2995 flux measurements, only 95 out of 304 were reported to have radio afterglows detected by VLA, corresponding to a detection rate of $\sim$30%, of which 1539 measurements are made in 8.5 GHz. They pointed out that the current detection rate of radio afterglows, much lower than in the X-ray ( 90%) or optical ( 75%) bands in the Swift era, may be seriously limited by instrumental sensitivity. Hancock, Gaensler & Murphy (2013) argued that the lower rate would be caused by two intrinsically different types of bursts, namely radio bright and radio faint sources. However, radio emissions from host galaxies of GRBs also make the radio afterglows more difficult to detect (e.g. Berger, Kulkarni & Frail 2001; Berger 2014; Li et al. 2014). The Chandra & Frail sample contains 33 short-hard bursts, 19 X-ray flashes, 26 GRBs/SNe candidates including low luminosity bursts and 4 high-reshift bursts, of which only a few radio afterglows are available owing to their lower detection rate. In general, short bursts with smaller istropic energy, similar to low luminosity ones, are thought to occur in a relatively neaby universe like supernova. On the other hand, afterglows of short bursts are usually much dimmer than those of long ones (see e.g. Rowlinson 2013 for a review). This motivates us to focus on contrasting the radio afterglows in Fig. 1 between short GRBs (050724, 051221A and 130603B), low luminosity GRBs (060218) and high-redshift GRBs (050904, 080913, 090423 and 090429B) with successful detections. Note that only GRB 130603B was not included in Chandra & Frail (2012) and its data are taken from Fong, Berger & Metzger et al (2014). As the largest next-generation single dish radio telescope, FAST is expected to bring us many important findings (Nan, Li & Jin et al. 2011; Li, Nan & Pan 2013). The FAST’s upper flux limits (see below) of detecting these kinds of bursts are also given in Fig. 1, showing that radio afterglows of the above-mentioned three kinds of special bursts would be easily detected by FAST. Interestingly, it is noted that high-redshift bursts, similar to short bursts except GRB 050724, have typical flux density less than 150 $\mu$Jy in radio band. Considering the above situations, we shall simulate theoretical radio afterglows of diffent types of bursts and probe the detectablity of FAST to them subsequently.
![Broad-band radio afterglows of high-redshift bursts (star: GRB 090429B; filled-squares: GRB 080913; filled-circles: GRB 090423; filled-triangles: GRB 050904) in panels (a)-(c), short bursts (filled-triangles: GRB 130603B; filled-diamonds: GRB 050724; empty squares: GRB 051221A) in panels (d)-(f) and SNe-associated bursts (empty circles: GRB 060218) in panels (g)-(i). The dotted and dashed lines respectively represent $3\sigma$ and $1\sigma$ sensitivity limit of FAST in an integral time of 10 minutes. Note that the data of GRB 130603B are taken from Fong et al. (2014) and **all the other radio data are collected from http://heasarc.gsfc.nasa.gov/W3Browse/all/rssgrbag.html (Chandra & Frail 2012) directly.**[]{data-label="Fig1"}](ms1766fig1.eps){width="14.5cm" height="12.5cm"}
Dynamical Model and FAST Sensitivity {#sect:Obs}
====================================
In terms of the generic dynamical model (Huang, Dai & Lu 1999, 2000a, 2000c), the overall evolution of the ejected outflows in either ultra-relativistic or non-relativistic (Newtonian) phase can be generally described as
$$\frac{dR}{dt}=\beta c\gamma(\gamma+\sqrt{\gamma^2-1}),$$
$$\frac{dm}{dR}=2\pi(1-cos\theta)R^{2}nm_p,$$
$$\frac{d\theta}{dt}=\frac{c_s}{R}(\gamma+\sqrt{\gamma^2-1}),$$
$$\frac{d\gamma}{dm}=-\frac{\gamma^2-1}{M_{ej}+\varepsilon m+2(1-\varepsilon)\gamma m},$$
where $R$ is the radial distance measured in the source frame from the initiation point; $m$ is the rest mass of the swept-up circumburst medium; $\theta$ is the half-opening angle of the ejecta; $\gamma$ is the bulk Lorentz factor of the moving material; $t$ is the arrival time of photons measured in the observer frame; $\beta=\sqrt{1-\gamma^{-2}}$, $c$ is the speed of light. $n$ is the number density of Interstellar Medium; $\varepsilon$ is the general radiative efficiency and would evolve from 1 (high radiative case) to 0 (adiabatic case) within several hours after a burst; $M_{ej}$ is the initial rest mass of the ejecta. $c_s$ is the comoving sound speed determined by $c_s^{2}=\hat{\gamma}(\hat{\gamma}-1)(\gamma-1)c^2/[1+\hat{\gamma}(\gamma-1)]$ with the adiabatic index $\hat{\gamma}=4/3$ in the ultra-relativistic limit and $\hat{\gamma}=1$ in the Newtonian limit (Huang et al. 2000b). We further define $\xi_e$ and $\xi_B^2$ as the energy equipartition factors for electrons and the comoving magnetic field, respectively.
Denoting $\Theta$ as the viewing angle between the velocity of ejecta and the line of sight and $\mu=cos\Theta$, in the burst comoving frame, we can obtain synchrotron radiation power of electrons at frequency $\nu '$ as (Rybicki & Lightman 1979) $$P'(\nu')=\frac{\sqrt{3}e^3B'}{m_ec^2}\int_{min(\gamma_{e,min}, \gamma_c)}^{\gamma_{e,max}}(\frac{dN_e^{'}}{d\gamma_e})F(\frac{\nu'}{\nu_c^{'}})d\gamma_e,$$ in which $dN_e^{'}/d\gamma_e\propto(\gamma_e-1)^{-p}$, with the typical value of the electron distribution index $p$ between 2 and 3 (Huang & Cheng 2003); $\nu_c^{'}=3\gamma_e^2eB'(4\pi m_ec)^{-1}$ is the characteristic frequency of electrons with charge $e$; $\gamma_c=6\pi m_ec/(\sigma_T\gamma B'^2t)$ is the typical Lorentz factor of electrons which cool rapidly due to synchrotron radiation, with $\sigma_T$ being the Thompson cross-section; $\gamma_{e,min}=\xi_e(\gamma-1)m_p(p-2)/[m_e(p-1)]+1$ and $\gamma_{e,max}\simeq10^8(B'/1G)^{-1/2}$ are respectively the minimum and the maximum of Lorentz factors of electrons, and $F(x)=x\int_x^{+\infty}K_{5/3}(x')dx'$ of which $K_{5/3}(x')$ is the Bessel function. Owing to cosmological expansion, the observed frequency $\nu$ should be (Wang, Huang & Kong 2009) $$\nu=[\gamma(1-\beta\mu)]^{-1}\nu'/(1+z)=\delta\nu'/(1+z),$$ here $\delta=[\gamma(1-\beta\mu)]^{-1}$ is the Doppler factor. For low-frequency radiation, the effect of synchrotron self-absorption on observation should be considered and hence the observed flux density radiated from a cosmological point source would be (Huang et al. 2000b; Wang, Huang & Kong 2009) $$F_{\nu}(t)=\frac{(1+z)\delta^3}{4\pi D_L^2}f(\tau)P'[ (1+z)\nu/\delta],$$ where $D_L$ is the luminosity distance and $f(\tau)=(1-e^{-\tau_{\nu'}})/\tau_{\nu'}$ is a reduction factor of synchrotron self-absorption with an optical depth $\tau_{\nu'}$. In order to calculate the total observed flux densities, we should integrate Eq. (7) over the equal arrival time surface determined by $$t=(1+z)\int\frac{1-\beta\mu}{\beta c}=const,$$ within the jet boundaries.
Basically, the above model can give a good description for the external shocks and GRB afterglows. For example, the dynamical model has been applied to many GRBs and can well explain the observations (Huang, Cheng & Gao 2006; Kong, Huang & Cheng 2009; Xu, Huang & Lu 2009; Kong, Wong, Huang & Cheng 2010; Xu, Nagataki & Huang 2011; Yu & Huang 2013; Geng, Wu, Huang et al. 2013). Here in Fig. 2, we display some exemplar afterglow light curves at X-ray band (0.3-10 keV) and optical R-band calculated by using this model. It can be seen that this model basically matches the general behaviors of multi-band afterglows (e.g., Zhang et al. 2006; Nousek et al. 2006; Zhang, Liang & Zhang 2007; Liang et al. 2008; Troja et al. 2007). Besides, afterglows from the extreme GRBs such as high luminosity, low luminosity and failed bursts are also predicted to shed new light on the future observations in the era of FAST.
![Plots of afterglow light curves for the standard (thick solid line), high luminosity (dashed line), low luminosity (dash-dot-dotted line) and failed (thin solid line) GRBs at X-ray energy band (0.3-10 keV) in upper panel and optical R band in lower panel, respectively, for a typical redshift z=0.5 and density n=1 $cm^{-3}$. See the text for more details.[]{data-label="Fig2"}](ms1766fig2.eps){width="12cm" height="10cm"}
In order to explore the detectability of FAST, we here adopt the following sensitivity to estimate the RMS noise in a position-switching observing mode as
$$F_{lim}=\frac{(S/N)2 K_b T_{sys}}{A_e \sqrt{\Delta\tau \Delta \nu}}\simeq (12\mu Jy)(\frac{0.77\times10^3 m^2/K}{A_{e}/T_{sys}})(\frac{S/N}{3})(\frac{1 hour}{\Delta \tau})^{1/2}(\frac{100 MHz}{\Delta \nu})^{1/2},$$
where $A_e$ is the effective area defined by $A_e=\eta_A A_g$ with an aperture efficiency $\eta_A=0.65$ (Yue et al. 2013), $\Delta\tau$ and $\Delta\nu$ are respectively the bandwidth and the integral time, the illuminated geometric area is $A_g=\pi\times(300/2)^2$ m$^2$, and the system temperature is $T_{sys}$ for the FAST. Table \[tab1\] gives the values of sensitivity for different frequencies of FAST. $S/N$ stands for the signal-to-noise ratio and is generally taken as no less than 3. More detailed properties of FAST can be found in Nan et al. (2011). Note that we have calculated FAST’s sensitivities by assuming that its temperature template is similar to that of VLA’s receivers. We acknowledge here the difficulty of the bandpass calibration for a single dish, which is more difficult than that for interferometers. The detection of faint, broadband continuum signal, however, has been achieved by single dishes before. FAST is also looking into new calibration technologies, such as injecting an artificial flat spectrum signal into the optical path.
[|c|c|c|c|c|c|c|c|c|c|c|c|]{} No.&Bands$^{\dag}$&$\nu_c^{\dag}$& $\Delta\nu^{\dag}$& $T_{sys}^{\dag}$ & $F_{lim,1}$& $F_{lim,2}$& $F_{lim,3}$& $T_{sys}'$& $F_{lim,1}'$& $F_{lim,2}'$& $F_{lim,3}'$\
&$GHz$&$GHz$&$MHz$&$K$&$\mu Jy$&$\mu Jy$&$\mu Jy$&$K$&$\mu Jy$&$\mu Jy$&$\mu Jy$\
1&0.07-0.14&0.1&70&1000&293.6&168.7&119.4&140&41.0&23.5&16.9\
2&0.14-0.28&0.2&140&400&82.5&47.6&33.8&140&28.8&16.6&11.9\
3&0.28-0.56&0.4&280&150&21.9&12.7&8.9&130&19.1&11.1&7.8\
4&0.56-1.02&0.8&560&60&6.1&3.6&2.5&97&9.9&5.8&4.2\
5&0.32-0.334&0.328&14&200&130.7&75.3&53.5&125&81.7&47.1&33.2\
6&0.55-0.64&0.6&90&60&15.5&8.9&6.4&112&28.9&16.6&11.9\
7&1.15-1.72&1.45&550&25&2.5&1.4&1.1&61&6.3&3.6&2.5\
8&1.23-1.53&1.38&300&25&3.6&1.9&1.4&60&8.6&5.0&3.6\
9&2.00-3.00&2.5&1000&25&1.9&1.1&0.8&30&2.2&1.4&0.8\
Results {#sect:Results}
=======
To determine the sensitivity of FAST to GRB radio afterglows, we have chosen the standard, high-luminosity, low-luminosity and failed GRBs for a comparative study. For convenience, let us define the initial values or parameters of these bursts as follows. (1) Standard GRBs: initial isotropic energy $E_0=10^{52}$ ergs and $\gamma_0=300$; (2) High luminosity GRBs: $E_0=10^{54}$ ergs and $\gamma_0=300$; (3) Low luminosity GRBs: $E_0=10^{49}$ ergs and $\gamma_0=300$; (4) Failed GRBs: $E_0=10^{52}$ ergs and $\gamma_0=30$. Apart from these differences, we assume all the bursts have the same values for other parameters, namely $n=1$ cm$^{-3}$, $p=2.5$, $\xi_e=0.1$, $\xi_B^2=0.001$, $\theta=0.1$ and $\Theta=0$ throughout this paper. Also, an assumption of the radiative efficiency $\varepsilon=0$ has been made because the relativistic fireball becomes fully adiabatic in about several hours after a burst. Light-curves of radio afterglows at different redshifts, i.e. $z$=0.5, 1, 5, 10, 15, have been derived in order to study FAST’s capability of probing bursts in the early universe.
Light-Curves of Radio Afterglows in the FAST Window
---------------------------------------------------
Figs. 3-7 show that almost all radio afterglow light curves, except the failed GRB afterglows have the same general characteristics of slow rise and fast decay (SRFD) although they may have distinct physical origins. We also find a common and interesting phenomenon that with the increase of observing frequency, the light curve peaks earlier, and the peak flux is also higher. At the same time, radio afterglows of the standard and the failed GRBs have almost the same peak time and the same peak flux density due to their similar kinetic energies. The two kinds of bursts decay congruously after their peak times. At higher frequencies and larger redshifts, the peak flux densities of failed GRBs are slightly weaker than those of the standard ones. Both the peak flux density and the peak time of radio afterglows sensitively depend on the initial energy injection. For the failed GRBs, the rising part is largely affected by the small initial Lorentz factors of the ejecta from the central engine. In any cases, we notice that radio afterglows of the low luminosity GRBs have the lowest brightness except that they are stronger than those of the failed GRBs at early stage of less than 1hr. The intersection point would be postponed when the observing frequency is relatively lower for a farther burst. Note that it seems unlikely for FAST to detect any kinds of radio afterglow emission at the extremely lower frequency of $\leq$ 0.1 GHz. For bursts at the same distance or redshift, their radio flux density in higher frequency is usually stronger than that in lower frequency when they are observed at the same time and could be detected at a very early stage. Furthermore, it is found that the radio flux densities are relatively insensitive to the redshift as seen in Figs 5-7, which is consistent with previous investigations (e.g. Ciardi & Loeb 2000; Gou et al. 2004; Frail et al. 2006; Chandra & Frail 2012).
### Standard GRBs
The thick solid lines in Figs. 3-7 denote the case of a standard fireball in different energy bands and redshifts. For the redshift $z=0.5$, the peaked radio emissions at $\nu>$ 0.4 GHz can be easily detected by FAST. With the increase of observing frequency, the bursts gradually brighten. FAST can even detect very early radio afterglows in the prompt phase for a time of less than 10 second at 2.5 GHz. It is interesting that in high frequency bands, the radio afterglow can typically be observable for $\sim115$ days. The peak flux densities at 1.4 GHz and 2.5 GHz can reach 70 $\mu$Jy and 200 $\mu$Jy, respectively. For the reshift $z=1$, FAST can detect the radio emission in the frequency band of $\nu>$0.6 GHz, especially $\nu=$2.5 GHz from 20 seconds to 62 days since a burst for 1 hour integration time. The earliest detection time may start as early as 10 seconds. For the redshift $z=5$, radio afterglows under 0.8 GHz are undetectable for FAST. The peak flux densities at 0.8 GHz and 2.5 GHz can respectively reach 3 $\mu$Jy and 6 $\mu$Jy, which will be observed up to 52 days from the earliest starting time of 30 minutes. For much higher redshifts of $z=10$ and 15, the radio flux densities have a peak value of $\sim2\mu$Jy and only radio afterglows above 1.4 GHz can be marginally detected. Note that the GRB radio afterglows in the standard case usually peak at 10-100 days, which is well consistent with the observations described in Chandra & Frail (2012).
### High Luminosity GRBs
The high luminosity GRBs marked with thick dashed lines in Figs. 3-7 are driven by the largest energy ejection and their radio afterglows naturally hold the strongest brightness when they are located at the same distance. For nearby bursts with a redshift of $z=0.5$, FAST has the capability of detecting all radio afterglows at a frequency of no less than 100 MHz. The typical peak flux densities are 100 $\mu$Jy at 0.2 GHz and $7\times10^3\mu$Jy at 2.5 GHz respectively. The former can be detected from 3 days to 3.5 years and the later can be observed from several seconds to 9 years after the burst. For a redshift $z=1$, FAST may detect the weak peak flux in the post-burst time of 2 hours to 1200 days in channel 3 at 0.4 GHz and 30 seconds to 6 years in a frequency of $\nu=1.4$ GHz. The radio afterglow flux densities for redshift higher than 5 are undetectable under $\nu\simeq$300 MHz but can be safely detected from 1 day to 2.5 years at 0.4 GHz and from several tens seconds to 4 years in high frequency bands. The peak radio flux densities in 0.4 -2.5 GHz for $z=15$ can reach 50-200 $\mu$Jy which is much higher than the threshold of FAST.
### Low Luminosity GRBs
In contrast, the low luminosity GRBs denoted by dash-dot-dot lines in Figs. 3-7 consist of some special bursts with lower energy input. This leads to smaller kinetic energy of outflows and then much weaker radio afterglows peaking at earlier time around 1 day after the GRB trigger. The weakest radio brightness is only $\sim10^{-4}\mu$Jy at 100 MHz for redshift $z=15$. The strongest flux density can approach $0.3\mu$Jy at 2.5 GHz for $z=0.5$ and is just close to the detection limit of FAST for 1 hour integration time. It is obvious that FAST can hardly detect radio afterglows from these low luminosity GRBs. In the future, if FAST’s passband can be expanded to 8 GHz, then low luminosity GRBs may also be detected. We can also consider to increase the integration time to tell apart the radio afterglow from a low level background.
### Failed GRBs
Such kinds of bursts are thought to be produced by an isotropic fireball with kinetic energy like that of the standard bursts but with much lower Lorentz factors of several tens. As shown in Fig. 3-7, radio afterglows of failed GRBs and the standard ones nearly peak simultaneously and they decay with time in the same way after the peak time. Compared with standard bursts, radio afterglows of failed bursts can be detected mainly at later times. For $z=0.5$, FAST can hardly detect them at a frequency lower than 0.6 GHz in 3-$\sigma$ levels with 1 hour integration time. But, we can detect the radio afterglows up to 85 days at higher frequencies above $0.8$ GHz. The afterglow is observable from a time of 1000 minutes, 70 minutes, 20 minutes, 15 minutes and 8 minutes for 0.6 GHz, 0.8 GHz, 1.38 GHz, 1.45 GHz and 2.5 GHz after the burst, respectively. For the redshift $z$=1, only radio emissions at a frequency larger than 0.6 GHz are detectable from 1 day to 60 days. Radio afterglows at a redshift $z=$5 for a frequency larger than 1.4 GHz can be observed. The radio flux densities at 0.8 GHz, 1.38 GHz, 1.45 GHz and 2.5 GHz are in a lower level of 1-2 $\mu$Jy. For $z>10$, radio afterglows of the failed GRBs become very difficult to be detected by FAST with its current sensitivity of 1 hour integration time.
![Radio flux density of GRBs at a redshift $z=0.5$ versus observation time $t$ in the observer frame at various frequencies within the FAST’s window. The radio light curves of standard, high-luminosity, low-luminosity and failed GRBs are marked with thick solid, dashed, dash-dot-dot and thin solid lines, respectively, and have been symbolized in panel 1. Three horizontal dotted lines from upper to bottom represent 1$\sigma$ ($S/N$=1) limiting flux density of the FAST for 10, 30 and 60 minutes integration time correspondingly. See the text for details.[]{data-label="Fig3"}](ms1766fig3.eps){width="16cm" height="14cm"}
![Radio flux density of GRBs at a redshift $z=1$ versus observation time $t$ in the observer frame at various frequencies within the FAST’s window. Symbols are the same as in Fig. 1.[]{data-label="Fig4"}](ms1766fig4.eps){width="14.5cm" height="12cm"}
![Radio flux density of GRBs at a redshift $z=5$ versus observation time $t$ in the observer frame at various frequencies within the FAST’s window. Symbols are the same as in Fig. 1.[]{data-label="Fig5"}](ms1766fig5.eps){width="14.5cm" height="12cm"}
![Radio flux density of GRBs at a redshift $z=10$ versus observation time $t$ in the observer frame at various frequencies within the FAST’s window. Symbols are the same as in Fig. 1.[]{data-label="Fig6"}](ms1766fig6.eps){width="14.5cm" height="12cm"}
![Radio flux density of GRBs at a redshift $z=15$ versus observation time $t$ in the observer frame at various frequencies within the FAST’s window. Symbols are the same as in Fig. 1.[]{data-label="Fig7"}](ms1766fig7.eps){width="14.5cm" height="12cm"}
Peak Spectra of Radio Afterglows in the FAST’s Window
-----------------------------------------------------
To investigate the sensitivity of FAST’s receiver at different frequencies, we plot the peak frequency against the peak flux density for the above-mentioned four types of bursts at different redshifts in Fig. 8. The data utilized here are extracted from the above calculations. Radio emission would be steeply cut off by the self-absorption effect at lower frequencies below several GHz. This frequency range covers the FAST’s frequency bands of 70 MHZ-3 GHz, which causes the radio flux density $F$ to be a power-law function of $F\propto\nu^{2}$ (e.g. Sari, Piran & Narayan 1998; Wu et al. 2005) if the observation frequency $\nu$ is less than the synchrotron self-absorption frequency $\nu_a$. Take the systematic temperature as $T_{sys}$=20 K, the bandwidth as $\Delta \nu=100$ MHz and the aperture efficiency as $\eta_A=0.65$ for FAST, we have plotted the 1$\sigma$ and 3$\sigma$ limiting flux density for a 10-minute integration time for comparison (see Fig. 8).
![Peak radio flux density versus peak frequency for the failed, low luminosity, high luminosity and standard GRBs at five representative redshifts. The two horizontal lines from top to bottom respectively stand for 3$\sigma$ and 1$\sigma$ limiting flux density of FAST in a 10-minute integration time. The different symbols for various redshifts are given in the first panel. Note that a systematic temperature of $T_{sys}$=20K and a bandwidth of $\Delta\nu$=100 MHz as typical parameters for FAST have been used for our estimation here. See the text for details.[]{data-label="Fig8"}](ms1766fig8.eps){width="16cm" height="13cm"}
We can see from Panel (a) in Fig. 8 that radio afterglows of failed bursts are detectable only for nearby sources with redshifts less than 5 and at relatively high frequencies. Panel (b) shows that the radio peaks of low luminosity bursts are far below the detection limits in all our cases and thus are difficult to detect by FAST. On the contrary, we see from Panel (c) that the peak flux densities of high luminosity GRBs are always above the detection limits, making them the best candidates for monitoring at almost any redshifts and frequencies except very high redshift at lower frequency of $\nu<$ 200 MHz. Panel (d) displays the radio peaks of the standard GRBs. They can be observed by FAST up to very high redshift of $z=$10 at higher frequency, very different from the failed bursts with relatively lower flux density at higher frequency, although they exhibit similar observational properties at lower redshifts in all the frequency bands of FAST. Another interesting phenomenon is that radio spectrum evolves with the cosmological redshift in the observer frame. In addition, the spectral shape of low luminosity bursts clearly differs from that of other classes of GRBs which hints they may be of distinct physical origin.
Recent numerical simulations by a few other authors have shown that sideways expansion, edge effect, **even off-axis effect (Zhang et al. 2014a) of jets**, together with its microphysical process from ultra-relativistic to non-relativistic phase, may play an in-negligible role on the expected light curves and spectra of afterglows (van Eerten, Leventis, Meliani et al. 2010; van Eerten & MacFadyen 2012). However, the results given by the above updated blastwave models do not significantly differ from those predicted by the generic afterglow model utilized in this paper, especially for the late-time radio afterglows, of which the peak flux density, the peak time and the post-peak decay are much comparable as a whole. Furthermore, it is worth pointing out that our numerical radio afterglows at 1.43 GHz is fairly consistent with those calculated with external shock model by Chandra & Frail (2012).
Discussion and Conclusion {#sect:discussion}
=========================
It is interesting to notice that the fractions of high, low and medium isotropic energy GRBs in the pre-Swift era are 4%, 16% and 80% respectively (Friedman & Bloom 2005). As Swift/BAT is more sensitive to long-soft bursts than pre-Swift missions, the percentages are 32%, 3% and 65% respectively for the high, low and medium isotropic energy GRBs in the Swift era. Obviously, the fraction of high luminosity GRBs detected by Swift/BAT is much larger than that of pre-Swift detectors, while the fraction of low luminosity GRBs is just the opposite. The on-going Swift satellite favors long bursts with higher redshifts on average. This naturally makes more and more super-long bursts observed with duration up to $10^4-10^6$s (Zhang et al. 2014b). How the radio afterglows of these GRBs will behave is a very important problem. We believe that their radio afterglows should be detectable to FAST. It will promote the study of early cosmology especially with the help of future FAST observations. This advantage is mainly attributed to the fact that long GRBs with high-redshifts are generally thought to be produced by higher luminosity sources, e.g. collapse of very massive stars.
Rhoads (1997) had pointed out that $\gamma$-ray radiation from some jetted GRBs can not be observed owing to relativistic beaming effects, but the corresponding late time afterglow emission is less beamed and can safely reach us. These lower frequency radiations are called as orphan afterglows, since they are not associated with any detectable GRBs. However, Huang, Dai & Lu (2002) pointed out another possibility that orphan afterglows can also be produced by failed GRBs (or a dirty fireballs). They argued that the number of failed GRBs may be much larger than that of normal bursts. It is very difficult to distinguish the two different origins of orphan afterglows. The initial Lorentz factor of ejecta is a key parameter that makes the failed GRBs be different from other kinds of bursts in principle. Compactness limit was thought to be a robust method for estimating the initial Lorentz factor (Zou, Fan & Piran 2011). Unfortunately, current Lorentz factor estimates are still controversial although extensive attempts had been made both theoretically and observationally (e.g. Zhang et al. 2007, 2011; Li 2010; Liang et al. 2010, 2011; Zou & Piran 2010; Zou, Fan & Piran 2011; Zhao, Li & Bai 2011; Chang et al. 2012; Hascoet et al. 2013). It is helpful to discriminate between failed and standard GRBs from their radio afterglows because such low frequency emission can be observed for quite a long time (Huang, Dai & Lu 2002). FAST may make important contribution in the aspect.
Note that contributions of host galaxies to radio fluxes have been neglected in our numerical calculations for simplification. This effect will add difficulties for afterglow observations with FAST and the detection rate of radio afterglows may be less optimistic than our current study (Li et al. 2014). On the other hand, the system temperature $T_{sys}$ is sensitively dependent on a variety of realistic factors and will be measured only after the radio telescope is built. As a single dish antenna, FAST will operate in a lower frequency range, i.e. from 70 MHz to 3 GHz in its first phase, and may extend to 8GHz in its second phase. The Low Frequency Array (LOFAR) covers relatively lower observation frequencies of 10-240 MHz (van Haarlem et al. 2013), partly overlapped with that of FAST. Thompson et al. (2007) gave the theoretical equation, $T_{sky}=60\lambda^{2.55}$ K, as the estimation of $T_{sys}$. It is illustrated that the LOFAR will be sky-noise dominated under 65 MHz, close to the FAST’s lower frequency limit of 70 MHz. In contrast, at a frequency of 200 MHz, FAST’s detection sensitivity is about one order of magnitude higher than LOFAR. In addition, we stress that the limiting sensitivity of FAST at $\nu >0.4$ GHz for 1-hour integration time is under 10 $\mu$Jy, which is much better than EVLA if FAST’s instrumental noise is ideally controlled to be below the values adopted in this work. In this case, we deduce that FAST will be the most powerful new-generation radio telescope for studying radio afterglows. If the international Very-Long-Baseline Interferometry operation is applied, FAST would have more powerful capability of detecting radio emission from low luminosity GRBs or very distant (Chandra et al. 2010) GRBs, which should largely increase the detection rate of radio afterglows in the near future. Considering that GRBs with $E_{iso} = 10^{51} - 10^{53}$ erg at $z\geq20$ will be observed by the next generation instruments in near infrared and radio bands (Mesler et al. 2014), our results would be very valuable for making further survey plans of GRB radio afterglows with the upcoming radio telescopes such as FAST, SKA, and so on.
In conclusion, we summarize our major conclusions in the following.
1. We presented a quantitative prediction for the detectability of GRB radio afterglows with FAST, based on the generic dynamical afterglow model. Our calculations are carried out for four kinds of bursts, i.e. failed GRBs, low luminosity GRBs, high luminosity GRBs and standard GRBs. We found that the radio afterglow detection rate sensitively depends on the model parameters of Lorentz factor and isotropic energy.
2. We predicted that radio afterglows of all the above types of bursts except the low luminosity ones should be detected by FAST in wide ranges of time, frequency and redshift. The detectabilities descend in order for high luminosity, standard, failed and low luminosity GRBs.
3. FAST is able to detect radio afterglows of GRBs at redshift up to $z\sim$10 or even more, which will be very helpful for the studies of GRB event rate and GRB cosmology.
4. Radio afterglows of low luminosity GRBs will be detectable in the 2nd phase of FAST or if the integration time is extended enough.
We thank the anonymous referee for helpful comments and suggestions. We appreciate D. A. Frail and P. Chandra for delivering us their invaluable observation data of radio afterglows. We acknowledge R. D. Nan, B. Zhang, X. F. Wu, Y. Z. Fan, C. S. Choi and H. Y. Chang for helpful discussions. This work was supported by the National Basic Research Program of China (973 Programs, Grant No. 2014CB845800, 2012CB821800), the National Natural Science Foundation of China (Grant No. 11033002; 11263002; 11311140248) and Guizhou Natural Science Foundations (20134021; 20134005). SWK acknowledges support by China Postdoctoral science foundation under grant 2012M520382.
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Takeshi <span style="font-variant:small-caps;">Hirama</span> and Koji <span style="font-variant:small-caps;">Hukushima</span>'
title: ' On-line Learning of an Unlearnable True Teacher through Mobile Ensemble Teachers '
---
Introduction
============
Learning is an inference problem of inhered rules from a given set of examples which consist of input data and corresponding output data generated by the rules. In practice, the examples are often supplied inexhaustibly and then the learning must proceed by using each example just once. Such learning is called on-line learning[@2; @3; @1]. On the contrary, the learning in which all the examples are presented repeatedly at anytime is called off-line or batch learning.
The on-line learning as well as the off-line one has been extensively studied by using statistical-mechanical methods so far and many extensions of the on-line learning scheme have been made in order to improve a generalization performance.[@1; @2] Recently, Miyoshi and Okada [@6] and Urakami, Miyoshi and Okada [@4] analyzed the generalization performance of a student supervised by a moving teacher that goes around a fixed true teacher in a framework of the on-line learning using the statistical mechanical method. In their model, the student is not directly given the outputs by the true teacher. The moving teacher learns from the true teacher and provides its output to the student. In this sense, the model is a kind of hierarchical learning. In ref. , the true teacher is a non-monotonic perceptron, while the moving teacher and the student are simple perceptron using perceptron learning, which could not infer the true teacher completely in principle. The theoretical bound of the generalization error of a simple perceptron learner has been obtained.[@InoueNishimori] In that case, the moving teacher goes around the true teacher with a fixed distance between them. Interestingly, it turned out that when the student’s learning rate is relatively small, the student’s generalization error can temporally become smaller than that of the moving teacher, even if the student only uses the examples from the moving teacher.
Subsequently, Miyoshi and Okada [@7] and Utsumi, Miyoshi and Okada [@5] analyzed the generalization performance of an extended model of the on-line learning with multiple teachers, which would be called ensemble-teachers learning model. This model is also regarded as an extension of the ensemble learning[@Urbanczik; @Miyoshi05] because the ensemble teachers and the student in the ensemble-teachers model can be interpreted as the ensemble students and their integrating mechanism, respectively. In particular, ref. discussed the model in which the true teacher, the ensemble teachers and the student are all simple perceptrons. In this model the true teacher and the ensemble teachers are fixed. The student adopts the Hebbian learning or the perceptron learning as a learning rule and uses examples from the ensemble teachers in turn or randomly. As a result, it was clarified that the Hebbian learning and the perceptron learning show qualitatively different behavior from each other. In the Hebbian learning, the generalization error monotonically decreases during the learning process and its asymptotic value is independent of the learning rate. The asymptotic value is reduced as the number of the ensemble teachers increases since the ensemble teachers have more variety in their representations. On the other hand, in the perceptron learning, the generalization error shows non-monotonic behavior and exhibits a minimum at a certain step in the learning. The minimum value of the generalization error decreases as the learning rate decreases and the total number of the teachers increases.
In ref. and ref. , it was shown that the generalization error of a student could be smaller than that of a moving teacher or fixed ensemble teachers. A comparison between the generalization performance with a fixed teacher and that with a mobile teacher, however, has not been made directly. Furthermore, in the on-line learning with the ensemble teachers it is not trivial that either the mobility or the multiplicity of the ensemble teachers is effective for the learning performance of the student. In this paper, we study the on-line learning for the ensemble teachers which can move around a true teacher. We discuss a model in which the fixed true teacher is non-monotonic perceptron and the ensemble moving teachers and the student are a simple perceptron. This is a generalized version of the model studied in ref. . Adopting the perceptron learning as a learning rule for the ensemble teachers, they go around the true teacher with constant order parameters in the steady state. Then we analyze the generalization performance of the student which learns from the mobile ensemble teachers using the Hebbian and the perceptron rules. We also study the model with the ensemble teachers fixed in their steady state. It is thus clarified that the movement of the ensemble teachers , in comparison with the fixed ensemble case, significantly improves the generalization performance of the student as a transient state in the learning process.
The paper is organized as follows: In sec. $2$, we introduce the model with the ensemble moving teachers going around the unlearnable true teacher. In sec. $3$, based on the statistical-mechanical idea, we theoretically derive the ordinal differential equations of order parameters and an explicit formula of the generalization error of our model in terms of the order parameters. In sec. $4$, we show the theoretical and numerical results of the generalization performance of the student with the Hebbian and perceptron rules. The last section is devoted to our conclusion. In the appendixes, the derivations of the differential equations discussed in sec. 3 are presented in detail.
Model
=====
In this paper, we consider a true teacher, $K$ ensemble moving teachers and a student, whose connection weights are expressed as $N$ dimensional vectors, , $_{k}$ and , respectively, with $k=1,2,\cdots,K$. For simplicity, each component $A_{i}$ of with $i=1,\cdots, N$ is assumed to be drawn from $\mathcal{N}(0,1)$ independently and fixed, where $\mathcal{N}(m,\sigma^2)$ denotes the Gaussian distribution with $m$ and $\sigma^2$ being a mean and variance, respectively. As an initial condition of the learning process, each of the components $B_{ki}^{0}$ and $J_{i}^{0}$ of $_{k}^0$, $^0$ are also assumed to be drawn from $\mathcal{N}(0,1)$ independently. Input $\textrm{\boldmath $x$}$ is also the $N$-dimensional vector and the component $x_{i}$ follows from $\mathcal{N}(0,1/N)$ independently. Thus, we have $$\left\langle A_i \right\rangle = \left\langle B_{ki}^0 \right\rangle =
\left\langle J_{i}^{0} \right\rangle = \left\langle x_i \right\rangle = 0,$$ $$\left\langle (A_i)^2 \right\rangle =
\left\langle \left(B_{ki}^0\right)^2 \right\rangle =
\left\langle \left(J_{i}^{0}\right)^2 \right\rangle =1,$$ and $$\left\langle (x_i)^2 \right\rangle =\frac{1}{N},$$ where $\langle \cdots \rangle$ denotes an average over the Gaussian distribution.
In the statistical mechanics of the learning,[@2; @3] we are interested in asymptotic behavior of ${\textrm{\boldmath $A$}}$, and in a thermodynamics limit $N\to \infty$. Then, one finds that the norms of the vectors are $$\Vert \textrm{\boldmath $A$}\Vert=\sqrt{N}, \,\Vert \textrm{\boldmath $B$}_{k}^0 \Vert =\sqrt{N}, \, \Vert \textrm{\boldmath $J$}^0 \Vert =\sqrt{N}, \, \Vert \textrm{\boldmath $x$} \Vert=1.$$ The norms, $\Vert \textrm{\boldmath $B$}_k \Vert$ and $\Vert \textrm{\boldmath $J$}
\Vert$, of the ensemble moving teachers and the student change during the learning process from their initial values. The normalized length of these vectors is introduced as $l_{B_k}=\Vert
\textrm{\boldmath $B$ }_k\Vert/\Vert \textrm{\boldmath$B$}_k^{0}\Vert$ for the ensemble teachers and $l_{J}=\Vert\textrm{\boldmath$J$}\Vert/\Vert\textrm{\boldmath$J$}^{0}\Vert$ for the student. In the thermodynamic limit, the direction cosines between these vectors are a relevant extensive quantity, denoted for and $_{k}$, and , $_{k}$ and $_{k'}$, and $_{k}$ and respectively as $$\begin{aligned}
R_{B_k}&=\frac{\textrm{\boldmath $A$}\cdot\textrm{\boldmath $B$}_k}{\Vert \textrm{\boldmath $A$}\Vert \Vert\textrm{\boldmath $B$}_k \Vert}, \, R_{J}=\frac{\textrm{\boldmath $A$}\cdot\textrm{\boldmath $J$}}{\Vert \textrm{\boldmath $A$}\Vert \Vert\textrm{\boldmath $J$} \Vert}, \\
q_{kk'}&=\frac{\textrm{\boldmath $B$}_k\cdot\textrm{\boldmath
$B$}_k'}{\Vert \textrm{\boldmath $B$}_k\Vert \Vert\textrm{\boldmath
$B$}_k' \Vert}, \, R_{B_{k}J}=\frac{\textrm{\boldmath
$B$}_{k}\cdot\textrm{\boldmath $J$}}{\Vert \textrm{\boldmath
$B$}_{k}\Vert \Vert\textrm{\boldmath $J$} \Vert}. \end{aligned}$$
In the present study, we assume that the true teacher is a non-monotonic perceptron and the ensemble moving teachers and the student are a simple perceptron. The output for a given input of the true teacher is defined by a non-monotonic function $$o=\text{sgn}\left(\left(\textrm{\boldmath $A$}\cdot\textrm{\boldmath $x$}-a\right)\textrm{\boldmath $A$}\cdot\textrm{\boldmath $x$}\left(\textrm{\boldmath $A$}\cdot\textrm{\boldmath $x$}+a\right)\right)$$ with a fixed threshold $a$, while those of the ensemble moving teachers and the student are simply given by sgn$\left(\textrm{\boldmath $B$}_k\cdot\textrm{\boldmath $x$}\right)$ and sgn$\left(\textrm{\boldmath $J$}\cdot\textrm{\boldmath $x$}\right)$, respectively. Here, sgn$(\cdot)$ is the sign function defined as $$\text{sgn}(s)=\left\{
\begin{array}{ll}
+1, &\quad s\geq 0, \\
-1, &\quad s < 0.
\end{array}
\right.$$ A measure of dissimilarity between the true teacher and the ensemble teachers or the student is defined by using their outputs as $$\epsilon_{B_{k}}\equiv\Theta\left(-o\cdot\text{sgn}\left(\textrm{\boldmath $B$}_k\cdot\textrm{\boldmath $x$}\right)\right)
\label{eqn:errorB}$$ for $k$th ensemble teacher and $$\epsilon_{J}\equiv\Theta\left(-o\cdot\text{sgn}\left(\textrm{\boldmath $J$}\cdot\textrm{\boldmath $x$}\right)\right)
\label{eqn:errorJ}$$ for the student, where $\Theta(\cdot)$ is the step function defined as $$\Theta(s)=\left\{
\begin{array}{ll}
+1, &\quad s\geq 0, \\
\,\,\,\,0, &\quad s < 0.
\end{array}
\right.$$ One of the main purposes of the statistical learning theory is to obtain theoretically the generalization errors $\epsilon_{B_{k}}^{g}$ and $\epsilon_{J}^{g}$, which are defined as the average of the errors, $\epsilon_{B_{k}}$ and $\epsilon_{J}$ over the whole set of possible inputs $\textrm{\boldmath $x$}$. Since the input appears in Eq. (\[eqn:errorB\]) and Eq. (\[eqn:errorJ\]) as inner products ${\textrm{\boldmath $A$}}\cdots{\textrm{\boldmath $x$}}$, ${\textrm{\boldmath $B$}}_k\cdot{\textrm{\boldmath $x$}}$ and ${\textrm{\boldmath $J$}}\cdot{\textrm{\boldmath $x$}}$, the average over Gaussian vector ${\textrm{\boldmath $x$}}$ could be reduced to an average over correlated Gaussian variables. When one defines a set of variables, $v$, $v_{B_{k}}$ and $u$ as $$\begin{aligned}
v&= \textrm{\boldmath $A$}\cdot \textrm{\boldmath $x$},\\
v_{B_{k}}l_{B_{k}}&= \textrm{\boldmath $B$}_{k}\cdot \textrm{\boldmath $x$},\\
ul_{J}&= \textrm{\boldmath $J$}\cdot \textrm{\boldmath $x$},\end{aligned}$$ they obey the multiple Gaussian distribution $$P(v, \{v_{B_{k}}\}, u) =\frac{1}{(2\pi)^{(K+2)/2}\vert \Sigma \vert
^{1/2}}\exp{\left(-\frac{(v, \{v_{B_{k}}\}, u)\Sigma^{-1}(v,
\{v_{B_{k}}\}, u)^{T}}{2}\right)},
\label{eqn:multiG}$$ with zero means and the covariance matrix $\Sigma$ $$\begin{aligned}
\Sigma &=\left(
\begin{array}{cccccc}
1 & R_{B_{1}}& R_{B_{2}}& \cdots & R_{B_{K}}& R_{J} \\
R_{B_{1}}& 1& q_{1,2}&\cdots & q_{1,K}& R_{B_{1}J} \\
R_{B_{2}}& q_{2,1}& 1& \ddots & \vdots& \vdots \\
\vdots& \vdots& \ddots & \ddots& q_{K-1,K}& R_{B_{K-1}J} \\
R_{B_{K}}& q_{K,1}& \cdots& q_{K,K-1}& 1& R_{B_{K}J}\\
R_{J}&R_{B_{1}J} &\cdots &R_{B_{K-1}J} & R_{B_{K}J}& 1
\end{array}
\right).\label{covariance}\end{aligned}$$ Evaluating the correlated Gaussian integrations, the generalization errors $\epsilon_{B_{k}}^{g}$ and $\epsilon_{J}^{g}$ are obtained as $$\begin{aligned}
\epsilon_{B_{k}}^{g}&=2\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv H\left(-\frac{R_{B_{k}}v}{\sqrt{1-R_{B_{k}}^2}}\right), \label{gerrorAB}\end{aligned}$$ and $$\begin{aligned}
\epsilon_{J}^{g}&=2\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv
H\left(-\frac{R_{J}v}{\sqrt{1-R_{J}^2}}\right), \label{gerrorAJ}\end{aligned}$$ where $Ds$ is the Gaussian measure defined as $$Ds\equiv \frac{ds}{\sqrt{2\pi}}\exp{\left(-\frac{s^2}{2}\right)},$$ and $H(\cdot)$ is the error function defined as $$H(s)\equiv \int_{s}^{\infty} Dx.$$ It should be noted that the dynamical effect of the generalization errors appears only through $R_{B_{k}}$ and $R_{J}$. This implies that the generalization errors have a fundamental minimum as a function of $R_{B_{k}}$ and $R_{J}$, irrespective of the matter if the values of $R_{B_{k}}$ and $R_{J}$ which give the minimum value of the generalization error appear in a particular chosen learning rule of the student and the ensemble teachers. An efficient learning rule might realize the fundamental minimum for a given learning model.
Let us defined the update rule in the on-line learning. The ensemble moving teachers $_{k}$ are updated from the current state $_k^{m'}$ using an input $\textrm{\boldmath $x$}$ and output of the true teacher for the input $^{m'}$, independently as $$\begin{aligned}
\textrm{\boldmath $B$}_{k}^{m'+1}= \textrm{\boldmath $B$}_{k}^{m'}+f_{k}^{m'}
(\textrm{\boldmath $x$}^{m'}, \textrm{\boldmath $B$}_{k}^{m'},
o^{m'})\textrm{\boldmath $x$}^{m'},
\label{eqn:updateB}\end{aligned}$$ where $f^{m'}$ is an update function of the ensemble moving teachers and $m'$ denotes the time step of the ensemble moving teachers. In particular, we choose the perceptron learning for the update function $f_k$, which is given by $$\begin{aligned}
f_{k}^{m'}=\eta_{B}\Theta\left(-v_{B_{k}}^{m'}o^{m'}\right)o^{m'}. $$ Here, $\eta_{B}$ is the learning rate of the ensemble moving teachers. In our analysis, the learning rate $\eta_{B}$ is independent of the teachers and is fixed during the learning process. After a sufficient long learning process using the perceptron rule, the ensemble moving teachers reach steady state with $R_{B_{k}}$, $l_{B_{k}}$ and $q_{kk'}$ fixed. In the present study, we focus our attention to dynamical effect of the ensemble teachers for the learning performance of the student. In order to separate off a transient effect of the ensemble teachers, the student learns from the ensemble teachers in the steady state. The student is updated using an input and an output of one of the $K$ ensemble moving teachers $_{k}$ chosen randomly. The explicit recursion formula for $\textrm{\boldmath $J$}^{m}$ with $m$ being the time step of the student is given by $$\begin{aligned}
\textrm{\boldmath $J$}^{m+1}= \textrm{\boldmath $J$}^{m}+g_{k}^{m}
(\textrm{\boldmath $x$}^{m}, \textrm{\boldmath $J$}^{m},
\text{sgn}(v_{B_{k}}l_{B_{k}}))\textrm{\boldmath $x$}^{m},
\label{eqn:updateJ}\end{aligned}$$ where $g_{k}^{m}$ is an update function of the student and $k$ is a uniform random integer chosen from $1$ to $K$. Note that the ensemble moving teachers are also updated using the same input. We particularly discuss two different learning rules for the student, which are the Hebbian learning $$g_{k}^{m}=\eta
\text{sgn}\left(v_{B_{k}}^{m}l_{B_{k}}^{m}\right), \label{eqn:Hrule}$$ and the perceptron learning $$g_{k}^{m}=\eta
\Theta\left(-v_{B_{k}}^{m}u^{m}\right)\text{sgn}\left(v_{B_{k}}^{m}l_{B_{k}}^{m}\right). \label{eqn:Prule}$$ The learning rate of the student $\eta$ is also constant during the learning process.
Order-parameter theory
======================
As shown in the previous section, the generalization errors of the ensemble teachers and the student are expressed in terms of the parameter $R_{B_k}$ and $R_{J}$ and evolve only trough a few parameters associated with the learning of $\textrm{\boldmath $B$}_{k}$ and $\textrm{\boldmath $J$}$ in the thermodynamic limit. It has been shown that a class of the on-line learning can be characterized by a few extensive parameters, called order parameter. In this section, following ref. , a set of ordinal differential equations of the order parameters are obtained in our model by taking the thermodynamic limit.
The learning process of the ensemble moving teachers are described by the three order parameter $R_{B_k}$, $l_k$ and $q_{kk'}$, which are assumed to be self-averaging. It is sufficient to consider the evolution of $R_{B_k}$ and $l_k$ in order to describe the dynamics of the ensemble teachers, but that of the overlap $q_{kk'}$ between two different teachers is necessary for the student dynamics as seen later. From the update rules of the ensemble teachers in eq. (\[eqn:updateB\]), one finds a closed formula of the ordinal differential equations of the order parameters as,
$$\begin{aligned}
\frac{dl_{B}}{dt'}&=
\frac{\eta_B}{\sqrt{2\pi}}
\left[R_{B}\left\{2\exp{\left(-\frac{a^2}{2}\right)}-1\right\}-1\right]+
\frac{1}{2l_B}\left(
2\eta_B^2\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv H\left(-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}\right)
\right)
, \label{eqn:opeL} \\
\frac{dR_{B}}{dt'}&=
-\frac{R_{B}}{l_{B}}\frac{dl_{B}}{dt'}+\frac{1}{l_{B}}\left(
\frac{\eta_B}{\sqrt{2\pi}}\left\{2\exp{\left(-\frac{a^2}{2}\right)}-R_{B}-1\right\}
\right)
, \label{eqn:opeR} \\
\frac{dq}{dt'}&=
-\frac{q}{l_{B}}\frac{dl_{B}}{dt'}-\frac{q}{l_{B}}\frac{dl_{B}}{dt'}
+\frac{2}{ l_{B}}\left(
\frac{\eta_B}{\sqrt{2\pi}}\left[R_{B}\left\{2\exp{\left(-\frac{a^2}{2}\right)}-1\right\}-q\right]
\right)\nonumber \\
&
+\frac{1}{l_{B}^2}\left(
2\eta_{B}^2\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv\int_{-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}}^{\infty}DxH(z)
\right), \label{eqn:opeQ} \end{aligned}$$
where $$z\equiv-\frac{(q-R_{B}^2)x+R_{B}\sqrt{1-R_{B}^2}v}{\sqrt{(1-q)(1+q-2R_{B}^2)}},$$ and $t'$ denotes continuous time. We omit the subscript $k$ from the order parameters, because the differential equations including their initial conditions have a permutation symmetry for the subscript $k$. Derivation of the differential equation is given in the appendix\[sec:A\].
From these equation one easily obtain the steady solutions of $R_{B}$, $l_{B}$ and of $q$ as follows: $$\begin{aligned}
R_{B}&=2\exp{\left(-\frac{a^2}{2}\right)}-1,\\
l_{B}&=\frac{\sqrt{2\pi}\eta_{B}}{1-R_{B}^2}\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv H\left(-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}\right), \\
q&=R_{B}^2+\frac{\displaystyle (1-R_{B}^2)\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv\int_{-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}}^{\infty}DxH(z)}{\displaystyle\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv H\left(-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}\right)}.\end{aligned}$$ Note that $R_{B}$, $q$ and $l_{B}/\eta_{B}$ depend only on the threshold $a$ of the true teacher. In our study, the ensemble teachers are assumed to take the steady state before the student begins to learn in order to make the dynamical effect of the ensemble teachers clear. Therefore these solutions of $R_B$, $l_B$ and $q$ are used as an initial condition of the learning dynamics of the student discussed below.
The learning dynamics of the student is also described by a set of ordinal differential equations of a few order parameters, which is derived from the update functions for the Hebbian rule (\[eqn:Hrule\]) and the perceptron one (\[eqn:Prule\]). We refer to the appendix\[sec:B\] for the derivation of the dynamical equations. A straightforward calculation for the Hebbian rule leads to $$\begin{aligned}
\frac{dl}{dt} & = & \eta\left(\sqrt{\frac{2}{\pi}}R_B+\frac{\eta}{2l}\right), \label{eqn:opeSH1}\\
\frac{dR_J}{dt} & = & -\frac{R_J}{l}\frac{dl}{dt}+\frac{\eta}{l}\sqrt{\frac{2}{\pi}}R_B, \label{eqn:opeSH2}\\
\frac{dR_{BJ}}{dt} & = &-R_{BJ}\left(\frac{1}{l}\frac{dl}{dt}+\frac{1}{l_B}\frac{dl_B}{dt}\right)
+
\frac{\eta_B}{l_B\sqrt{2\pi}}\left\{R_J\left(2e^{-\frac{a^2}{2}}-1\right)-R_{BJ}\right\}\nonumber \\
&+&\frac{\eta}{lK}\sqrt{\frac{2}{\pi}}q -\frac{2\eta\eta_B}{Kl_Bl}\left(
(K-1)\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv
\int_{-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}}^{\infty} Dx
\left\{2H(z)-1\right\} \right. \nonumber \\
&+& \left.\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv
H\left(-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}\right)
\right)
\label{eqn:opeSH3}. \end{aligned}$$ Corresponding differential equations for the perceptron rule are given as $$\begin{aligned}
\frac{dl}{dt} & = & \eta\left(\frac{R_{BJ}-1}{\sqrt{2\pi}}+\frac{\eta}{\pi}\tan^{-1}\left(\frac{\sqrt{1-R_{BJ}^2}}{R_{BJ}}\right)\right),\label{eqn:opeSP1}\\
\frac{dR_J}{dt} & = & -\frac{R_J}{l}\frac{dl}{dt}+\frac{\eta}{l\sqrt{2\pi}}(R_B-R_J), \label{eqn:opeSP2}\\
\frac{dR_{BJ}}{dt} & = & -R_{BJ}\left(\frac{1}{l}\frac{dl}{dt}+\frac{1}{l_B}\frac{dl_B}{dt}\right)+
\frac{\eta_B}{l_B\sqrt{2\pi}}\left\{R_J\left(2e^{-\frac{a^2}{2}}-1\right)-R_{BJ}\right\}\nonumber \\
&+&\frac{\eta q}{lK\sqrt{2\pi}}\left(\frac{q}{K}-R_{BJ}\right)\nonumber \\
&+&\frac{2\eta\eta_B}{Kl_Bl}\left((K-1)
\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv \int_{-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}}^{\infty} Dx \left\{-\int_{z}^{\infty}DyH\left(-z_1\right)+\int_{-\infty}^{z}DyH\left(z_1\right)\right\}
\right. \nonumber \\
&+& \left.\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv
\int_{-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}}^{\infty} Dx
\left\{2H(z_2)-1\right\}\right)
\label{eqn:opeSP3}\end{aligned}$$ Solving these differential equations for the student and the ensemble teachers, we can obtain the generalization errors $\epsilon_J^g$ and $R_J$ as a function of time step.
Results and Discussion
======================
In this section we present dynamical behavior of the order parameter $R_J$ and the generalization error $\epsilon_J^g$ obtained by solving numerically the set of the differential equations obtained in the previous section. In order to study “dynamical” effect of the ensemble teachers, we compare results of two different cases; one with the teachers fixed to a steady state and the other with the teachers kept to learn in the steady state sharing the same inputs with the student. In this study, we choose the threshold value $a=0.5$ of the non-monotonic perceptron for the true teacher, yielding $l_{B}/\eta_{B}\simeq 0.93$, $R_{B}\simeq 0.76$ and $q\simeq 0.91$ in the steady state for the ensemble teachers. We also perform direct simulations of the given update rules for the finite-size perceptrons. In the simulations we use the dimension of vectors $N=10^4$ and perform $10^5$ trajectories of the learning process for taking the average over the random inputs. As shown in figures below, although a limited case with $\eta=0.1$ is only shown for avoiding crowded plots, the results of $R_J$ and $\epsilon_J^g$ obtained by the simulations for all the parameter studied agree with the theoretical ones by the order-parameter differential equations, This confirms that the assumption of the self-averaging is appropriate in our model.
Figure \[f1\] shows time dependence of $R_J$ for the Hebbian learning when the ensemble teachers stop to learn and take a steady-state vector. The transient process of $R_J$ depends on the learning rate $\eta$ of the student and the number $K$ of the ensemble teachers. The value of $R_J$ gets larger with increasing the number $K$ and the learning rate $\eta$, meaning that the student comes close to the true teacher. As the time $t$ goes on, it approaches monotonically a steady value, which increases as $K$ increases. Interestingly, the steady value of $R_J$ exceeds the value of $R_B$ when the number $K$ of the ensemble teachers is greater than 1. This is similar to that shown in ref. . Figure \[f2\], on the other hand, shows the corresponding time dependence of $R_J$ when the ensemble teachers continue to learn in their steady state. While at the very beginning of the learning process the value of $R_J$ shows monotonic time development similar to the case that the ensemble teachers are fixed, it is larger than that with the fixed teachers after a certain time and eventually approaches unity, which is independent of the learning rate, even if the number $K$ is one. It should be noted that the value of $R_B$ is common in two cases of Figs. \[f1\] and \[f2\]. This implies that the number $K$ of the ensemble teachers is not efficient for the learning of the student, but their continuous learning even with a fixed similarity to the true teacher is significantly important.
Figure \[f3\] shows dynamical behavior of the generalization error of the student for the Hebbian learning, which monotonically decreases and eventually converges to the steady value when the ensemble teachers are fixed. The steady value of $\epsilon_J^g$ only depends on the number $K$ and not the learning rate $\eta$. As $K$ increases, the value decreases and furthermore it can be smaller than that of the generalization error $\epsilon_B^g$ of the ensemble teachers when $K$ is larger than one, reflecting the behavior of $R_J$. This means that the performance of the student becomes better than the ensemble teachers when $K\geq 2$. The obtained value of $\epsilon_J^g$, however, does not reach the fundamental minimum value of the generalization error in this case even when $K$ increases to infinity. In Fig. \[f4\] the dynamical behavior of $\epsilon_J^g$ is shown in the case where the ensemble teachers are moving. In contrast to the case of the fixed ensemble teachers, $\epsilon_J^g$ shows non-monotonic behavior in the learning process and the steady value of independent of both $K$ and $\eta$ while it is quite larger than $\epsilon_B^g$. The minimum value of $\epsilon_J^g$ reaches the fundamental minimum value at a certain time step, depending on the learning rate $\eta$. In a sense, the mobile ensemble teachers is a better on-line learning model, while the best performance occurs only at a transient state unfortunately.
Let us turn to the perceptron learning of the student. We show the time development of $R_J$ for the fixed and moving ensemble teachers in Figs. \[f5\] and \[f6\], respectively. The steady values of $R_J$ coincide with $R_B$ both for the two cases and it is independent of $K$ and $\eta$. Further non-monotonic behavior is found for small $\eta$ and large $K$ and then the value of $R_J$ takes a maximum value at a certain time step, which exceeds $R_B$ certainly as a transient state. Moving the ensemble teachers enhances significantly the maximum value, meaning that the student is closer to the true teacher. In particular, for small value of $\eta$ the maximum value of $R_J$ for the unique moving teacher is larger than that for the $K=\infty$ fixed ensemble teachers.
Fugues \[f7\] and \[f8\] show the corresponding dynamical behavior of the generalization errors $\epsilon_J^g$ of the perceptron-learning student with the fixed and mobile ensemble teachers, respectively. As expected from the behavior of $R_J$ in Figs. \[f5\] and \[f6\], the steady value of $\epsilon_J^g$ for all the case is the same as that of the ensemble teachers. However, an essential difference is found in transient behavior of $\epsilon_J^g$. Although the minimum value does not necessarily achieve the fundamental minimum value of $\epsilon_J^g$ in the case of the fixed ensemble teachers, it does for small value of $\eta$ in the moving ensemble teachers with a finite time interval as shown in Fig. \[f8\]. This means again that moving the ensemble teachers plays an important role for the learning performance of the student.
¡¡
Conclusion
==========
We have analyzed the generalization performance of a student supervised by ensemble moving teachers in the framework of on-line learning. In this paper we adopted a non-monotonic perceptron as a true teacher and a simple perceptron as the ensemble moving teachers and the student. We have treated the Hebbian learning and the perceptron learning as a learning rule for the student and have calculated the generalization error of the student with some order parameters analytically or numerically. In this study, we particularly focus on the effect of mobile ensemble teachers on the learning performance of the student. Therefore, it is assumed that the ensemble teachers learn only from the true teacher by using the perceptron learning and reach a steady state before the student begins to learn. This is helpful for separating a transient learning effect of the ensemble teachers from an intrinsic effect.
In the Hebbian learning, it has been proven that the number $K$ of the ensemble teachers is not efficient, but their continuous learning in their steady state is significantly important for the student to come close to the true teacher. In the case that the ensemble teachers continue to learn, the value of $R_{J}$ eventually approaches unity, which is independent of the learning rate, even if the number $K$ is one. Although the student with $R_J=1$ does not always mean a best learning performance in the Hebbian learning, the minimum value of $\epsilon_J^g$ reaches the fundamental minimum value as a transient state, regardless of the number $K$. This is sharp contrast to the case of the fixed ensemble teachers, in which the fundamental minimum value of $\epsilon_J^g$ never occurs. The time step at which $\epsilon_J^g$ has a minimum value decreases with increasing the learning rate $\eta$, but its precise step has not been predicted theoretically at the present moment.
In the perceptron learning, in contrast to the Hebbian learning, no significant difference has been found in the steady states. The steady values of $R_J$ and $\epsilon_J^g$ coincide with those of $R_B$ and $\epsilon_B^g$ in both of the fixed and mobile ensemble teachers. However, the effect of the movement of the ensemble teachers appears in the transient state in the learning process, where, in particular for the small value of the learning rate $\eta$, the maximum value of $R_J$ exceeds the value of $R_B$ and then the minimum value of $\epsilon_J^g$ reaches the fundamental minimum value even if the number $K$ is one. In the case of the fixed ensemble teachers, while the former is found only for the large $K$ and small $\eta$, the latter is hardly seen for any parameter observed. It would be interesting to see that the result of the mobile ensemble teachers weakly depends on the number of the ensemble teachers. Further, the minimum value of $\epsilon_J^g$ for the $K=1$ mobile ensemble teacher is smaller than that for $K=\infty$ fixed ensemble teachers. Our study suggests that the movement of the ensemble teachers, rather than the number $K$, is important for the student learning in our model.
One of the drawbacks of the present model is that the minimum of $\epsilon_J^g$ is given as the transient state in the learning process and that no algorithm is found to stop the learning at the transient state. We point out that the perceptron learning shows a finite time interval of the transient state which gives the minimum of $\epsilon_J^g$ as shown in Fig. \[f8\]. This might be convenient in comparison to the Hebbian learning, but the explicit construction of the stopping algorithm, including a practical way, still remains to be solved in further work.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to S. Miyoshi for a critical reading of this manuscript and fruitful discussions. This work was supported by the Grant-in-Aid for Scientific Research on the Priority Area “Deepening and Expansion of Statistical Mechanical Informatics” (No. 1807004) by Ministry of Education, Culture, Sports, Science and Technology.
Derivation of the learning dynamics for the ensemble teachers {#sec:A}
=============================================================
In this appendix, we derive a set of the ordinal differential equations (\[eqn:opeL\]), (\[eqn:opeR\]) and (\[eqn:opeQ\]) of the order parameters for the ensemble moving teachers in our model. From the update rules of the ensemble teachers of eq. (\[eqn:updateB\]), a standard calculus[@1] leads to the following ordinal differential equations in terms of the average over the correlated Gaussian variables, $$\begin{aligned}
\frac{dl_{B_{k}}}{dt'}=&\langle f_{k}v_{B_{k}} \rangle+\frac{\langle f_{k}^2 \rangle}{2l_{B_{k}}}, \label{c1}\\
\frac{dR_{B_{k}}}{dt'}&=-\frac{R_{B_{k}}}{l_{B_{k}}}\frac{dl_{B_{k}}}{dt'}+\frac{\langle f_{k}v \rangle}{l_{B_{k}}}, \label{c2}\\
\frac{dq_{kk'}}{dt'}&=-\frac{q_{kk'}}{l_{B_{k}}}\frac{dl_{B_{k}}}{dt'}-\frac{q_{kk'}}{l_{B_{k'}}}\frac{dl_{B_{k'}}}{dt'} \nonumber \\
& \quad \, +\frac{\langle f_{k'} v_{B_{k}}\rangle}{ l_{B_{k'}}}+\frac{\langle f_k v_{B_{k'}}\rangle}{l_{B_{k}}} +\frac{\langle f_k f_{k'}\rangle}{l_{B_{k'}}l_{B_{k}}} \label{c3},\end{aligned}$$ where the continuous time $t'$ is defined by the thermodynamic limit of $m'/N$ with $m'$ being the time step of the ensemble teachers in eq. (\[eqn:updateB\]). The bracket $\langle\cdots\rangle$ denotes the average with respect to the multiple Gaussian distribution given in eq. (\[eqn:multiG\]). Since each component of and $_{k}^{0}$ are generated independently from the Gaussian distribution, and $_{k}^{0}$ with any $k$ are orthogonal to each other in the thermodynamic limit. Then, the initial conditions of the differential equations for $R_{B_k}$ and $q_{kk'}$ are given by $$\begin{aligned}
R_{B_{k}}^{0}&=0, \quad q_{kk'}^{0}=0, \label{initial1}\end{aligned}$$ One easily finds that from eqs. (\[c1\])-(\[c3\]) and (\[initial1\]) that the order parameters $R_{B_{k}}$, $l_{B_{k}}$ and $q_{kk'}$ are invariant under a permutation of the index $k$ of the ensemble teachers. Because of the symmetry, we omit the subscripts $k$ from the order parameters. We can calculate sample averages in eqs. (\[c1\])-(\[c3\]) and obtain $$\begin{aligned}
\langle f_{k}v_{B_{k}} \rangle =& \frac{\eta_B}{\sqrt{2\pi}} \left[R_{B}\left\{2\exp{\left(-\frac{a^2}{2}\right)}-1\right\}-1\right],\\
\langle f_{k}^2 \rangle =& 2\eta_B^2\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv H\left(-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}\right) ,\\
\langle f_{k}v \rangle =& \frac{\eta_B}{\sqrt{2\pi}}\left\{2\exp{\left(-\frac{a^2}{2}\right)}-R_{B}-1\right\}, \\
\langle f_{k'} v_{B_{k}} \rangle =&\langle f_{k} v_{B_{k'}} \rangle =\frac{\eta_B}{\sqrt{2\pi}}\left[R_{B}\left\{2\exp{\left(-\frac{a^2}{2}\right)}-1\right\}-q\right],\\
\langle f_{k} f_{k'} \rangle =&2\eta_{B}^2\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv\int_{-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}}^{\infty}DxH(z), \end{aligned}$$ where $$z\equiv-\frac{(q-R_{B}^2)x+R_{B}\sqrt{1-R_{B}^2}v}{\sqrt{(1-q)(1+q-2R_{B}^2)}}.$$ Substituting them into eqs. (\[c1\]), (\[c2\]) and (\[c3\]), the differential equations (\[eqn:opeL\]), (\[eqn:opeR\]) and (\[eqn:opeQ\]) are derived.
Derivation of the learning dynamics for the student {#sec:B}
===================================================
As in the appendix\[sec:A\], a set of the differential equations for the student dynamics is derived in this appendix. From the update rule (\[eqn:updateJ\]) of the student, the standard calculus again leads to the following equations: $$\begin{aligned}
\frac{dl}{dt}&=\frac{1}{K}\sum_{k=1}^{K}\left(\langle g_{k}u \rangle+\frac{\langle g_{k}^2 \rangle}{2l}\right), \label{s1}\\
\frac{dR_{J}}{dt}&=-\frac{R_{J}}{l}\frac{dl}{dt}+\frac{1}{K}\sum_{k=1}^{K}\frac{\langle g_{k}v \rangle}{l},\label{s2}\\
\frac{dR_{B_{k}J}}{dt}&= -\frac{R_{B_{k}J}}{l}\frac{dl}{dt}-\frac{R_{B_{k}J}}{l_{B_{k}}}\frac{dl_{B_{k}}}{dt} \nonumber \\
&+\frac{1}{K}\sum_{k'=1}^{K}\left(\frac{\langle f_k u \rangle}{l_{B_{k}}}+\frac{\langle g_{k'} v_{B_{k}}\rangle}{l}+\frac{\langle f_k g_{k'}\rangle}{l_{B_{k}}l}\right), \label{s3}\end{aligned}$$ where $t$ denotes a continuous time defined by $t=m/N$. As an initial condition of eqs. (\[s2\]) and (\[s3\]), we take $$R_{J}^{0}=0, \quad R_{B_{k}J}^{0}=0, \label{initial2}$$ since , $_{k}^{0}$ and $^{0}$ are orthogonal to each other in the thermodynamic limit. It is shown from eqs. (\[initial2\]) and (\[s3\]) that the order parameter $R_{B_{k}J}$ does not depend on the index $k$. Then, one can omit the subscript $k$ from the order parameter without loss of the generality. By substituting the two update functions $g$ of the Hebbian and the perceptron learning respectively, one calculates the Gaussian averages in eqs. (\[s1\])-(\[s3\]) in the case of the Hebbian learning as $$\begin{aligned}
\langle g_{k}u \rangle&=\eta\sqrt{\frac{2}{\pi}}R_{BJ}, \label{eqn:Hebb-b}\\
\langle g_{k}^2 \rangle&=\eta^2 ,\\
\langle g_{k}v \rangle &= \eta\sqrt{\frac{2}{\pi}}R_{B}, \\
\langle f_{k}u \rangle&=\frac{\eta_B}{\sqrt{2\pi}}\left[R_{J}\left\{2\exp{\left(-\frac{a^2}{2}\right)}-1\right\}-R_{BJ}\right],\\
\langle g_{k'}v_{B_{k}}\rangle&= \eta\sqrt{\frac{2}{\pi}}q\delta_{k,k'}, \\
\langle f_{k}g_{k'} \rangle&=-2\eta\eta_B\left[\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv \int_{-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}}^{\infty} Dx \left\{2H(z)-1\right\}\right], \\
\langle f_{k}g_{k} \rangle&=
-2\eta\eta_B\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv
H\left(-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}\right), \label{eqn:Hebb-f}\end{aligned}$$ and in the case of the perceptron learning as $$\begin{aligned}
\langle g_{k}u \rangle&=\frac{\eta}{\sqrt{2\pi}}(R_{BJ}-1), \label{eqn:percep-b}\\
\langle g_{k}^2 \rangle&=\frac{\eta^2}{\pi}\tan^{-1}\left(\frac{\sqrt{1-R_{BJ}^2}}{R_{BJ}}\right),\\
\langle g_{k}v \rangle &= \frac{\eta}{\sqrt{2\pi}}(R_{B}-R_{J}), \\
\langle f_{k}u \rangle&=\frac{\eta_B}{\sqrt{2\pi}}\left[R_{J}\left\{2\exp{\left(-\frac{a^2}{2}\right)}-1\right\}-R_{BJ}\right],\\
\langle g_{k'}v_{B_{k}}\rangle&= \frac{\eta}{\sqrt{2\pi}}(q\delta_{k,k'}-R_{BJ}), \\
\langle f_{k}g_{k'} \rangle&=2\eta\eta_B \left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv \int_{-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}}^{\infty} Dx \left\{-\int_{z}^{\infty}DyH\left(-z_1\right)+\int_{-\infty}^{z}DyH\left(z_1\right)\right\}, \nonumber \\
&&\\
\langle f_{k}g_{k}
\rangle&=2\eta\eta_B\left(\int_{-\infty}^{-a}+\int_{0}^{a}\right)Dv
\int_{-\frac{R_{B}v}{\sqrt{1-R_{B}^2}}}^{\infty} Dx
\left\{2H(z_2)-1\right\}. \label{eqn:percep-f}\end{aligned}$$ Here, $z_{1}$ and $z_{2}$ are defined as $$z_{1}\equiv-\frac{(R_{BJ}-R_{B}R_{J})\left(\sqrt{1-q}y+\sqrt{1+q-2R_{B}^2}x\right)+R_{J}\sqrt{(1-R_{B}^2)(1+q-2R_{B}^2)}v}{\sqrt{(1-R_{B}^2)\left\{(1+q)(1-R_{J}^2)-2(R_{B}^2-2R_{B}R_{J}R_{BJ}+R_{BJ}^2)\right\}}}$$ and $$z_{2}\equiv-\frac{(R_{BJ}-R_{B}R_{J})x+R_{J}\sqrt{1-R_{B}^2}v}{\sqrt{1-R_{J}^2-R_{B}^2-R_{BJ}^2+2R_{B}R_{J}R_{BJ}}},$$ and $\delta_{k,k'}$ is the Kronecker delta defined by $$\delta_{k,k'}=\left\{
\begin{array}{ll}
+1, &\quad k=k', \\
\,\,\,\,0, &\quad k\neq k'.
\end{array}
\right.$$ Inserting (\[eqn:Hebb-b\])-(\[eqn:Hebb-f\]) and (\[eqn:percep-b\])-(\[eqn:percep-f\]) into (\[s1\])-(\[s3\]) gives the dynamical equations (\[eqn:opeSH1\])-(\[eqn:opeSH3\]) for the Hebbian rule and those (\[eqn:opeSP1\])-(\[eqn:opeSP3\]) for the perceptron one, respectively.
[99]{} D.Saad, (ed.): On-line Learning in Neural Networks (Cambridge University Press, Cambridge, 1998). A.Engel and C.Van den Broeck: Statistical Mechanics of Learning (Cambridge University Press, Cambridge, 2001). H.Nishimori: Statistical Physics of Spin Glasses and Information Processing: An Introduction (Oxford University Press, Oxford, 2001).
S.Miyoshi and M.Okada: J. Phys. Soc. Jpn. **75** (2006) 024003. M.Urakami, S.Miyoshi and M.Okada: J. Phys. Soc. Jpn. **76** (2007) 044003. J.Inoue, H.Nishimori and Y.Kabashima: J. Phys A **30** (1997) 3795. S.Miyoshi and M.Okada: J. Phys. Soc. Jpn. **75** (2006) 044002. H.Utsumi, S.Miyoshi and M.Okada: J. Phys. Soc. Jpn. **76** (2007) 114001. R. Urbanczik: Phys. Rev. E **62** (2000) 1448. S. Miyoshi, K. Hara and M. Okada: Phys. Rev. E **71** (2005) 036116.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Francesco Orabona
- Dávid Pál
bibliography:
- 'biblio.bib'
title: 'Scale-Free Algorithms for Online Linear Optimization'
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We introduce a new class of complex Hadamard matrices which have not been studied previously. We use these matrices to construct a new infinite family of parity proofs of the Kochen-Specker theorem. We show that the recently discovered simple parity proof of the Kochen-Specker theorem is the initial member of this infinite family.'
author:
- |
Petr Lisoněk[^1]\
Department of Mathematics\
Simon Fraser University\
Burnaby, BC, V5A 1S6\
Canada\
\
[[email protected]]{}
title: 'Kochen-Specker sets and Hadamard matrices'
---
Introduction
============
Kochen-Specker theorem is an important result in quantum mechanics [@KS-paper]. It demonstrates the contextuality of quantum mechanics, which is one of its properties that may become crucial in quantum information theory [@How]. In this paper we focus on proofs of Kochen-Specker theorem that are given by showing that, for $n\ge 3$, there does not exist a function $f:\C^n\rightarrow \{0,1\}$ such that for every orthogonal basis $B$ of $\C^n$ there exists [*exactly one*]{} vector $x\in B$ such that $f(x)=1$ (where $\C^n$ denotes the $n$-dimensional vector space over the field of complex numbers). This particular approach has been used in many publications, see for example [@Adan-18; @Lis-PRA; @WA; @WA-preprint] and many references cited therein. The following definition formalizes one common way of constructing such proofs.
\[def-KS-pair\] We say that $(\V,\B)$ is a [*Kochen-Specker pair in $\C^n$*]{} if it meets the following conditions:
- $\V$ is a finite set of vectors in $\C^n$.
- $\B=(B_1,\ldots,B_k)$ where $k$ is odd, and for all $i=1,\ldots,k$ we have that $B_i$ is an orthogonal basis of $\C^n$ and $B_i\subset \V$.
- For each $v\in\V$ the number of $i$ such that $v\in B_i$ is even.
Let us show that the existence of a Kochen-Specker pair demonstrates the non-existence of a function $f$ with the properties given above. Towards a contradiction suppose that $(\V,\B)$ satisfies Definition \[def-KS-pair\] and $f:\C^n\rightarrow \{0,1\}$ has the properties specified above. Denote $V_1=\{ x\in\V : f(x)=1\}$. By conditions (2) and (3) and by properties of $f$, the number of $i$ such that $|B_i\cap V_1|=1$ is even. Since the length of the list $\B$ is odd, there exists an $i$ such that $|B_i\cap V_1|\neq 1$, in contradiction to the required properties of $f$. Since this contradiction is based on a parity argument, the Kochen-Specker pairs introduced in Definition \[def-KS-pair\] are often called “parity proofs of the Kochen-Specker theorem.”
It is quite common in the literature [@T7-experiment; @Lis-PRA; @WA-preprint] to refer to a Kochen-Specker pair as [*Kochen-Specker set,*]{} and we will do so sometimes in this paper. Kochen-Specker sets are key tools for proving some fundamental results in quantum theory and they also have various potential applications in quantum information processing [@T7-experiment]. In Section \[sec-KS-construction\] we give a construction of a family of Kochen-Specker sets in infinitely many different dimensions. Before that, in Section \[sec-SL-Had\] we introduce a new class of complex Hadamard matrices, which are used in our construction, and they may be also an interesting object of study on their own. In Section \[sec-conclusion\] we draw some conclusions from our results.
S-Hadamard matrices {#sec-SL-Had}
===================
For $z\in\C$ let $\overline z$ denote its complex conjugate. We work with the usual inner product on $\C^n$ defined by $\la x,y\ra=\sum_{i=1}^n x_i\overline{y_i}$. For a complex matrix $H$, let $H^*$ denote its conjugate transpose. Let $I_n$ denote the $n\times n$ identity matrix. We say that a complex number $z$ is [*unimodular*]{} if $|z|=1$. We say that a vector $x\in\C^n$ is [*unimodular*]{} if each coordinate of $x$ is unimodular.
\[def-SL-Had-m\] An $n\times n$ matrix $H=(h_{i,j})$ whose entries are complex numbers is called [*S-Hadamard matrix of order $n$*]{} if it meets the following conditions:
- $HH^*=nI_n$
- $|h_{i,j}|=1$ for all $1\le i,j\le n$
- for each $1\le k, \ell \le n$, $k\neq \ell$, we have $\sum_{j=1}^n h_{k,j}^2 \overline{h_{\ell,j}^2} =0$.
Conditions (1) and (2) are the definition of Hadamard matrices which are prominent objects in combinatorial design theory that have been studied since the 19th century [@Horadam], first as matrices over $\{-1,1\}$ and then more generally as matrices over complex numbers. Condition (3) appears to be a new condition not previously seen in the literature. The proposed name [*S-Hadamard matrix*]{} reflects the form of this new condition, which involves [****]{}quares of the entries of $H$. All three conditions in Definition \[def-SL-Had-m\] are required for our construction of Kochen-Specker pairs given in Theorem \[thm-Had-KS\].
Throughout this paper we use additive notation for group operations. By $\Z_g$ we denote the cyclic group of integers modulo $g$.
[@handbook-GH Definition 5.1] \[def-GH-mat\] Let $G$ be a group of order $g$ and let $\lambda$ be a positive integer. A [*generalized Hadamard matrix over $G$*]{} is a $g\lambda \times g\lambda$ matrix $M=(m_{i,j})$ whose entries are elements of $G$ and for each $1\le k<\ell\le g\lambda$, each element of $G$ occurs exactly $\lambda$ times among the differences $m_{k,j}-m_{\ell,j}$, $1\le j\le g\lambda$. Such matrix is denoted [GH]{}$(g,\lambda)$.
\[proposition-GH-SH\] Suppose that $g>2$ and [GH]{}$(g,\lambda)$ over $\Z_g$ exists. Then there exists an S-Hadamard matrix of order $g\lambda$.
Assume that $M=(m_{i,j})$ is a GH$(g,\lambda)$ over $\Z_g$. Let $\z_g=e^{2\pi\sqrt{-1}/g}$ be the primitive $g$-th root of unity in $\C$. Define the $g\lambda \times g\lambda$ matrix $H=(h_{i,j})$ by $h_{i,j}=\z_g^{m_{i,j}}$ for all $i,j$. We claim that $H$ is an S-Hadamard matrix of order $g\lambda$. We will verify that the three conditions in Definition \[def-SL-Had-m\] are satisfied. Let $h_k$ and $h_{\ell}$ be two rows of $H$, $k\neq \ell$. Then $$\la h_k,h_{\ell}\ra=\sum_{j=1}^{g\lambda} \z_g^{m_{k,j}-m_{\ell,j}}
=\lambda\sum_{i=0}^{g-1}\z_g^i=0$$ and $\la h_k,h_k\ra =\sum_{j=1}^{g\lambda} |h_{k,j}|^2=g\lambda$, thus condition (1) is satisfied. Condition (2) follows from the construction of $H$. To verify condition (3) we compute $$\sum_{j=1}^{g\lambda} h_{k,j}^2 \overline{h_{\ell,j}^2}
=\sum_{j=1}^{g\lambda} \z_g^{2(m_{k,j}-m_{\ell,j})}
=\lambda\sum_{i=0}^{g-1}\z_g^{2i}
=\lambda\frac{\z_g^{2g}-1}{\z_g^2-1}=0$$ where we used the assumption that $g>2$, hence $\z_g^2\neq 1$.
It would be interesting to find other constructions of S-Hadamard matrices, and we pose this as an open problem.
\[example-GH-exist\] The construction given in our main result (Theorem \[thm-Had-KS\]) requires the existence an S-Hadamard matrix of [*even*]{} order. For illustration let us consider small even orders. Proposition \[proposition-GH-SH\] allows one to construct an S-Hadamard matrix of order $n$ for all even $0<n\le 100$ except $n=2, 4, 8, 32, 40, 42, 60, 64, 66, 70, 78, 84, 88$. The underlying generalized Hadamard matrices are obtained by constructions given in [@handbook-GH], moreover for $n=16$ examples of GH(4,4) over $\Z_4$ are given in [@Harada]. By consulting Table 6.1 in [@Lampio] we see that no other suitable GH$(g,\lambda)$ over $\Z_g$ are known for $g\le 7$ and $\lambda\le 10$. For some of the values of $n$ which we excluded above, the existence of a suitable generalized Hadamard matrix is an open problem, and it is possible that S-Hadamard matrices of those orders may exist.
An infinite family of Kochen-Specker sets {#sec-KS-construction}
=========================================
S-Hadamard matrices introduced in the previous section will now be used to construct an infinite family of Kochen-Specker sets.
\[thm-Had-KS\] Suppose that there exists an S-Hadamard matrix of order $n$ where $n$ is even. Then there exists a Kochen-Specker pair $(\V,\B)$ in $\C^n$ such that $|\V| \le {{n+1}\choose 2}$ and $|\B|=n+1$.
First we construct the set $\V$. Let the elements of $\V$ be denoted $v^{\{r,s\}}$ where $1\le r,s\le n+1$, $r\neq s$. Note that we use the standard convention that sets are unordered, hence $v^{\{r,s\}}$ and $v^{\{s,r\}}$ denote the same element of $\V$, for all $r\neq s$. For $x,y\in \C^n$ we define $x\circ y=(x_1y_1,\ldots,x_ny_n)$.
Let $H=(h_{i,j})$ be the S-Hadamard matrix of order $n$ whose existence is assumed, and let $h_i$ denote the $i$-th row of $H$. Without loss of generality we can assume that $h_1$ is the all-one vector, denoted $\bf 1$. If this is not the case, then replace each entry $h_{ij}$ of $H$ with $h_{ij}h_{1j}^{-1}$; this operation preserves all conditions of Definition \[def-SL-Had-m\].
We construct the elements of $\V$ as follows:
- For $1 < s\le n+1$ let $v^{\{1,s\}}=h_{s-1}$.
- For $2 < s\le n+1$ let $v^{\{2,s\}}=h_{s-1}\circ h_{s-1}$.
- For $2 < r <s \le n+1$ let $v^{\{r,s\}}=h_{r-1}\circ h_{s-1}$.
For $1\le r\le n+1$ let $B_r=\{ v^{\{r,i\}}\;:\; 1\le i\le n+1,\; i\neq r\}$, and let $\B=(B_1,\ldots,B_{n+1})$. We will now prove that each $B_r$ is an orthogonal basis of $\C^n$. Note that for $x,y,z\in\C^n$ such that $z$ is unimodular we have $$\label{eq-circ-orth}
\la z\circ x, z\circ y\ra =
\la x\circ z, y\circ z\ra = \sum_{i=1}^n x_iz_i\overline{y_iz_i}=\la x,y \ra.$$ Since distinct rows of $H$ are orthogonal and all rows of $H$ are unimodular, equation (\[eq-circ-orth\]) proves $$\la v^{\{r,s\}},v^{\{r,t\}}\ra
=
\la h_{r-1}\circ h_{s-1}, h_{r-1}\circ h_{t-1}\ra
=
\la h_{s-1}, h_{t-1}\ra
=0
$$ whenever $$2< r,s,t\le n+1
\mbox{\ \ and\ \ } r,s,t \mbox{\ are\ distinct.}
\label{eq-easy-rst}$$
We will now prove the desired orthogonality relations $\la v^{\{r,s\}},v^{\{r,t\}}\ra =0$ for those pairs of vectors $v^{\{r,s\}},v^{\{r,t\}}$ which are not covered by condition (\[eq-easy-rst\]). We will split the proof into cases according to the value of $r$.
Let $r=1$. For $1< s < t \le n+1$ we have $$\la v^{\{1,s\}},v^{\{1,t\}}\ra
=
\la h_{s-1},h_{t-1}\ra
=
0.$$
Now let $r=2$. For $2< s < t \le n+1$ we have $$\la v^{\{2,s\}},v^{\{2,t\}}\ra
=\la h_{s-1}\circ h_{s-1}, h_{t-1}\circ h_{t-1}\ra =
\sum_{j=1}^n h_{s-1,j}^2 \overline{h_{t-1,j}^2} =0$$ by condition (3) in Definition \[def-SL-Had-m\]. For $2< t \le n+1$ we have $$\la v^{\{2,1\}},v^{\{2,t\}}\ra
= \la {\bf 1} , h_{t-1}\circ h_{t-1} \ra
= \sum_{j=1}^n \overline{h_{t-1,j}^2} =0$$ by condition (3) in Definition \[def-SL-Had-m\], applied with $k=1$.
Now let $2< r \le n+1$. For $t>2$, $t\neq r$ we have $$\begin{aligned}
\la v^{\{r,1\}},v^{\{r,t\}}\ra
&=&
\la h_{r-1} , h_{r-1}\circ h_{t-1} \ra
= \la h_{r-1}\circ h_1 , h_{r-1}\circ h_{t-1} \ra
= \\
&=&\la h_1, h_{t-1} \ra =0\end{aligned}$$ as well as $$\la v^{\{r,2\}},v^{\{r,t\}}\ra
=
\la h_{r-1}\circ h_{r-1} , h_{r-1}\circ h_{t-1} \ra
= \la h_{r-1},h_{t-1}\ra =0.$$ Finally we have $$\begin{aligned}
\la v^{\{r,1\}},v^{\{r,2\}}\ra
&=& \la h_{r-1} , h_{r-1}\circ h_{r-1} \ra
\\
&=& \la h_1 \circ h_{r-1} , h_{r-1}\circ h_{r-1} \ra =
\la h_1, h_{r-1} \ra = 0.\end{aligned}$$
We note that $|\B|=n+1$ is odd since $n$ is assumed to be even. We will complete the proof by verifying that condition (3) in Definition \[def-KS-pair\] is satisfied. If the mapping $\{i,j\}\mapsto v^{\{i,j\}}$ is injective, then each $v^{\{i,j\}}$ belongs to exactly two entries of $\B$, namely $B_i$ and $B_j$. If the list $(v^{\{i,j\}})_{1\le i<j\le n+1}$ contains repeated vectors, then let $x$ be a vector that occurs exactly $t$ times in this list. Then by the previous argument $x$ belongs to exactly $2t$ entries of $\B$, since for distinct $i,j,k$ we have $v^{\{i,j\}}\neq v^{\{i,k\}}$ as $\la v^{\{i,j\}}, v^{\{i,k\}}\ra =0$.
There are infinitely many dimensions $n$ to which Theorem \[thm-Had-KS\] applies. By considering Proposition \[proposition-GH-SH\], a sufficient condition for applying Theorem \[thm-Had-KS\] in an even dimension $n$ is the existence of GH$(g,n/g)$ over $\Z_g$ for some $g>2$. The simplest forms of an infinite family for which this condition is satisfied are $n=2^kp^m$ where $k\in\{1,2\}$, $p$ is an odd prime and $m\ge 1$, or $n=8p$ where $p>19$ is prime. The constructions for the underlying generalized Hadamard matrices can be found by consulting Table 5.10 in [@handbook-GH]. Furthermore, any known generalized Hadamard matrices can be used as ingredients to recursive constructions given in Theorems 5.11 and 5.12 in [@handbook-GH]. As these recursive constructions can be applied repeatedly, it is impossible to give a closed form for all $n$ to which Theorem \[thm-Had-KS\] applies. For $n\le 100$ such $n$ are listed in Example \[example-GH-exist\] above.
Conclusion {#sec-conclusion}
==========
A Kochen-Specker pair $(\V,\B)$ in $\C^6$ with $|\V|=21$ and $|\B|=7$ was recently discovered [@Lis-PRA]. It was noted [@T7-experiment] as the [*simplest*]{} Kochen-Specker pair (KS pair) since it strictly minimizes the cardinality of $\B$ among all known KS pairs $(\V,\B)$, see [@WA-preprint]. This KS pair was originally found by computer search and its internal structure has not been fully studied yet. In this paper we have revealed the structure of this KS pair, since it is obtained by applying Theorem \[thm-Had-KS\] in the smallest possible dimension $n=6$. We have discovered an application of generalized Hadamard matrices to the construction of KS pairs, and we have proposed a new class of Hadamard matrices as the suitable domain for such constructions.
[10]{}
A. Cabello, A proof with 18 vectors of the Bell-Kochen-Specker theorem. In: M. Ferrero and A. van der Merwe (Eds.), New Developments on Fundamental Problems in Quantum Physics. Kluwer Academic, Dordrecht, Holland, 1997, pp. 59–62.
G. Cañas, M. Arias, S. Etcheverry, E.S. Gómez, A. Cabello, G.B. Xavier, G. Lima, Applying the simplest Kochen-Specker set for quantum information processing. Phys. Rev. Lett. [**113**]{} (2014), 090404.
M. Harada, C. Lam, V.D. Tonchev, Symmetric (4,4)-nets and generalized Hadamard matrices over groups of order 4. Des. Codes Cryptogr. [**34**]{} (2005), 71–87.
K.J. Horadam, Hadamard Matrices and Their Applications. Princeton University Press, Princeton, NJ, 2007.
M. Howard, J. Wallman, V. Veitch, J. Emerson, Contextuality supplies the ‘magic’ for quantum computation. Nature [**510**]{} (2014), 351–355.
S. Kochen, E.P. Specker, The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics [**17**]{} (1967), 59–87.
P.H.J. Lampio, Classification of difference matrices and complex Hadamard matrices. Dissertation, Aalto University, Helsinki, Finland, 2015. https://aaltodoc.aalto.fi/handle/123456789/18228 (Retrieved 30 November 2017)
W. de Launey, Generalized Hadamard matrices. In: C.J. Colbourn and J.H. Dinitz (Eds.), Handbook of Combinatorial Designs. Second edition. Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 301–306.
P. Lisoněk, P. Badziag, J.R. Portillo, A. Cabello, Kochen-Specker set with seven contexts. Phys. Rev. A [**89**]{} (2014), 042101.
M. Waegell, P.K. Aravind, Parity proofs of the Kochen-Specker theorem based on the Lie algebra E8. J. Phys. A: Math. Theor. [**48**]{} (2015), 225301.
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[^1]: Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Molecular Communication as the most potential methods to solve the communication in nano scale, for it’s derived from nature, and it becomes more and more prevalent. Though molecular communication happens in three dimensional situation, there are also some situation that are in the one dimensional situation, especially when considering the transmitters and the receivers are in extremely short distance or in long slim pipe. In this paper, we introduce the one dimensional situation, and studied how the continuous information molecules transmitted in this situation, also introduced how to encode and decode the information molecules, and based on the molecular communication model, we studied some metrics of it, such as the distance between transmitter and receiver, the emitting frequency of transmitter. Through the research we know that the distance and frequency are important metrics to the successful communication, which can direct us how to place the nano transmitters and receivers in the future nano network environment.'
author:
-
-
title: Continuous Molecular Communication in one dimensional situation
---
Introduction
============
With the quickly development of nano technology, the scale of electrical devices becoming smaller and smaller, and the ability of single nano machine is very limited due to its size, therefore, sharing information among nano machines is good for the use of single nano machine through the connection of them, which organise the nano network. Achieving the communication of nano machines has two main ways which are nano electrical communication through Carbon nanotubes and molecular communication through information molecules. Among them, molecular communication(MC) is the most potential way to achieve, which relies on the bio-inspired method and fits for the biological environment.\
At the open space of biological environment, the three dimensional space is fit for molecular communication[@2013-Nakano-p-] to model channel. In Mahfuz et al’s articles, they elaborated on the spatiotemporal distribution of information messages under the binary concentration-encoded modulate methods. Mahfuz’s researches have a great impact on the study of molecular communication, which gives us a new modulate methods. For the binary concentration-encoded modulate methods, it’s much fit for the biological environment, and without considering the inner structures of the molecules. Also in Akan’s paper, he studied the ligand and receptor process of molecular communication, in order to study the details of how molecules bind with the receptors, which is based on the binary concentration-encoded modulate methods.\
All of their works help us understand the molecular communication clearly. As we know in molecular communication different modulation methods or different transportation environment can make different channel. Most studies are based on the three dimensional environmental, and fews are concentrate on the one dimensional environment. Comparing with the three dimensional , one dimensional can make sure the receiver receive all the information molecules for a long time, and the one dimensional situation is simple enough, therefore, it is worth for us to study it and it has a wealth of applications. For instance, in Nariman’s molecular communication test-bed, which is studied based on the one dimensional environment. Also in extremely tiny pipe, the transmitter and receiver nano machines on sides of the pipe can regard as in the same line. Additionally, when transmitter and receiver nano machines are in extremely small distance, we can consider they are in one dimensional environment.\
In this paper, we mainly considered in the tiny slim pipe, how the continuous information molecules are transmitted. And how the information transmission is affected by the transmission frequency and the distance of transmitter and receiver. The study can help us know how the frequency affect the communication between nano machines. Also can help us to place the nano machines in the future.\
The paper is organised as follows: in Section \[sec\_mc\_model\], we produced a molecular comunication model; in Section \[sec\_sample\_threshold\] we introduced the sample method and the threshold of the receiver; in Section \[sec\_analysis\], we analyzed some parameters which affect the model; in Section \[sec\_conclusion\], we make a conclusion of our work.
Molecular Communication via Diffusion {#sec_mc_model}
=====================================
We model a communication system composed of a pair of devices which are connected by an extremely slim pipe, each called a nanonetworking-enabled node(NeN,ie., nano node or nano robot)[@2014-Yilmaz-p929-932]. The communication system is described as Fig.\[fig\_mc\_model\]. In molecular communication via diffusion(MCvD), the NeNs communicate with each other through the propagation of certain molecules via diffusion[@2010-Pierobon-p602-611]. The TN transmits the encoded information molecules in the propagation medium, the communication phase being known as the sending phase. The transmitted molecules propageated by the channel, the bind with the RN ,which is known as the ligand-receptor binding[@2010-Mahfuz-p289-300]. Since the TN and RN nano machine are connected by the extremely slim pipe, surely, we can consider the TN and RN in the one dimensional environment, then we can study its spatiotemporal distribution of signal strength in one dimensional environment.
The emission of information can be an instantaneous or a continuous emission. As the instantaneous release of information molecules, in a single puff, would be ideal design for communications requiring rapid fade-out. On the other hand, the source emitting message molecules continuously at a constant rate, to some extent might be useful for status telemetry, navigational beacons or periodic sampling monitors[@2009-Lacasa-p-]. In this paper we regard the emission is continuous.\
The spatiotemporal distribution of molecules transmitted by the TN that will be available at RN is calculated by the well-known Roberts equation explained in [@1963-Bossert-p443-469]. We assume that a point source type TN continuously emitting molecules at a rate $Q(t)$ molecules per second, x is taken as the distance down the pipe from the source, $A$ is the cross-sectional area of the long and slim pipe,and the concentration of molecules $U(x,t)$ at distance $x$ and time instant $t$, then in one dimensional situation, $U(x,t)$ is described as follows Eq.\[equation\_spatiotemporal\_1D\], $$\label{equation_spatiotemporal_1D}
U(x,t)=\int_0^t{\frac{Q(t)}{A\sqrt{4\pi{D}(t-t^{\star})}}e^{\frac{-x^2}{4D(t-t^{\star})}}}dt^{\star} \quad t>0.$$ where $t^{\star}$ as the dummy variable of integration, D is the diffusion constant in $cm^2/sec$.\
We use the ON-OFF modulation method to transmit information. By transmit 1 bit, the TN will emitting a quantity of molecules $Q(Q>0)$, and transmit 0 bit, the TN will emitting nothing to the channel. We utilize $f_b$ stands fot the emitting frequency of TN, and $T_b=1/f_b$ is the emitting period. In the emitting period, we can emitting molecules continuously, or transmitting nothing as described in Eq.\[equation\_on\_off\] or Fig.\[fig\_on\_off\_modulation\]. $$\label{equation_on_off}
Q(t)=\left\{
\begin{aligned}
Q_{average}; \quad \quad \text{for bit '1'}\\
0; \quad \quad \text{for bit '0'}
\end{aligned}
\right.$$
Sample method and concentration threshold {#sec_sample_threshold}
=========================================
In this paper, we regard $Q(t)$ as a constant value $Q_{average}$ which is the average molecules in the emitting period. As aforementioned we transmit “1” when emitting a quantity of molecules, and transmit “0” when emitting nothing. In this situation, the distance between TN and RN is x, as Fig.\[fig\_single\_duration\_spray\] describes, we try to transmit bit sequences “10110”.
When molecules were transmitted to the receiver(RN), the RN needs to decide the information is “1” or “0. Therefore, we need to set a threshold for the receiver, by comparing the available information molecules(AIM) to the threshold, we can know the information is ”1“ if the AIM is great or equal than the threshold, and in vice versa, the information is ”0". In the RN side, it’s no need to sample the value of AIM for each time instant, according to the Nyquist–Shannon sampling theorem, we can get the sampling value at time instant of $\frac{(2n-1)T_b}{2}$ is $Z_{SD}$, where $n$ is the length of the bit information, we only need to compare the $Z_{SD}$ to the threshold. In Fig.\[fig\_single\_duration\_spray\], the red arrow is the sampling instant, and the interference is the noise information molecules from the previous molecules informations, in this paper, for a simple description, we only need to consider one bit information of the previous information molecules.
Analysis {#sec_analysis}
========
In this section, we study how the distance and emitting frequency affect the transmission under the environment of air medium. And at last, we do the simulation to testify the difference of information transmission in one dimensional(1-D) situation and three dimensional(3-D) situation. In this section, we only transmit two bit information “10”, for it can help us analysis the AIM in the first bit “1”, and interference in the second bit “0”.
How distance affect the concentration
-------------------------------------
We assume that the $Q_{average}$ is 10000 molecules/sec, $T_b$ is 30$sec$, the diffusion coefficient in air medium is $D$=0.43$cm^2/s$, we utilize the model in Eq.\[equation\_spatiotemporal\_1D\] to do the simulation at different distances, such as $x$ are 0.05$cm$,1$cm$,10$cm$ respectively.
Through the Fig.\[fig\_distance\_varies\], we find that with the increase of distance between TN an RN, the AIM in the RN side is become smaller and smaller, but the interference is becoming larger and larger. Therefore, we need to set the distance at short distance, in order to transmit enough information molecules, and improve the probability of successful information transmission.
How frequency affect the concentration
--------------------------------------
Assuming that the $Q_{average}$ is 10000 molecules/sec, the distance between TN and RN is $d=1cm$, and the diffusion coefficient in air medium is $D=0.43cm^2/s$. Using the model in Eq.\[equation\_spatiotemporal\_1D\], we do the simulation at different emitting frequencies, such as $\frac{1}{10}Hz$,$\frac{1}{20}Hz$ and $\frac{1}{30}Hz$.
Through the Fig.\[fig\_distance\_varies\], we find that with the increase of $f_b$, the AIM in the RN side is become smaller and smaller, also the interference is becoming smaller and smaller. Since the AIM and interference increase or decrease simultaneously, we need to make a compromise for the change of frequency, in order to transmit enough information molecules, and improve the probability of successful information transmission.
The comparison available of molecule concentration in 1-D and 3-D
-----------------------------------------------------------------
We need to do some researches to study the differences between one dimensional situation and three dimensional situation. We use the same basic values of the parameters described before, except for the distance $x=0.5$, the frequency $f=\frac{1}{20}$. The three dimensional model of molecules transmission is adapted from [@2015-Mahfuz-p67-83]. The simulation result shows in Fig.\[fig\_compare\_1D\_3D\].
From Fig.\[fig\_compare\_1D\_3D\], we know that the 3-D line higher than 1-D, and the interference is almost the same. In one dimensional situation can make sure the receiver the information molecules completely in certain time, however, the available information in three dimensional situation is higher. We think the reason is that in three dimensional situation, there are a large quantities of molecules in certain time slot, but to the one dimensional situation the information molecules needs to transmit sequencely which result the through out is smaller.
Conclusion {#sec_conclusion}
==========
In this paper, we studied continuous molecules were transmitted in one dimensional molecular communication. We use binary concentration-encoded method to encode information molecules into bit “1” or “0”, also we introduce sample theorem and threshold for the receiver to decode information molecules. To study the molecular communication deeper, we studied that how the metrics of distance and frequency affect the molecules concentration, and we learned that the longer the distance the lower of the available information molecules in RN, but higher in interferences, and the frequency has two sides on the available molecular communication, either positive or negative, therefore, we should make a compromise between the available information molecules and the interferences from the former information bits. Finally, we made a comparison between one dimensional situation and three dimensional situation, and we find that in three dimensional situation, the RN side has a higher available information molecules, but cannot make sure the RN receives the molecular information for one hundred percent.\
Still there are many areas need to study, for the future research work, we can promote our research from three aspects. For the first,we can study how the variation of different emit rates affect the molecules concentration. For the second , we should find a method to make a cut-off between the frequency and the interference, in order to make the communication more accurate and high efficiency. For the third, we should consider to combine the advantages of the three dimensional and one dimensional situation to improve the available information molecules.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by the National Natural Science Foundation of China (61173190) and the Fundamental Research Funds for the Central Universities(GK201501008).
[1]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
T. Nakano, A. W. Eckford, and T. Haraguchi, *Molecular Communication*.1em plus 0.5em minus 0.4emCambridge University Press, 2013.
H. B. Yilmaz, A. C. Heren, T. Tugcu, and C.-B. Chae, “Three-dimensional channel characteristics for molecular communications with an absorbing receiver,” *Communications Letters, IEEE*, vol. 18, no. 6, pp. 929–932, 2014.
M. Pierobon and I. F. Akyildiz, “A physical end-to-end model for molecular communication in nanonetworks,” *IEEE Journal on Selected Areas in Communications*, vol. 28, no. 4, pp. 602–611, 2010.
M. U. Mahfuz, D. Makrakis, and H. T. Mouftah, “On the characterization of binary concentration-encoded molecular communication in nanonetworks,” *Nano Communication Networks*, vol. 1, no. 4, pp. 289–300, 2010.
N. R. Lacasa, “Modeling the molecular communication nanonetworks,” Master’s thesis, The Universitat Polit[è]{}cnica de Catalunya (UPC), Spain, 2009.
W. H. Bossert and E. O. Wilson, “The analysis of olfactory communication among animals,” *Journal of theoretical biology*, vol. 5, no. 3, pp. 443–469, 1963.
M. U. Mahfuz, D. Makrakis, and H. T. Mouftah, “On the detection of binary concentration-encoded unicast molecular communication in nanonetworks,” in *BIOSIGNALS*, 2011, pp. 446–449.
M. Mahfuz, D. Makrakis, and H. Mouftah, “A comprehensive analysis of strength-based optimum signal detection in concentration-encoded molecular communication with spike transmission,” *NanoBioscience, IEEE Transactions on*, vol. 14, no. 1, pp. 67–83, Jan 2015.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We will consider about some inequalities on operator means for more than three operators, for instance, ALM and BMP geometric means will be considered. Moreover, log-Euclidean and logarithmic means for several operators will be treated.'
address:
- |
Department of Information and Computer Engineering,\
Kisarazu National College of Technology,\
2-11-1 Kiyomidai-Higashi, Kisarazu,\
Chiba 292-0041, Japan
- |
Department of Electrical, Electronic and Computer Engineering\
Toyo University\
Kawagoe 350-8585, Japan
author:
- Shuhei Wada
- Takeaki Yamazaki
title: |
Equivalence relations among\
some inequalities on operator means
---
Introduction
============
Let $\mathcal{H}$ be a complex Hilbert space, and $B(\mathcal{H})$ be the algebra of all bounded linear operators on $\mathcal{H}$. An operator $A$ is said to be positive semi-definite (resp. positive definite) if and only if $\langle Ax,x\rangle \geq 0$ for all $x\in \mathcal{H}$ (resp. $\langle Ax,x\rangle > 0$ for all non-zero $x\in \mathcal{H}$). We denote positive semi-definite operator $A\in B(\mathcal{H})$ by $A\geq 0$. Let $B(\mathcal{H})_{+}$ and $B(\mathcal{H})_{sa}$ be the sets of all positive definite and self-adjoint operators, respectively. We can consider the order among $B(\mathcal{H})_{sa}$, i.e., for $A, B\in B(\mathcal{H})_{sa}$, $$A\leq B \quad \text{if and only if} \quad 0\leq B-A.$$ A real valued function $f$ on an interval $J\subset \mathbb{R}$ is called an [*operator monotone function*]{} if and only if $$A\leq B \quad \mbox{implies}\quad
f(A)\leq f(B)$$ for all $A, B\in B(\mathcal{H})_{sa}$ whose spectral are contained in $J$.
For two positive definite operators, the operator mean is important in the operator theory.
A binary operation $\sigma: B(\mathcal{H})_{+}^{2} \to B(\mathcal{H})_{+}$ is called an [*operator mean*]{} if and only if the following conditions are satisfied.
1. If $A\leq C$ and $B\leq D$, then $A\sigma B\leq C\sigma D$,
2. $X^{*} (A\sigma B)X\leq (X^{*}AX)\sigma (X^{*}BX)$ for $X\in B(\mathcal{H})$,
3. $A_{n}\sigma B_{n}\downarrow A\sigma B$ when $A_{n}\downarrow A$ and $B_{n}\downarrow B$ in the strong operator topology,
4. $I\sigma I=I$, where $I$ means the identity operator on $\mathcal{H}$.
We notice that operator means can be defined for positive semi-definite operators by (3) in Definition 1. Kubo-Ando [@KA1980] have shown the following important result:
\[thm:Kubo-Ando\] For each operator mean $\sigma$, there exists the unique operator monotone function $f: (0,\infty) \longrightarrow (0,\infty)$ such that $f(1)=1$ and $$f(t)I=I\sigma (tI) \quad \text{for all $t\in (0,\infty)$.}$$ Moreover for $A\in B(\mathcal{H})_{+}$ and $B\geq 0$, the formula $$A\sigma B=A^{\frac{1}{2}}f(A^{\frac{-1}{2}}BA^{\frac{-1}{2}})A^{\frac{1}{2}}$$ holds, where the right hand side is defined via the analytic functional calculus. An operator monotone function $f$ is called the representing function of $\sigma$.
Typical examples of operator means are weighted harmonic, geometric and arithmetic means denoted by $!_{w}$, $\sharp_{w}$ and $\nabla_{w}$ for $w\in [0,1]$, respectively. Their representing functions are $[ (1-w)+wt^{-1}]^{-1}$, $t^{w}$ and $1-w+wt$, respectively. In fact, we can define $A!_{w} B=[(1-w)A^{-1}+wB^{-1}]^{-1}$, $A\sharp_{w}B=A^{\frac{1}{2}}(A^{\frac{-1}{2}}BA^{\frac{-1}{2}})^{w}A^{\frac{1}{2}}$ and $A\nabla_{w}B=(1-w)A+wB$.
Extending Kubo-Ando theory to the theory for three or more operators was a long standing problem, in particular, we did not have any nice definition of geometric mean for three operators. Recently, Ando-Li-Mathias have given a nice definition of geometric mean for $n$-tuples of positive definite matrices in [@ALM2004]. Then many authors study about operator means for $n$-tuples of positive definite operators, and now we have three definitions of geometric means which are called ALM, BMP and the Karcher means. Moreover, we have an extension of the Karcher mean which is called the power mean.
M. Uchiyama and one of the authors have obtained equivalence relations between inequalities for the power and arithmetic means as extensions of a converse of Loewner-Heinz inequality [@UY2014].
In this paper, we shall investigate the previous research [@UY2014] to other operator means for $n$-tuples of operators. In fact, we shall treat ALM and BMP means, moreover we shall discuss about some types of logarithmic means of several operators. This paper is organized as follows. In Section 2, we will introduce some definitions and notations which will be used in this paper. Then we shall consider about weighted operator means in the view point of their representing functions in Section 3. In Section 4, we shall consider about generalizations of the results by M. Uchiyama and one of the authors [@UY2014]. Especially, we shall consider about the log-Euclidean mean which is a kind of geometric mean for $n$-tuples of positive definite operators. In the last section, we shall introduce some properties of the $M$-logarithmic mean which is generated from an arbitrary operator mean via an integration.
Primarily
=========
Let $OM$ be the set of all operator monotone functions on $(0,\infty)$, and let $OM_{1}=\{ f\in OM :\ f(1)=1\}$. For $f\in OM_{1}$, there exists an operator mean $\sigma_{f}$ such that $$A\sigma_{f}B=
A^{\frac{1}{2}} f(A^{\frac{-1}{2}}BA^{\frac{-1}{2}})
A^{\frac{1}{2}}$$ for $A, B\in B(\mathcal{H})_{+}$. It is well known that for $w\in [0,1]$, if $$A !_{w} B\leq A\sigma_{f} B\leq A\nabla_{w} B$$ holds for all $A, B\in B(\mathcal{H})_{+}$, then $$\left[(1-w)+wt^{-1}\right]^{-1}
\leq f(t) \leq (1-w)+wt$$ holds for all $t>0$.
Let $A,B\in B(\mathcal{H})_{+}$. The Thompson metric $d(A,B)$ is defined by $$d(A,B)=\max\{ \log M(A/B), \log M(B/A)\},$$ where $M(A/B)=\inf \{\alpha>0\ |\ B\leq \alpha A\}$. It is known that a cone of positive definite operators is a complete metric space for the Thompson metric. In what follows, we will consider about “limit” of operator sequences or “continuous” of operator valued functions in the Thompson metric without any explanation.
For $n$-tuples of positive definite operators, the ALM and BMP (geometric) means are defined as follows.
\[thqt1\] For $\mathbb{A}=(A_{1}, A_{2})\in B(\mathcal{H})_{+}^{2}$, the ALM (geometric) mean $\mathfrak{G}_{ALM}(\mathbb{A})$ of $\mathbb{A}$ is defined by $\mathfrak{G}_{ALM}(\mathbb{A})=A_{1}\sharp_{1/2} A_{2}$. Assume that the ALM (geometric) mean $\mathfrak{G}_{ALM}(\cdot)$ on $B(\mathcal{H})_{+}^{n-1}$ is defined. Let $\mathbb{A}=(A_{1}, \dots, A_{n})\in B(\mathcal{H})_{+}^{n}$ and $\{A_i^{(r)}\}_{r=0}^{\infty}$ $(i=1,...,n)$ be the sequences of positive definite operators defined by $$\begin{aligned}
A_i^{(0)}=A_i\quad\mbox{and}\quad
A_i^{(r+1)}=\mathfrak{G}_{ALM}\left((A_j^{(r)})_{j\neq i}\right),\end{aligned}$$ where $(A_j^{(r)})_{j\neq i}=(A_{1}^{(r)},...,A_{i-1}^{(r)},A_{i+1}^{(r)},...,A_{n}^{(r)})$. Then there exists $\lim_{r\rightarrow\infty}A_i^{(r)}$ $(i=1,...,n)$ and it does not depend on $i$. The ALM (geometric) mean $\mathfrak{G}_{ALM}(\mathbb{A})$ for $n$-tuples of positive definite operators $\mathbb{A}\in
B(\mathcal{H})_{+}^{n}$ is defined by $\lim_{r\rightarrow\infty}A_i^{(r)}$.
A vector $\omega=(w_{1},...,w_{n})\in (0,1)^{n}$ is said to be a [*probability vector*]{} if and only if $\sum_{k}w_{k}=1$. Let $\Delta_{n}$ be the set of all probability vectors in $(0,1)^{n}$.
\[thqt2\] For $\mathbb{A}=(A_{1}, A_{2})\in B(\mathcal{H})_{+}^{2}$ and $\omega=(1-w, w) \in \Delta_{2}$, the BMP (geometric) mean $\mathfrak{G}_{BMP}(\omega; \mathbb{A})$ of $\mathbb{A}$ is defined by $\mathfrak{G}_{BMP}(\omega; \mathbb{A})=A_{1}\sharp_{w} A_{2}$. Assume that the BMP (geometric) mean $\mathfrak{G}_{BMP}(\cdot; \cdot)$ on $\Delta_{n-1}\times B(\mathcal{H})_{+}^{n-1}$ is defined. Let $\mathbb{A}=(A_{1}, \dots, A_{n})\in B(\mathcal{H})_{+}^{n}$ and $\omega=(w_{1},...,w_{n})\in \Delta_{n}$. Define the sequences of positive definite operators $\{A_i^{(r)}\}_{r=0}^{\infty}$ $(i=1,...,n)$ by $$\begin{aligned}
A_i^{(0)}=A_i\quad\mbox{and}\quad
A_i^{(r+1)}=\mathfrak{G}_{BMP}
\left(\hat{\omega}_{\neq i}; (A_j^{(r)})_{j\neq i}\right)\sharp_{w_{i}}A_{i}^{(r)},\end{aligned}$$ where $\hat{\omega}_{\neq i}=\frac{1}{\sum_{j\neq i}w_{j}}(w_{j})_{j\neq i}$. Then there exists $\lim_{r\rightarrow\infty}A_i^{(r)}$ $(i=1,...,n)$ and it does not depend on $i$. The BMP (geometric) mean $\mathfrak{G}_{BMP}(\omega; \mathbb{A})$ for $n$-tuples of positive definite operators $\mathbb{A}\in B(\mathcal{H})_{+}^{n}$ is defined by $\lim_{r\rightarrow\infty}A_i^{(r)}$.
We remark that it is not known any weighted ALM mean. Let $\mathbb{A}=(A_{1},...,A_{n}), \mathbb{B}=(B_{1},...,B_{n})
\in B(\mathcal{H})_{+}^{n}$ and $\omega=(w_{1},...,w_{n})\in \Delta_{n}$. Here we denote the above geometric means of $\mathbb{A}$ for the weight $\omega$ by $\frak{G}(\omega; \mathbb{A})$, and they have at least 10 basic properties [@ALM2004; @BMP2010; @IN2009; @LLY2011] as follows (in the ALM mean case, we consider just only $\omega=(\frac{1}{n},...,\frac{1}{n})$ case).
- If $A_{1},...,A_{n}$ commute with each other, then $$\frak{G}(\omega; \mathbb{A})=
\prod_{k=1}^{n}A_{k}^{w_{k}}.$$
- For positive numbers $a_{1},...,a_{n}$, $$\frak{G}(\omega;a_{1}A_{1},...,a_{n}A_{n})
=\frak{G}(\omega;a_{1},...,a_{n})
\frak{G}(\omega;\mathbb{A})=
\left(\prod_{k=1}^{n} a_{k}^{w_{k}}\right)\frak{G}(\omega;\mathbb{A}).$$
- For any permutation $\sigma$ on $\{1,2,...,n\}$, $$\frak{G}(w_{\sigma(1)},...,w_{\sigma(n)};
A_{\sigma(1)},...,A_{\sigma(n)})=
\frak{G}(\omega; \mathbb{A}).$$
- If $A_{i}\leq B_{i}$ for $i=1,...,n$, then $ \frak{G}(\omega; \mathbb{A})
\leq \frak{G}(\omega; \mathbb{B}).$
- $\frak{G}(\omega;\cdot)$ is continuous on each operators. Especially, $$d(\frak{G}(\omega;\mathbb{A}), \frak{G}(\omega;\mathbb{B}))
\leq \sum_{i=1}^{n}w_{i}d(A_{i}, B_{i}).$$
- For each $t\in [0,1]$, $ (1-t)\frak{G}(\omega; \mathbb{A})+
t\frak{G}(\omega; \mathbb{B})
\leq
\frak{G}(\omega; (1-t)\mathbb{A}+t\mathbb{B}).
$
- For any invertible $X\in B(\mathcal{H})$, $ \frak{G}(\omega; X^{*}A_{1}X,...,X^{*}A_{n}X)=
X^{*}\frak{G}(\omega; \mathbb{A})X. $
- $\frak{G}(\omega; \mathbb{A}^{-1})^{-1}=
\frak{G}(\omega; \mathbb{A}), $ where $\mathbb{A}^{-1}=(A_{1}^{-1},..., A_{n}^{-1})$.
- If every $A_{i}$ is a positive definite matrix, then $\det \frak{G}(\omega; \mathbb{A})=
\prod_{i=1}^{n} \det A_{i}^{w_{i}}. $
- $$\left[ \sum_{i=1}^{n}w_{i}A_{i}^{-1}\right]^{-1}
\leq \frak{G}(\omega; \mathbb{A})
\leq
\sum_{i=1}^{n}w_{i}A_{i}.$$
Operator means of two variables
===============================
In this section, we shall consider the weighted operator means in the view point of their weight.
\[prop1\] Let $\Phi, f\in OM_{1}$ be non-constant, and let $\sigma$ be an operator mean whose representing function is $\Phi$. If $\Phi'(1)=w\in (0,1)$, then for $A, B\in B(\mathcal{H})_{sa}$, they are mutually equivalent:
1. $(1-w)A\leq wB$,
2. $f(\lambda A+I)\sigma f(-\lambda B+I)\leq I$ holds for all sufficiently small $\lambda \geq 0$.
Theorem \[prop1\] is an extension of the following Theorem \[th:2\] in [@UY2014] by Lemma \[lem2\] introduced in the below. It was shown as a converse of Loewner-Heinz inequality.
\[th:2\] Let $f(t)\in OM_{1}$ be non-constant, and let $A,B\in B(\mathcal{H})_{sa}$. Let $\sigma$ be an operator mean satisfying $! \leq_{1/2} \sigma\leq \nabla_{1/2}$. Then $A\leq B$ if and only if $f(\lambda A+I) \sigma f(-\lambda B+I) \leq I $ for all sufficiently small $\lambda\geq 0$.
To prove Theorem \[prop1\], we need the following lemma.
\[lem2\] Let $\Phi\in OM_{1}$. Then for each $w\in (0,1)$, they are mutually equivalent:
1. $\Phi'(1)=w$,
2. $[ (1-w)+wt^{-1}]^{-1}\leq \Phi(t)\leq (1-w)+wt $ for all $t\in (0,\infty)$.
Proof of (1) $\Longrightarrow$ (2) has been given in [@Ppreprint2014 Lemma 2.2]. But we shall introduce its proof for the reader’s convenience. Since every operator monotone function is operator concave, $\Phi$ is a concave function. We have $$\Phi(t)\leq \Phi(1)+\Phi'(1)(t-1)=(1-w)+wt.$$ On the other hand, $\frac{t}{\Phi(t)}$ is also an operator monotone function, and $$\frac{d}{dt}\frac{t}{\Phi(t)} \bigg|_{t=1}=1-w.$$ Then by the same argument as above, we have $$\frac{t}{\Phi(t)}\leq w+(1-w)t,$$ that is, $$[ (1-w)+wt^{-1}]^{-1}\leq \Phi(t).$$ Conversely, we shall prove (2) $\Longrightarrow$ (1). Since the tangent line of $ f(t)=[ (1-w)+wt^{-1}]^{-1}$ at $t=1$ is $y=(1-w)+wt$, $\Phi(t)$ has the same tangent line of $ [ (1-w)+wt^{-1}]^{-1}$ at $t=1$. Therefore $\Phi'(1)=w$.
Before proving Theorem \[prop1\], we introduce the following formulas. For any differential function $f$ on $1$ and $w\in (0,1)$, the following hold in the norm topology. $$\begin{aligned}
& \lim_{\lambda\to 0}f(\lambda A+I)^{\frac{1}{\lambda}}=
e^{f'(1)A} \text{ for $A\in B(\mathcal{H})_{sa}$},
\label{eq: limit-f(1)}\\
& \lim_{p\to 0} \left[(1-w)A^{p}+wB^{p}\right]^{\frac{1}{p}}=
\exp\left[ (1-w)\log A+w\log B\right]
\text{ for $A,B\in B(\mathcal{H})_{+}$},
\label{eq:limit-exp}\end{aligned}$$ where can be obtained by $\lim_{\lambda\to 0}f(\lambda a+1)^{\frac{1}{\lambda}}=e^{f'(1)a}$ for $a\in \mathbb{R}$, and is introduced in [@NC1988 (2.4)], for example.
By Lemma \[lem2\], $\Phi'(1)=w$ is equivalent to $$[(1-w)+wt^{-1}]^{-1}\leq \Phi(t) \leq (1-w)+wt
\quad\mbox{for all } t>0.
\label{eq:Harmonic-Arithmetic means}$$
We shall prove (1) $\Longrightarrow$ (2). If $(1-w)A\leq wB$, then it is equivalent to $(1-w)(\lambda A+I)+w(-\lambda B+I)\leq I$ for all $\lambda \geq 0$. Since $f$ is an operator concave function with $f(1)=1$, we have $$\begin{aligned}
I=f(I) & \geq f\left( (1-w)(\lambda A+I)+w(-\lambda B+I)\right) \\
& \geq (1-w) f(\lambda A+I)+wf(-\lambda B+I) \\
& \geq f(\lambda A+I)\sigma f(-\lambda B+I),\end{aligned}$$ where the last inequality holds by .
Conversely, assume that $f(\lambda A+I)\sigma f(-\lambda B+I)\leq I$ for all sufficiently small $\lambda \geq 0$. By , we have $$\begin{aligned}
I & \geq f(\lambda A+I)\sigma f(-\lambda B+I) \\
& \geq
\left[ (1-w)f(\lambda A+I)^{-1}+wf(-\lambda B +I)^{-1}\right]^{-1} \\
& \geq
\left[ (1-w)f(\lambda A+I)^{\frac{-p}{\lambda}}+wf(-\lambda B+I)^{\frac{-p}{\lambda}}\right]^{\frac{-\lambda}{p}}\end{aligned}$$ for all $0<\lambda \leq p$, where the last inequality follows from the operator concavity of $t^{\alpha}$ for $\alpha\in [0,1]$. Then we have $$\left[ (1-w)f(\lambda A+I)^{\frac{-p}{\lambda}}+wf(-\lambda B+I)^{\frac{-p}{\lambda}}\right]^{\frac{-1}{p}}\leq I.$$ By letting $\lambda \to 0$ and , we have $$\left[ (1-w)e^{-pf'(1)A}+we^{pf'(1)B}\right]^{\frac{-1}{p}}\leq I,$$ and $p\to 0$, we have $$\exp\left( -(1-w)f'(1)A+wf'(1)B\right) \geq I$$ by . It is equivalent to $(1-w)A\leq wB$.
A kind of a converse of Theorem \[prop1\] can be considered as follows.
\[prop-converse\] Let $\Phi, f\in OM_{1}$ be non-constant, and let $\sigma$ be an operator mean whose representing function is $\Phi$. For $A, B\in B(\mathcal{H})_{sa}$ and $w\in (0,1)$, if $f(\lambda A+I)\sigma f(-\lambda B+I)\leq I$ holds for all sufficiently small $\lambda \geq 0$ whenever $(1-w)A\leq wB$. Then $\Phi'(1)=w$.
We may assume $f'(1)>0$. Let $A=wtI$ and $B=(1-w)tI$ for a real number $t$. Then we have $(1-w)A\leq wB$. By the assumption, we have $f(\lambda tw+1)\sigma f(-\lambda t(1-w)+1)\leq 1 $ holds for all sufficiently small $\lambda \geq 0$. It is equivalent to $$\Phi \left(\frac{f(-\lambda t(1-w)+1)}{f(\lambda tw+1)}\right)\leq
\frac{1}{f(\lambda tw+1)}.$$ For each $\lambda >0$, we have $$\frac{ \Phi \left(\frac{f(-\lambda (1-w)t+1)}{f(\lambda tw+1)}\right)-1}
{\lambda}\leq
\frac{\frac{1}{f(\lambda wt+1)}-1}{\lambda}.$$ Letting $\lambda \to 0$, the right-hand side of the above inequality converges to $$\frac{\partial}{\partial \lambda} \frac{1}{f(\lambda wt+1)}\biggl|_{\lambda =0}
=
\frac{-wtf'(\lambda wt+1)}{f(\lambda wt+1)^{2}} \biggl|_{\lambda =0}=
-wtf'(1)$$ by the assumption $f(1)=1$. On the other hand, the left-hand side is $$\frac{\partial}{\partial \lambda}
\Phi \left(\frac{f(-\lambda (1-w)t+1)}{f(\lambda tw+1)}\right) \biggl|_{\lambda=0}=
-t\Phi'(1)f'(1)$$ by the assumption $\Phi(1)=1$. Hence we have $t\Phi'(1)\geq wt$ for all real number $t$. Hence we have $\Phi'(1)=w$.
More than three operators case
==============================
Let $\mathbb{A}=(A_{1},...,A_{n})\in B(\mathcal{H})_{+}^{n}$ and $\omega=(w_{1},...,w_{n})\in \Delta_{n}$. Define $$\displaystyle \frak{A}(\omega; \mathbb{A})=\sum_{i=1}^{n}w_{i}A_{i}\quad \mbox{and}\quad
\displaystyle \frak{H}(\omega; \mathbb{A})=
\left(\sum_{i=1}^{n}w_{i} A_{i}^{-1}\right)^{-1}.$$ As an extension of the Karcher mean, the power mean is given by Lim-Páifia [@LP2012] as follows. Let $\mathbb{A}=(A_{1},...,A_{n})\in B(\mathcal{H})_{+}^{n}$ and $\omega=(w_{1},...,w_{n})\in \Delta_{n}$. For $t\in (0,1]$, the power mean $P_{t}(\omega; \mathbb{A})$ is defined by the unique positive definite solution of $$X=\sum_{k=1}^{n}w_{k} X\sharp_{t}A_{k},$$ and for $t\in[-1,0)$, the power mean $P_{t}(\omega; \mathbb{A})$ is defined by $P_{t}(\omega; \mathbb{A})=P_{-t}
(\omega; \mathbb{A}^{-1})^{-1}$ (see also [@LL2014]). We remark that $P_{t}(\omega;\mathbb{A})$ converges to the Karcher mean $\Lambda(\omega; \mathbb{A})$ as $t\to 0$, strongly. So we can consider $P_{0}(\omega; \mathbb{A})$ as $\Lambda(\omega; \mathbb{A})$. It is known that the Karcher mean also satisfies all properties (P1) – (P10) in Section 2 (cf. [@BH2006; @LL2010; @LL2014]). It is easy to see that $P_{1}(\omega; \mathbb{A})=
\frak{A}(\omega; \mathbb{A})$ and $P_{-1}(\omega; \mathbb{A})=
\frak{H}(\omega; \mathbb{A})$. Moreover $P_{t}(\omega; \mathbb{A})$ is increasing on $t\in [-1,1]$. Hence the power mean interpolates arithmetic-geometric-harmonic means. In [@UY2014], we had a generalization of Theorem \[th:2\] as follows.
\[thm: power mean\] Let $T_{1},..., T_{n}$ be Hermitian matrices, and $\omega=(w_{1},...,w_{n})\in \Delta_{n}$. Let $f \in OM_{1}$ be non-constant. Then the following assertions are equivalent:
1. $\displaystyle \sum_{i=1}^{n} w_{i} T_{i}\leq 0$,
2. $\displaystyle P_{1}(\omega; f(\lambda T_{1}+I),...,f(\lambda T_{n}+I))=\sum_{i=1}^{n}w_{i} f(\lambda T_{i}+I)\leq I$ for all sufficiently small $\lambda\geq 0$,
3. for each $t\in [-1,1]$, $P_{t} (\omega; f(\lambda T_{1}+I),..., f(\lambda T_{n}+I))\leq I$ for all sufficiently small $\lambda \geq 0$.
Here we shall generalize the above result into the following Theorem \[thm-extension\].
\[thm-extension\] Let $f\in OM_{1}$ be non-constant, and let $\Phi:\ \Delta_{n}\times B(\mathcal{H})_{+}^{n}\times \mathcal{H}\to \mathbb{R}^{+}$ satisfying $$\| \frak{H}(\omega; \mathbb{A})\| \leq
\sup_{\|x\|=1} \Phi(\omega; \mathbb{A}; x) \leq \| \frak{A}(\omega; \mathbb{A})\|
\label{eq:assumption-extension}$$ for all $\mathbb{A}\in B(\mathcal{H})_{+}^{n}$ and $\omega\in \Delta_{n}$. Then for $\mathbb{T}=(T_{1},...,T_{n})\in B(\mathcal{H})_{sa}^{n}$ and $\omega=(w_{1},...,w_{n})\in \Delta_{n}$, they are mutually equivalent:
1. $\displaystyle \sum_{i=1}^{n}w_{i}T_{i}\leq 0$,
2. $\Phi(\omega; f(\lambda T_{1}+I),..., f(\lambda T_{n}+I); x)\leq 1$ for all sufficiently small $\lambda \geq 0$ and all unit vector $x\in \mathcal{H}$.
In fact, we obtain Theorem \[thm: power mean\] by putting $ \Phi (\omega; \mathbb{A}; x)=\langle P_{t}(\omega; \mathbb{A})x,x\rangle $ in Theorem \[thm-extension\].
First of all, we may assume $f'(1)>0$. Firstly, we shall prove (1) $\Longrightarrow$ (2). For each $\lambda > 0$, (1) is equivalent to $$\sum_{i=1}^{n}w_{i}(\lambda T_{i}+I)\leq I.$$ Since operator concavity of $f$ and $f(1)=1$, we have $$\begin{aligned}
I=f(I)
& \geq
f\left( \sum_{i=1}^{n}w_{i}(\lambda T_{i}+I)\right) \\
& \geq
\sum_{i=1}^{n}w_{i}f(\lambda T_{i}+I)
=
\frak{A}(\omega; f(\lambda T_{1}+I),...,f(\lambda T_{n}+I)).\end{aligned}$$ Here by , $$\begin{aligned}
1 & \geq \| \frak{A}(\omega; f(\lambda T_{1}+I),...,f(\lambda T_{n}+I))\| \\
& \geq
\sup_{\|x\|=1} \Phi(\omega; f(\lambda T_{1}+I),...,f(\lambda T_{n}+I); x),\end{aligned}$$ we have $$1\geq \Phi(\omega; f(\lambda T_{1}+I),..., f(\lambda T_{n}+I); x)$$ for all unit vector $x\in \mathcal{H}$, i.e., (2).
Conversely, we shall prove (2) $\Longrightarrow$ (1). By , we have $$\begin{aligned}
1 & \geq
\sup_{\|x\|=1} \Phi(\omega; f(\lambda T_{1}+I),..., f(\lambda T_{n}+I); x)\\
& \geq
\| \frak{H}(\omega; f(\lambda T_{1}+I),...,f(\lambda T_{n}+I))\| .\end{aligned}$$ Then $$\begin{aligned}
I & \geq
\frak{H}(\omega; f(\lambda T_{1}+I),...,f(\lambda T_{n}+I))\\
& =
\left[\sum_{i=1}^{n}w_{i}f(\lambda T_{i}+I)^{-1}\right]^{-1}
\geq
\left[\sum_{i=1}^{n}w_{i}f(\lambda T_{i}+I)^{\frac{-p}{\lambda}}\right]^{\frac{-\lambda}{p}}\end{aligned}$$ for all $0<\lambda\leq p$ since $t^{\alpha}$ is operator concave for $\alpha\in [0,1]$. Hence we have $$\left[\sum_{i=1}^{n}w_{i}f(\lambda T_{i}+I)^{\frac{-p}{\lambda}}\right]^{\frac{-1}{p}}
\leq I.$$ By letting $\lambda \to 0$ and , we obtain $$\left[\sum_{i=1}^{n}w_{i}e^{-pf'(1)T_{i}}\right]^{\frac{-1}{p}}
\leq I,$$ and $p\to 0$, we have $ f'(1)\sum_{i=1}^{n}w_{i}T_{i}\leq 0$, that is, (1).
\[Corollary norm\] Let $f\in OM_{1}$ be non-constant. Then for $\mathbb{T}=(T_{1},...,T_{n})\in B(\mathcal{H})_{sa}^{n}$ and $\omega=(w_{1},...,w_{n})\in \Delta_{n}$, they are mutually equivalent:
1. $\displaystyle \sum_{i=1}^{n}w_{i}T_{i}\leq 0$,
2. $\prod_{i=1}^{n} \| f(\lambda T_{i}+I)^{\frac{1}{2}}x\|^{w_{i}} \leq 1$ for all sufficiently small $\lambda > 0$ and all unit vector $x\in \mathcal{H}$,
For $\mathbb{A}=(A_{1},...,A_{n})\in B(\mathcal{H})_{+}^{n}$, let $\Phi(\omega; \mathbb{A}; x)=\prod_{i=1}^{n}
\| A_{i}^{\frac{1}{2}}x\|^{2w_{i}}$. We shall only check $$\| \frak{H}(\omega; \mathbb{A})\| \leq \sup_{\|x\|=1}\prod_{i=1}^{n} \| A_{i}^{\frac{1}{2}}x\|^{2w_{i}} \leq
\| \frak{A}(\omega; \mathbb{A})\|$$ for all $\mathbb{A}=(A_{1},...,A_{n})\in B(\mathcal{H})_{+}^{n}$ and $\omega=(w_{1},...,w_{n})\in \Delta_{n}$. Firstly, we shall show $\sup_{\|x\|=1}\prod_{i=1}^{n} \| A_{i}^{\frac{1}{2}}x\|^{2w_{i}} \leq
\| \frak{A}(\omega; \mathbb{A})\|$. $$\begin{aligned}
\prod_{i=1}^{n} \| A_{i}^{\frac{1}{2}}x\|^{2w_{i}}
=
\prod_{i=1}^{n} \langle A_{i}x,x\rangle^{w_{i}}
\leq
\sum_{i=1}^{n} w_{i} \langle A_{i}x,x\rangle
=
\langle \frak{A}(\omega; \mathbb{A})x,x\rangle. \end{aligned}$$ Hence, we have $\sup_{\|x\|=1}\prod_{i=1}^{n} \| A_{i}^{\frac{1}{2}}x\|^{2w_{i}} \leq
\| \frak{A}(\omega; \mathbb{A})\|$.
Next, we shall prove $ \| \frak{H}(\omega; \mathbb{A})\| \leq \sup_{\|x\|=1}\prod_{i=1}^{n} \| A_{i}^{\frac{1}{2}}x\|^{2w_{i}}$. $$\begin{aligned}
\prod_{i=1}^{n} \| A_{i}^{\frac{1}{2}}x\|^{2w_{i}}
& =
\prod_{i=1}^{n} \langle A_{i}x,x \rangle^{w_{i}} \\
& \geq
\langle \Lambda(\omega; \mathbb{A}) x,x\rangle
\quad \text{(by \cite{Y2013})}\\
& \geq
\langle \frak{H}(\omega; \mathbb{A})x,x\rangle.\end{aligned}$$ Therefore the proof is completed by Theorem \[thm-extension\].
Corollary \[Corollary norm\] is an extension of the following result:
\[thm: Karcher mean non-weighted\] Let $T_{1},..., T_{n}$ be Hermitian matrices, and let $f\in OM_{1}$ be non-constant. Then the following are equivalent:
1. $\displaystyle \sum_{i=1}^{n}T_{i}\leq 0$,
2. $\displaystyle \|x\|^{n} \leq \prod_{i=1}^{n} \|f(\lambda T_{i}+I)^{\frac{-1}{2}} x\|$ for all sufficiently small $\lambda\geq 0$ and all $x\in \mathcal{H}$.
From here we shall consider another geometric mean for $n$-tuples of positive definite operators which is called the log-Euclidean mean $\frak{G}_{E}(\omega; \mathbb{A})$ for $\mathbb{A}=(A_{1},...,A_{n})\in B(\mathcal{H})_{+}^{n}$ and $\omega=(w_{1},...,w_{n})\in \Delta_{n}$. It is defined by $$\frak{G}_{E}(\omega; \mathbb{A})=\exp\left(\sum_{i=1}^{n}w_{i}\log A_{i}\right).$$ Log-Euclidean mean satisfies some of properties (P1)–(P10) in Section 2. However, log-Euclidean mean does not satisfy important properties (P4) and (P10).
\[cor1\] Let $f\in OM_{1}$ be non-constant. For $\mathbb{A}\in B(\mathcal{H})_{+}^{n}$ and $\omega\in \Delta_{n}$, let $M(\omega; \mathbb{A})$ be ALM or weighted BMP or log-Euclidean mean (in the ALM mean case, $\omega$ should be $\omega=(\frac{1}{n},...,\frac{1}{n})$). Then for $\mathbb{T}=(T_{1},...,T_{n})\in B(\mathcal{H})_{sa}^{n}$ and $\omega=(w_{1},...,w_{n})\in \Delta_{n}$, the following assertions are equivalent:
1. $\displaystyle \sum_{i=1}^{n}w_{i}T_{i}\leq 0$,
2. $\displaystyle
M(\omega; f(\lambda T_{1}+I),...,f(\lambda T_{n}+I))\leq I$ for all sufficiently small $\lambda\geq 0$.
The cases of ALM and BMP means. Put $\Phi(\omega; \mathbb{A};x)=\langle M(\omega; \mathbb{A})x,x\rangle$. Then by (P10), $\Phi(\omega; \mathbb{A};x)$ satisfies the condition . So that we can prove the cases of ALM and BMP means by Theorem \[thm-extension\].
By the way, log-Euclidean mean satisfies $$\log \frak{H}(\omega; \mathbb{A}) \leq
\log \frak{G}_{E}(\omega; \mathbb{A}) \leq
\log \frak{A}(\omega; \mathbb{A})
\label{log-inequality log-Euclidean mean}$$ for $\mathbb{A}\in B(\mathcal{H})_{+}^{n}$ and $\omega\in \Delta_{n}$. In fact, by the operator concavity of $\log t$, we have $$\begin{aligned}
\log \frak{G}_{E}(\omega; \mathbb{A}) & =
\log \left[ \exp\left(\sum_{i=1}^{n}w_{i}\log A_{i}\right) \right]\\
& =
\sum_{i=1}^{n}w_{i}\log A_{i} \\
& \leq
\log \left(\sum_{i=1}^{n}w_{i}A_{i}\right) =\log \frak{A}(\omega; \mathbb{A}).\end{aligned}$$
On the other hand, we have $$\begin{aligned}
\log \frak{H}(\omega; \mathbb{A}) & =
\log \frak{A}(\omega; \mathbb{A}^{-1})^{-1} \\
& =
- \log \frak{A}(\omega; \mathbb{A}^{-1}) \\
& \leq
- \log \frak{G}_{E}(\omega; \mathbb{A}^{-1}) \\
& =
\log \frak{G}_{E}(\omega; \mathbb{A}).\end{aligned}$$ Hence we have . We remark that if $\log A\leq \log B$ for $A,B\in B(\mathcal{H})_{+}$, then for each $p>0$, there is a unitary operator $U_{p}$ such that $A^{p}\leq U_{p}^{*}B^{p}U_{p}$ in [@F1997]. Hence we have $\|A\|\leq \|B\|$. By using this fact to , we have $$\| \frak{H}(\omega; \mathbb{A})\| \leq
\| \frak{G}_{E}(\omega; \mathbb{A}) \| \leq
\| \frak{A}(\omega; \mathbb{A})\|.$$ Hence we can prove Corollary \[cor1\] by putting $\Phi(\omega; \mathbb{A};x)=\langle \frak{G}_{E}(\omega; \mathbb{A})x,x\rangle$ in Theorem \[thm-extension\].
Logarithmic means
=================
We shall consider some logarithmic means for $n$-tuples of positive definite operators. Since the representing function of logarithmic mean is $\frac{t-1}{\log t}$, logarithmic mean $A\lambda B$ of $A, B\in B(\mathcal{H})_{+}$ can be considered as $$A\lambda B =\int_{0}^{1} A\sharp_{t} B dt.$$ So it is quite natural to consider the similar type of integrated means as follows.
Let $M:\ \Delta_{n}\times B(\mathcal{H})_{+}^{n}\to B(\mathcal{H})_{+}$. Then for $\mathbb{A}\in B(\mathcal{H})_{+}^{n}$, the $M$-logarithmic mean $\frak{L}(M)(\mathbb{A})$ of $\mathbb{A}\in B(\mathcal{H})_{+}^{n}$ is defined by $$\frak{L}(M)(\mathbb{A}):=\int_{\Delta_{n}}
M(\omega; \mathbb{A}) dp(\omega)$$ if there exists, where $dp(\omega)$ means an arbitrary probability measure on $\Delta_{n}$.
In what follows, we consider the case of $dp(\omega)=(n-1)!d\omega$.
\[prop:logarithmic mean\] Let $M: \Delta_{n}\times B(\mathcal{H})_{+}^{n}\to B(\mathcal{H})_{+}$ satisfying (P3), (P7), (P8) and (P10). Then $M$-logarithmic mean $$\frak{L}(M)(\mathbb{A})= (n-1)! \int_{\Delta_{n}} M(\omega; \mathbb{A})d \omega$$ satisfies (P3) and (P7) if it exists. Especially, $\frak{L}(M)$ satisfies (P10), i.e., $$\frak{H}(\mathbb{A})\leq
\frak{L}(M)(\mathbb{A})\leq \frak{A}(\mathbb{A}).$$
We remark that $\frak{L}(\frak{A})(\mathbb{A})=\frak{A}(\mathbb{A})$. As for the preparation, we define some notations. Let $S$ be the cyclic shift operator on $\mathbb{C}^{n}$ and let $\mathbb{S}$ be also the cyclic shift operator on $B(\mathcal{H})^{n}$; namely, $$\begin{aligned}
S(w_{1},w_{2},...,w_{n}) & = (w_{2},w_{3},...,w_{n}, w_{1}). \\
\mathbb{S}(A_{1},A_{2},...,A_{n}) & = (A_{2},A_{3},...,A_{n}, A_{1}).\end{aligned}$$ We claim that if $M$ satisfies (P3), then $M(S\omega; \mathbb{A})=M(\omega; \mathbb{S}^{*}\mathbb{A})$.
It is clear that $\frak{L}(M)$ satisfies (P3) and (P7). The remain is to show (P10). Let $\mathbb{M}$ be the set of all maps $M: \Delta_{n}\times B(\mathcal{H})_{+}^{n}\to B(\mathcal{H})_{+}$. It is easy to show that $\frak{L}$ is a linear map on $\mathbb{M}$, and $\frak{L}(M)(\mathbb{A})\geq 0$ for all $\mathbb{A}\in B(\mathcal{H})^{n}_{+}$ if $M\in \mathbb{M}$. Hence for $N_{1}, M, N_{2}\in \mathbb{M}$, if $N_{1}(\omega; \mathbb{A})\leq M(\omega; \mathbb{A}) \leq
N_{2}(\omega; \mathbb{A})$ holds for all $\omega\in \Delta_{n}$ and $\mathbb{A}\in B(\mathcal{H})_{+}^{n}$, then $$\frak{L}(N_{1})(\mathbb{A}) \leq \frak{L}(M)(\mathbb{A}) \leq \frak{L}(N_{2})(\mathbb{A})$$ holds for all $\mathbb{A}\in B(\mathcal{H})^{n}_{+}$. Since $M(\omega;\mathbb{A})$ satisfies (P10), we have $$\begin{aligned}
\frak{L}(M)(\mathbb{A}) & =
(n-1)!\int_{\Delta_{n}}M(\omega; \mathbb{A})d \omega \\
& \leq
(n-1)!\int_{\Delta_{n}}\frak{A}(\omega; \mathbb{A})d \omega \\
& =
\frak{L}(\frak{A})(\mathbb{A})=
\frak{A}(\mathbb{A}).\end{aligned}$$ On the other hand, we have $$\begin{aligned}
\frak{H}(\mathbb{A})
& =
\frak{A}(\mathbb{A}^{-1})^{-1} \\
& \leq
\{\frak{L}(M)(\mathbb{A}^{-1})\}^{-1} \\
& =
\left( (n-1)!\int_{\Delta_{n}} M(\omega; \mathbb{A}^{-1})d \omega\right)^{-1} \\
& =
\left( (n-1) \int_{\Delta_{n}} M(\omega; \mathbb{A})^{-1}d \omega \right)^{-1}
\quad\mbox{ by (P8)} \\
& \leq
(n-1)! \int_{\Delta_{n}} M(\omega; \mathbb{A})d \omega
=
\frak{L}(M)(\mathbb{A}).\end{aligned}$$
The above theorem is valid for an arbitrary permutation (shift)- invariant probability measure $p$.
Let $M:\ \Delta_{n}\times B(\mathcal{H})_{+}^{n}\to B(\mathcal{H})_{+}$ be a map satisfying (P3), (P7), (P8) and (P10). We put $$M_{0}(\omega; \mathbb{A}):=
M\left( (\frac{1}{n},...,\frac{1}{n});
M(\omega; \mathbb{A}), M(S\omega; \mathbb{A}),...,
M(S^{n-1}\omega; \mathbb{A})\right).$$ Then $M_{0}$ satisfies the assumption of Proposition \[prop:logarithmic mean\]. So $\frak{L}(M_{0})$ also satisfies (P10). Moreover, the following inequalities hold $$\frak{H}(\mathbb{A})\leq \frak{L}(M_{0})(\mathbb{A})\leq
\frak{L}(M)(\mathbb{A})\leq \frak{A}(\mathbb{A}).$$ The second inequality can be shown as follows. Since $M(\omega; \mathbb{A})$ satisfies (P10), we have $$M_{0}(\omega; \mathbb{A})\leq \sum_{k=0}^{n-1}\frac{1}{n}M(S^{k}\omega;\mathbb{A}).$$ Then we obtain $$\begin{aligned}
\frak{L}(M_{0})(\mathbb{A}) & =
(n-1)!\int_{\Delta_{n}} M_{0}(\omega;\mathbb{A})d\omega \\
& \leq
(n-1)!\int_{\Delta_{n}} \left\{ \sum_{k=0}^{n-1}\frac{1}{n}M(S^{k}\omega;\mathbb{A})\right\} d\omega \\
& =
\frac{(n-1)!}{n} \sum_{k=0}^{n-1}\int_{\Delta_{n}} M(S^{k}\omega;\mathbb{A}) d\omega \\
& =
\frac{1}{n} \sum_{k=0}^{n-1}\frak{L}(M)(\mathbb{A})
= \frak{L}(M)(\mathbb{A}).\end{aligned}$$
Since the weighted Karcher mean $\Lambda(\omega;\mathbb{A})$ is continuous on the probability vector in the Thompson metric [@LL2014], so $\frak{L}(\Lambda)(\mathbb{A})$ exists.
\[prop4\] $$\frak{H}(\mathbb{A})\leq \frak{L}(\Lambda)(\mathbb{A})\leq
\frak{A}(\mathbb{A}).$$
Since the weighted Karcher mean satisfies (P1)–(P10) in Section 2 [@BH2006; @LL2010; @LL2014], it is easy by Proposition \[prop:logarithmic mean\].
\[cor5\] Logarithmic mean $\frak{L}(\Lambda)(\mathbb{A})$ satisfies the same assertion to Corollary \[cor1\], too.
We can prove Corollary \[cor5\] by the same way to the proof of Corollary \[cor1\].
[99]{} T. Ando, C.-K. Li and R. Mathias, [*Geometric means*]{}, Linear Algebra Appl., [**385**]{} (2004), 305–334. R. Bhatia and J. Holbrook, *Riemannian geometry and matrix geometric means*, Linear Algebra Appl., [**413**]{} (2006), 594–618. D.A. Bini, B. Meini and F. Poloni, *An effective matrix geometric mean satisfying the Ando-Li-Mathias properties*, Math. Comp., [**79**]{} (2010), 437–452. T. Furuta, [*Characterizations of chaotic order via generalized Furuta inequality*]{}, J. Inequal. Appl., [**1**]{} (1997), 11–24. S. Izumino and N. Nakamura, *Geometric means of positive operators II*, Sci. Math. Jpn., [**69**]{} (2009), 35–44. F. Kubo and T. Ando, [*Means of positive linear operators*]{}, Math. Ann., [**246**]{} (1979/80), 205–224. J.D. Lawson and Y. Lim, *Monotonic properties of the least squares mean*, Math. Ann. 351 (2011), 267–279. J.D. Lawson and Y. Lim, [*Karcher means and Karcher equations of positive definite operators.*]{} Trans. Amer. Math. Soc. Ser. B, [**1**]{} (2014), 1–22. H. Lee, Y. Lim and T. Yamazaki, [*Multi-variable weighted geometric means of positive definite matrices*]{}, Linear Algebra Appl., [**435**]{} (2011), 307–322. Y. Lim and M. Pálfia, [*Matrix power means and the Karcher mean*]{} J. Funct. Anal., [**262**]{} (2012), 1498–1514. R. D. Nussbaum and J. E. Cohen, [*The arithmetic-geometric mean and its generalizations for noncommuting linear operators*]{}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) [**15**]{} (1988), 239–308. M. Pálfia and D. Petz, [*Weighted multivariable operator means of positive definite operators*]{}, Linear Algebra Appl., [**463**]{} (2014), 134–153. M. Uchiyama and T. Yamazaki, [*A converse of Loewner-Heinz inequality and applications to operator means*]{}, J. Math. Anal. Appl., [**413**]{} (2014), 422–429. T. Yamazaki, [*An elementary proof of arithmetic-geometric mean inequality of the weighted Riemannian mean of positive definite matrices*]{}, Linear Algebra Appl., [**438**]{} (2013), 1564–1569.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider Toeplitz operators in different Bergman type spaces, having radial symbols with variable sign. We show that if the symbol has compact support or decays rapidly, the eigenvalues of such operators cannot decay too fast, essentially faster than for a sign-definite symbol with the same kind. On the other hand, if the symbol decays not sufficiently rapidly, the eigenvalues of the corresponding operator may decay faster than for the operator corresponding to the absolute value of the symbol.'
address: |
1. Department of Mathematics\
Chalmers University of Technology\
2.Department of Mathematics University of Gothenburg\
Chalmers Tvärgatan, 3, S-412 96 Gothenburg Sweden
author:
- Grigori Rozenblum
title: On lower eigenvalue bounds for Toeplitz operators with radial symbols in Bergman spaces
---
Introduction {#intro}
============
Toeplitz operators arise in many fields of Analysis. The general setting is the following. Let ${{\mathcal H}}$ be a Hilbert space of functions and ${{\mathcal B}}$ be a closed subspace in ${{\mathcal H}}$. For a function $V$, called the *symbol* further on, the Toeplitz operator $T_V:{{\mathcal B}}\to{{\mathcal B}}$ acts as $T_V: u\mapsto PVu$, where $P$ is the orthogonal projection from ${{\mathcal H}}$ onto ${{\mathcal B}}$. Of course, it is supposed that the operator of multiplication by $V$ maps ${{\mathcal B}}$ into ${{\mathcal H}}$.
In the present paper we consider Toeplitz operators in some Bergman type spaces. Let ${\Omega}$ be a domain in ${{\mathbb R}}^d$, or ${{\mathbb C}}^d$, ${{\mathcal H}}$ be the space $L^2({\Omega})$ with respect to some measure ${\mu}$ and ${{\mathcal B}}$ be the subspace in ${{\mathcal H}}$ consisting of solutions of some elliptic equation or system. The leading example here is provided by Bergman-Toeplitz operators, where ${\Omega}$ is a bounded domain with nice boundary and ${{\mathcal B}}$ consists of harmonic functions in ${{\mathcal H}}$ (the harmonic Bergman space), and, in the complex case, ${{\mathcal B}}$ consists of analytical functions in ${\Omega}$ (the analytical Bergman space). Another series of examples is given by Bargmann-Toeplitz operators, where ${\Omega}$ is the whole (real or complex) space, ${{\mathcal H}}=L^2({\Omega})$ with respect to the Gaussian measure and ${{\mathcal B}}$ consists of harmonic or (in the complex case) analytical functions in ${{\mathcal H}}$.
We are interested in the spectral properties of Toeplitz operators for the case when the symbol $V$, which is supposed to be real and bounded, has compact support (when ${\Omega}$ is a ball) or decays rapidly at infinity (when ${\Omega}$ is the whole space). One can easily see that such operator is compact, and our question is about determining how fast the eigenvalues of $T_V$ tend to zero. The interest to this topic grew recently due to the close relation of the spectral properties of Toeplitz operators to the spectral analysis of the perturbed Landau Hamiltonian describing the quantum particle in a homogeneous magnetic field.
For a sign-definite symbol, in the complex Bargmann case, rather complete results were obtained in [@RaiWar], [@MelRoz] and, in dimension $d=1$, improved in [@FilPush], see also references therein. Even earlier, the case of complex Bergman spaces in dimension $d=1$ has been studied in [@Parf]. It was proved that the eigenvalues of the Toeplitz operator follow an asymptotic law, of an exponential type for the Bergman case and super-exponential type for the Bargmann one.
For the case of the symbol $V$ having variable sign, it was for a long time unclear, even whether it is possible that the positive and negative parts $V_\pm$ of $V$ can compensate each other almost completely, so that the spectrum of $T_V$ is finite. It has been proved only recently, see [@Lue2], that such complete cancelation is impossible, in other words, for a nontrivial symbol $V$ with compact support the Toeplitz operator cannot have finite rank; see [@RozToepl] for the most complete results on the finite rank problem and related references. A further analysis in [@PushRoz2] has shown that the (infinite now) spectrum of the Toeplitz operator depends essentially on the geometry of the support of $V_\pm$. In particular, if, say, the support of $V_+$ surrounds the support of $V_-$ (in a proper meaning) then the negative spectrum of $T_V$ is finite and the asymptotic behavior of the positive eigenvalues is the same as if $V_-$ were absent. On the other hand, if $V_\pm$ are supported in geometrically well separated sets, then both the positive and the negative spectra of $T_V$ are infinite, and, taken together, obey the same asymptotic law as if $V$ were sign-definite (in the Bargmann case); these results can be understood that no cancelation of $V_\pm$ takes place for this class of symbols, as it concerns the properties of the spectrum.
In the present paper we continue the study of the spectrum of Toeplitz operators with non-sign-definite symbol. We consider the model case of the symbol $V$ being radial, i.e., depending only on the the distance to the origin; no restrictions on the supports of $V_\pm$ are imposed. We find out that there is essential difference in the spectral properties of Toeplitz operators with rapidly decaying symbols (including compactly supported ones, the exact definitions are given in the paper), on the one hand, and symbols decaying not that rapidly, on the other. In the former case we establish that, although no information on the positive and negative spectra of $T_V$ separately can be obtained, the distribution function of the positive and negative eigenvalues counted together, i.e. of singular numbers of the operator, is subject to lower asymptotical bounds that have the same order and even the same coefficient as if the symbol were sign definite. This is expressed, in a general form, by the relation . So, again, no cancelation happens. On the other hand, in the latter case it is possible that the eigenvalues of the Toeplitz operator decay considerably faster than for the operator with the corresponding sign-definite symbol.
We start in Section \[Sect2\] with describing the Bergman type spaces under consideration (we consider operators in the Bergman and Bargmann spaces of analytical and harmonic functions as well as in the spaces of solutions of the Helmholtz equation) and finding the expression for the eigenvalues. These expressions are quite explicit. The asymptotic formulas for eigenvalues are obtained, first for $V$ being the characteristic function of a ball, and then these results are carried over to general sign-definite compactly supported case by means of simple monotonicity arguments. For Bargmann spaces a class of symbols with rapid decay at infinity is considered as well. Some of these results are well known, the remaining are obtained in a more or less standard way – we, however, present them all here for further reference. In the end of Section \[Sect2\] we show that, similarly to the results in [@PushRoz2], the same asymptotics holds even for non-sign-definite symbols, as long as geometrically the support of $V_+$ surrounds the support of $V_-$ (or the other way round).
Passing to general non-sign-definite radial symbols, we encounter a serious inconvenience. We still can write the explicit expression for the eigenvalues, however the numbering of the eigenvalues in the non-increasing or the non-decreasing order does not coincide with their natural numbering, stemming from the one in the separation of variables, and the relation between two numberings is rather hard to control. To handle this circumstance, we need certain considerations from infinite combinatorics (Proposition \[Prop.reord.2\]). In order to prove that the eigenvalues cannot decay too fast, we need rather advanced results in complex analysis, and these results are also presented in Sect \[Sect3\]. The main results of the paper on the lower eigenvalue estimates for general non-sign-definite radial symbols are presented in Sect.\[Sect4\], see Theorems \[ThMainBergmanC\], \[ThEstBargC\]. Finally, in Sect.\[Sect5\], we describe examples showing that a considerable cancelation may take place for not sufficiently rapidly decaying fast oscillatory symbols.
Eigenvalues of Toeplitz operators with radial weight {#Sect2}
====================================================
In this Section we calculate the eigenvalue asymptotics for Toeplitz operators with radial symbols in the spaces under consideration. For sign-definite symbols some of these results are known, others are obtained in a standard way using the explicit expression for eigenvalues. Further on, we extend these results to a class of symbols, not necessarily sign-definite, but having a constant sign at the periphery of the support.
Eigenvalues and re-ordering – 1 {#SSreordering1}
-------------------------------
For each of operators $T_V$ under consideration in the paper, we are going to find explicitly the sequence of eigenvalues ${\Lambda}_k$ having multiplicities ${{\mathbf d}}_k$. To describe the ordered set of eigenvalues, counting multiplicities, one should consider the set of the numbers ${\Lambda}_k$, each ${\Lambda}_k$ counted ${{\mathbf d}}_k$ times, and then re-order the positive numbers in the non-increasing way and the negative ones in the nondecreasing way. Thus we obtain two (finite or infinite) sequences ${\lambda}_n^{\pm}$ of eigenvalues of $T_V$. The union of the the sequences $\pm{\lambda}_n^{\pm}$ is the sequence of $s$-numbers, numerated in the nonincreasing order $s_n=s_n(T_V)$.
For problems under consideration, it is often more convenient to describe the spectrum by means of the counting functions defined as $$\label{counting}
n_\pm({\lambda})=\#\{n:\pm{\lambda}_n^{\pm}>{\lambda}\}, \ n({\lambda})=n_+({\lambda})+n_-({\lambda})=\#\{n:s_n>{\lambda}\};$$ we will include the designation of the operator and the space in question in the notation, when needed. In the terms of multiplicities, we, obviously, have $$\label{counting multi}
n_\pm({\lambda})=\sum_{\pm{\Lambda}_k>{\lambda}}{{\mathbf d}}_k, \ n({\lambda})=\sum_{|{\Lambda}_k|>{\lambda}}{{\mathbf d}}_k.$$
In most simple cases below, the sequence ${\Lambda}_k$ is non-negative and already non-increasing, and thus no re-ordering is needed. However, generally, we should not expect that the sequence $|{\Lambda}_k|$ is monotonous, and thus the question arises, how the estimates for $|{\Lambda}_k|$ are related to estimates of this sequence monotonously re-ordered. In one direction, the result is obvious. We, however, formulate it in order to be able to refer to it later on.
\[PropReorderTriv\]Let $a_k, b_k, \ k=0,1,\dots ,$ be two sequence of real numbers, so that $b_k>0,$ $b_k\to 0$ monotonously. Suppose that $|a_k|\le b_k$. Denote by $a^*_k$ the sequence obtained by the non-increasing re-ordering of the sequence $|a_k|$. Then $a^*_k\le b_k.$
Of course, generally, one should not expect the direct conversion of Proposition \[PropReorderTriv\] to be correct: an estimate for the monotonously re-ordered sequence cannot be carried over to the initial sequence. It turns out, however, that in a certain sense Proposition \[PropReorderTriv\] can be partially conversed, see Proposition \[prop.reord.major\].
The operators we consider in this paper have very fast, exponential or even super-exponential rate of decay of eigenvalues. Since these eigenvalues have very high multiplicity, the eigenvalues ${\lambda}_n^{\pm}$, counting multiplicity, do not follow any regular asymptotic law, unlike the well-studied case of elliptic operators. Therefore it is more convenient to consider the behavior of the eigenvalues in the logarithmic scale, where the oscillations caused by high multiplicities are suppressed and a regular asymptotics exists. In this scale, in particular, the leading term of the asymptotics does not change when the symbol $V$ (and thus the eigenvalues of the Toeplitz operator) is multiplied by a positive constant. Alternatively, the asymptotical behavior of eigenvalues can be described by their counting function. Unlike the case of power-like asymptotics, typical for elliptic boundary problems, the asymptotic formula for the counting function is not equivalent to the one for the eigenvalues, however it is equivalent to the asymptotic eigenvalue formula in the logarithmic scale. We will use this equivalence persistently.
We will use the following notation. For functions $f$ and $g$ of a real or integer argument $t$, the symbol $f(t)\sim g(t)$ means, as usual, $f(t)/g(t)\to 1$ as $t\to\infty$ or $t\to 0$, which is always clear from the context. The relation $f(t)\lesssim g(t)$ means that $\limsup f(t)/g(t)\le1$, the obvious meaning has the notation $f(t)\gtrsim g(t)$. Finally, $f(t)\asymp g(t)$ is used when $c f(t)\le g(t)\le C f(t)$ for sufficiently large (or small) $t$ and for some positive constants $c,C.$
Bergman type spaces and quadratic forms
---------------------------------------
### The spaces.
The following Bergman type spaces will be considered in this paper.
1. Bergman spaces.
- The Bergman spaces of analytical functions in the ball. Let $D^{2d}\subset{{\mathbb C}}^d, d\ge1$ be the ball with radius ${{\mathbf R}}$. The space ${{\mathcal H}}$ is the space $L^2(D^{2d})$ with respect to the Lebesgue measure and ${{\mathcal B}}={{\mathcal B}}_{{\mathbf R}}^{{{\mathbb C}}}\subset{{\mathcal H}}$ consists of analytical functions.
\[Another Weight\] In the literature, Bergman spaces with the Lebesgue measure with weight $(1-(|z|/{{\mathbf R}})^2)^{\alpha}$ are considered as well. The results of the paper extend to this case almost automatically.
- The Bergman spaces of harmonic functions in the ball. Let $D^d\subset {{\mathbb R}}^d, d>1,$ be the ball with radius ${{\mathbf R}}$. The space ${{\mathcal H}}$ is $L^2(D^d)$ with respect to the Lebesgue measure and ${{\mathcal B}}={{\mathcal B}}_{{\mathbf R}}^{{{\mathbb R}}}\subset{{\mathcal H}}$ consists of harmonic functions.
- The Bergman spaces of solutions of the Helmholtz equation. The space ${{\mathcal H}}$ is, again, $L^2(D^d)$ and ${{\mathcal B}}={{\mathcal B}}_{{\mathbf R}}^{{{\mathbf H}}}\subset{{\mathcal H}}$ consists of solutions of the Helmholtz equation ${\Delta}u+ u=0$.
2. Bargmann spaces.
- The Bargmann spaces of analytical functions in ${{\mathbb C}}^d$. Here ${{\mathcal H}}$ is $L^2({{\mathbb C}}^d)$ with respect to the Lebesgue measure with Gaussian weight $e^{-|z|^2}$ and ${{\mathcal B}}={{\mathcal B}}^{{\mathbb C}}\subset{{\mathcal H}}$ consists of analytical functions.
- The Bargmann spaces of harmonic functions in ${{\mathbb R}}^d$. Here ${{\mathcal H}}$ is $L^2({{\mathbb R}}^d)$ with respect to the Lebesgue measure with Gaussian weight $e^{-|x|^2}$ and ${{\mathcal B}}={{\mathcal B}}^{{\mathbb R}}\subset{{\mathcal H}}$ consists of harmonic functions.
- The Bargmann spaces of solutions of the Helmholtz equation in ${{\mathbb R}}^d$. Here ${{\mathcal H}}$ is $L^2({{\mathbb R}}^d)$ with respect to the Lebesgue measure with Gaussian weight $e^{-|x|^2}$ and ${{\mathcal B}}={{\mathcal B}}^{{{\mathbf H}}}\subset{{\mathcal H}}$ consists of solutions of the Helmholtz equation.
3. The Agmon-Hörmander space ${{\mathcal H}}=B^*$, see [@AgmonHorm], [@Strich], is defined as consisting of (equivalence classes of) functions $u\in L^2_{{\operatorname{loc}}}({{\mathbb R}}^d)$ such that the norm $$\label{AHspace}
\|u\|_{B^*}=\left(\sup_{r\in(0,\infty)}r^{-1}\int_{|x|<r}|u|^2dx\right)^{\frac12}$$ is finite. All functions $u$ in this space, satisfying the Helmholtz equation, form a closed subspace ${{\mathcal B}}={{\mathcal B}}^{{{\mathbf A}}{{\mathbf H}}}$. For $u\in {{\mathcal B}}^{{{\mathbf A}}{{\mathbf H}}}$, the limit $$\label{AKlim}
|||u|||^2=\lim_{r\to\infty}r^{-1}\int_{|x|<r}|u|^2dx$$ exists, and it defines the norm equivalent to (see [@Strich], Lemma 3.2.)
### Toeplitz operators and quadratic forms
The convenient way to study the eigenvalues of Toeplitz operators is by using the quadratic form setting. Let ${{\mathcal B}}\subset{{\mathcal H}}$ be the Bergman type space under study and $<\cdot,\cdot>$ and $\|\cdot\|$ be the corresponding scalar product and norm. The Toeplitz operator $T_V$ in ${{\mathcal B}}$ is defined by the quadratic form $h_V[u]=<Vu,u>,\ u\in{{\mathcal B}}$. It is convenient to use this definition even in the case when one does not consider the embracing space, as, for example, for ${{\mathcal B}}={{\mathcal B}}^{{{\mathbf A}}{{\mathbf H}}}$: having a Bergman type space ${{\mathcal B}}$ we will still call the operator $T_V$ defined by the quadratic form $\int V|u|^2 d{\mu}$ in ${{\mathcal B}}$ the Toeplitz operator in ${{\mathcal B}}$ with symbol $V$. As soon as a complete system of functions $u_n\in {{\mathcal B}}$ is found, which diagonalizes both quadratic forms $\|u\|^2$ and $h_V[u]$, this system can serve as a complete system of eigenfunctions of $T_V$, with eigenvalues $h_V[u_n]/\|u_n\|^2$. We emphasize here again that these eigenvalues should be properly re-ordered.
In this paper we are going to study Toeplitz operators with radial symbols, i.e., $V(z)=V(|z|)$ in the complex case and $V(x)=V(|x|)$ in the real case. For such symbols the eigenfunctions and eigenvalues of the Toeplitz operator can be found explicitly by means of passing to spherical co-ordinates.
Operators in Bergman spaces
---------------------------
In the sections to follow we collect the results on the eigenvalue asymptotic formulas for Toeplitz operators in Bergman spaces. Some of them are known, the rest are obtained in a standard way, using the explicit expressions for the eigenvalues.
### Complex Bergman spaces. {#ss.cBergman}
Denote by ${{\mathcal P}}^{{\mathbb C}}_k$ the space of homogeneous polynomials of degree $k$ of variables $z_1,\dots, z_d$. It has dimension ${{\mathbf d}}_k^{{{\mathbb C}}}=\binom{k+d-1}{d-1}= \frac{k^{d-1}}{(d-1)!}(1+O(k^{-1}))$. In the space of functions of the form $Z({\omega})=p(z)|z|^{-k}$, $p\in{{\mathcal P}}^{{\mathbb C}}_k$, ${\omega}=z|z|^{-1}\in S^{2d-1}$, we choose a basis $Z_{k,j}({\omega})$, $j\in[1,{{\mathbf d}}_k^{{\mathbb C}}]$, orthonormal with respect to the Lebesgue measure on the sphere $S^{2d-1}$ (complex spherical functions). The functions $u_{k,j}(z)=|z|^kZ_{k,j}({\omega}), \ k=0,1,\dots, \ j\in[1,{{\mathbf d}}_k^{{\mathbb C}}],$ form an orthogonal basis in the space ${{\mathcal B}}_{{\mathbf R}}^{{{\mathbb C}}}$. For a radial function $V(|z|)$, this system of functions diagonalize also the quadratic form $\int V(|z|)|u(z)|^2 d{\mu}$. Therefore, the functions $u_{k,j}$ form a complete system of eigenfunctions of the Toeplitz operator $T_V$ in ${{\mathcal B}}_{{\mathbf R}}^{{{\mathbb C}}}$ with eigenvalues $$\label{Bergman C}
{\Lambda}_k={\Lambda}_k^{{\mathbb C}}(V)=\frac{<Vu_{k,j},u_{k,j}>}{\|u_{k,j}\|^2_{{{\mathcal B}}_{{{\mathbf R}}}^{{{\mathbb C}}}}}=(2k+2d){{\mathbf R}}^{-(2k+2d)}\int_0^{{\mathbf R}}V(r)r^{2k+2d-1}dr,$$ having multiplicity ${{\mathbf d}}_k^{{{\mathbb C}}}$. By re-ordering, see Sect.\[PropReorderTriv\], we obtain two sequences ${\lambda}_n^{\pm}$ of eigenvalues of $T_V$. By the results of [@AlexRoz] (see also [@RozToepl]), at least one of these sequences is infinite.
Let $V$ be the characteristic function of the ball $D_b$ with center at the origin and radius $b\in(0,{{\mathbf R}})$. Then there are no negative eigenvalues, and the positive eigenvalues, by , are ${\Lambda}_k=(b/{{\mathbf R}})^{2k+2d}$ with multiplicities ${{\mathbf d}}_k^{{{\mathbb C}}}$. Taking into account the asymptotics ${{\mathbf d}}_k^{{{\mathbb C}}}\sim k^{d-1 }/(d-1)!$, we have in terms of the counting function, $$\label{asymp.compl.count}
n({\lambda}; T_V,{{\mathcal B}}^{{\mathbb C}}_{{\mathbf R}})\sim \sum\limits_{(b/{{\mathbf R}})^{2k+2d}>{\lambda}}\frac{k^{d-1}}{(d-1)!}\sim (d!)^{-1}(2|\log(b/{{\mathbf R}})|)^{-d}|\log{\lambda}|^d, \ {\lambda}\to 0,$$ or, in the logarithmic scale, $$\label{asymp.compl}
\log({\lambda}_n^+)=\log(s_n(T_V))\sim2 (nd!)^{\frac1d}\log (b/{{\mathbf R}}).$$
### Harmonic Bergman spaces {#ss.hBergman}
Denote by ${{\mathcal P}}^{{\mathbb R}}_k$ the space of degree $k$ homogeneous harmonic polynomials of the variables $x_1,\dots, x_d$. This space has dimension ${{\mathbf d}}_k^{{\mathbb R}}=\binom{d+k-1}{d-1}-\binom{d+k-2}{d-2}= \frac{2}{(d-2)!}k^{d-2}(1+O(k^{-1}))$ (see, e.g., calculations in [@Shubin], Sect.22). In the space of functions of the form $Y({\omega})=p(x)|x|^{-k}$, $p\in{{\mathcal P}}^{{\mathbb R}}_k$, ${\omega}=x|x|^{-1}\in S^{d}$, we choose a basis $Y_{k,j}({\omega})$, $j\in[1,{{\mathbf d}}_k^{{\mathbb R}}]$, orthonormal with respect to the Lebesgue measure on the sphere $S^{d-1}$, i.e., the usual spherical functions. The functions $u_{k,j}(x)=|x|^kY_{k,j}({\omega}), k=0,1,\dots, \ j\in[1,{{\mathbf d}}_k^{{\mathbb R}}],$ form an orthogonal basis in the space ${{\mathcal B}}_{{\mathbf R}}^{{{\mathbb R}}}$. For a radial function $V(|x|)$, this system of functions diagonalizes also the quadratic form $\int V(|x|)|u(x)|^2 d{\mu}$. Therefore, the functions $u_{k,j}$ form a complete system of eigenfunctions of the Toeplitz operator $T_V$ in ${{\mathcal B}}_{{\mathbf R}}^{{{\mathbb R}}}$ with eigenvalues ${\Lambda}_k^{{\mathbb R}}$ given by $$\label{Bergman R}
{\Lambda}_k={\Lambda}_k^{{\mathbb R}}(V)=(2k+d){{\mathbf R}}^{-2k-d}\int_0^{{\mathbf R}}V(r)r^{2k+d-1}dr$$ and multiplicities ${{\mathbf d}}_k^{{\mathbb R}}$. Taking into account the multiplicities and reordering, as in Section \[ss.cBergman\], we obtain the eigenvalue sequences ${\lambda}_n^{\pm}$ and the sequence of $s-$numbers $s_n$. Again, as it is shown in [@AlexRoz], the sequence $s_n$ and at least one of the sequences ${\lambda}_n^{\pm}$ are infinite.
For $V$ being the characteristic function of the ball $D_b$, the eigenvalues ${\Lambda}_k^{{\mathbb R}}$ are equal to ${\Lambda}_k^{{\mathbb R}}=(b/{{\mathbf R}})^{2k+d}$, by . Thus, there are no negative eigenvalues ${\lambda}_n^-$, while for the positive eigenvalues ${\lambda}_n^+$ we have the asymptotics $$\label{asymp.harm}
n({\lambda}; T_V,{{\mathcal B}}^{{\mathbb R}}_{{\mathbf R}})\sim \sum\limits_{(b/{{\mathbf R}})^{2k+d}>{\lambda}}2\frac{k^{d-2}}{(d-2)!}\sim 2((d-1)!)^{-1}(2|\log(b/{{\mathbf R}})|)^{-d+1}|\log{\lambda}|^{d-1}, \ {\lambda}\to 0,$$ and, in the logarithmic scale,
$$\label{asympt.Harm.l}
\log{\lambda}_n^+\sim 2\log(b/{{\mathbf R}})((d-1)!/2)^{\frac{1}{d-1}}n^{\frac{1}{d-1}}, \ n\to\infty.$$
### Helmholtz Bergman spaces {#ss.HelmBergman}
After passing to spherical co-ordinates in the Helmholtz equation, we arrive at the orthogonal system of functions $$\label{SystemHelm}
u_{k,j}(x)=Y_{k,j}({\omega})|x|^{-\frac{d-2}{2}}J_{k+\frac{d-2}{2}}(|x|); \ {\omega}=x|x|^{-1}\in S^{d-1}, k = 0,1,\dots, \ j=1,\dots, {{\mathbf d}}_k^{{\mathbb R}},$$ where $J_{\nu}(r)$ are the Bessel functions and $Y_{k,j}$ are the real spherical functions as in Sect.\[ss.hBergman\]. For a radial symbol $V(|x|)$, the eigenvalues of the Toeplitz operator equal $$\label{EigenHelm}
{\Lambda}_k={\Lambda}_k^{{{\mathbf H}}}(T_V)=\frac{\int_0^{{\mathbf R}}V(r)J^2_{k+\frac{d-2}{2}}(r)rdr}{\int_0^{{\mathbf R}}J^2_{k+\frac{d-2}{2}}(r)rdr},$$ with multiplicity ${{\mathbf d}}_k^{{\mathbb R}}$. The integral in the denominator in is estimated by means of the identity (see, e.g., [@Watson]) $$\label{BesselIntegr}
\int_0^R J^2_{\nu}(r) r dr=\frac{R^2}{2}[J^2_{\nu}(R)-J_{{\nu}-1}(R)J_{{\nu}+1}(R)],$$ and the asymptotics (see, again [@Watson]), uniform in $r$ on any finite interval $[a,b]\subset[0,\infty)$: $$\label{BesselAs}
J_{\nu}(r)\sim\left(\frac{r^2}{2}\right)^{{\nu}}({\Gamma}({\nu}+1))^{-1}, \ |{\nu}|\to +\infty, \operatorname{{\rm Re}\,}{\nu}\ge0.$$ So, we obtain $$\label{BesselintAs}
\int_0^{{\mathbf R}}J^2_{k+\frac{d-2}{2}}(r)rdr\sim \left(\frac{{{\mathbf R}}^2}2\right)^{k+d/2}\frac{1}{{\Gamma}(k+\frac{d}{2}){\Gamma}(k+\frac{d+2}{2})}, \ k\to \infty.$$
Again, as before, the numbers $ {\Lambda}_k^{{{\mathbf H}}},$ counted with multiplicities ${{\mathbf d}}_k^{{\mathbb R}}$ and properly re-ordered, form the sequences ${\lambda}_n^\pm={\lambda}_n^\pm(T_V)$ of eigenvalues of $T_V$, and the union of the sequences $\pm{\lambda}_n^\pm$ is the sequence of $s$-numbers $s_n=s_n(T_V)$. It is proved in [@RozToepl] that for $d>2$ at least one of the sequences ${\lambda}_n^\pm$ is infinite. The proof in [@RozToepl] does not cover the case $d=2$, and the above infiniteness will follow from the results of the present paper.
For $V$ being the characteristic function of the ball $|x|\le b<{{\mathbf R}}$, the numbers ${\Lambda}_k^{{{\mathbf H}}}(V)$ have, by , , and , the asymptotics $$\label{HelmCharAs}
{\Lambda}_k^{{{\mathbf H}}}\sim (b/{{\mathbf R}})^{2k+d }.$$ Therefore, taking into account multiplicities, the eigenvalues ${\lambda}_n^+$ obey the asymptotic law , , the same as for the harmonic Bergman space.
Operators in Bargmann and AH spaces
-----------------------------------
### Complex Bargmann spaces {#ss.cBargmann}
The functions $u_{k,j}(z)=Z_{k,j}({\omega})|z|^k, $ $j\in[1,{{\mathbf d}}_k^{{\mathbb C}}],$ form an orthogonal basis in the Bargmann space ${{\mathcal B}}^{{{\mathbb C}}}.$ Thus, the eigenvalues of the Toeplitz operator $T_V$ in ${{\mathcal B}}^{{{\mathbb C}}}$ equal $$\label{CBargm}
{\Lambda}_k^{{\mathbb C}}=\frac{\int_0^\infty V(r)r^{2k+2d-1}e^{-r^2}dr}{\int_0^\infty r^{2k+2d-1}e^{-r^2}dr}=2\frac{\int_0^\infty V(r)r^{2k+2d-1}e^{-r^2}dr}{{\Gamma}(k+d)}.$$ For the case of $V(r)$ being the characteristic function of $D_b$, $0<b<\infty$, obviously, $$\left|\log\int_0^\infty V(r)r^{2k+2d-1}e^{-r^2}dr\right|\asymp k,$$ and, therefore, by the Stirling formula, $$\label{BargLambda}
| \log {\Lambda}_k^{{{\mathbb C}}}|\sim k\log k.$$ Taking into account the multiplicities, we obtain for the eigenvalues of $T_V$: $$\label{CBargm.eigenvaluesN}
n({\lambda})=\sum_{{\Lambda}_k^{{\mathbb C}}>{\lambda}}{{\mathbf d}}_k^{{\mathbb C}}\sim \frac1{d!}\left(\frac{|\log {\lambda}|}{\log|\log {\lambda}|}\right)^d,$$ or, in the logarithmic scale, inverting : $$\label{CBargm.eigenvalues}
\log({\lambda}_n^+)=\log(s_n)\sim - d^{-1}(d!)^{\frac1d} n^{\frac1d}\log n.$$
\[remWithout b\]The asymptotic relation was found in [@MelRoz] (in [@RaiWar] for $d=1$); it was discovered there, in particular, that the leading term in the eigenvalue asymptotics of Bargmann-Toeplitz operators does not depend on the symbol $V\ge 0$ with compact support (of course, provided it is not identically zero). In [@FilPush], for $d=1$, the second term of the asymptotics in was found, depending on the logarithmic capacity of ${\hbox{{\rm supp}}\,}V$.
### Harmonic Bargmann spaces {#ss.hBargmann}
The functions $u_{k,j}(x)=Y_{k,j}({\omega})|x|^k, $ $j\in[1,{{\mathbf d}}_k^{{\mathbb R}}],$ form an orthogonal basis in the Bargmann space ${{\mathcal B}}^{{{\mathbb R}}}.$ Thus, the eigenvalues of the Toeplitz operator $T_V$ in ${{\mathcal B}}^{{{\mathbb R}}}$ equal $$\label{RBargm}
{\Lambda}_k^{{\mathbb R}}=\frac{\int_0^\infty V(r)r^{2k+d-1}e^{-r^2}dr}{\int_0^\infty r^{2k+d-1}e^{-r^2}dr}=2\frac{\int_0^\infty V(r)r^{2k+d-1}e^{-r^2}dr}{{\Gamma}(k+\frac{d}{2})}.$$ For $V$ being the characteristic function of the interval ball $D_b$ we obtain for the eigenvalues of $T_V$, taking into account the multiplicities: $$\label{RBargm.eigenvalues}
n({\lambda})=\sum_{{\Lambda}_k^{{\mathbb R}}>{\lambda}}{{\mathbf d}}_k^{{\mathbb R}}\sim2((d-1)!)^{-1}\left(\frac{|\log{\lambda}|}{\log|\log{\lambda}|}\right)^{d-1},$$ or, in the logarithmic scale, inverting , $$\label{RBargm.eigenvaluesL}
\log({\lambda}_n^+)=\log(s_n)\sim -(d-1)^{-1}((d-1)!/2)^{\frac{1}{d-1}}(n^{\frac1{d-1}}\log n), \ n\to\infty.$$
### Helmholtz Bargmann spaces {#ss.HelmBargmann}
The functions $$u_{k,j}(x)=Y_{k,j}({\omega})|x|^{-\frac{d-2}{2}}J_{k+\frac{d-2}{2}}(|x|); \ k = 0,1,\dots, \ j=1,\dots, {{\mathbf d}}_k^{{\mathbb R}},$$ form an orthogonal basis in the space ${{\mathcal B}}^{{{\mathbf H}}}.$ Thus, the eigenvalues of $T_V$ in ${{\mathcal B}}^{{{\mathbf H}}}$ equal $$\label{HarmBargm}
{\Lambda}_k={\Lambda}_k^{{{\mathbf H}}}(V)=\frac{\int_0^\infty V(r)J_{k+\frac{d-2}{2}}(r)^2re^{-r^2}dr}{\int_0^\infty J_{k+\frac{d-2}{2}}(r)^2 e^{-r^2}rdr},$$ with multiplicity ${{\mathbf d}}_k^{{\mathbb R}}.$ The denominator in equals $\frac12\exp(-\frac12)I_{k+\frac{d-1}{2}}(\frac12)$, where $I_{\nu}$ is the modified Bessel function (see [@Grad], 6.663.2). By , this denominator has the asymptotics $\left(\frac{1}2\right)^{k+\frac{d-1}{2}}\exp(-\frac12){\Gamma}(k+\frac{d+1}{2})^{-1}.$
For $V$ being the characteristic function of the ball $D_b$, the numerator in is estimated from above and from below by constants times $b^{2k+d}\left({\Gamma}(k+\frac{d}{2}){\Gamma}(k+\frac{d+1}{2})\right)^{-1}.$ Therefore, in this case, the eigenvalues ${\Lambda}_k^{{{\mathbf H}}}$ obey two-sided asymptotic estimates $$\label{L Helm Barg}
{\Lambda}_k^{{{\mathbf H}}}\asymp \left(\frac{1}2\right)^{k+\frac{d}{2}}b^{2k+d}({\Gamma}(k+\frac{d+1}{2}))^{-1}.$$ Taking into account multiplicities, the eigenvalues of the Toeplitz operator $T_V$ in the space ${{\mathcal B}}^{{\mathbf H}}$ have the same asymptotics , as for the harmonic Bargmann space.
### The Agmon-Hörmander space {#ssAHspace}
The functions $$u_{k,j}(x)=Y_{k,j}({\omega})|x|^{-\frac{d-2}{2}}J_{k+\frac{d-2}{2}}(|x|); \ k = 0,1,\dots, \ j=1,\dots, {{\mathbf d}}_k^{{\mathbb R}},$$ form an orthogonal basis in the space ${{\mathcal B}}^{{{\mathbf A}}{{\mathbf H}}}$. The ${{\mathbf A}}{{\mathbf H}}$ norm of these functions equals $\frac1\pi$ (see, e.g., [@Strich], p. 63). Thus, the eigenvalues of the Toeplitz operator $T_V$ in ${{\mathcal B}}^{{{\mathbf A}}{{\mathbf H}}}$ equal $$\label{EigenvAH}
{\Lambda}_k^{{{\mathbf A}}{{\mathbf H}}}=\pi \int_0^\infty V(r)J_{k+\frac{d-2}{2}}(|r|)^2 rdr.$$ For $V$ being the characteristic function of $D_b$, these eigenvalues have the asymptotics $$\label{Eigenv.AH}
{\Lambda}_k^{{{\mathbf A}}{{\mathbf H}}}\sim\pi\left(\frac{b^2}{k}\right)^{k+\frac{d-2}{2}}({\Gamma}(k+\frac{d}{2}))^{-2}.$$ So, the eigenvalues of the Toeplitz operator in the space ${{\mathbf A}}{{\mathbf H}}$ decay considerably faster than in the space ${{\mathcal B}}^{{\mathbf H}}$, with the same symbol. Counting multiplicities, we obtain for the eigenvalues of $T_V$ the asymptotics $$\label{Eigenv.AH.log}
\log {\lambda}_n=\log(s_n)\sim -\frac1{d-1}((d-1)!/2)^{\frac{2}{d-1}}(n^{\frac2{d-1}}\log n), \ n\to\infty$$ and $$\label{Eigenv.AH.N}
n({\lambda})\sim 2((d-1)!)^{-1}\left(\frac{|\log{\lambda}|}{\log|\log{\lambda}|}\right)^{\frac{d-1}2}.$$
Sign-definite radial symbols with compact support {#compsupp}
-------------------------------------------------
We introduce the following notion.
\[exactSupport\] Let the function $V(r), \ r\ge0,$ have compact support. The number $b$ is called the *exact support radius* (ESR) for $V$ if $V(r)=0$ for $r>b$, while for any $b'\in(0,b)$, $$\int_{b'}^b|V(r)| dr >0.$$
\[PropNonneg\] Suppose that $b>0$ is the ESR for $V\ge0$. Then for the operator $T_V$ in the spaces ${{\mathcal B}}_{{\mathbf R}}^{{\mathbb C}}, {{\mathcal B}}_{{\mathbf R}}^{{\mathbb R}}, {{\mathcal B}}_{{\mathbf R}}^{{\mathbf H}}, {{\mathcal B}}^{{\mathbb C}}, {{\mathcal B}}^{{\mathbb R}}, {{\mathcal B}}^{{\mathbf H}}, {{\mathcal B}}^{{{\mathbf A}}{{\mathbf H}}}$ hold the asymptotic formulas , respectively, , , , , , and (as well as the corresponding asymptotic formulas for the counting function.)
In all cases, the asymptotic estimate from above is trivial, since the numerator in the expression for the eigenvalues ${\Lambda}_k$ increases when $V$ is replaced by the characteristic function of the ball with radius $b$, multiplied by some positive constant, and this constant is not felt in the logarithmic scale. As for the lower estimates, the reasoning is similar for all cases. We present it, as an example, for operator in the space ${{\mathcal B}}_{{\mathbf R}}^{{\mathbb C}}$.
For the operator $T_V$ in ${{\mathcal B}}_{{\mathbf R}}^{{\mathbb C}}$, fix some $b'<b.$ We have $$\label{lower.posit.C.Be}
\int_0^b V(r)r^{2k+2d-1}dr\ge \int_{b'}^b V(r) r^{2k+2d-1}dr\ge
(b')^{2k+2d-1}\int_{b'}^b V(r)dr.$$ Passing to the logarithmic scale, we obtain $$\log({\lambda}_n^+)=\log(s_n(T_V))\gtrsim n^{1/d}(d!)^{\frac1d}\log (b'/{{\mathbf R}}),$$ which gives the required lower asymptotic estimate, due to the arbitrariness of $b'$.
Rapidly decaying sign-definite symbols
--------------------------------------
\[RapidDecayDef\] A bounded function $V(r), \ r\in[0,\infty),$ is called rapidly decaying, $V\in{{\mathcal R}}{{\mathcal D}},$ if $$\label{RD1}
V(r)=o(\exp(-r^{{\varsigma}})), \ r\to\infty \mbox{ for any } {\varsigma}>0,$$ or, equivalently, $\log |V(r)|<-Cr^{\varsigma}, C>0,$ for any ${\varsigma}>0.$
For further reference, we formulate here an important property of functions in ${{\mathcal R}}{{\mathcal D}}$, which is easily established by a proper change of variables.
\[LemRD\]If $V\in{{\mathcal R}}{{\mathcal D}}$, then $$\label{RD2}
\left|\int_0^\infty V(r)r^s dr\right|=O({\Gamma}({\epsilon}s)), \ s\to\infty,$$ for any ${\epsilon}>0$.
Let $V\ge0$, $V\in {{\mathcal R}}{{\mathcal D}}$. Then for the eigenvalues of the operator $T_V$ in the spaces ${{\mathcal B}}^{{\mathbb C}}, {{\mathcal B}}^{{\mathbb R}}, {{\mathcal B}}^{{\mathbf H}}, {{\mathcal B}}^{{{\mathbf A}}{{\mathbf H}}}$ the eigenvalue asymptotic formulas , respectively, , , and hold (as well as the corresponding asymptotic formulas for the counting function.)
Consider the complex Bargmann space first. If $V\in{{\mathcal R}}{{\mathcal D}}$, $V\ge0$, the numbers ${\Lambda}_k^{{{\mathbb C}}}$ are given by the same formula . By monotonicity and Lemma \[LemRD\], $$\label{RD3C}
|\log \left|\int_0^\infty V(r)r^k dr\right||=o(k\log k), \ k\to\infty,$$ and therefore, for the numbers $\log {\Lambda}_k^{{{\mathbb C}}}$ we have the same asymptotics as for a compactly supported symbol, which leads to the asymptotics for the counting function of the operator $T_V$. The same reasoning takes care of the space ${{\mathcal B}}^{{\mathbb R}}$.
Consider now the operator in the space ${{\mathcal B}}^{{\mathbf H}}.$ For a general $V\in {{\mathcal R}}{{\mathcal D}}$, for estimating the numerator in , we use the representation formula for Bessel functions, $$\label{BeselIntRepr}
J_{\nu}(r)=\left(\frac{r}2\right)^{{\nu}}\left[{\Gamma}({\nu}+1/2){\Gamma}(1/2)\right]^{-1}\int_{-1}^1 (1-t^2)^{{\nu}-1/2}\cos(rt) dt, \ \operatorname{{\rm Re}\,}{\nu}>-\frac12,$$ see, e.g., [@Grad], 8.411.8. It follows from that $|J_{k+\frac{d-2}{2}}(r)|\le C r^{k+\frac{d-2}{2}}{\Gamma}(k+\frac{d-1}{2})^{-1}$; substituting this bound into , we obtain $$\label{HelmEigenv}
{\Lambda}_k\le C 2^k ({\Gamma}(k+\frac{d+1}{2}))^{-1}\left[\int_0^\infty V(r) r^{k+d}dr\right]^2.$$ By Lemma \[LemRD\], the integral in is majorated by ${\Gamma}({\epsilon}(k+d))$ for any ${\epsilon}>0$. So, in logarithmic scale, $$\label{HelmEigenv1}
\log {\Lambda}_k\sim-\log({\Gamma}(k+\frac{d+1}{2}))+o( k\log k),$$ which gives the same asymptotics for $\log {\Lambda}_k$, as in the case of $V$ with compact support. So, for $V\in {{\mathcal R}}{{\mathcal D}}$ the same asymptotics , hold.
The same reasonings takes care of operators in the Agmon-Hörmander spaces.
\[RDRem\] So, the asymptotic formulas for $n({\lambda})$ are the same for a compactly supported $V$ and for $V\in{{\mathcal R}}{{\mathcal D}}$. On the other hand, it was established in [@RaiWar] (see Theorem 2.1 there) that if a reasonably regular $V$ *does not* belong to ${{\mathcal R}}{{\mathcal D}}$, the asymptotics of $n({\lambda}; {{\mathcal B}}^{{\mathbb C}})$ is different. This circumstance justifies the introduction of the class ${{\mathcal R}}{{\mathcal D}}$. Further on, in Section \[Sect5\] we consider oscillating symbols not belonging to ${{\mathcal R}}{{\mathcal D}}.$
Radial symbols, sign-definite at the periphery
----------------------------------------------
As it was found in [@PushRoz2] for Toeplitz operator in ${{\mathcal B}}^{{\mathbb C}}$ in dimension $d=1$, the asymptotics of eigenvalues is determined only by the sign of $V$ at the periphery of its support. It turns out that such effect is present in other dimensions and other spaces as well. We explain the corresponding results for the case of a radial symbol, however, with a proper formulation, they hold also in much more general case.
\[PropPeriph\] Suppose that the ESR for the function $V(r)$ equals $b>0$, $0<b\le\infty$, and for some $b_0<b$, $V(r)\ge0$ for $r\in(b_0,b).$ Then for such $V$ there are only finitely many negative eigenvalues and the assertion of Proposition \[PropNonneg\] holds true.
The proof follows the ideas of Theorem 1.1 in [@PushRoz2]. All cases are treated in a similar way, so we consider only the operator in ${{\mathcal B}}^{{{\mathbf H}}}$ as an example.
The upper asymptotic estimate is, again, trivial. For the lower estimate, fix $b'\in(b_0,b)$. Let $G_R(x,y)$ be the Green function for the Dirichlet problem in the ball $D_R:r<R$ for the Helmholtz equation. Such function exists as long as zero is not an eigenvalue of the Helmholtz operator with Dirichlet boundary conditions in $D_R$. Such exceptional values of $R$ form a discrete set, therefore we can find an interval $(b_1,b_2)\subset(b',b),$ such that $G_R$ exists for all $R\in(b_1,b_2)$. By our condition, the interval $(b_1,b_2)$ can be also chosen in such way that $\int_{b_1}^{b_2}V(r)dr>0$. Note that the function $G_R(x,y)$ is smooth for $x\ne y$.
Let $u(x)$ be a solution of the Helmholtz equation in ${{\mathbb R}}^d$. For $x\in D_{b_0}$ and $R\in(b_1,b_2)$, the following integral representation is valid $$\label{integrRepr}
u(x)=\int_{\partial D_R}u(y)K(x,y;R)dS_R(y),$$ where $K(x,y;R)=G_{{\nu}(y)}(x,y)$ is the derivative of $G$ in the direction of the outer normal to $\partial D_R$ at the point $y\in \partial D_R$ and $dS_R(y)$ is the normalized surface measure on $\partial S_R(y)$. We multiply by $V(R)R^{d-1}$ and integrate in $R\in(b_1,b_2)$. Thus we obtain the integral representation $$\label{integrRepr2}
u(x)= \left(\int_{b_1}^{b_2} V(R)R^{d-1} dR\right)^{-1}\int_{|y|\in(b_1,b_2)}K(x,y;R)u(y)V(|y|)dy.$$ Since $K(x,y;R)$ is a smooth bounded function for $|x|\le b_0$ and $|y|\ge b_2$, the integral operator assigning the function $u(x)$, restricted to $D_{b_0} $ and considered as an element in $L^2(D_{b_0})$, to the same function considered as an element of $L^2(D_{b_0})$ with weight $V$, is compact. Therefore, the quadratic form $$\label{QformPlusMinus}
{{\mathbf a}}_{b_0}[u]\equiv \int_{D_{b_0}}V(|x|)|u(x)|^2dx$$ is compact with respect to the quadratic form $$\label{Qformplus}
\int_{D_{b}\setminus D_{b_0}}V(|x|)|u(x)|^2dx,$$ all forms, recall, being considered on the space of solutions of the Helmholtz equation.
Now we represent the quadratic forms ratio for the operator $T_V$ as $$\label{qFormRatio}
\frac{<T_V u,u>}{<u,u>}=\left[1+\frac{\int_{D_{b_0}}V|u|^2 dx}{\int_{|x|>b_0} V|u|^2 dx}\right]\frac{\int_{D_b}V_{b_0}|u|^2 dx}{<u,u>}, \ u\in{{\mathcal B}}^{{{\mathbf H}}},$$ where $V_{b_0}(r)=0,\ r<b_0,$ $V_{b_0}(r)=V(r)$ otherwise.
Due to the compactness, explained above, for any ${\epsilon}>0$, there exists a subspace ${{\mathcal L}}_{\epsilon}\subset {{\mathcal B}}^{{{\mathbf H}}}$, having finite dimension ${\kappa}({\epsilon})<\infty$, and such that $$\label{qFormRatio2}
\left|\frac{\int_{D_{b_0}}V|u|^2 dx}{\int_{|x|>b_0} V|u|^2 dx}\right|<{\epsilon}\ {\rm{for\ }} u\in {{\mathcal B}}^{{{\mathbf H}}}, {\rm{ orthogonal\ to\ }} {{\mathcal L}}_{\epsilon}.$$ For ${\epsilon}=1/2$, this means that there are no more than ${\kappa}({1/2})$ negative eigenvalues of $T_V$. Further on, by the variational principle, for the positive eigenvalues of $T_V$ the estimate holds $${\lambda}_n^+(T_V)\ge \frac12{\lambda}_{n+{\kappa}({1/2)}}(T_{V_{b_0}}).$$ Now the required lower estimate follows from Proposition \[PropNonneg\] applied to the nonnegative symbol $V_{b_0}$.
Auxiliary theorems {#Sect3}
==================
Re-ordering – 2 {#ss.Reordering2}
---------------
As it was explained in the Introduction, the main complication for proving lower estimates for eigenvalues lies in the need of reordering of the sequence of eigenvalues ${\Lambda}_k$ obtained by the explicit formulas in Sect. \[Sect2\]. So, supposing that the lower estimate is wrong, and thus a contradicting upper estimate holds, we can obtain a bound for the re-ordered sequence of the numbers ${\Lambda}_k$, which, however, does not imply directly any estimate for the numbers ${\Lambda}_k$ themselves. In order to deal with this circumstance, we need the following statement which plays a key role in the sequel.
\[Prop.reord.2\] Let $k\mapsto m_k$ be a bijection of the set of nonnegative integers ${{\mathbb Z}}_+$. For ${\beta}>1$, we denote by $E_{\beta}$ the set $\{k\in{{\mathbb Z}}_+:m_k\le{\beta}k\}$, $F_{\beta}=\{m_k: k\in E_{\beta}.\}$. Then $$\label{reorder}
\#\{F_{\beta}\cap[0,N]\}\ge \frac{{\beta}-1}{{\beta}} N,$$ for any natural $N.$
In other words, the Proposition states that, under a bijection, a controllably nonzero share of integers $m_k$ are not too large, compared with $k.$
Suppose that $m_k\in[0,N]\setminus F_{\beta}$. Then $k<m_k/\beta\le N/\beta$. And therefore $\#\{F_{\beta}\cap[0,N]\}> N+1-N/\beta\ge \frac{{\beta}-1}{{\beta}} (N+1) -\frac{1}{\beta}.$
\[RemReorder2\] The constant $\frac{{\beta}-1}{{\beta}}$ in Proposition \[Prop.reord.2\] is sharp. In fact, set $m_k=[\beta k]+1$ for $k\in\mathbb N\setminus2^{\mathbb N}$, while for integers powers of $2$ we define $m_k$ so that to obtain a bijection. Then $\limsup N^{-1}\#\{F_{\beta}\cap[0,N]\}=\frac{{\beta}-1}{{\beta}}.$
Proposition \[Prop.reord.2\] leads to the following partial conversion of Proposition \[PropReorderTriv\], mentioned in Sect. \[SSreordering1\].
\[prop.reord.major\] Suppose that $a_k, b_k$ are real sequences, $b_k>0$ is non-increasing, and for the non-increasing permutation $a^*_k$ of $|a_k|$, we have $$\label{reord.major}
a^*_k\le b_k.$$ Then for any ${\beta}>1$ there exists a subsequence $a_{k_l}$, $k_1<k_2<\dots$, such that $$\label{reord.major.1}
|a_{k_l}|\le b_{[k_l/{\beta}]}$$ and $k_l\le [\frac{{\beta}}{{\beta}-1}l]+1$.
The statement means that if the sequence $a_k$, after being non-increasingly reordered, satisfies some sort of monotonous estimate, then in the initial sequence there exists a controllably dense subsequence, for which a similar but slightly weaker estimate holds.
Let the non-increasing permutation of the sequence $|a_k|$ be given by the bijection $j\mapsto m_j:$ $|a_{m_j}|=a^*_j,$ so that $|a_{m_j}|\le b_j$. Thus, for any $m_j\in F_{\beta}$, we have $m_j\le j{\beta}$, therefore $j\ge m_j/{\beta}$ and $b_j\le b_{[m_j/{\beta}]}$. Now we take as the subsequence $k_l$, the elements $m_j\in F_{\beta}$ taken in the increasing order. The inequality is therefore fulfilled. By Proposition \[Prop.reord.2\], $\#\{F_{\beta}\cap[0,N]\} \ge \frac{{\beta}}{{\beta}-1} N$ for any $N$, which is equivalent to the second inequality we need.
We will also need a simple consequence of Proposition \[Prop.reord.2\] concerning the rate of divergence of the series composed of the inverse values of $m\in F_{\beta}$.
\[PropInverse\] Under the conditions of Proposition \[Prop.reord.2\], $$\label{inversesum}
\limsup_{N\to\infty}{(\log N)^{-1}}{\sum\limits_{m\in F_{\beta}\cap [0,N]}m^{-1}}\ge\frac{{\beta}-1}{{\beta}}.$$
Estimates for functions analytical in a half-plane
--------------------------------------------------
There are a number of results in the classical complex analysis relating the estimates along the real axis of a function analytical in the half-plane $\operatorname{{\rm Re}\,}{\zeta}>0, {\zeta}={\xi}+i{\eta}$, with estimates of its values at some sequence of points. The first of such results we need, with ideas originating in [@Levinson], was obtained in [@Boas], p. 200.
\[ThBoas\] Let $f({\zeta})$ be a function, analytical in the right half-plane, of exponential type, satisfying $$\label{BoasCond}
\int_{-\infty}^\infty \frac{\log_+|f(i{\eta})|}{1+{\eta}^2}d{\eta}<\infty.$$ Suppose that ${\mu}_l$ is a monotone sequence of real points tending to infinity so that $|{\mu}_l-{\mu}_{l-1}|\ge{\delta}>0$ and $\sum {\mu}_l^{-1}=\infty.$ Then $$\label{BoasResult}
\limsup_{l\to\infty}\frac{\log |f({\mu}_l)|}{{\mu}_l}=\limsup_{{\xi}\to+\infty}\frac{\log|f({\xi})|}{{\xi}}.$$
\[BoasRemark\] In [@Levinson] and in [@Boas] the additional condition $\lim l {\mu}_l^{-1}=0$ was imposed. However, it was shown in [@Leont] that this condition is excessive and can be deleted.
Theorem \[ThBoas\] will be used for the study of the spectrum of operators in Bergman spaces. For the case of Bargmann spaces another result about estimates of functions, *not* of exponential type, will be used. We cite its version from [@Eiderman], see also [@EidermanEssen], with an obvious typo corrected.
\[thmEider\]Let the function $f({\zeta})$ be analytical in the half-plane $\operatorname{{\rm Re}\,}{\zeta}={\xi}>0$ and satisfy the estimate
$$\label{Cond.Eider1}
|f({\rho}e^{i{\varphi}})|=O(\exp[{\rho}(a\log {\rho}\cos{\varphi}+\pi c |\sin {\varphi}|+b\cos {\varphi})]), \ |{\varphi}|<\pi/2, {\rho}\to\infty,$$
for some $a\ge 0, c\ge -a/2.$ Suppose also that the growing sequence ${\mu}_l$ of positive numbers satisfies ${\mu}_{l+1}-{\mu}_l\ge {\delta}>0$ and $$\label{Cond.Eider2}
\limsup_{N\to\infty}\left[\sum\limits_{{\mu}_l\le N}{\mu}_l^{-1}-(c+a/2)\log N\right]=\infty.$$ Then the bound $$\label{Cond.Eider3}
\limsup_{l\to\infty}\frac{\log|f({\mu}_l)|}{{\mu}_l\log{\mu}_l}<-2c$$ implies that $f({\zeta})\equiv 0.$
This theorem improves the classical result by N. Levinson, see [@Levinson], Theorem XLI, in the sense that it does not require any regularity of the sequence ${\mu}_l.$
Eigenvalues of Toeplitz operators with non-sign-definite weight {#Sect4}
===============================================================
This Section contains the main results of the paper. These results can be expressed in the following way: $$\label{EstimGeneral}
\limsup_{{\lambda}\to 0}\frac{n({\lambda}; T_V)}{n({\lambda};T_{|V|})}=1,$$ in all spaces under consideration, where the radial function $V$ has compact support for ${\Omega}=D_{{\mathbf R}}$, and $V\in{{\mathcal R}}{{\mathcal D}}$ for the case of ${\Omega}={{\mathbb C}}^d$ or ${\Omega}={{\mathbb R}}^d$. Further on, we consider the concrete cases in detail.
Operators in Bergman spaces
---------------------------
In this section the symbol $V$ is supposed to be an arbitrary real bounded radial function with ESR $b<{{\mathbf R}}.$
\[ThMainBergmanC\] For the singular numbers of the operator $T_V$ in the Bergman spaces the following asymptotic formulas hold.\
For the complex space ${{\mathcal B}}^{{\mathbb C}}_{{{\mathbf R}}},$ $$\label{MainBergmanC1}
\limsup_{{\lambda}\to0} n({\lambda})|\log{\lambda}|^{-d}= (d!)^{-1}(2|\log(b/{{\mathbf R}})|)^{-d};\ n({\lambda})=n({\lambda}; T_V,{{\mathcal B}}^{{\mathbb C}}_{{\mathbf R}}).$$ For the harmonic and Helmholtz spaces ${{\mathcal B}}_{{\mathbf R}}^{{\mathbb R}}$ and ${{\mathcal B}}_{{\mathbf R}}^{{\mathbf H}}:$ $$\begin{gathered}
\label{MainBergmanR1}
\limsup_{{\lambda}\to0} n({\lambda})|\log{\lambda}|^{-d}=2 ((d-1)!)^{-1}(2|\log(b/{{\mathbf R}})|)^{-d+1};\\
{\rm for}\ n({\lambda})=n({\lambda}; T_V,{{\mathcal B}}^{{\mathbb C}}_{{\mathbf R}}) \ {\rm{or }} \ n({\lambda})=n({\lambda}; T_V,{{\mathcal B}}^{{\mathbf H}}_{{\mathbf R}}).\nonumber
\end{gathered}$$
The proofs for the analytical and harmonic cases are almost identical; we present the first one.
The upper estimate in follows from Proposition \[PropNonneg\] by monotonicity and Proposition \[PropReorderTriv\]. We will prove the lower estimate. Suppose that it is wrong; this means that $$\label{MainBergmanC22}
n({\lambda}; T_V,{{\mathcal B}}^{{\mathbb C}}_{{\mathbf R}})<{\gamma}(d!)^{-1}(2|\log(b/{{\mathbf R}})|)^{-d}|\log{\lambda}|^{d}$$ for some ${\gamma}<1$ and for ${\lambda}$ small enough. The singular numbers of $T_V$ are equal to the numbers $|{\Lambda}_k|$ defined in , permuted in the non-increasing order (we denote by ${\sigma}_m$ this permuted sequence), with multiplicities ${{\mathbf d}}_k^{{\mathbb C}}$ given in Sect.\[Sect2\]. So, ${\sigma}_{m_k} =|{\Lambda}_{k}|$, where $k\mapsto m_k$ is some bijection of ${{\mathbb Z}}_+$.
By , $$\label{MainBergmanC3}
n({\lambda})=\sum_{|{\Lambda}_k|>{\lambda}}{{\mathbf d}}_k^{{{\mathbb C}}}=\sum_{{\sigma}_{m_k}>{\lambda}}{{\mathbf d}}_k^{{{\mathbb C}}}.$$ Since the numbers ${{\mathbf d}}_k^{{{\mathbb C}}}$ increase with $k$ growing, the quantity in can only decrease if we replace in the values of $k$ by their smallest possible values, i.e., $$\label{MainBergmanC4}
n({\lambda})\ge \sum_{k=0}^{n_0({\lambda})}{{\mathbf d}}_k^{{{\mathbb C}}},$$ where ${n_0({\lambda})}=\#\{j:{\sigma}_j>{\lambda}\}.$
So, since ${{\mathbf d}}_k^{{{\mathbb C}}}=(1+O(k^{-1}))k^{d-1}((d-1)!)^{-1}$ for large $k$, we have $$\label{MainBergmanC5}
n({\lambda})\ge \sum_{k=0}^{n_0({\lambda})} \frac{k^{d-1}(1+O(k^{-1}))}{(d-1)!}\ge(1+O(n_0({\lambda})^{-1}))\frac{n_0({\lambda})^d}{d!}.$$
Substituting into , we obtain $$\label{MainBergmanC6}
n_0({\lambda})\le (1+O(|\log {\lambda}|^{-1})){\gamma}^{\frac1d}|\log {\lambda}|, \ \mbox{or} {\sigma}_m\lesssim \left({\gamma}'\frac{b}{{{\mathbf R}}}\right)^{2m},$$ for ${\lambda}$ small enough, resp., $m$ large enough and some ${\gamma}'<1$. Our next aim is to derive an estimate for ${\Lambda}_k$ from .
We fix some ${\beta}>1$, to be determined later, and apply Proposition \[prop.reord.major\] to the sequences $a_k=|{\Lambda}_k|$, $b_k=\left({\gamma}'\frac{{\beta}}{{{\mathbf R}}}\right)^{2m}.$ Thus there exists a subsequence ${\Lambda}_{k_l}$ such that $$\label{MainBergmanC7}
|{\Lambda}_{k_l}|\le C\left({\gamma}'\frac{b}{{{\mathbf R}}}\right)^{{2}k_l{\beta}^{-1}}.$$
We are going to show now that the inequality holds not only for the subsequence ${\Lambda}_{k_l}$ but for all ${\Lambda}_k$, probably, with slightly worse constants. To do this, we introduce the complex variable ${\zeta}={\xi}+i{\eta}$ and consider the function
$$\label{MainBergmanC8}
f({\zeta})=(2{\zeta}+2d)R^{-(2{\zeta}+2d)}\int_{0}^{{{\mathbf R}}}V(r)r^{2{\zeta}+2d-1}dr.$$
The function $f({\zeta})$ is analytical and bounded in the half-plane $\operatorname{{\rm Re}\,}{\zeta}={\xi}>0$, so the condition is satisfied. The values of $f$ at integer points $k$ coincide with the numbers ${\Lambda}_k$, due to . By the second inequality in , the series $\sum (k_l)^{-1}$ diverges. So, all conditions or Theorem \[ThBoas\] are fulfilled and, therefore, $$ \limsup_{k\to\infty}\frac{\log |{\Lambda}_{k}|}{k}= \limsup_{l\to\infty}\frac{\log|{\Lambda}_{k_l}|}{k_l},$$ or, returning back from the logarithmic scale, $$\label{MainBergmanC9}
{\Lambda}_k\le C\left({\gamma}'\frac{b}{{{\mathbf R}}}\right)^{{2}k/{\beta}'}$$ for any ${\beta}'>{\beta}$. Since ${\gamma}'<1$, we can choose the parameter ${\beta}$ and then ${\beta}'>1$ in the above reasoning so close to $1$ that $$\label{MainBergmanC10}
\left({\gamma}'\frac{b}{{{\mathbf R}}}\right)^{{2/{\beta}'}}< \left({\theta}\frac{b}{{{\mathbf R}}}\right)^{{2}}$$ for some ${\theta}<1.$ We substitute into and obtain $$\label{MainBergmanC11}
\int\limits_0^{{\mathbf R}}V(r)r^{2k+2d-1}dr =O(({\theta}b)^{2k}).$$ It remains to apply a classical theorem about the properties of the moments problem, say, Theorem 6.9.5 in [@Boas], saying that implies ${\hbox{{\rm supp}}\,}V\subset[0,{\theta}b]$, and this inclusion contradicts our condition that $b$ is the ESR for $V$.
Now we pass to the proof for the Helmholtz case. Again, suppose that is wrong; this means that $$\label{MainBergmanH2}
n({\lambda}; T_V,{{\mathcal B}}^{{\mathbf H}}_{{\mathbf R}})|\log{\lambda}|^{-d+1}<2{\gamma}((d-1)!)^{-1}(2|\log(b/{{\mathbf R}})|)^{-d+1}$$ with some ${\gamma}<1$, for small ${\lambda}.$ In the same way as for the case of the complex spaces, implies the estimate for the numbers ${\sigma}_m$, the monotonically re-ordered sequence of the numbers ${\Lambda}_k={\Lambda}_k^{{\mathbf H}}:$
$$\label{MainBergmanH3}
{\sigma}_m\le \left({\gamma}\frac{b}{{{\mathbf R}}}\right)^{2m}.$$
We can further proceed as before, to derive from the estimate for a sufficiently dense subsequence in ${\Lambda}_m:$ $$\label{MainBergmanH4}
|{\Lambda}_{k_l}|\le C\left({\gamma}\frac{b}{{{\mathbf R}}}\right)^{{2}k_l/{\beta}'},$$ with $k_l\le \frac{{\beta}}{{\beta}-1}l.$
Next, as before, we need to carry over the estimate from the subsequence to the whole sequence ${\Lambda}_k$, and even to the fractional $k$. To achieve this, we consider the auxiliary function $f({\zeta})$ analytical in the half-plane ${\xi}=\operatorname{{\rm Re}\,}{\zeta}>0:$ $$\label{MainBergmanH5}
f({\zeta})={\Gamma}({\zeta}+d/2){\Gamma}({\zeta}+(d+2)/2)){\int_0^{{\mathbf R}}V(r)J^2_{{\zeta}+\frac{d-2}{2}}(r)rdr}.$$ By the known asymptotics of Bessel functions for the large index (see ), the function $f({\zeta}) $ is bounded in the half-plane $\operatorname{{\rm Re}\,}{\zeta}>0$, moreover, by , , and , its values the integer points ${\zeta}=k$ are asymptotically equal to the numbers ${\Lambda}_k$.
Therefore, we can apply Theorem \[ThBoas\] to the function $f({\zeta})$, similarly to the reasoning for the complex case above, thus obtaining for real ${\xi}>0$ and some ${\gamma}'\in{({\gamma},1)}$ $$\label{MainBergmanH4a}
|f({\xi})|\le C\left({\gamma}'\frac{b}{{{\mathbf R}}}\right)^{{2}{\xi}/{\beta}}, \ {\xi}\to\infty,$$ which means $$\label{MainBergmanH5aa}
\int_0^{{\mathbf R}}V(r)J^2_{{\xi}+\frac{d-2}{2}}(r)rdr\le C[{\Gamma}({\xi}+d/2){\Gamma}({\xi}+(d+2)/2)]^{-1}\left({\gamma}'\frac{b}{{{\mathbf R}}}\right)^{{2{\xi}/{\beta}}}, \ {\xi}\to\infty.$$ Now we need to consider the cases of even and odd $d$ separately.
For the case of an even dimension $d$, we write for integer values ${\xi}=k:$ $$\label{MainBergmanH5a}
\int_0^{{\mathbf R}}V(r)J_{k+\frac{d-2}{2}}(r)^2rdr\le C[{\Gamma}({\kappa}+d/2){\Gamma}(k+(d+2)/2)]^{-1}\left({\gamma}'\frac{b}{{{\mathbf R}}}\right)^{{2k/{\beta}}}.$$ We use now C.Neumann’s formula, see [@Watson], 2.72(2), or [@Grad], 8.536.2: $$\label{Besselseries}
\sum_{j=m}^{\infty}\frac{{\Gamma}(m+j)}{j{\Gamma}(j-m+1)}J_j^2(r)=\frac{(2m)!}{(m!)^2}\left(\frac{r}2\right)^{2m}, \ m\in{{\mathbb Z}}_+.$$ It is easy to see from the Stirling formula that the series in converges uniformly on finite intervals, and, asymptotically in $m$, the leading term prevails. We substitute the expression for $r^{2m}$ with $m=k+\frac{d-2}{2}$ from into $\int_0^{{\mathbf R}}r^{2(k+\frac{d-2}{2})}V(r)dr:$ $$\label{MainBergmanH6}
\int_0^{{\mathbf R}}r^{2m}V(r)rdr=2^{2m}\frac{(m!)^2}{(2m)!} \sum_{j=m}^{\infty}\frac{j{\Gamma}(m+j)}{{\Gamma}(j-m+1)}\int_0^{{\mathbf R}}V(r)J^2_j(r)r dr.$$ For each term in , we apply the estimate . Calculations with ${\Gamma}$-functions show that $$\label{MainBergmanH7}
\int_0^{{\mathbf R}}r^{2k}V(r)r^{d-1}dr=O(\left({\gamma}'{b}\right)^{{2k/{\beta}}}).$$ Finally, since ${\gamma}'<1$, we can choose ${\beta}>1$ so close to $1$ that $ \left({\gamma}'{b}\right)^{1/{{\beta}}}<{\theta}b, \ {\theta}<1$. Now we can again apply Theorem 6.9.5 in [@Boas], which gives ${\hbox{{\rm supp}}\,}V\subset[0,{\theta}b]$, ${\theta}<1$ which contradicts our assumption that $b$ is the ESR for $V$.
A similar reasoning works for case of an odd dimension $d.$ We however consider for half-integer ${\xi}$, ${\xi}=k+1/2$. Then, since $(d-2)/2$ is half-integer, gives an estimate of integrals containing Bessel functions with integer index. The final step in the proof is the same.
Operators in Bargmann and AH-spaces
-----------------------------------
For the case of Bargmann and AH spaces, the consideration follows the same idea as for the Bergman spaces, with minor modifications.
\[ThEstBargC\] Let the radial symbol $V$ belong to ${{\mathcal R}}{{\mathcal D}}.$ Then (i) For ${{\mathcal B}}={{\mathcal B}}^{{\mathbb C}}({{\mathbb C}}^d)$, $$\label{SnizuCompl}\limsup_{{\lambda}\to 0}\left[n({\lambda}) \left(\frac{|\log{\lambda}|}{\log|\log{\lambda}|}\right)^{-d}\right]= (d!)^{-1} .$$ (ii) For ${{\mathcal B}}={{\mathcal B}}^{{\mathbb R}}({{\mathbb R}}^d)$, or ${{\mathcal B}}^{{\mathbf H}}({{\mathbb R}}^d)$ $$\label{SnizuReal}
\limsup_{{\lambda}\to 0}\left[n({\lambda}) \left(\frac{|\log{\lambda}|}{\log|\log{\lambda}|}\right)^{-d+1}\right]= ((d-1)!)^{-1}.$$ (iii) For $Bc={{\mathcal B}}^{{{\mathbf A}}{{\mathbf H}}}({{\mathbb R}}^d)$ $$\label{SnizuAH}
\limsup_{{\lambda}\to 0}\left[n({\lambda}) \left(\frac{|\log{\lambda}|}{\log|\log{\lambda}|}\right)^{-d+1}\right]=2 ((d-1)!)^{-1}.$$
All cases are proved in a similar manner. We give the proof of the part (i) of Theorem \[ThEstBargC\] in detail and then explain the changes needed for other cases.
The proof starts in the same way, as for Theorem \[ThMainBergmanC\]. The estimate from above in is already established. Suppose that the lower bound in is wrong. This means that for some ${\gamma}<1$, the inequality $$\label{SnizuC.1}
n({\lambda})<{\gamma}(d!)^{-1}\left(\frac{|\log{\lambda}|}{\log|\log{\lambda}|}\right)^{d}$$ holds for all sufficiently small ${\lambda}>0$. The singular numbers of the operator $T_V$ are equal to the numbers $|{\Lambda}_k|$, see , permuted in the non-increasing order, with multiplicities ${{\mathbf d}}^{{\mathbb C}}_k $ (defined in Sect. 2.3.1). So, these singular numbers equal ${\sigma}_{m_k}=|{\Lambda}_k|$, where $k\mapsto m_k$ is some bijection of ${{\mathbb Z}}_+.$ By , $$\label{SnizuC.2}
n({\lambda})=\sum_{|{\Lambda}_k|>{\lambda}}{{\mathbf d}}^{{\mathbb C}}_k=\sum_{{\sigma}_{m_k}>{\lambda}}{{\mathbf d}}^{{\mathbb C}}_k.$$ Since the multiplicities ${{\mathbf d}}^{{\mathbb C}}_k$ are non-decreasing as $k$ grows, the quantity can only decrease if we replace in the sum in the values of $k$ by their smallest possible values, $$\label{SnizuC.3}
n({\lambda})\ge \sum_{k=0}^{n_0({\lambda})}{{\mathbf d}}^{{\mathbb C}}_k,$$ where $n_0({\lambda})=\#\{{\sigma}_m>{\lambda}\}.$ So, since ${{\mathbf d}}^{{\mathbb C}}_k= k^{d-1}((d-1)!)^{-1}(1+O(k^{-1}))$, we have $$\label{SnizuC.4}
n({\lambda})\ge\sum_{k=0}^{n_0({\lambda})}\frac{k^d}{(d-1)!}(1+O(k^{-1}))=(1+O(n_0({\lambda})^{-1}))\frac{n_0({\lambda})^d}{d!}.$$ We substitute into and obtain $$\label{SnizuC.5}
{\gamma}'n_0({\lambda})\le (1+o(1)) \frac{|\log{\lambda}|}{\log|\log{\lambda}|}, \ {\gamma}'\in (1, {\gamma}^{-\frac1d}).$$
We rewrite in terms of an estimate for ${\sigma}_m:$ $$\label{SnizuC.6}
{\sigma}_m\lesssim {\Gamma}({\gamma}' m)^{-1}.$$ We derive now an estimate for $\log|{\Lambda}_k|$ from . Fix some ${\beta}>1$, to be determined later, and apply Proposition \[prop.reord.major\] to the sequences $a_k=|{\Lambda}_k|, b_k={\Gamma}({\gamma}' m_k)^{-1}.$ Thus there exists a subsequence ${\Lambda}_{k_l}$ such that
$$\label{SnizuC.7a}
|{\Lambda}_{k_l}|\lesssim {\Gamma}({\gamma}' k_l/{\beta})^{-1},$$
or, in the logarithmic scale, $$\label{SnizuC.7}
\log|{\Lambda}_{k_l}|\le -{\gamma}'{\beta}^{-1} k_l \log ( k_l/{\beta})(1+o(1)).$$
Now we introduce the complex variable ${\zeta}={\xi}+i{\eta}$ and consider in the half-plane ${\xi}>0$ the function $$\label{SnizuC.8}
f({\zeta})=2{\int_0^\infty V(r)r^{2{\zeta}+2d-1} e^{-r^2}dr}.$$ This function coincides with ${\Lambda}_k{{\Gamma}(k+d)}$ at the integer points ${\zeta}=k$. Therefore, by at the points ${\zeta}=k_l$, the function $ f({\zeta})$ satisfies $$\label{SnizuC.80}
\log|f(k_l)|<\log{\Gamma}( k_l/{\beta})-{\gamma}'{\beta}^{-1} k_l \log ( k_l/{\beta})(1+o(1))\lesssim(1-{\gamma}'/{\beta})\log{\Gamma}(k_l/{\beta}).$$
By Lemma \[LemRD\], for ${\zeta}={\rho}e^{i{\varphi}},$ $$\label{SnizuC81}
|f({\zeta})|=O\left(\int_0^\infty |V(r)|r^{2{\rho}\cos{\varphi}+2d-1} e^{-r^2}dr\right)=O({\Gamma}({\epsilon}{\rho}\cos{\varphi})), \ {\rho}\to +\infty$$ for any ${\epsilon}>0. $ Taking into account the asymptotics for the ${\Gamma}$-function for large real values of argument, we obtain that the estimate is satisfied for any $a>0, c>0.$ Now, we choose ${\beta}>1$ so that ${\gamma}'/{\beta}>1.$ After this, we fix $a,c>0$ so small that $\frac{{\beta}-1}{\beta}>a+c/2$ and $2c<{\gamma}'/{\beta}-1$. Then, by Proposition \[PropInverse\], the sequence of integers $\{{\mu}_l\}$ obtained by the increasing reordering of the sequence $\{k_l\}$, satisfies the conditions and . So, all conditions of Theorem \[thmEider\] are satisfied, and we can conclude that $f({\zeta})=0$ for all ${\zeta}.$
It remain to notice that $f(-{\zeta})$ is the Mellin transform of the function $V(r)r^{2d-1}e^{-r^2}$, and by the inversion theorem for the Mellin transform we conclude that $V(r)\equiv0$, and this takes care of the proof for the complex Bargmann space.
The proof for the space ${{\mathcal B}}^{{\mathbb R}}$ follows the reasoning above, only with $d$ replaced by $d-1$.
Now we consider the operator in the Helmholtz Bargmann space, where some more changes are needed.
Similar to , we suppose that is wrong, and this would mean that $n({\lambda})\le {\gamma}((d-1)!)^{-1}\left(\frac{|\log{\lambda}|}{\log|\log{\lambda}}\right)^{d-1}$ for small ${\lambda}$ and some ${\gamma}<1$. By repeating the calculations in -, we obtain for the distribution function $n_0({\lambda})$ (now, of the numbers ${\Lambda}_k={\Lambda}_k^{{{\mathbf H}}}$) an estimate of the form , with some ${\gamma}'>1.$ After this, we apply again Proposition \[prop.reord.major\] to obtain for a (sufficiently dense) subsequence ${\Lambda}_{k_l}$ the estimate . Starting from this point, the reasoning is somewhat different.
We introduce the complex variable ${\zeta}={\xi}+i{\eta},\ {\xi}\ge0,$ and consider the function $$\label{sniHelm1}
f({\zeta})={\Gamma}^2\left({\zeta}+({d-2})/{2}\right)\int_0^\infty J^2_{{\zeta}+\frac{d-2}{2}}V(r)e^{-r^2}rdr.$$ This function is analytical in the half-plane ${\xi}>-\frac12$. It follows also from that $$\label{sniHelm2}
| f({\zeta})|\le C\int_0^\infty r^{2({\zeta}+d/{2})}|V(r)|e^{-r^2}dr,$$ and by , $f({\zeta})$ satisfies $$\label{sniHelm3}
|f({\zeta})|=O(|{\Gamma}({\epsilon}{\zeta})|)$$ for any ${\epsilon}>0$ as $|{\zeta}|\to \infty$, $\operatorname{{\rm Re}\,}{\zeta}>-\frac12$. It follows that the function $f({\zeta})$ satisfies condition of Theorem \[thmEider\], with arbitrarily small positive values of $a,c$.
After this property has been established, the proof follows the one for the complex Bargmann space. For real integer ${\zeta}=k$ we have $|f(k)|\asymp|{\Lambda}_{k}| {\Gamma}(k+(d-2)/2)$. Therefore, by , the values of $f({\zeta}) $ at the points ${\zeta}=k_l$ satisfy . We take ${\beta}>1$ so close to $1$ that ${\gamma}'/{\beta}>1$ and then fix $a,c$ so small that $\frac{{\beta}-1}{\beta}>a+c/2$ and $2c<{\gamma}'/{\beta}-1$. Then the sequence of integers ${\mu}_l$ obtained by the increasing re-ordering of the sequence $k_l$, satisfies all conditions of Theorem \[thmEider\], and therefore $f({\zeta})\equiv 0.$ This means, in particular, that all numbers ${\Lambda}_k$ are zeros, therefore $T_V=0.$ To prove that this implies $V=0$, we use to express $\int_0^\infty r^{2m} V(r)e^{-r^2} dr$ as a linear combination of the numbers ${\Lambda}_k$, all of them being equal to zero. Therefore we obtain that the function $g({\zeta})=\int_0^\infty r^{2{\zeta}+2d} V(r)e^{-r^2} dr$ takes zero values at all integer points. Finally, we note that, again by Lemma \[LemRD\], function $g({\zeta})$ satisfies the conditions of Theorem \[thmEider\] with arbitrarily small positive $a,c$, and therefore $g\equiv 0$. The proof that $V\equiv 0$ concludes again by the inversion theorem for the Mellin transform.
The case of the space ${{\mathcal B}}^{{{\mathbf A}}{{\mathbf H}}}$ is proved in the same way, with minimal changes..
Not that rapidly decaying symbols. A counterexample {#Sect5}
===================================================
The results of Sect.\[Sect4\] might lead to the impression that, probably, one should expect the absence of cancelation of the positive and negative parts of the symbol in a more general situation as well. The author was of such opinion for a certain time. However, the example given in this Section shows that such impression is wrong.
We present here a construction of symbols that decay at infinity rather fast but not sufficiently fast to get into the ${{\mathcal R}}{{\mathcal D}}$ class. This symbols oscillate very rapidly at infinity. We show that the s-numbers of the Toeplitz operator with symbol $V$ decay essentially faster than the ones for the operator with symbol $|V|,$ so an analogy with theorems in Sect. \[Sect4\] does not hold. In order to simplify the calculations, we restrict ourselves here to the operators in the space ${{\mathcal B}}^{{\mathbb C}}$ in the one-dimensional case, $d=1.$ The same constructions work in any dimension, and for the spaces ${{\mathcal B}}^{{\mathbb R}}$ as well. We suppose that for other spaces under consideration a similar construction produces proper examples.
We consider the symbol $$\label{Ncounter1}
V(r)= V_{p,q}(r)=e^{-r^{2p}+r^2}\sin( r^{2q})$$ with $p>1$, $q>p$.
\[ThmNcounter\] For the operator $T_V$ $$\label{NCounter2}
\limsup_{{\lambda}\to 0}n({\lambda}, T_V) \frac{\log|\log{\lambda}|}{|\log {\lambda}|}\le\frac{q}{q-1},$$ while $$\label{NCounter3}
n({\lambda}, T_{|V|})\sim n({\lambda}, T_{V_+})\sim n({\lambda},T_{V_+})\sim\frac{p}{p-1}\frac{|\log {\lambda}|}{\log|\log{\lambda}|}, \ {\lambda}\to 0.$$
The difference in coefficients in front of $\frac{|\log {\lambda}|}{\log|\log{\lambda}|}$ in , transforms into a large difference in the decay order of the eigenvalues: $$\label{NCounterLog}
\log s_n(T_V)\lesssim \frac{q(p-1)}{p(q-1)}\log s_n(T_{|V|}),$$ so, in fact, a rather strong cancelation takes place.
In order to prove , we need to estimate the numerator in , i.e., the integral $$\label{NCounter4}
{{\mathbf I}}(k)=\int_0^\infty e^{-r^{2p}+r^2}\sin\left(r^{2q}\right) e^{-r^2}r^{2k+1}dr.$$ After the change of variables $t=r^{2q}$, transforms to $$\label{NCounter5}
{{\mathbf I}}(k)=(2q+1)^{-1}\int_0^\infty t^{\frac{k+2}{q}-1}e^{-t^{p/q}}\sin t dt=\Im (2q+1)^{-1}\int_0^\infty t^{\frac{k+2}{q}-1}e^{-t^{p/q}}\exp(it) dt .$$ We consider here $t$ as complex variable living on the positive real half-line and the whole expression as the integral in along this half-line.
Now we replace the integration line by means of rotating it to the line $\arg t=\pi/2$. The integral in does not change due to the factor $e^{-t^{p/q}}$ which decays fast in the whole first quarter. So, by setting $t=i{\tau}$, ${\tau}\in(0,1)$, we have $$\label{NCounter6}
|{{\mathbf I}}(k)|\le (2q+1)^{-1}\int_0^\infty {\tau}^{\frac{k+2}{q}-1}e^{-{\tau}} d{\tau}={\Gamma}((k+2)/q).$$ Finally, taking into account the expression for the denominator in and Stirling’s formula we arrive at .
To prove , we estimate the integral, say, $${{\mathbf I}}_+(k)=\int_0^\infty t^{\frac{k+2}{q}-1}e^{-t^{p/q}}\sin_+(t) dt$$ from below. To do this we consider the intervals $I_j=(2\pi/3+4\pi j, 4\pi/3+4\pi j)$. On each of these intervals, for $j$ large enough, $\sin(t)>\frac12$. This inequality easily implies a lower estimate for ${{\mathbf I}}_+(k):$ $${{\mathbf I}}_+(k)\ge\sum_j \frac12 \int_{I_j}{\frac{k+2}{q}-1}e^{-t^{p/q}}\sin_+(t) dt \ge C\int _0^\infty t^{\frac{k+2}{q}-1}e^{-t^{p/q}}dt=C {\Gamma}\left(\frac{k+2}{p}\right),$$ which leads to .
[333]{}
Agmon, S., Hörmander, L., Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math. 30 (1976), 1-38. Alexandrov, A., Rozenblum, G. Finite rank Toeplitz operators: some extensions of D. Luecking’s theorem, J. Funct. Anal. 256 (2009), no. 7, 2291–2303. Boas, R.P., Entire functions. New York, Academic Press, 1954
Eiderman, V., Decrease on a sequence of points of a function holomorphic on a half plane. Sibirsk. Mat. Zh. 24 (1983) (in Russian); English translation in Siberian Math. J. 24 (1983), no. 2, 304–315. Eiderman, V., Essen, M., Uniqueness theorems for analytic and subharmonic functions. St. Petersburg Math. J. 14 (2003), no. 6, 889–952 Filonov, N., Pushnitski, A.: Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains. Comm. Math. Phys. 264 (2006), 759 – 772 Gradshtein I.S., Ryzhik I.M., Tables of Integrals, Series and Products. Acad. Press, 2007. Luecking, D. Finite rank Toeplitz operators on the Bergman space, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1717–1723. Leontiev, A.F., On the growth of an entire function on a sequence of points., Mat.Sborn., 96 (1975), 601-613 (in Russian); English transl. in Math. USSR–Sb. 25 (1975), 4, 567-578. Levinson, G., Gap and Density Theorems, AMS, 1940.
Melgaard, M., Rozenblum, G.: Eigenvalue asymptotics for weakly perturbed Dirac and Schrödinger operators with constant magnetic fields of full rank. Comm. Partial Differential Equations 28 (2003), 2, 180–192 697–736
Parfenov, O.: Asymptotics of singular numbers of imbedding operators for certain class of analytic functions. Mat. Sb. 115(157) (1981), no. 4, 632–641 (in Russian), English translation in Mathematics of the USSR-Sbornik (1982),43(4):563 Pushnitski, A., Rozenblum, G., On the spectrum of Bargmann-Toeplitz operators with symbols of a variable sign, `arXiv:0912.4486`, to appear in J. Analyse Math. Raikov, G. Warzel, S., Quasi-classical versus non-classical spectral asymptotics for magnetic Schroödinger operators with decreasing electric potentials. Rev. Math. Phys. 14, 1051–1072 (2002). Rozenblum, G., Finite rank Toeplitz operators in Bergman spaces, in: Around the Research of Vladimir Maz’ya. III: Analysis and Applications.", Springer 2010, 331–358, `arXiv:0904.0171` Shubin, M. Pseudodifferential Operators and Spectral Theory. Second edition. Springer-Verlag, Berlin, 2001. Strichartz, R., Harmonic analysis as spectral theory of Laplacians, Journal of Functional Analysis **87**, 51-148 (1989). Watson, G.N., A treatise on the theory of Bessel functions, Cambridge University Press, 1995
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The graph isomorphism problem is of practical importance, as well as being a theoretical curiosity in computational complexity theory in that it is not known whether it is $NP$-complete or $P$. However, for many graphs, the problem is tractable, and related to the problem of finding the automorphism group of the graph. Perhaps the most well known state-of-the art implementation for finding the automorphism group is Nauty. However, Nauty is particularly susceptible to poor performance on star configurations, where the spokes of the star are isomorphic with each other. In this work, I present an algorithm that explodes these star configurations, reducing the problem to a sequence of simpler automorphism group calculations.'
author:
- |
RUSSELL K. STANDISH\
Mathematics and Statistics\
The University of New South Wales
bibliography:
- 'rus.bib'
title: 'SuperNOVA: a novel algorithm for graph automorphism calculations'
---
\[Computations on Discrete Structures\] \[Graph Algorithms\]
Introduction
============
Given two graphs $g_1=\{V,E_1\}$ and $g_2=\{V,E_2\}$ where $V$ is a set of labelled vertices, and $E_{\{1,2\}}\subset V\times V$ are sets of edges, the [*graph isomorphism*]{} problem is the problem of finding a permutation $\sigma: V\longrightarrow V$ of the vertices such that $\forall (i,j)\in E_1, (\sigma(i),\sigma(j))\in E_2$. The map $\sigma$ is known as an [*isomorphism*]{}. The graph isomorphism problem is of practical importance in applications such as storing and retrieving molecular structure data from a database[@Kuramochi-Karypis06] or verification of printed circuit layout with respect to a schematic[@Ebeling-Zajicek83]. It is also interesting, because it is not known whether the problem in general can be solved in polynomial time, or whether it is $NP$-complete.
A [*graph automorphism*]{} is an isomorphism of a graph onto itself. The set of graph automorphisms of a graph forms a group under composition. The graph automorphism problem is the problem of finding whether a graph has any automorphism other than the identity automorphism, which like the graph isomorphism problem has unknown computational complexity[@Lubiw81]. More generally, one is interested in the size of the automorphism group[@Standish05a], and the orbits of the group.
The graph isomorphism problem can be reduced to the problem of finding a canonical labeling of the vertices of a graph. If the adjacency matrices of two graphs under their canonical labeling are equal, then they are isomorphic. For many graphs, finding a canonical labeling is tractable, and Nauty[@McKay81] is probably considered one of the best-of-breed implementations. Nauty will also return the size of the automorphism group of a graph.
Unfortunately, Nauty struggles with star-like graphs, ie graphs where several isomorphic graphs are attached to each other via a single hub vertex. In this paper, I present the SuperNOVA, or “star exploder” algorithm, which can handle these sorts of graphs efficiently.
The algorithm
=============
Canonical Ranges {#canonRange}
----------------
The algorithm proceeds by defining an ordering relation on the graph vertices, sorting the vertices according to that ordering relation and the assigning a range of possible canonical labels to each vertex according to its position in the sorted list. For example, if the following sorted list was returned: $$n_3 < n_2 = n_4 = n_5 < n_1 = n_0$$ then the vector of canonical ranges will look like $$[4,5), [4,5), [1,4), [0,1), [1,4), [1,4).$$
1. Initially, the ordering relation used is vertex degree (both in-degree and out-degree, and the number of bidirectional edges).
2. Once all vertices have been assigned a canonical range, we can compare the canonical ranges of the nearest neighbours of a pair of vertices. If two vertices have the same canonical range, yet their neighbourhoods differ, we can further discriminate between the vertices, enabling a refinement of the canonical ranges. This step is repeated until no further refinement is possible.
The computational complexity lies between $O(n\log n)$ (the computational complexity of a sort) and $O(n^2\log n)$ as at most $n$ iterations can occur in step 2.
If the result of this algorithm is that every vertex has a canonical range of size 1, then we are done. The canonical labeling is given by the lower bounds of the canonical ranges, and there is only one automorphism (the identity). However, if some of the vertices have non-unit ranges, then the graph may have symmetries. Unfortunately, we cannot just take the product of the ranges as the size of the automorphism group, as not all such relabelings are automorphisms.
Symmetry Breaker
----------------
If the graph has symmetries, then at least two vertices will have identical canonical ranges. We need to determine which of the possible labelings is a canonical labeling. There may be more than one canonical labeling, but each such labeling produces an identical adjacency matrix. To compute a canonical labeling, we induce an ordering over adjacency matrices, and pick a labeling having the least adjacency matrix.
The outline of the symmetry breaker algorithm is:
XX = XX = XX = If all canonical ranges are of size 1, then\
return an automorphism count of 1, and\
an adjacency matrix for that labeling,\
otherwise\
Find first non-unit canonical range $[m,M)$\
For each vertex $j$ having canonical range $[m,M)$,\
set vertex $j$’s canonical range to $[m,m+1)$\
apply step 2 of §\[canonRange\]\
recursively apply the symmetry breaker algorithm to the new\
canonical ranges.\
add the returned automorphism count to the map entry indexed by\
the returned adjacency matrix\
return the least adjacency matrix and its automorphism count\
The worst case scenario for this algorithm is when the sorting algorithm in §\[canonRange\] fails to discriminate vertices, in which case the complexity is $O(n!)$, as each permutation of vertices will be tried by the symmetry breaker. This will occur for the fully connected graph, which will always be a worst case, but also for the empty graph and star configurations. A star graph of order $n+1$ containing a single hub of degree $n$, and $n$ leaf vertices, will cause the symmetry breaker algorithm to have complexity $O(n!)$. Given that this is the same problem that afflicts Nauty, this leads naturally to the star exploder algorithm.
Star Exploder
-------------
If we have a simple star topology, with $c$ isomorphic graphs attached to a central hub, then the automorphism group size is given by $c!r$, where $r$ is the automorphism group size of each of the spokes of the star, since there are $c!$ ways of relabeling the spokes. A slightly more general case occurs where there are $c_0$ spokes isomorphic to each other, another group of $c_1$ spokes isomorphic to each other and so on. In this case, the resulting automorphism group size is given by $$\label{stareqn}
r=\prod_ic_i!r_i.$$
To establish whether an arbitrary graph has a star-like topology, we remove all vertices with unit canonical range, which we call “fixed vertices”. A graph colouring algorithm can be used to find the different contiguous subgraphs. If the graph breaks into more than one contiguous piece, then we can recursively apply the complete automorphism algorithm to each piece to obtain $r_i$, and count each piece using a map indexed by the canonical adjacency matrix. Then the overall automorphism group size can be found from the individual size by using equation (\[stareqn\]). The overall canonical labeling can be found by using the algorithm described in §\[canonRange\], but with a modified ordering that includes information about which subgraph the vertices belong to (subgraphs sorted according to their canonical adjacency matrix order), and the canonical label of the vertex within the subgraph. If two vertices belong to different, but isomorphic subgraphs, and further that they have the same canonical label within their respective subgraph, then they are ordered simply by their original label, as in this case it wouldn’t matter which way they were labeled, the adjacency matrix would be identical. This allows a canonical labelling to be generated.
A subtle twist to be considered here is that a subgraph connected to one fixed vertex, and another subgraph connected to a different fixed vertex are not equivalent, even thought they may be isomorphic. To deal with this issue, we attach the vertex’s canonical range from the original graph as an attribute to the equivalent vertex in the subgraph. Only isomorphic subgraphs whose attributes are identical are equivalent, otherwise they’re counted as distinct graphs.
Star exploder fails when either there are no fixed vertices, or when removing all the fixed vertices does not partition the graph. Because the symmetry breaker algorithm gradually fixes more and more vertices at each level of recursion, the star exploder algorithm is applied at each level of recursion of the symmetry breaker algorithm, and will eventually succeed in breaking the graph into disjoint pieces. The worst case scenario is not so much the full graph (which being the dual of the empty graph is trivially transformed), but a digraph where each vertex is connected to every other vertex, and arranged so that the indegree and outdegree of each vertex is the same (the order of the graph must be odd for this to occur). Each vertex is the same as any other, so symmetry breaker must iterate over all $n!$ permutations of vertex labels.
Implementation and results
==========================
The algorithm was implemented in C++ as part of the open-source [[ ** ]{}]{}[@Standish-Leow03] simulation environment, from ecolab.4.D31. onwards. It makes use of the C++ standard library sort() algorithm, and the standard associative containers map and set.
The algorithm was tested by comparing its calculated automorphism group size with that given by Nauty. If $S(g)$ is the canonical representation of $g$ calculated by SuperNOVA, and $N(g)$ the canonical representation calculated by Nauty, a second important check is that $S(N(g))=S(g)$ and that $N(S(g))=N(g)$.
A database of 48940 symmetric graphs obtained from Brendan McKay’s website (http://cs.anu.edu.au/\~bdm/data/graphs.html) was used to check the equivalence of SuperNOVA with Nauty. Exhaustively generating all digraphs of a certain number of vertices and edges provided an independent test to ensure the algorithm worked for digraphs, including some with a star-like nature, however this was only feasible up to order 9 or so. Certain star-like digraphs extracted from the wiring diagram of a [*C. elegans*]{} brain was used to test the performance of SuperNOVA on graphs that proved intractable with Nauty. Attempting to run Nauty on these digraphs was unsuccessful, as Nauty didn’t complete after several days of running, and had to be killed. By contrast, SuperNOVA computed these examples in seconds.
=
Figure \[randomGraph\] shows 1000 randomly generated Erdös-Rényi graphs with order $10\le n<100$ and edge count $0<l\le n(n-1)/2$. Both SuperNOVA and Nauty were timed, and the times plotted as a function of order and edge count. Because some graphs can potentially take a very long to compute the canonical labeling, a maximum of 10 minutes was imposed on the computation by using CPU resource limit functionality of Linux. These examples appear in the data as having an execution time of 10 minutes ($6\times10^8\mu $s), and all from Nauty, and appear when $l/n<10$.
=
Figure \[worstCase\] shows the performance of SuperNOVA versus Nauty for the worst case scenario of a fully connected digraph, with the indegree and outdegree of each vertex being the same. As expected, SuperNOVA’s execution time blows up very rapidly, but interestingly Nauty performs well, with polynomial time complexity.
Discussion
==========
SuperNOVA effectively handles the star-like configurations that give trouble to Nauty. On a sampling of 1000 Erdös-Rényi random graphs of order between 10 and 100, SuperNOVA computed the automorphism group size of all of the whole set within 10 minutes on a quad core Intel Core 2. By contrast, Nauty failed to complete the calculations on several of the graphs within the 10 minute time limit. Interestingly, all of these examples lie in the range $0<l/n<10$. Overall, Nauty is an order of magnitude faster than SuperNOVA on those examples it handles well, which is a reflection of the intense effort that has goine into the optimisation of that code. With more optimisation, SuperNOVA should be able to close the gap somewhat.
On the artificially constructed worst-case scenario, SuperNOVA performs poorly as expected, but Nauty performs well, executing in polynomial time.
All of this suggests the possibility of a hybrid algorithm, leaving the sparse examples to SuperNOVA, and the denser examples to Nauty. The precise heuristic for determining which algorithm will need to be determined by future research. Furthermore, the precise canonical form returned differs in the two algorithms, so combining the algorithms will need to take this into account. For the isomorphism problem, it should not matter, so long as the same algorithm (SuperNOVA or Nauty) is applied to the two graphs being compared.
Finally, one may speculate as to whether the hybrid algorithm is truly polynomial complexity. Further work will be needed to try and identify worst case scenarios for both algorithms.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Classical and path integral molecular dynamics (PIMD) simulations are used to study $\alpha$ and $\beta$ quartz in a large range of temperatures at zero external stress. PIMD account for quantum fluctuations of atomic vibrations, which can modify material properties at temperatures below the Debye temperature. The difference between classical and quantum mechanical results for bond lengths, bond angles, elastic modulii, and some dynamical properties is calculated and comparison to experimental data is done. Only quantum mechanical simulations are able to reflect the correct thermomechanical properties below room temperature. It is discussed in how far classical and PIMD simulations can be helpful in constructing improved potential energy surfaces for silica.'
address: |
Institut für Physik, WA 331; Johannes Guntenberg-Universität\
55099 Mainz; Germany
author:
- 'Martin H. Müser'
title: 'Simulation of Material Properties Below the Debye Temperature: A Path-Integral Molecular Dynamics Case Study of Quartz'
---
[2]{}
Introduction
============
Quartz is one of the most abundant and best studied minerals on Earth; the structure and many other physical properties are well understood [@heaney94]. Classical molecular dynamics (MD) simulations have been particularly successful in connecting atomic interactions between silicon and oxgen atoms in the condensed phase with structural and elastic properties of quartz [@tsuneyuki88; @beest90; @tse91; @somayazulu93; @muser00] and other silica polymorphs at finite temperatures [@tsuneyuki88; @tse91; @tsuneyuki90; @tse92]. At zero temperature, this connection has also been done by mere first-principle studies [@liu94; @boer95; @swainson95] or ab-inito methods incorporating bulk-system information [@beest90].
Some of the numerical approaches reach a nearly perfect agreement with experiment, especially for structural properties. None of the calculations, however, include quantum effects of the ionic motion. Quantum effects lead to an equilibrium structure that is different from the “classical” equilibrium structure as soon as the interactions between the atoms constituting a crystal are not purely harmonic. Path integral simulations are a convenient tool to compute such quantum mechanical effects [@muser95; @martonak98; @herrero00], e.g., the zero-temperature lattice constant $a$(Ne$^{\rm cla})$ of classical Neon is 0.18$~\AA$ smaller than the one of Ne$^{22}$, which again is about 0.19% smaller than $a$(Ne$^{20})$. [@muser95] Treating atomic motion quantum mechanically even effects covalent bonds in systems such as crystalline polyethylene [@martonak98] and silicon [@herrero00].
Quantum effects are certainly less strong in silica than in rare gas solids such as in neon. Thermal expansion can nevertheless be observed for $\alpha$-quartz below the Debye temperature and quantum effects have to be taken into account properly, if we want to relate simulations and experiments in a meaningful way. Correct quantum simulations will converge to a zero expansion coefficient as absolute zero is approached, while classical MD simulations will (nearly) always result in a finite expansion coefficient even at $T = 0$ K thus violating the third law of thermodynamics. It is obvious that structure and elastic constants are reflected well by the model potential if these properties have been used to fit the free parameters of the potential energy surface - as done in the case of the so-called BKS potential [@beest90]. It is therefore a much harder test for a model potential surface to yield the correct thermal expansion at low temperature, because the anharmonic interactions must be reflected accurately by the model potential. Overall, ab-initio calculations, MD simulations, and experimental data have partially achieved such good agreement that estimating quantum effects will play an important role in determining the real merits of a potential energy surface.
In this study, path-integral molecular dynamics (PIMD) are used to determine the quantum effects on physical properties of quartz. Due to the large computational demand of path integral simulations, we confine ourselves to the use of only one potential energy surface, namely the BKS potential [@beest90]. It has been particularly successful in reproducing silica properties not only of $\alpha$-quartz [@beest90; @tse91], but also of other silica polymorphs [@tse91; @tse92] and of the glassy state [@vollmayr96; @badro98; @horbach99]. It is therefore rather plausible that the BKS potential predicts accurately the shifts from classical results to quantum mechanical results.
Recently, $\beta$-quartz and $\beta$-cristobalite have been investigated by means of path integral Monte Carlo (PIMC) simulations [@rickwardt]. The PIMC study, however, was only done for phases that are stable at temperatures above $800$ K. Quantum effects are relatively small at such high temperatures. Moreover, the resolution of our PIMD simulations exceeds that of the emploied PIMC algorithm by orders of magnitudes. E.g., the PIMD approach makes it possible to calculate the ground-state equilibrium lattice constants with a resolution of more than $0.001\,\AA$ for a given potential energy surface, while the PIMC simulations had an uncertainty of typically $0.1\,\AA$. This improvement in resolution is possible despite a strong reduction in CPU time. Furthermore, arbitrary parallelepiped simulation cells are permitted in this PIMD study allowing to calculate all elastic constants. The PIMC studies were confined to orthorhombic geometries and did not allow calculations of any elastic constants due to large statistical error bars.
The remainder of this paper is organized as follows: In Sec. \[sec:method\], the PIMD method used in this study is described along with some specific, technical details. In Sec. \[sec:results\], results are presented for structural data, elastic constants, and dynamic properties. Quantum mechanical results are compared to classical simulations and experiment. Conclusions are drawn in Sec. \[sec:conclusions\].
Method {#sec:method}
======
Path Integral Molecular Dynamics in the Constant Stress Ensemble
----------------------------------------------------------------
Although path integral Monte Carlo (PIMC) is usually used to estimate quantum effects in solids [@chakravarty97; @nielaba97; @marx99], path integral molecular dynamics (PIMD) [@tuckerman93] arise as a more natural choice for long-range Coulomb interactions. The Ewald sum [@frenkel96] has to be evaluated only once per time step in PIMD as opposed to each single local move in PIMC. Moreover, global moves in which the shape and size of the simulation cell are varied, are at basically no extra cost in molecular dynamics simulations, while Monte Carlo requires the evaluation of the net energy for each single trial move of the strain tensor.
In the path integral formulation of quantum statistical mechanics [@feynman65], a quantum point particle at temperature $T$ is represented by a closed polymer at temperature $P\,T$, in which adjacent beads interact via harmonic springs. The stiffness of the springs increases with decreasing thermal de Broglie wavelength $\lambda$ that a free particle would have at temperature $P\,T$. Studying this model in a molecular dynamics simulation, would require small time steps in the quantum limit $P \to \infty$, if the dynamical masses of the polymer beads were chosen to be identical with the physical mass $m$ of the quantum particle. However, it is possible to adjust all intra-molecular vibrations to similar time scales if the equations of motion are expressed in a convenient representation and appropriate “dynamical” masses are attributed to the beads. [@tuckerman93] In this study, the coordinates are represented in terms of eigenmodes of the free particle. The dynamical masses $\tilde{m}_\omega$ attributed to the motion of the eigenmode $\omega$ are usually chosen according to $\tilde{m}_\omega/m = k_{\rm E} / (k_{\rm E} + k_\omega)$, with $k_{\rm E}$ the coupling of an atom to its lattice site in the Einstein model of solids and $k_\omega$ the spring constant associated with eigenmode $\omega$. This choice of dynamical masses allows for efficient sampling of all degrees of freedom, because all modes move essentially on the same time scale. The dynamical mass of the center-of-mass motion of the polymer $\tilde{m}_0$ is of course identical with the real mass $m$.
The motion of the simulation cell is constrained to symmetric strain tensors, but otherwise done as is in the classical Parrinello-Rahman method [@parrinello80]. The dynamical mass $W$ associated with the motion of the simulation cell geometry is again chosen such that a typical oscillation time of the box is close to a typical oscillation time of a silicon or oxygen atom. The choice of $W$ merely controls the efficiency of the sampling but leaves meaningful observables uneffected.
One advantage of the Parrinello-Rahman method is the possibility to determine all elastic constants at zero external stress.[@parrinello82] This is done by using appropriate relations between strain fluctuations and mechanical compliances. It is important to note that only [*isothermal*]{} strain fluctuations are accessible in PIMD simulations. A constant enthalpy simulation of the isomorphic classical representation would not translate into conserved enthalpy of the quantum crystal.
In principle, our method is closely related to a recently proposed PIMD scheme for constant-strain constant-temperature simulations [@tuckerman98; @martyna99]. The special representation used here as well as omitting the thermostat included in the equations of motions in Ref. [@tuckerman98; @martyna99], anticipates to briefly review the final result. In order to do this, we represent the coordinate ${\bf R}_{i t}$ of particle $i$ at imaginary time $t$ as a product of a scaled coordinate ${\bf r}_{i t}$ and the time-dependent (symmetrical) matrix $h$, which contains the shape and the volume $V = \det h$ of the simulation cell: $$R_{i t \alpha} = h_{\alpha\beta} r_{i t \beta}.$$ The values of $r_{i t \alpha}$ are constrained to values $ 0 \le r_{i t \alpha} < 1$. The components of the metric tensor $G$ are defined as $G_{\alpha\beta} = h_{\alpha\mu} h_{\mu\beta}$ where summation convention over Greek indices enumerating spatial dimensions is implicitly assumed. It is then convenient to express the equations of motion for the scaled coordinates in reciprocal Fourier space, namely in terms of coordinates $$\label{eq:fourier}
\tilde{\bf r}_{i\omega} = {1\over \sqrt{P}}
\sum_{t=1}^P {\bf r}_{it} \exp\left({2\pi{\bf i}\over P}\omega t\right)$$ for which the motion of the free particles is diagonalized. Introducing $k_{i\omega} = 4m_i \sin^2(\pi\omega/P)/(\beta\hbar/P)^2$, allows to represent the equations of motion in a rather condensed form: $$\begin{aligned}
\label{eq:eqa_mot_1}
m_{i\omega} \ddot{\tilde{r}}_{i \omega \mu} & = &
\tilde{m}_{i\omega} ({{\bf G}}^{-1})_{\mu\nu} \dot{G}_{\nu\sigma}
\dot{\tilde{r}}_{i \omega \sigma} - k_{i\omega}
\tilde{r}_{i \omega \mu} + \nonumber \\
& & {1 \over \sqrt{P}}
\sum_{t} e^{{2\pi {\bf i}\over P} \omega t}
\sum_{j \ne i}
{\partial v_{ij} \over \partial R_{(ij)t}}
{r_{i t \mu} - r_{j t \mu} \over R_{(ij)t}}
\\
\label{eq:eqa_mot_2}
PW \ddot{h}_{\mu\nu} & = &
\sum_{i\omega} \tilde{m}_{i\omega} \dot{\tilde{r}}_{i\omega\nu}
\dot{\tilde{R}}_{i\omega\mu} -
\nonumber \\ & &
\sum_{it}
{m_iP^2\over \beta^2\hbar^2}
(r_{i t \nu}-r_{i\, t-1\, \nu})(R_{i t \mu}-R_{i\, t-1\, \mu}) +
\nonumber\\ & &
\sum_{it} \sum_{j>i}
{\partial v_{ij} \over \partial R_{(ij)t}}
R_{(ij)t\mu} R_{(ij)t\nu}\end{aligned}$$ with $v_{ij}$ a two-particle interaction potential between particle $i$ and $j$ and ${\bf R}_{(ij)t}$ the vector connecting particle $i$ and $j$ at imaginary time $t$ emploing minimum image conventions.
Despite the well-known disadvantages of Langevin-type thermostats [@frenkel96; @schneider78], correlation times turned out to be particulary small when all degrees of freedom (including the geometry of the simulation cell) were weakly coupled to a friction force linear in velocity and to a corresponding random force. [@schneider78] Chosing the damping term $\gamma$ of the Langevin dynamics to be $\gamma = 0.01 \,dt$ and $dt = 0.04 t_{\rm char}$ with $t_{\rm char}$ the (smallest) characteristic time-scale of the system, systematic errors were made much smaller than statistical errors. Moreover, the ergodicity problems, which is inherent to some PIMD algorithms [@tuckerman93], can be most easily overcome with a Langevin thermostat. The average “dynamic” kinetic energy $\langle T_{\rm dyn}\rangle$ and the fluctuation of $T_{\rm dyn}$ in the PIMD approach described above, are utterly sensitive to bad choices of $\gamma$ and $dt$. In the correct limit, one obtains $\langle T_{\rm dyn}\rangle/N = d k_{\rm B}TP/2$ per degree of freedom and the associated specific heat (fluctuation) $\langle \delta T^2_{\rm dyn}\rangle/Nk_{\rm b}T^2P^2 = d k_{\rm B}/2$ with $d$ the spatial dimension of the system. This sensitivity can be used to determine the accuracy of the simulation resulting in reliable thermodynamic expectation values of other observables. Note that only independent components of the Fourier transform $\tilde{{\bf r}}_{i\omega}$ need to be thermostated and considered for the calculation of the dynamic kinetic energy.
Dynamical information
---------------------
Imaginary-time path-integral methods do not allow direct calculation of dynamical properties. While the complete dynamical information is obtained in imaginary-time correlation functions in principle [@baym61], the inverse Laplace transform that one needs to carry out in order to assess real-time correlation functions is numerically unstable. A generalization of the PIMD method, the centroid molecular dynamics (CMD) method [@cao93], appears to give a more direct link between exact quantum dynamics and CMD. The basic idea of CMD is to propagate the center of mass of a PIMD quantum chain such that the internal degrees of freedom of the quantum chain are in their thermodynamical equilibrium. This makes the center of mass move on an effective classical potential surface. Time correlation functions can then be formulated in terms of centroid coordinates and averaged during the simulation. The Fourier transform of the centroid correlation function $I_{\rm c}(\omega)$ and the real quantum mechanical spectral function $I(\omega)$ are related linearly by $$\label{eq:centroid_spec}
I(\omega) =
n(\beta\hbar\omega) I_{\rm c}(\omega)$$ with $$n(\beta\hbar\omega) = {\hbar\beta\omega\over 2}
\left(1 + \coth{\hbar\beta\omega\over 2} \right).
\label{eq:norm}$$ Eq. (\[eq:centroid\_spec\]) is exact for harmonic systems as well as in the classical limit. Recently, much progress has been made in formulating rigorous relations between the dynamics of path-integral centroid variables and true dynamics as well as systematic correction to the CMD method sketched above. [@jang99]
Model Specific Information
--------------------------
In the present simulation, $v_{ij}$ in Eqs. (\[eq:eqa\_mot\_1\],\[eq:eqa\_mot\_2\]) corresponds to the BKS interaction potential [@beest90] between particles $i$ and $j$ including the Coulomb energy among other contributions. For the evaluation of the Ewald sum in arbitrarily shaped parallelepiped simulation cells, a recently proposed algorithm [@wheeler97] was used and constrained to symmetric matrices $h$ resulting in a 30% reduction of CPU time to evaluate the Ewald sum. The non-Coulombic interactions were cut off at a distance $r_{\rm c} = 9.5$ $\AA$.
In the following, it will be distinguished between classical and quantum mechanical results. Classical results are obtained by simply chosing $P=1$ in a PIMD simulation. Exact quantum results require taking the limit $P\to\infty$ in principle. It is well known that the so-called primitive decomposition of the density matrix, upon which the PIMD algorithm is based, invokes systematic errors in the partition function and observables that vanish proportionally to $1/(TP)^2$ [@muser95]. It is therefore important to chose $P$ large enough so that quantum effects are well reflected. On the other hand, $P$ should not be too large, which would result in large statistical error bars.
In order to assess at what values of $TP$ one can expect convergence to the quantum limit, the average potential energy $\langle V_{\rm pot} \rangle$ is calculated for various values of $P$ at the temperature $T = 300$ K. The results are shown in Fig. \[fig:vpot\_room\]. For $PT > 1200$ K, extrapolation to the quantum limit is possible with correction in the order $1/(TP)^2$. Combinations satisfying $PT > 4,800$ K basically correspond to the quantum limit.
Depending on the property of interest, it is necessary to extrapolate to the quantum limit with a $1/P^2$ corrections. This concerns particulary structural data, such as lattice constants, which can be obtained with high resolution. Elastic constants and dynamical information, however, are plagued with relatively large statistical error bars. For these observables, statistical uncertainties are much larger than systematic deviations and results obtained with $PT > 2,400$ K are referred to as quantum limit.
The total number of atoms used in the simulations was $N = 1080$. The linear box dimensions typically are $24.9\,\AA$, $25.9\,\AA$, and $21.9\,\AA$ along the $a,b$, and $c$ axis, respectively. In order to obtain elastic constants with an accuracy of about $3$ GPa, 60,000 MD steps of length 1 fs have to be performed, which takes about one day on an Intel II processor. For quantum simulations, the numerical effort has to be multiplied with Trotter number $P$.
Results {#sec:results}
=======
Structural Properties
---------------------
Two temperature regimes are particularly interesting to study, namely room temperature and temperatures near absolute zero. We first start with a discussion of quantum effects at room temperature, $T = 300$ K. This temperature is already well below the Debye temperature $T_{\rm D}$ of quartz. $T_{\rm D}$ of $\alpha$-quartz as determined by specific heat measurements is a strongly temperature-dependent function [@striefler75]: At $T = 0$, $T_{\rm D} \approx 550$ K, while at room temperature $T_{\rm D} \approx 1,000$ K. Some quantum effects can therefore be expected to be relatively strong at room temperature. In Fig. \[fig:bond\_room\], one can see that the main quantum effect at room temperature is the quantum mechanical freezing of the Si-O bond. This freezing is reflected by the fact that the distribution $p_{\rm SiO}(r)$ of the Si-O bond is much broader for the quantum mechanical study than for the classical study (Fig. \[fig:bond\_room\]a). In fact, the quantum mechanical $p_{\rm SiO}(r)$ barely changes its form when the temperature is lowered further. This is an indication that Si-O bonds are in their quantum mechanical ground state. On the other hand, Si-O-Si bond angles as well as O-Si-O bond angles do not differ considerably between classical and PIMD simulations (Fig. \[fig:bond\_room\]b).
Fig. \[fig:bond\] and Fig. \[fig:latt\_const\] show the effect of the quantum mechanical ionic motion on simple structural properties. The average Si-O bond length $r_{\rm SiO}$ is shown in Fig. \[fig:bond\] as a function of temperature, while the lattice constants $a$ and $c$ are shown in Fig. \[fig:latt\_const\]. In all cases, it is noticeable that the quantum mechanical values are larger than the classical equilibrium lengths. The effects are relatively small, but clearly within the resolution of the simulations. While $r_{\rm SiO}$ only differ by 0.19% at $T = 150$ K, the lattice constants differ by 0.35% in the case of both the $a$ axis and $c$ axis of $\alpha$-quartz. This means that the “excess” quantum volume can be attributed to both the SiO bond length and quantum fluctuations of the so-called rigid unit modes [@dove97; @welche98].
In the case of the lattice constants, Fig. \[fig:latt\_const\], direct comparison can be made to experimental data [@carpenter98]. The difference in the lattice constants between quantum mechanical calculations and experiment is about $0.06\,\AA$ for the $a$ axis and $0.07\,\AA$ for the $c$ axis. This difference is much larger than the discrepancy between classical and quantum mechanical simulations. It is, however, obvious that only quantum mechanical simulations reflect qualitatively the right low-temperature behaviour: While classical simulations result in a finite expansion coefficient even at absolute zero, the PIMD simulations lead to a vanishing expansion coefficient.
There is, however, a very good agreement in the low-temperature thermal expansion along the $c$ axis between PIMD simulation and experiment up to about 600 K, which shows that even anharmonic effects are well reflected by the BKS potential as long as the system is still far away from the $\alpha$-$\beta$ transition. Near the transition, however, the agreement becomes siginficantly less good. The “jump” in $c$ at the transition is absent in the simulation. Thus, an important ingredient in the potential energy surface is missing. Note that classical simulations based on both the BKs and the TTAM potential do not reflect the anomaly in the $c/a$ ratio, which has been observed experimentally [@muser00]. Thus, while harmonic and low-temperature anharmonic effects are well described by the BKS potential, there seems to be the need to reflect many-body effects as well. Fig. \[fig:angle\_80\] and Fig. \[fig:angle\_temp\] give detailed information on the bond angle distribution and their quantum effects. In Fig. \[fig:angle\_80\], the Si-O-Si and O-Si-O angle distributions are shown exemplarily at a temperature $T = 80$ K. Fig. \[fig:angle\_80\] confirms the picture that the local structure in $\alpha$-quartz does not correspond to tetrahedra. This can be concluded from the existence of two peaks in the classical O-Si-O bond angle distribution and a broadened shoulder in the quantum mechanical O-Si-O bond angle distribution. The difference in quantum mechanical and classical mean bond angles is rather small, yet, noticeable, e.g., the classical average O-Si-O bond angles approaches the ideal tetrahedra angle of $109.471^o$ much closer than the quantum mechanical simulation. The difference of about $0.04^o$ between the two approaches can be clearly resolved. The effect is much larger for the Si-O-Si bond angle, namely $0.2^o$ at $T = 80$ K, but less obvious (Fig. \[fig:angle\_temp\]b) because of the strong temperature dependence of $\langle\alpha_{\rm SiOSi}\rangle$.
Elastic properties {#subsec:ela_const}
------------------
Just like other properties can elastic constants be expected to differ between classical and quantum mechanical treatments. In order to calculate classical constants at zero temperature, it is sufficient to calculate the second derivative of the ground state (potential) energy with respect to the stress tensor, resulting in the so-called Born expression for elastic constants: [@born54] $$C_{\alpha\beta} =\partial^2 \langle V(T,\epsilon) \rangle /
\partial \epsilon_\alpha \partial \epsilon_\beta,
\label{eq:born}$$ where the derivative is evaluated at zero strain. At finite temperatures, it is not sufficient to generalize this expression by simpling taking the thermal expectation value of the right hand side. It has been pointed out correctly [@squire69] that the free energy surface ${\cal F}(T,\epsilon)$ should be considered instead of $V(T,\epsilon)$. This generalization leads to different estimators of the elastic constants when evaluated in the (NVT) ensemble. The main effect of this generalization is that fluctuations of the stress tensor need to be considered on top of the Born term described in Eq. \[eq:born\]. These fluctuations usually lead to a reduction of the elastic constants. Unlike classical fluctuation terms, quantum mechanically calculated terms will not vanish as the temperature approaches absolute zero. Among other effects, this will lead to different elastic constants for quantum mechanical and classical systems [@schoffel01]. In the case of silicates, however, it turns out that it is more efficient to calculate elastic constants $C_{ij}$ by exploiting the relations [@parrinello82] between $C_{ij}$ and the thermal fluctuations of the strain tensor.
Experimental, classical, and quantum mechanical elastic constants are compared in Fig. \[fig:ela\_aq\]. Elastic constants can be expected to show larger (relative) quantum corrections than lattice constants and heat of formation [@schoffel01]. For quartz, the reduction of about 5 GPa in $C_{33}$ seems to be the most dramatic effect. At $T = 300$ K, classical and quantum mechanical elastic constants agree within the statistical error bars. Below 300 K, the classical $C_{33}$ shows a stronger temperature dependence than the quantum mechanical $C_{33}$. This effect should be taken into account when trying to optimize potential energy surfaces: $C_{33}$ predicted by the force field parameters for $T = 0$ K should be a little larger than $C_{33}$ measured at a temperature of 300 K. For other $C_{ij}$ the same comment applies in principle, but quantitatively, the effects are less dramatic.
Dynamical Properties {#subsec:dyn_prop}
--------------------
As a generic dynamical property we consider the (classical) inverse-mass weighted momentum autocorrelation function $C(t)$ $$C(t) = \sum_i m_i^{-1} \left\langle \vec{p}_{i}(t)
\vec{p}_{i}(0) \right\rangle,
\label{eq:time_cor}$$ where $t$ denotes the real time. $C(t)$’s Fourier transform $\tilde{C}(\omega)$ can be used to define an effective density of states $g_{\rm eff}(\nu)$ $$g_{\rm eff}(\nu) = { \tilde{C}(2\pi\nu) \over Nk_BTn(\beta h\nu)}
\label{eq:dos}$$ with $n(\beta h\nu)$ being introduced in Eq. (\[eq:norm\]). $g_{\rm eff}(\nu)$ is identical with the real density of states (DOS) if the harmonic approximation is valid. $\tilde{C}(\omega)$ and hence the effective DOS can be exactly related to the imaginary-time correlation function $G(\tau)$ $$G(\tau) = \sum_i m_i \left\langle \left(\vec{R}_i(\tau) - \vec{R}_i(0)
\right)^2\right\rangle$$ via the two-sided Laplace transform $$\begin{aligned}
G(\tau) & = & \int_{-\infty}^{\infty} d\omega\,
\exp\left({-\hbar \omega \beta / 2}\right)
\nonumber\\ & &\, \times
{\tilde{C}(\omega)\over \omega^2}
\left[ \cosh\left\{\hbar\omega\left({\beta\over 2}-\tau\right)\right\} -
\cosh{\hbar\omega\beta\over 2}\right].
\label{eq:twosided}\end{aligned}$$ Note that the imaginary time $\tau$ has to be considered within the interval $0 \le \tau < \beta$. Outside of this interval, imaginary-time correlation functions are repeated periodically.
Eq. (\[eq:twosided\]) is useful to check the validity of the centroid molecular dynamics (CMD) method and hence to establish the validity of spectral functions as obtained by CMD. If $C(t)$ is determined in terms of the mass-weighted autocorrelation function of the centroid velocities and use is made of Eq. (\[eq:centroid\_spec\]), the effective DOS and hence $G(\tau)$ can be estimated in terms of centroid dynamics. If CMD is applicable, $G(\tau)$ as obtained by direct sampling and $G(\tau)$ as estimated via CMD have to agree. As shown in Fig. \[fig:imacorr\], the agreement is perfect within our statistical error bars for $\alpha$-quartz at very low temperatures. Of course, this agreement could be expected as the dynamics are dominated by the harmonic interactions unlike the thermal expansion coefficients. Note that a purely classical simulation leads to a similarly good agreement. CMD and classical velocity autocorrelation functions can barely be distinguished in the case of quartz.
Fig. \[fig:dos\] shows the density of states as calculated via the centroid PIMD. Of course, a degree of (meaningful) complexity in a spectrum such as shown in Fig. \[fig:dos\] can never be obtained by inverting imaginary-time correlation functions as shown in Fig. \[fig:imacorr\]. Thus, centroid PIMD are a useful tool to obtain DOS of silica.
Conclusions
===========
This study shows that path integral molecular dynamics (PIMD) are an efficient tool to calculate low-temperature properties of solids even if the complexity is larger than in rare gas crystals or other monoatomic solids. PIMD turns out to be particulary useful (as compared to path-integral Monte Carlo) when long-range forces have to be evaluated such as it is the case for the simulations of silica. Structural properties can be evaluated with high resolution and the shift from properties that are obtained if atomic motion is treated classically to the “real” quantum mechanical properties can be assessed. This shift can also be calculated for elastic constants, which are notoriously hard to compute even in classical simulations.
The result of the PIMD simulations anticipate that path integral techniques may not only become an important way of evaluating the merits and failures of potential energy surfaces, but PIMD might give valuable input to [*construct*]{} reliable model potentials. Here, the PIMD calculations of the thermomechanical properties of $\alpha$-quartz were based on the BKS potential [@beest90]. The construction of the BKS potential was pioneering in the sense that ab-initio calculations were combined with bulk properties in order to fit the free model parameters. In the latter part, lattice constants and elastic constants were calculated for a classical system at $T = 0$ K from the (fit) parameters and adjusted such that agreement with experimental “quantum mechanical” (finite temperatures) data was optimum. The PIMD results in combination with the classical MD results presented in this paper, show that this part of adjusting the parameters of the BKS potential allows for further optimization. Of course, one can not necessarily expect to find a two-body potential energy surface for silica that describes interactions much better than the BKS potential. \[sec:conclusions\]
We thank Kurt Binder for useful discussions. Support from the BMBF through Grant 03N6015 and from the Materialwissenschaftliche Forschungszentrum Rheinland-Pfalz is gratefully acknowledged.
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Chris S. Hanson , Laurent Gizon, Zhi-Chao Liang'
bibliography:
- 'References.bib'
title: |
Solar Rossby waves\
observed in GONG++ ring-diagram flow maps
---
[Solar sectoral Rossby waves have only recently been unambiguously identified in Helioseimsic and Magnetic Imager (HMI) and Michelson Doppler Imager (MDI) maps of flows near the solar surface. So far this has not been done with the Global Oscillation Network Group (GONG) ground-based observations, which have different noise properties. ]{} [We utilize 17 years of GONG++ data, to identify and characterize solar Rossby waves using ring-diagram helioseismology. We compare directly with HMI ring-diagram analysis. ]{}
Maps of the radial vorticity are obtained for flows within the top 2 Mm of the surface for 17 years of GONG++. The data is corrected for [systematic effects including the annual periodicity related to the $B_0$ angle.]{} We then compute the Fourier components of the radial vorticity of the flows in the co-rotating frame. We perform the same analysis on the HMI data that overlap in time.
[We find that the solar Rossby waves have measurable amplitudes in the GONG++ sectoral power spectra for azimuthal orders between $m=3$ and $m=15$. The measured mode characteristics (frequencies, lifetimes and amplitudes) from GONG++ are consistent with the HMI measurements in the overlap period from 2010 to 2018 [for $m\le9$. For higher-$m$ modes the amplitudes and frequencies agree within two sigmas.]{} The signal-to-noise ratio of modes in GONG++ power spectra [ is comparable to HMI for $8\le m\le11$, but is lower by a factor of two for other modes.]{} ]{} [The GONG++ data provide a long and uniform data set to study solar global-scale Rossby waves from 2001. ]{}
Introduction
============
@loeptien_etal_2018 recently discovered solar global-scale Rossby waves (or r modes) in the SDO/HMI surface flow field using correlation tracking of granulation and helioseismic ring-diagram analysis. The Rossby waves are potentially important as they could be probes of the deep convection zone. @liang_etal_2019 confirmed this result using time-distance helioseismology applied to both HMI and SOHO/MDI observations. It has yet to be seen how well the r modes show in the GONG++ ring-diagram data. This is the goal of this study.
In the first Rossby wave findings, @loeptien_etal_2018 generated radial vorticity maps from flows maps measured by local correlation tracking of granules [LCT, @welsch_etal_2004; @fisher_welsch_2008] and ring-diagram analysis [RDA, @hill_1988]. They found that the sectoral r-mode spectrum closely follows the standard theoretical dispersion relation $\omega = -2\Omega_{\rm eq}/(m+1)$, where $\Omega_{\rm eq}/2\pi=453.1$ nHz is the equatorial rotation rate and $m$ is the azimuthal order [see, e.g., @saio_1982]. @liang_etal_2019 confirmed the results of @loeptien_etal_2018 using time-distance helioseismology on the meridional component $u_y$ of the horizontal flow near the equator. Unlike @loeptien_etal_2018, who imaged near surface layers, @liang_etal_2019 imaged at a depth of $0.91$ R$_\odot$. They used 21 years of data spanning both the observation sets of MDI and HMI from 1996 to 2017. Meanwhile, @hanasoge_mandal_2019 provided another independent means of detecting solar r modes, through the use of normal-mode coupling in two years of HMI data. Additionally, @proxauf_etal_2019 investigated the latitudinal and depth dependence of the r modes using HMI ring-diagram analysis. Finally, in an effort to assess the accuracy of machine learning techniques for ring-diagram analysis, @alshehhi_etal_2019 used r modes as a litmus test on the suitability of this new technique for helioseismic inversions.
In this study we aim to contribute to this growing literature on solar Rossby waves, with an analysis of the 17 years of data from GONG++ and compare a subset of it to the overlapping observations of HMI. We will focus on using the RDA products from both the GONG++ and HMI pipelines. The GONG++ data is interesting for solar r-mode characterization for two reasons. Firstly, the GONG data and HMI data overlap since 2010, enabling the direct comparison of two data sets produced by similar pipelines, but with different noise properties. Secondly, while @liang_etal_2019 merged the MDI (14 years) and HMI (7 years) data to create a combined 21 years of data, the GONG++ data should be uniform for 17 years. In this study we explore the signature of the solar r modes in the GONG++ data and report differences with the HMI data.
Data analysis and results {#sec.dataAnalysis}
=========================
Ring-diagram analysis
---------------------
We use ring-diagram flow maps generated by the GONG++ pipeline from September 2001 to February 2019. The pipeline follows the method outlined in @cobard_etal_2003, whereby flows are computed from the p-mode frequency shifts extracted from tracked patches (tiles) of the Doppler velocity. These patches are tracked across a transverse cylindrical equidistant projection of the solar disk for approximately 27 hours, following the Snodgrass rotation profile [e.g. Eq. 3 of @cobard_etal_2003]. For every day of tracking there are 189 tiles across the solar disk.
For each tile, the two horizontal components of the flows in the $x$ (prograde) and $y$ (northward) directions, $(u_x,u_y)$, are computed as follows:
- A 2D cosine bell apodization is applied to a $16^\circ\times16^\circ$ square patch, to obtain a circular tile of radius $15^\circ$.
- For each data cube of the Doppler velocity $\Psi(x,y,t)$, a 3D Fourier transform is performed to generate spectral cubes $\widehat{\Psi}(k_x,k_y,\omega)$, from which the power spectrum $\mathcal{P}(k_x,k_y,\omega) = |\widehat{\Psi}(k_x,k_y,\omega)|^2$ is computed.
- For each fixed wave number $k=(k_x^2+k_y^2)^{1/2}$, a cylindrical cut through the power spectrum is made to generate $\mathcal{P}_k(\vartheta, \omega)$, where $\vartheta$ is the angle between the wave vector and the prograde direction [@bogart_etal_1995].
- In the absence of flows, acoustic modes appear as horizontal ridges in $\mathcal{P}_k({\vartheta},\omega)$. The presence of flows causes Doppler frequency shifts that depend on the magnitude and direction of flow. The observed ridges are fit with a six parameter Lorentzian-like model [e.g. @haber_etal_2000].
- For each p-mode radial order $n$ and wave number $k$, two flow parameters $(U_x,U_y)$ are extracted from the data. The physical flow $(u_x, u_y)$ at a particular depth in the interior is inferred by inverting the set of measured parameters. In the current study we restrict our attention to the flows at a depth of 2 Mm.
Sectoral power spectra of radial vorticity
------------------------------------------
With the horizontal flow components for each tile computed, we then construct longitude-latitude maps following the procedure outlined by @loeptien_etal_2018. We remove the one year periodicity from each tile which arises primarily from center-to-limb effects that vary with the $B_0$, [and from a small error in the accepted inclination angle of the solar rotation axis [@beck_giles_2005; @hathaway_rightmire_2010]]{}, using Eq. [A.5]{} of @liang_etal_2019. The model consists of five components, accounting for time-invariant effects and effects with one-year periods. With these systematics removed, we then remap the flow maps in a longitude-latitude frame that rotates at the equatorial rotation rate (453.1 nHz), as was done by @loeptien_etal_2018. Finally, the radial vorticity, $\zeta$, is computed.
The sectoral power spectrum of the radial vorticity $\zeta$ for the azimuthal order $m$ is computed through $$P(m,\omega) =
\left|
\int_{0}^{T}{\rm d}t\int_0^{2\pi}{\rm d}\phi \,
e^{-\ii m\phi+\ii\omega t}
\int_0^{\pi} {\rm d}\theta \, (\sin\theta)^{m+1} \zeta(\theta,\phi,t)
\right|^2 ,$$ where $T$ is the observation duration, $\theta$ is the colatitude, and $\phi$ is the longitude. In obtaining the above expression we used the fact that the sectoral spherical harmonic $Y^m_m(\theta,\phi)$ is proportional to $(\sin\theta)^m e^{\ii m\phi} $.
Figure \[fig.rossby\_powspect\] shows the power spectrum computed from the 17 years of GONG++ data. We have shown the power spectrum rebinned to one third of the frequency resolution. Similar to previous studies, we clearly identify the r modes which follow the theoretical dispersion relation.
\
Interestingly, while $m=2$ mode is absent in the spectrum [see also @liang_etal_2019 for a discussion], it appears that there is a peak near the expected frequency of the $m=1$ mode in Figure \[fig.rossby\_powspect\]. In order to further investigate this, we perform the same preprocessing on synthetic data (see Appendix \[sec.synthetics\]). The synthetics show that any yearly variation signal will leak into $m=1$ due to the partial coverage of the solar surface (window function). Using the annual variation model of @liang_etal_2019, this leaked signal is removed in our processing. The apparent signal present in $m=1$ of Fig. \[fig.rossby\_powspect\] coincides with leakage from low frequency power in $m=0$ and is likely not a true $m=1$ r mode.
[cccccc]{} $m$ & $\omega_0/2\pi$ & $\Gamma/2\pi$ & $S$ & $N$ & $S/N$\
& \[nHz\] & \[nHz\] & \[$10^{-2}$\] & \[$10^{-2}$\]\
Rossby Mode parameters.
-----------------------
In this section we seek to characterize the r-mode spectrum. The vorticity power spectrum is fit for each of the Rossby modes $m$ assuming the functional form of the Lorentzian, $$\mathcal{L}_m(\omega,\pmb{\lambda}) = \frac{S}{1 + (\omega-\omega_0)^2/(\Gamma/2)^2} + N ,$$ where $S$ is the mode amplitude, $\omega_0$ is the mode frequency, $\Gamma$ is the full width at half maximum, $N$ is the background noise of the spectrum and $\pmb{\lambda}=\{S,\omega_0,\Gamma,N\}$. The probability density function of the vorticity power spectrum is exponential. As such, the estimates on the parameters $\pmb{\lambda}$ for each mode are determined by minimizing the negative of the log-likelihood function, $$J_m(\pmb{\lambda}) = \sum_{k=1}^K \ln{\mathcal{L}_m(\omega_k,\pmb{\lambda})} + P(m,\omega_k)/\mathcal{L}_m(\omega_k,\pmb{\lambda}) ,$$ where $\omega_k$ is the $k$-th frequency bin within the frequency range of interest, [consisting of $K$ bins]{}. We use a semi-empirical approach to compute the Hessian and thus the errors on the fit [see @toutain_appourchaux_1994]. [The fit for each mode is performed in the frequency range $\pm300$ nHz from the theoretical dispersion relation.]{}
Table \[tab.mode\_details\] lists the fit parameters and their error for the 17 year GONG++ RDA r-mode power spectrum. The $m=1$ peak is not listed in the table. Its frequency coincides where low frequency $m=0$ power should leak through ($421.41$ nHz) and has a much smaller amplitude than the other modes. The fit for $m=2$ was not performed due to the absence of any signal.
Comparing GONG++ and HMI
------------------------
In this section we compare the r-mode power spectrum derived from the respective RDA flow map pipelines of HMI and GONG++, for a shared observation period between May 2010 and December 2018. Figure \[fig.comparing\_gonghmi\_spectra\] compares the GONG++ and HMI r-mode power spectra for the shared observational period. These results show that while the noise is different, the mode power spectra are in general agreement. The amplitudes of GONG++ and HMI r-mode signals are close, with the GONG++ having a greater background noise. For $m>8$ the mode power in the GONG++ data becomes noticably smaller.
{width="0.95\linewidth"}
Figure \[fig.comparing\_gonghmi\] compares the measured frequencies, line widths, vorticity amplitudes and $S/N$ ratios between four different data sets. The data sets include GONG++ and HMI ring diagrams with overlapping observation time, the travel-time results of @liang_etal_2019 derived from time-distance helioseismology, and the LCT results of @loeptien_etal_2018. These results show that within error bars the reported mode frequencies and line widths agree for $m\le9$. For $m\ge10$ the mode frequencies tend not to agree, with the time-distance results of @liang_etal_2019 being shifted towards smaller negative frequencies, while the HMI RDA data analyzed here tends to have greater negative frequencies. In terms of line width, the four spectra tend to be in agreement for all $m$, though the large reported errors make it difficult to make any strong conclusions on this characteristic of the modes. [The HMI and GONG++ RDA mode amplitudes are within error limits for $m\le10$. But, for higher order modes HMI data has a greater amplitude by a factor two. ]{} [Comparing the $S/N$ ratios, shows that in general the ratio is greater in HMI RDA than GONG++ RDA by a factor of two for $m<8$, are similar from $m=8$ to $m=11$, and are greater again for higher order modes. Our results also show that the $S/N$ ratios are significantly higher in ring-diagram data than in travel-time measurements [@liang_etal_2019].]{}
\
Discussion and conclusions
==========================
We have utilized 17 years of observations from the ground based GONG program to characterize solar equatorial Rossby waves. Using ring diagram analysis from the GONG++ pipeline, we have clearly identified the solar Rossby waves in the sectoral power spectra of the radial vorticity maps.
Like previous studies we have found that the $m\le2$ r modes are absent from the data. An apparent $m=1$ peak appears in the power spectra but we conclude that this peak is leaked through from low frequency power in $m=0$. The synthetics show that a dipole r mode will appear as two peaks separated by $2\times 31.7$ nHz due to leakage. No such configuration is seen in the solar spectrum.
With 17 years of near continuous data, we measured the Rossby mode characteristics. We find that our results, for the period during which GONG++ and HMI data overlap, agree with both @loeptien_etal_2018 and @liang_etal_2019 for $4\le m\le9$. For $m>9$ modes the GONG++ r-mode frequencies are only in agreement with @loeptien_etal_2018. [This is due to the chosen method for computing the r-mode spectrum. We have found that if we also compute the spectrum through the Fourier transform of $u_y$ at the equator, the frequencies agree with @liang_etal_2019. These discrepancies arise for high $m$, because the latitudinal eigenfunctions of the r-modes are not sectoral spherical harmonic functions [@proxauf_etal_2019]. Our analysis also suggests that this choice in latitudinal basis function contributes to the small discrepancy for $m=3$. ]{} The amplitudes from the GONG++ and HMI RDA are within errorbars for $m\le10$. For higher-order modes the HMI RDA amplitudes are greater than GONG++ by a factor of two. The measured $S/N$ ratios of modes in GONG++ RDA power spectra are in the range between 5 and 45. This shows the suitability of the GONG++ data for future studies of global-scale Rossby waves. We note we do not detect any mode in the $m=2$ sectoral power spectrum, which is coherent with previous studies.
In this study we have focused on only one depth (2 Mm), and measured the mode characteristics for the entire 17 (or 8) year time series. We have found good agreement with HMI, despite the lower $S/N$ ratios in the GONG++ data. The advantage that GONG++ data has over other data sets is the long and uniform observation window. The next stages for Rossby wave studies should focus on the temporal variation of the waves. With the long time series of GONG shown in this study, and the combined HMI and MDI time series of @liang_etal_2019, the temporal dependence of r-modes could be investigated with two independent data sets.
Synthetics {#sec.synthetics}
==========
In order to quantify the effects of systematics, we generate synthetic Rossby waves and analyze the effects of our data processing (Sec. \[sec.dataAnalysis\]) and the window function due to partial coverage of the Sun. For the functional form of the Rossby waves, we assume they are purely horizontal and obey mass conservation. The flow components $(u^m_\theta,u^m_\phi)$ of a sectoral r mode of azimuthal order $m$ are defined by, $$\begin{aligned}
\label{eq.rossbySyn}
\begin{split}
u_\theta^m(\theta,\phi,t)&= -A (\sin\theta)^{m-1} \sin(m\phi-\sigma_m t) e^{-\Gamma t},\\
u_\phi^m(\theta,\phi,t) &= -A\cos\theta\ (\sin\theta)^{m-1} \cos( m \phi -\sigma_mt) e^{-\Gamma t},\\
\end{split}\end{aligned}$$ where $\sigma_m=-2\Omega_{\rm eq}/(m+1)$ and $A=1$ m/s. Here, $\Gamma/2\pi$ is chosen to be 10 nHz. [For a single ring-diagram tile the flow components $u_x$ and $u_y$ can be identified with $u_\phi$ and $-u_\theta$, respectively. The horizontal flow field within a single tile centered at co-latitude $\theta$ and longitude $\phi$ is then computed by summing the individual r modes:]{} $$\begin{aligned}
u_x(\theta,\phi,t) &= \sum_{m=1}^{M} {u_\phi^m(\theta,\phi,t)}, \\
u_y(\theta,\phi,t) &= - \sum_{m=1}^{M} {u_\theta^m(\theta,\phi,t)}. \end{aligned}$$ The radial vorticity $\zeta$ is then computed through; $$\zeta(\theta,\phi,t) = \frac{1}{R_\odot\sin\theta}\left[ \frac{\partial}{\partial \theta}(u_x\sin\theta) + \frac{\partial}{\partial \phi}u_y\right]
+ \frac{A}{R_\odot}\cos({\omega_\oplus t}) ,$$ where we added a signal with a yearly period ($2\pi/\omega_\oplus=1$ year) to emulate the annual systematic error that is present in RDA [e.g. @komm_etal_2015]. We limit ourselves to a sum of $M=15$ modes.
Using Eq. \[eq.rossbySyn\], we compute the flows in each $15^\circ$ tile, across the solar surface. We then examine three cases: 1) An ideal case where we have tiles across the entire solar surface at all times. 2) The realistic case when we only have tiles observed by GONG on the visible disk, but no annual variation is removed. 3) The same as the previous case, except after the performing the processing outlined in this paper (e.g. temporal systematic removal).
Figure \[fig.windowAnalysis\] shows the [sectoral]{} power spectra [of the radial vorticity]{} for these three cases. In the ideal case, where the entire sun is observed, we see a clean spectrum without aliasing. In the realistic case (2), there is leakage of modes into their neighbouring modes, albeit at frequencies $\pm 421.54$ nHz from the central frequency. For the ridge of concern to us, the most significant effect is the leakage into $m=1$.
![[Sectoral]{} power spectra for the synthetic [radial vorticity data]{} for the complete solar surface (full Sun, left panel), including the observational window function (pre-treatment, central panel), and after processing (post-treatment, right panel). Cyan arrows show the peaks at $\pm (1\; {\rm year})^{-1}=\pm 31.7$ nHz. Blue arrows show the leakage of $|m|=1$ into $m=0$ once the window function is applied. These leakages coincide with the annual variation. After treatment, the annual variation and its leakage disappears, leaving the $m=1$ r-modes and associated leakage. The green arrows highlight the effects the window has on $m=1$ power. []{data-label="fig.windowAnalysis"}](FIGURES/Spectrum_Compare_AllM.png){width="\linewidth"}
This work was supported by NYUAD Institute Grant G1502. LG acknowledges partial support from the European Research Council Synergy Grant WHOLE SUN 810218. This work utilizes data obtained by the Global Oscillation Network Group (GONG) Program, managed by the National Solar Observatory, which is operated by AURA, Inc. under a cooperative agreement with the National Science Foundation. The HMI data is courtesy of NASA/SDO and the HMI Science Team. CSH thanks Frank Hill and Andrew Marble for their assistance with the GONG data. We also thank Bastian Proxauf and Jishnu Bhattacharya for insightful discussions and Martin Bo Nielsen for performing consistency checks between the analytical fitting of this study and Markov Chain Monte Carlo.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In this article we study a cake cutting problem. More precisely, we study *symmetric* fair division algorithms, that is to say we study algorithms where the order of the players does not influence the value obtained by each player. In the first part of the article, we give a symmetric and envy-free fair division algorithm. More precisely, we show how to get a symmetric and envy-free fair division algorithm from an envy-free division algorithm.\
In the second part, we give a proportional and symmetric fair division algorithm with a complexity in $\bigO(n^3)$ in the Robertson-Webb model of complexity. This algorithm is based on Kuhn’s algorithm. Furthermore, our study has led us to study aristotelian fair division. This notion is an interpretation of Aristotle’s principle: give equal shares to equal people.\
We conclude this article with a discussion and some questions about the Robertson-Webb model of computation.
address: |
Guillaume Chèze: Institut de Mathématiques de Toulouse\
Université Paul Sabatier\
118 route de Narbonne\
31 062 TOULOUSE cedex 9, France
author:
- Guillaume Chèze
title: |
Don’t cry to be the first!\
Symmetric fair division algorithms exist.
---
Introduction {#introduction .unnumbered}
============
In this article we study the problem of fair resource allocation. It consists to share an heterogeneous good between different players or agents. This good can be for example: a cake, land, time or computer memory. This problem is old. For example, the Rhind mathematical papyrus contains problems about the division of loaves of bread and about partition of plots of land. In the Bible we find the famous “Cut and Choose" algorithm and in the greek mythology we find the trick at Mecone.\
The problem of fair division has been formulated in a scientific way by Steinhaus in 1948, see [@Steinhaus]. Nowadays, there exists several papers, see e.g. [@DubinsSpanier; @EvenPaz; @EdmondsPruhs; @BramsTaylorarticle; @RoberstonWebbarticle; @Pikhurko; @Thomson2006; @Procacciasurvey; @BJK; @AzizMackenzie], and books about this topic, see e.g. [@RobertsonWebb; @BramsTaylor; @Procacciachapter; @Barbanel]. These results appear in the mathematics, economics, political science, artificial intelligence and computer science literature. Recently, the cake cutting problem has been studied intensively by computer scientists for solving resource allocation problems in multi agents systems, see e.g. [@Chevaleyre06; @Chen; @Dynamic; @Branzei].\
Throughout this article, the cake will be an heterogeneous good represented by the interval $[0,1]$. We consider $n$ players and we associate to each player a non-atomic probability measure $\mu_i$ on the interval $X=[0,1]$. These measures represent the utility functions of the player. The set $X$ represents the cake and we have $\mu_i(X)=1$ for all $i$. The problem in this situation is to get a fair division of $X=X_1\sqcup \ldots \sqcup X_n$, where the $i$-th player get $X_i$.\
A practical problem is the computation of fair divisions. In order to describe algorithms we thus need a model of computation. There exist two main classes of cake cutting algorithms: discrete and continuous protocols (also called moving knife methods). Here, we study discrete algorithms. These kinds of algorithms can be described thanks to the classical model introduced by Robertson and Webb and formalized by Woeginger and Sgall in [@Woeg]. In this model we suppose that a mediator interacts with the agents. The mediator asks two type of queries: either cutting a piece with a given value, or evaluating a given piece. More precisely, the two type of queries allowed are:
1. $eval_i(x,y)$: Ask agent $i$ to evaluate the interval $[x,y]$. This means return $\mu_i([x,y])$.
2. $cut_i(x,a)$: Ask agent $i$ to cut a piece of cake $[x,y]$ such that $\mu_i([x,y])=a$. This means: for given $x$ and $a$, return $y$ such that $\mu_i([x,y])=a$.
In the Robertson-Webb model the mediator can adapt the queries from the previous answers given by the players. In this model, the complexity counts the finite number of queries necessary to get a fair division. For a rigourous description of this model we can consult: [@Woeg; @Branzei2017].\
When we design a cake cutting algorithm, we have to precise what is the meaning of a fair division. Indeed, there exists different notions of fair division.\
We say that a division is *proportional* when for all $i$, we have $\mu_i(X_i) \geq 1/n$.\
We say that a division is *envy-free* when for all $i \neq j$, we have $\mu_i(X_i) \geq \mu_i(X_j)$.\
We say that a division is *equitable* when for all $i\neq j$, we have $\mu_i(X_i)=\mu_j(X_j)$.\
The first studied notion of fair division has been proportional fair division, [@Steinhaus]. Proportional fair division is a simple and well understood notion. In [@Steinhaus] Steinhaus explains the Banach-Knaster algorithm, also called last diminisher algorithm, which gives a proportional fair division. There also exists an optimal algorithm to compute a proportional fair division in the Robertson-Webb model, see [@EvenPaz; @EdmondsPruhs]. The complexity of this algorithm is in $\bigO\big(n\log(n)\big)$. Furthermore, the portion $X_i$ given to the $i$-th player in this algorithm is an interval.\
It is more difficult to get an envy-free fair division. Indeed, whereas envy-free fair divisions where each $X_i$ is an interval exist, there does not exist an algorithm in the Robertson-Webb model computing such divisions. These results have been proved by Stromquist in [@Stromquistexist; @Stromquist]. The first envy-free algorithm has been given by Brams and Taylor in [@BramsTaylorarticle]. This algorithm has been given approximatively 50 years after the first algorithm computing a proportional fair division. The Brams-Taylor algorithm has an unbounded complexity in the Robertson-Webb model. This means that we cannot bound the complexity of this algorithm in terms of the number of players only. It is only recently that a finite and unbounded algorithm has been given to solve this problem [@AzizMackenzie]. The complexity of this algorithm is in $\bigO\Big(n^{n{^{n^{n^{n^{n}}}}}}\Big)$. A lower bound for envy-free division algorithm has been given by Proccacia in [@Procaccia-lowerbound]. This lower bound is in $\bigO(n^2)$.\
Equitable fair divisions have been less studied than proportional and envy-free divisions. However, there exist some results showing the difficulty to get such fair divisions. Indeed, there exist equitable fair divisions where each $X_i$ is an interval, see [@Cechexistence; @Segal-Halevi; @Chezeequitable]. However, there do not exist algorithms computing an equitable fair division, see [@Cech; @ProcWang; @ChezeBSSRW].\
In practice, a cake cutting algorithm $\mathcal{F}$ has in inputs a list of measures $\underline{\mu}=[\mu_1,\ldots,\mu_n]$, and returns a partition $X=\mathcal{F}(X,\underline{\mu},1) \sqcup \ldots \sqcup \mathcal{F}(X,\underline{\mu},n)$, where each $\mathcal{F}(X,\underline{\mu},i)$ is a finite union of disjoint intervals. The set $\mathcal{F}(X,\underline{\mu},i)$ is the part given to the $i$-th player appearing in the the list $\underline{\mu}$ when we apply the algorithm $\mathcal{F}$ to this list of measures.\
The definition of proportional or envy-free fair division is independent of the order of the players in the list $\underline{\mu}$. However, this order is important in cake-cutting algorithms. For example, the role of the two players in the “Cut and Choose" algorithm are not symmetric. This leads the definition of *symmetric fair division algorithm*.\
We denote by $\underline{\mu}^{\sigma}$ the list $\underline{\mu}^{\sigma}=[\mu_{\sigma(1)},\ldots,\mu_{\sigma(n)}]$, where $\sigma$ belongs to the permutation group $\mathfrak{S}_n$. A cake cutting algorithm $\mathcal{F}$ is *symmetric* when $$\forall i \in \{ 1, \ldots, n\}, \forall \sigma \in \mathfrak{S}_n, \, \mu_i\big(\mathcal{F}(X,\underline{\mu},i)\big)=\mu_i\big(\mathcal{F}(X,\underline{\mu}^{\sigma},\sigma^{-1}(i))\big).$$
For example, if $n=3$ and $\sigma=(1\,2\,3)$ then a symmetric fair division algorithm satisfies: $$\mu_1\big(\mathcal{F}(X,[\mu_1,\mu_2,\mu_3],1)\big)=\mu_1\big(\mathcal{F}(X,[\mu_2,\mu_3,\mu_1],3)\big).$$ A cake cutting algorithm is symmetric means whatever the order of the measure given in inputs, all players will receive the same value of the cake. Indeed, $\mathcal{F}(X,[\mu_2,\mu_3,\mu_1],3)$ is the portion given to the third player in the list $[\mu_2,\mu_3,\mu_1]$. Thus, this corresponds to the portion given to the player with measure $\mu_1$ when the algorithm $\mathcal{F}$ as in input the list $[\mu_2,\mu_3,\mu_1]$. Thus, if the player with associated measure $\mu_1$ is in the first or in the last position in the inputs he or she will get a portion with the same measure relatively to his or her preference $\mu_1$. Therefore, there is no advantage to be the first in the list $\underline{\mu}$. The measure of the received portion is independent of the position of a player in the list.\
This notion has been introduced by Manabe and Okamoto in [@ManabeOkamoto]. They call this kind of fair division *meta envy-free*. In this article we call this property *symmetric* in order to emphasize the role of the permutations of the players. In their paper Manabe and Okamoto have shown that classical algorithms such as Selfridge-Conway, and Brams-Taylor’s algorithms are not symmetric. Then they have given a symmetric and envy-free algorithm for 4 players and ask if it is possible to get such a division protocol for $n\geq 4$ players. Here, we answer to this question and we prove the following result:
There exists deterministic symmetric and envy-free cake cutting algorithms.
In order to prove this result we show how to construct such an algorithm from an envy-free algorithm. The idea is to use an already existing envy-free algorithm $f$, see e.g. [@BramsTaylorarticle; @RoberstonWebbarticle; @Pikhurko; @AzizMackenzie] and to construct from it a symmetric and envy-free algorithm $\mathcal{F}$. In order to get a symmetric algorithm we compute all $f(\underline{\mu}^{\sigma})$ and then we take the “best" one. Here “best" will mean : satisfy some topological conditions, e.g. we select a partition with the minimal number of cuts.\
Our approach computes $n!$ envy-free divisions, thus this gives an algorithm with a huge complexity in the Robertson-Webb model. Furthermore, our algorithm gives a proportional division since it gives an envy-free division. A natural question is then: Can we get a symmetric and proportional division algorithm with a polynomial complexity?\
We prove in Section \[sec:propvrac\] the following result:
There exists a deterministic symmetric and proportional algorithm which uses at most $\bigO(n^3)$ queries in the Robertson-Webb model.
The deterministic assumption is important. We do not want to get a situation where a player could think that he is unlucky.\
We can already remark that the Evan-Paz algorithm, see [@EvenPaz], and the last diminisher procedure are not deterministic and not symmetric. Indeed, if during these algorithms several players cut the cake at the same point, then this tie is usually breaked with a random process. Another way to break the tie is to use the order on the players. For example, if all the players in the first step of the Evan-Paz algorithm cut the cake at the same point, then we can give to the players $1, \ldots, \lfloor n/2 \rfloor$ the left part of the cake and to the other players the right part of the cake. This tie breaking method depends on the order the players and thus it does not give a symmetric procedure.\
At last, in this article we study also another fair division notion. This notion comes from the study of symmetric fair divisions in a particular case: Suppose that $\mathcal{F}$ is a symmetric fair division algorithm. Then we have $$\mu_1\big( \mathcal{F}(X,[\mu_1,\mu_2,\mu_3],1) \big) = \mu_1\big( \mathcal{F}(X,[\mu_2,\mu_1,\mu_3],2) \big).$$ Now, suppose that $\mu_1=\mu_2$, this gives $$\mu_1\big( \mathcal{F}(X,[\mu_1,\mu_2,\mu_3],1) \big) = \mu_2\big( \mathcal{F}(X,[\mu_1,\mu_2,\mu_3],2) \big).$$ This means that if two players have the same measure then they consider as equal the portions they get. We call a fair division satisfying this property an “*aristotelian fair division*".
We say that we have an aristotelian division when $\mu_i=\mu_j$ implies $\mu_i(X_i)=\mu_j(X_j)$.
We have given the name “aristotelian fair division" to this kind of fair divisions because in the Nicomachean Ethics by Aristotle (Book V) we find:\
*“…it is when equals possess or are allotted unequal shares, or persons not equal equal shares, that quarrels and complaints arise."*\
Therefore, aristotelian fair division is not a new notion. This notion has been already studied, see e.g. [@Maniquet; @MoulinBook; @MoulinSym]. In the literature this notion also appears as “Equal Treatment of Equals".\
We remark that symmetric fair division algorithms give aristotelian fair divisions. However, the converse is not true.\
As a first step towards the construction of a symmetric and proportional fair division algorithm, we describe in Section \[sec:propvrac\] an aristotelian and proportional fair division algorithm. This algorithm needs $\bigO(n^3)$ queries but less arithmetic operations than the symmetric and proportional algorithm.\
We remark easily that an envy-free division is always proportional and aristotelian, but a fair division which is aristotelian and proportional is less demanding than an envy-free division. However, to the author’s knowledge all existing aristotelian proportional fair division algorithms were envy-free algorithms.\
Thus our algorithm shows that if we just want an aristotelian proportional fair division it is not necessary to use an envy-free algorithm which uses an exponential number of queries.
Structure of the paper {#structure-of-the-paper .unnumbered}
----------------------
In Section \[sec:symenvyfree\], we give a symmetric and envy-free fair division algorithm. Then, we give some remarks about the complexity of this algorithm. In this first section, we also discuss the problem of symmetric and envy-free fair division in the approximate setting. In Section \[sec:propvrac\], we explain why the Evan-Paz and the last diminisher algorithm do not give aristotelian fair division. Then we give an aristotelian proportional fair division algorithm and next a symmetric and proportional fair division algorithm. In Section \[sec:conclusion\], we conclude this article with several questions about symmetric and aristotelian fair divisions and the Robertson-Webb model of computation.
An envy-free and symmetric cake cutting algorithm {#sec:symenvyfree}
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Two orders on partitions and one algorithm
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In this section we introduce two different orders on the partitions. These orders will be used to choose a “good" partition among the $n!$ possible fair divisions given by all $f(\underline{\mu}^{\sigma})$, where $f$ is a fair division procedure.\
In this section, when we study a partition $X=X_1 \sqcup \ldots \sqcup X_n$, $X_i$ will be the part given to the $i$-th player.\
For each partition $X=X_{1} \sqcup \ldots \sqcup X_{n}$ we set $$X_{i}=\bigsqcup\limits_{j \in I_i} [x_{i,j},x_{i,j+1}], \textrm{ where } I_i \textrm{ is a finite set.}$$ Thus $$X=\bigsqcup\limits_{i=1}^{n} \bigsqcup\limits_{j \in I_i} [x_{i,j},x_{i,j+1}]$$ and $$X=\bigsqcup\limits_{l=0}^{M}[z_l,z_{l+1}]$$ where $z_0=0$, $z_{M+1}=1$, $z_l=x_{i,j}$ and $z_l < z_{l+1}$. From this partition we construct a vector $(z_1,\ldots,z_M) \in \RR^M$. We say that $M+1$ is the size of the partition.
The graded order on $\sqcup_{k=1}^{\infty} \RR^k$ is the following:\
Let $(x_1,\ldots,x_M) \in \RR^M$ and $(y_1,\ldots, y_N) \in \RR^N$ we have: $$\begin{aligned}
(y_1,\ldots, y_N) \succ_{gr} (x_1,\ldots,x_M) &\iff & N>M\\
&&\textrm{ or } N=M \textrm{ and } y_1>x_1,\\
&& \textrm{ or }N=M, \exists j> 1 \textrm{ such that } y_i=x_i \textrm{ for } i< j\\
&& \hphantom{ or } \textrm{ and } y_j > x_j.\end{aligned}$$
The graded order gives thus an order on the partitions.\
Now, we give an algorithm which computes a word over the alphabet $a_1,\ldots,a_n$ from a partition. The $l$-th letter of the word $\omega$ is denoted by $\omega(l)$.\
`Word from partition`\
A partition $X=X_1\sqcup \ldots \sqcup X_n$, where $X_i=\sqcup_{j \in I_i} [x_{i,j},x_{i,j+1}]$, and\
$X=\sqcup_{l=1}^{M}[z_l;z_{l+1}]$ is the associated decomposition.\
A word $\omega$ constructed over the alphabet $a_1,\ldots,a_n$.
1. If $[z_0,z_1] \subset X_j$ then $a_1$ is associated to $X_j$ and $\alpha:=2$.
2. $\omega(1):=a_1$.
3. For $l$ from 1 to $M$ do
1. If $[z_l,z_{l+1}] \subset X_i$ and $X_i$ is associated to $a_k$ where $k<\alpha$\
Then $\omega(l+1):=a_k$,\
Else associate $a_{\alpha}$ to $X_i$, $\omega(l+1):=a_{\alpha}$, and $\alpha:=\alpha+1$.\
Now, we introduce a second order on the partitions.
Consider two partitions $X=X_1\sqcup \ldots \sqcup X_n$ and $X=X'_1\sqcup \ldots \sqcup X'_n$. With the previous algorithm we associate a word $\omega$ to the first partition and we associate a word $\omega'$ to the second partition.
If $\omega \succ_{lex} \omega'$, that is to say, if $\omega$ is bigger than $\omega'$ with the lexicographic order with $a_n \succ_{lex} a_{n-1} \succ_{lex} \ldots \succ_{lex} a_1$, then we say that the partition $X=X_1\sqcup \ldots \sqcup X_n$ is bigger than the partition $X=X'_1\sqcup \ldots \sqcup X'_n$ relatively to the lexicographic order.\
If two partitions gives the same word then we say that the partitions are equal relatively to the lexicographic order.
\[lem:permut\] Consider two partitions $X=X_1\sqcup \ldots \sqcup X_n$ and $X=X'_1\sqcup \ldots \sqcup X'_n$. If these partitions give the same vector $(z_1,\ldots,z_M)$ and if these partitions are equal relatively to the lexicographic order, then there exists a permutation $\sigma \in \mathfrak{S}_n$ such that: $$X_{\sigma(i)} =X'_{i}.$$
This follows from the construction of the lexicographic order on the partitions.
The two previous orders allow us to get a symmetric and envy-free fair division.\
`Symmetric and Envy-free`\
$\underline{\mu}=[\mu_1,\ldots,\mu_n]$, a deterministic envy-free cake cutting algorithm $f$.\
$X=\mathcal{F}(X,\underline{\mu},1) \sqcup \ldots \sqcup \mathcal{F}(X,\underline{\mu},n)$, where $\mathcal{F}(X,\underline{\mu},i)$ is a finite union of disjoint intervals and $\mathcal{F}(X,\underline{\mu},i)$ is given to the $i$-th player.
1. For all $\sigma \in \mathfrak{S}_n$, computes the partition $f(\underline{\mu}^{\sigma})$ and\
set $S:=\{f(\underline{\mu}^{\sigma}) \, | \, \sigma \in \mathfrak{S}_n \}$.
2. Let $S_1$ be the subset of $S$ of all partitions with a minimal graded order.
3. If $|S_1|=1$, then Return the unique partition in $S_1$, else go to the next step.
4. Let $S_2$ be the set of all the partitions in $S_1$ with a minimal lexicographic order.
5. \[step:fin\_envyfree\] Return a partition $f(\underline{\mu}^{\sigma}) \in S_2$.
The algorithm *`Symmetric and Envy-free`* is deterministic symmetric and envy-free.
This algorithm is envy free because we return a result coming from an envy-free protocol.\
We remark that if we apply the algorithm to the list $\underline{\mu}$ or $\underline{\mu}^{\rho}$ where $ \rho \in\mathfrak{S}_n$, then the set $S$ computed in the first step will always be the same. Therefore, we just have to study the situation where $S_2$ contains several partitions.\
Consider two distinct partitions in $S_2$, $X=X_1\sqcup \ldots \sqcup X_n$ and $X=X'_1\sqcup \ldots \sqcup X'_n$. Thanks to Lemma \[lem:permut\], there exists a permutation $\sigma \in \mathfrak{S}_n$ such that $X_{\sigma(i)}=X'_{i}$.\
As $f$ is an envy-free protocol, if the $i$-th player receives the portion $X_i$ then we have . Therefore, $\mu_i(X_i) \geq \mu_i(X_{\sigma(i)})=\mu_i(X'_i)$. In the same way, we show that $\mu_i(X'_i) \geq \mu_i(X_i)$. This gives $\mu_i(X'_i) =\mu_i(X_i)$. Then, for all partitions in $S_2$ each player will evaluate in the same way his or her portion. Thus the algorithm is symmetric.\
In step \[step:fin\_envyfree\] we have to choose a partition among all partitions in $S_2$. We can choose the first computed partition appearing in $S_2$. This last step depends on the order of the measures given in input. However, as explained before this choice does not have en effect on how the $i$-th player evaluate his or her part.
The idea of the algorithm is the following: if we have different possible partitions coming from all the $f(\underline{\mu}^{\sigma})$ then we prefer the ones with the fewest number of intervals and with the smallest leftmost part. It seems natural to prefer a partition with few intervals. The second condition can be interpreted as follows: If the different pieces of cake are given from left to right, thus in increasing order of the $x_{i,j}$, then our algorithm gives a first piece with small length to the first served player. If we imagine that a mediator is used to cut the cake then our convention means the following: if a player cooperates quickly with the mediator (the player accepts the leftmost part of the cake) then he gets quickly a piece of cake.
Some remarks about the complexity of symmetric and envy-free algorithm {#sec:complexity}
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Our algorithm relies on an envy-free division algorithm and needs to compute all fair divisions for all permutation orders. Suppose that this envy-free division algorithm has a complexity equals to $T(n)$ in the Robertson-Webb model, then our algorithm uses $n! \times T(n)$ queries. Indeed, our approach needs to compute all the fair divisions for all permutation orders. A natural question is the following: Is it necessary?\
Recently Aziz and Mackenzie have proposed in [@AzizMackenzie] the first envy-free algorithm with a complexity bounded in terms of the number of players. If we use this algorithm then we get a symmetric and envy-free algorithm with a complexity bounded in terms of the number of players.\
At last, we remark that if the envy-free algorithm $f$ uses a continuous protocol (a moving knife method) then our algorithm $\mathcal{F}$ gives a continuous protocol to compute a symmetric and envy-free division.
Approximate symmetric and envy-free fair division algorithm {#sub:sec-approx}
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Envy-free fair division has also been studied in an approximate setting. A division is said to be $\varepsilon$-envy-free when we have for all $i$ and $j$: $\mu_i(X_i)\geq \mu_i(X_j) -\varepsilon$, where $\varepsilon >0$. There exists an algorithm which gives such fair division, see [@Branzei2017]. The complexity of this algorithm is in $O(n/\varepsilon)$ in the Robertson-Webb model.\
In the approximate setting a new definition of symmetric fair division is required. We say that an algorithm $\mathcal{F}$ gives an $\varepsilon$-symmetric fair division when we have for all $i$ and all permutations $\sigma \in \mathfrak{S}_n$: $$\Big|\mu_i\big(\mathcal{F}(X,\underline{\mu},i) \big)- \mu_i\big(\mathcal{F}(X,\underline{\mu}^{\sigma},\sigma^{-1}(i)) \big)\Big| \leq \varepsilon.$$ This means that if we modify the order of the measures in the input of the algorithm then the perturbation on the new value obtained by the $i$-th player is bounded by $\varepsilon$.\
With these definitions it is natural to look for an $\varepsilon$-symmetric and $\varepsilon$-envy-free fair division. In this situation we do not need to repeat $n!$ times an $\varepsilon$-envy-free algorithm. Indeed, contrary to the exact setting there exists an algorithm computing an $\varepsilon$-perfect fair division, see [@Branzei]. This means that there exists an algorithm $\mathcal{F}$ such that $$\Big|\mu_i\big(\mathcal{F}(X,\underline{\mu},i)\big)-\dfrac{1}{n}\Big|\leq \varepsilon.$$ The complexity of this algorithm is in $O(n^2/\varepsilon)$.\
Thus the $\varepsilon$-perfect algorithm gives an $\varepsilon$-symmetric and $\varepsilon$-envy-free fair division without increasing the complexity of an $\varepsilon$-envy-free protocol by a factor $n!$. Unfortunately, this algorithm has an exponential time complexity in $n$ if we take into account the number of elementary operations (arithmetic operations and inequality tests). Indeed, in this algorithm we have to consider all subsets $Y$ with cardinal at most $n(n-1)$ in a set with cardinal $nK$ where $K=\lceil \frac{2n(n-1)}{\varepsilon}\rceil$. Therefore, the asymptotic formula $\binom{2n}{n} \approx \dfrac{4^n}{\sqrt{\pi n}}$ shows that we have to consider an exponential number of subsets.
Aristotelian, symmetric and proportional cake cutting algorithms {#sec:propvrac}
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In this section we first give an aristotelian and proportional fair division algorithm and then a symmetric and proportional one. These two algorithms are based on Kuhn’s algorithm, see [@Kuhn].
An aristotelian proportional cake cutting algorithm {#sec:aristote}
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### The Evan-Paz algorithm and the last diminisher procedure are not aristotelian
Before giving our aristotelian and proportional algorithm we show that the classical Evan-Paz algorithm and the last diminisher procedure do not give an aristotelian fair division.\
In the Evan-Paz algorithm we can have the following situation: We consider four players with associated measures $\mu_1$, $\mu_2$, $\mu_3$, $\mu_4$. Furthermore, we suppose that $\mu_1=\mu_4$ is the Lebesgue measure on $[0,1]$. We also suppose that $\mu_2([0,0.5])=\mu_3([0,0.5])=1/2$ and $\mu_3([0.5,0.51])=1/4$.\
In the first step of the Evan-Paz algorithm we ask each player to cut the cake in two equal parts. More precisely, we ask $cut_i(0,1/2)$. In our situation, each player give the same point: $y=0.5$. In the second step, the algorithm consider two sets of two players. The first part of the cake $[0,0.5]$ will be given to the first set of players and the second part $[0.5,1]$ will be given to the second set of players. Usually, when all players give the same answers the two sets are constructed randomly or in function of the order of the players. Thus we can suppose that in the second step we give $[0,0.5]$ to $\mu_1$ and $\mu_2$ and $[0.5,1]$ to $\mu_3$ and $\mu_4$. At last, the “Cut and Choose" algorithm is used to share $[0,0.5]$ (respectively $[0.5,1]$) between the two players $\mu_1$, $\mu_2$ (respectively $\mu_3$, $\mu_4$). Thus $\mu_1$ cut the interval $[0,0.5]$ and get $X_1$ such that $\mu_1(X_1)=1/4$, and $\mu_3$ cuts the interval $[0.5,1]$ and get $X_3=[0.5,0.51]$. Thus $X_4=[0.51,1]$ and $0.49=\mu_4(X_4)> \mu_1(X_1)=0.25$. As $\mu_1=\mu_4$, we deduce that the division is not aristotelian.\
In the last diminisher procedure we can have the following situation:\
We suppose that $\mu_1=\mu_2$ is the Lebesgue measure on $[0,1]$. Furthermore, we consider a measure $\mu_3$ such that $\mu_3([0,0.4])=1/3$, and $\mu_3([1/3,0.5])=1/3$.\
In the first step of the last diminisher procedure we ask each player the query $cut_i(0,1/3)$. The first and second player give $\mu_1([0,1/3])=\mu_2([0,1/3])=1/3$ and the third player gives $\mu_3([0,0.4])=1/3$. In the first step of this algorithm we give the portion $[0,1/3]$ to the first or to the second player. Suppose that we give this portion to the first player. In the second step of the last diminisher algorithm we ask $cut_2(1/3,1/3)$ and $cut_3(1/3,1/3)$. We get thus the following information $\mu_2([1/3, 2/3])=1/3$ and $\mu_3([1/3, 0.5])=1/3$. After the second step the algorithm gives $[1/3,0.5]$ to the third player. It follows that the second player get $[0.5,1]$ and $\mu_2([0.5,1])=0.5>1/3=\mu_1([0,1/3])$. Therefore, this is not an aristotelian division since $\mu_1=\mu_2$.
### An aristotelian proportional fair division algorithm
In this subsection, we recall Kuhn’s fair division algorithm, see [@Kuhn], and then we show how to modify it to get an aristotelian fair division algorithm. In order to state this algorithm we introduce the following definition:
Let $X=\sqcup_j A_j$ be a partition of $X$. An allocation relatively to this partition is a set $\{ (\mu_{i_1},A_{j_1}), \ldots, (\mu_{i_l},A_{j_l})\}$ such that for $k=1, \ldots , l$: $$\mu_{i_k}(A_{j_k})\geq \dfrac{\mu_{i_k}(X)}{n} \textrm{ and } \mu_{i}(A_{j_k})<\dfrac{\mu_i(X)}{n} \textrm{ if } i \neq i_1,\ldots, i_l.$$ A maximal allocation is an allocation whose cardinal is maximal.\
In the following we say that a piece of cake $A_k$ is acceptable for the $i$-th player if $\mu_i(A_k)\geq \mu_i(X)/n$.
In the previous definition the part $A_i$ is not necessarily given to the $i$-th player. The measurable sets $A_i$ do not play the same role than $X_i$ in the previous section. The partition $X=\sqcup_i A_i$ is just a partition of $X$, it is not necessarily the final result of a proportional fair division problem.\
For a given partition there always exists a maximal allocation.
With the Frobenius-König theorem, Kuhn has shown in [@Kuhn] that there always exists an allocation relatively to a given partition. This gives the existence of maximal allocations.
Kuhn’s algorithm proceeds as follows: The first player cuts the cake in $n$ parts with value $1/n=\mu_1(X)/n$ for his or her own measure. This gives a partition $X=\sqcup_i A_i$. Then we compute a maximal allocation relatively to this partition. Each player in the maximal allocation receives his or her associated portion. The remaining part of the cake is then divided between the rest of the players with the same method.\
Now, we can describe our aristotelian algorithm. The idea is the following:\
As before the first player cut the cake in $n$ parts with value $1/n$ for his or her own measure. This gives a partition $X=\sqcup_j A_j$ and we compute a maximal allocation relatively to this partition. Then each player $i_k$ in the maximal allocation receives his or her associated part if $\mu_{i_k}(A_{j})=1/n$ for all $A_j$ in the maximal allocation. In particular, all players with the same measure than the first player receive the same value. Then it remains two subcakes $X_1$ and $X_2$. We associate respectively these two subcakes to two set of players $\mathcal{E}_1$ and $\mathcal{E}_2$.\
First, the set $\mathcal{E}_1$ corresponds to the set of players with an index $i_k$ in the maximal allocation such that there exists $A_j$ in the maximal allocation with . Thus a player in $\mathcal{E}_1$ does not evaluate all portions $A_j$ as $\mu_1$. Then, we consider the set $\mathcal{L}_1$ constructed in the following way: $j_k \in \mathcal{L}_1$ if and only if $i_k \in \mathcal{E}_1$. At last, we set $X_1=\sqcup_{j \in \mathcal{L}_1}A_j$.
Then we put together the players in $\mathcal{E}_1$ which seem to have the same measure. More precisely, we consider a partition of $\mathcal{E}_1=\sqcup_{m=1}^d \mathcal{E}_{1,m}$ and $\mathcal{L}_1=\sqcup_{m=1}^d \mathcal{L}_{1,m}$ such that: $$(\star) \quad \begin{cases}
\forall i,i' \in \mathcal{E}_{1,m},\, \forall j, \quad \mu_i(A_j)=\mu_{i'}(A_{j}),\\
\mathcal{L}_{1,m}=\{j_k \, | \, i_k \in \mathcal{E}_{1,m} \}.\\
\end{cases}$$ This means that for all $i \in \mathcal{E}_{1,m}$, there exists a constant $c_{j,m}$ (independent of $i$) such that for all $j$ we have $\mu_i(A_j)=c_{j,m}$.\
In particular, as $\mu_{i_k}(A_{j_k}) \geq \mu_{i_k}(X)/n$, we have the following
\[rem:tech\] For all $i \in \mathcal{E}_{1,m}$ and $j \in \mathcal{L}_{1,m}$ we have $\mu_i(A_j)\geq \mu_i(X)/n$.
Then we consider $X_{1,m}=\sqcup_{j \in \mathcal{L}_{1,m}}A_j$ and we associate to these subcakes the players with indices in $\mathcal{E}_{1,m}$. Therefore, it will be possible to share $X_{1,m}$ between the players with indices in $\mathcal{E}_{1,m}$ because by construction they evaluate all $A_j$ in the same way with a value bigger than $1/n$.\
At last, we denote by $X_2$ the part of the cake not appearing in the maximal allocation. Then we can share $X_2$ between the players not appearing in the maximal allocation since by definition they do not find acceptable the portions in the maximal allocation.\
The algorithm will call recursively the algorithm on $X_{1,m}$ and $X_2$.\
In the following we will use queries for a “subcake" $\mathcal{X} \subsetneq [0,1]$. Indeed, as in Kuhn’s algorithm we are going to consider situations where the cake will be of the form $[0,1] \setminus Y$, where $Y$ will correspond to the part of the cake already given by the algorithm. We need thus the following notations:
1. $eval_i^{\mathcal{X}}(x,y)$: Ask agent $i$ to evaluate $[x,y]\cap \mathcal{X}$.\
This means return $\mu_i([x,y]\cap \mathcal{X})$.
2. $cut_i^{\mathcal{X}}(x,a)$: Ask agent $i$ to give $y$ such that $\mu_i([x,y]\cap \mathcal{X})=a$.
We will see that these queries do not introduce new operations. More precisely, during the algorithm these queries $eval_i^{\mathcal{X}}(x,y)$ and $cut_i^{\mathcal{X}}(x,a)$ can be compute thanks to $eval_i(x,y)$ and $cut_i(x,a)$.\
`AristoProp`\
$\underline{\mu}=[\mu_1,\ldots,\mu_{n}]$, $\mathcal{X} \subset [0;1]$.\
$\mathcal{X}=\mathcal{F}(\mathcal{X},\underline{\mu},1) \sqcup \ldots \sqcup \mathcal{F}(\mathcal{X},\underline{\mu},n)$, where $\mathcal{F}(\mathcal{X},\underline{\mu},i)$ is a finite union of disjoint intervals and $\mathcal{F}(\mathcal{X},\underline{\mu},i)$ is given to the $i$-th player.
1. \[step1:aristo\]%*Ask the first player to cut the cake in $n$ parts with values $\mu_1(\mathcal{X})/n$.* %\
%*This gives: $\mathcal{X}=\sqcup_i A_i$.*%\
$x_0:=\min_{x \in \mathcal{X}}(x)$\
For $j$ from 1 to $n$ do\
$x_j:=cut_1^{\mathcal{X}}\big(x_{j-1},\mu_1(\mathcal{X})/n\big),$\
Set $A_j:=[x_{j-1};x_{j}]\cap \mathcal{X}$.\
2. \[step2:aristo\] % *Ask each player to evaluate each $A_j$.*%\
For $i$ from 2 to $n$ do\
For $j$ from $1$ to $n$ do\
$eval_i^{\mathcal{X}}(x_{j-1},x_j)$.\
3. \[step3:aristo\] Compute a maximal allocation $\mathcal{A}:=\{ (\mu_{i_1},A_{j_1}), \ldots, (\mu_{i_l},A_{j_l})\}$ relatively to the partition $\mathcal{X}=\sqcup_i A_i$.\
4. \[step4:aristo\]% *If for all $j$ in $\{j_1, \ldots, j_l\}$, we have $\mu_{i_k}(A_{j})=\mu_1(A_1)$ then give the portion $A_{j_k}$ to the player with associated measure $\mu_{i_k}$.*%\
Set $\mathcal{E}:=\emptyset$, $\mathcal{E}_1:=\emptyset$, $\mathcal{L}_1:=\emptyset$, $\mathcal{X}_1:=\emptyset$.\
For $i_k$ in $\{ i_1, \ldots, i_l\}$ do\
t:=true;\
For $j$ in $\{ j_1, \ldots, j_l\}$ do\
If $\mu_{i_k}(A_{j}) \neq \mu_1(A_1)$ Then t:=false.\
If t=true Then $\mathcal{F}(X,\underline{\mu},i_k):=A_{j_k}$, $\mathcal{E}:=\mathcal{E} \cup \{i_k\}$,\
Else $\mathcal{E}_1:=\mathcal{E}_1 \cup \{i_k\}$, $\mathcal{L}_1:=\mathcal{L}_1 \cup \{j_k \}$, $\mathcal{X}_1:=\mathcal{X}_1 \cup A_{j_k}$.\
5. \[step5:aristo\] Construct a partition $\mathcal{E}_1=\sqcup_{m=1}^d \mathcal{E}_{1,m}$ and a partition $\mathcal{L}_1:=\sqcup_{m=1}^d \mathcal{L}_{1,m}$ sastisfying $(\star)$.\
Set $\underline{\mu}_{1,m}$ as the list of measures associated to players with index in $\mathcal{E}_{1,m}$.\
Set $\mathcal{X}_{1,m}:=\sqcup_{j \in \mathcal{L}_{1,m}} A_{j}$.\
Set $\mathcal{E}_2:=\{1,\ldots,n\} \setminus \{i_1, \ldots,i_l\}$, $\mathcal{X}_2:=\mathcal{X} \setminus \big(\sqcup_{k=1}^l A_{j_k}\big)$.\
Set $\underline{\mu}_2$ as the list of measures associated to players with index in $\mathcal{E}_2$.\
6. \[step6:aristo\] Return($\sqcup_{i \in \mathcal{E}} \mathcal{F}(\mathcal{X},\underline{\mu},i) \sqcup_{m=1}^d$ `AristoProp` $(\underline{\mu}_{1,m},\mathcal{X}_{1,m}) \sqcup$ `AristoProp` $(\underline{\mu}_2,\mathcal{X}_2)\big)$.
\[prop:aristo\_prop\] The algorithm `AristoProp` applied to $\underline{\mu}=[\mu_1,\ldots,\mu_n]$ and terminates and is aristotelian.
The algorithm terminates since after one call of the algorithm the number of player decreases strictly since the first player always get a part of the cake.\
Now, we are going to prove by induction that the algorithm is aristotelian.\
We consider the following claim:\
$(H_n)$: The algorithm `AristoProp` applied with $n$ measures is aristotelian.\
For $n=2$, $H_2$ is true. Indeed, if $\mu_1=\mu_2$ then $\mu_2$ belongs to the maximal allocation computed in Step \[step3:aristo\]. Furthermore, we can suppose without loss of generality that the maximal allocation has the following form $\mathcal{A}=\{(\mu_1, A_1),(\mu_2,A_2)\}$.\
By construction we have $\mu_1(A_1)=\mu_1(A_2)$. Thus $\mu_1(A_1)=\mu_2(A_1)=\mu_2(A_2)$ since $\mu_1=\mu_2$. As in Step \[step4:aristo\], $\mu_1$ gets the portion $A_1$ and $\mu_2$ gets the portion $A_2$, we deduce that $H_2$ is true.\
Now, we suppose that $H_k$ is true when $k \leq n$ and we are going to prove that $H_{n+1}$ is true.\
We suppose that we have $n+1$ measures $\mu_1, \ldots, \mu_{n+1}$, and that $\mu_{p}=\mu_{q}$, where $p,q \in\{1,\ldots,n+1\}$.\
First, we remark that if $(\mu_{p},A_{j_p})$ belongs to a maximal allocation then $\mu_{q}$ also belongs to the same maximal allocation. Indeed, if $\mu_q$ does not belong to the maximal allocation then $\mu_q(A_{j_p}) <\mu_q(\mathcal{X})/n$ but $\mu_q(A_{j_p})=\mu_p(A_{j_p})\geq \mu_p(\mathcal{X})/n$ because $\mu_p=\mu_q$ and $(\mu_p,A_{j_p})$ belongs to the maximal allocation. This gives the desired contradiction and proves our remark.\
Now two situations appear: In Step \[step3:aristo\], $\mu_p$ and $\mu_q$ belongs to the maximal allocation or they do not belong to it.\
If $\mu_p$ and $\mu_q$ do not belong to the maximal allocation then $\mu_p$ and $\mu_q$ belong to the list $\underline{\mu}_2$. Then, $\mu_p$ and $\mu_q$ get their portions when, in Step \[step6:aristo\], we apply `AristoProp` $(\underline{\mu}_2,\mathcal{X}_2)$. As the list $\underline{\mu}_2$ have at most $n$ measures and $H_n$ is true we deduce that $\mu_p$ and $\mu_q$ get the same value and then the algorithm is aristotelian in this case.\
If $\mu_p$ and $\mu_q$ belong to the maximal allocation then we have two cases:\
there exists an index $j_0$ in $\{j_1, \ldots, j_l\}$ such that $\mu_p(A_{j_0})\neq \mu_1(A_1)$ or\
for all $j_k$ in $\{j_1, \ldots, j_l\}$ we have $\mu_p(A_{j_k})=\mu_1(A_1)$.\
In the first case, as $\mu_p=\mu_q$ then we also have $\mu_q(A_{j_0})\neq \mu_1(A_1)$. Then, $p$ and $q$ belong to $\mathcal{E}_1$. Furthermore, as $\mu_p=\mu_q$, for all $j$ we have $\mu_p(A_j)=\mu_q(A_j)$. Then $\mu_p$ and $\mu_q$ belong to the same list $\underline{\mu}_{1,m}$. As the list $\underline{\mu}_{1,m}$ have at most $n$ measures and $H_n$ is true we deduce that $\mu_p$ and $\mu_q$ get the same value and then the algorithm is also aristotelian in this case.\
In the second case, we have $p,q \in \mathcal{E}$ and $\mu_p(A_{j_k})=\mu_q(A_{j_k})=\mu_1(A_1)$ for all $j_k$ in $\{j_1, \ldots, j_l\}$. Thus the $p$-th and $q$-th player evaluate in the same way the portion they get. Thus, the algorithm is also aristotelian in this case and this concludes the proof.
\[prop:aristo\_prop2\] The algorithm `AristoProp` applied to $\underline{\mu}=[\mu_1,\ldots,\mu_n]$ gives a proportional fair division of $[0,1]$.
We are going to prove this result by induction. We consider the following claim:\
$(H_n)$: The algorithm `AristoProp` applied to $n$ measures gives a proportional fair division of $[0,1]$.\
For $n=1$, $H_1$ is true.
Now, we suppose that $H_k$ is true for $k \leq n$ and we are going to prove that $H_{n+1}$ is true.\
We consider $n+1$ measures $\mu_1$, …, $\mu_{n+1}$.\
If $i \in \mathcal{E}$ then the $i$-th player receive a portion $\mathcal{F}(X,\underline{\mu}, i)$ that he or she consider to have a value equal to $\mu_1(A_1)$. As $\mu_1(A_1)=\mu_1(X)/n$, we get $\mu_i\big( \mathcal{F}(X,\underline{\mu}, i) \big) \geq \mu_1(X)/n$. Thus, the algorithm is proportional in this case.\
If $i \not \in \mathcal{E}$ then we have the following situation: $\mu_i$ belongs to a list $\underline{\mu}_{1,m}$ or to the list $\underline{\mu}_2$.\
If $\mu_i$ belongs to the list $\underline{\mu}_2$, then $\mu_i$ does belong to the maximal allocation considered and for all $j_k \in \{j_1, \ldots,j_l\}$ we have $\mu_i(A_{j_k})<\mu_1(A_1)=\mu_i(X)/n$. Therefore, we have $$(\sharp) \quad \mu_i(\mathcal{X}_2)=\mu_i(X)-\sum_{k=1}^l \mu_i(A_{j_k})\geq \mu_i(X)-\dfrac{l\mu_i(X)}{n}=\dfrac{(n-l) \mu_i(X)}{n},$$ where $l$ is the size of the maximal allocation computed in Step \[step3:aristo\].\
Thanks to our induction hypothesis, in Step \[step6:aristo\] the $i$-th player receives at least for his or her own measure. Thus, by $(\sharp)$ the $i$-th player gets at least $\mu_i(X)/n$. Thus, in this case, the algorithm is proportional.\
If $\mu_i$ belongs to a list $\underline{\mu}_{1,m}$, by Remark \[rem:tech\], we have $\mu_i(A_j) \geq \mu_i(X)/n$, for all $j \in \mathcal{L}_{1,m}$. Thus $$(\sharp \sharp) \quad \mu_i(\mathcal{X}_{1,m})=\mu_i(\sqcup_{j \in\mathcal{L}_{1,m}} A_j) \geq \dfrac{|\mathcal{L}_{1,m}|\mu_i(X)}{n},$$ where $|\mathcal{L}_{1,m}|$ is the number of elements in $\mathcal{L}_{1,m}$ and this number is equal to $\mathcal{E}_{1,m}$ the number of measures in the list $\underline{\mu}_{1,m}$.\
Thanks to our induction hypothesis, in Step \[step6:aristo\] the $i$-th player receives at least $\mu_i(\mathcal{X}_{1,m})/|\mathcal{E}_{1,m}|$ for his or her own measure. Thus, by $(\sharp \sharp)$ the $i$-th player gets at least $\mu_i(X)/n$ and this concludes the proof.
\[prop:complexityaristot\] The algorithm `AristoProp` applied to $\underline{\mu}=[\mu_1,\ldots,\mu_n]$, and $X=[0,1]$ uses at most $\bigO(n^3)$ queries in the Robertson-Webb model.
In Step \[step1:aristo\] we use $n$ $cut_i$ queries. In Step \[step2:aristo\] we use $n(n-1)$, $eval_i$ queries.\
During the first call of the algorithm we use the $cut_i$ and $eval_i$ queries. In the next calls the algorithm uses the $cut_i^{\mathcal{X}}$ and $eval_i^{\mathcal{X}}$ queries where $\mathcal{X}=\sqcup_j A_j$ and the measures of $A_j$ are known by the players thanks to Step \[step2:aristo\] of the algorithm. We remark that the situation $\mathcal{X}_2=\mathcal{X}\setminus \Big( \sqcup_{k=1}^l A_{j_k} \Big)$ of Step \[step5:aristo\] corresponds to $\mathcal{X}_2=\sqcup_{j \not \in \mathcal{L}\sqcup \mathcal{L}_1} A_j$. It follows that we can write $\mathcal{X}$ in the following form: $\mathcal{X}=\sqcup_{j=1}^k [s_j;t_j]$, where $s_1<t_1<s_2<t_2<\cdots<s_k<t_k$ and the measures of $[s_j;t_j]$ and $[t_j;s_{j+1}]$ are known thanks to the previous calls of the algorithm.\
Now, we define a function $\mathsf{f}$ in order to explain how we compute $eval_i^{\mathcal{X}}$ from $eval_i$: If $s_{j_0}<x< t_{j_0}$ then we set $\mathsf{f}(x)=j_0$.\
If $\mathsf{f}(x)=\mathsf{f}(y)$ then $[x;y]\subset [s_{j_0};t_{j_0}]$ and $eval_i^{\mathcal{X}}(x,y)=eval_i(x,y)$,\
else we have $$\begin{aligned}
eval_i^{\mathcal{X}}(x,y)&=&\mu_i\big([x,y]\cap \mathcal{X}\big)=\mu_i\Big([x,y] \cap \big( \sqcup_{j=1}^k [s_j,t_j]\big) \Big)\\
&=&eval_i(x,y)-\sum_{j=\mathsf{f}(x)}^{\mathsf{f}(y)-1} eval_i(t_{j},s_{j+1}).\end{aligned}$$ As $\mu_i([t_{j},s_{j+1}])=eval_i(t_{j},s_{j+1})$ is known thanks to the previous calls of the algorithm, the query $eval_i^{\mathcal{X}}(x,y)$ needs just one new query: $eval_i(x,y)$.\
For the $cut_i^{\mathcal{X}}$ query we proceed in the following way:\
Suppose that we want to compute $cut_i^{\mathcal{X}}(x,a)$.\
First, compute $eval_i(x,t_{\mathsf{f}(x)})$. As we know $\mu_i([s_j;t_j])$ for $j=1,\ldots,k$ then with all these values we can deduce in which interval $[s_{1},t_1]$,…, $[s_{k},t_{k}]$ is the cutpoint $y$. We denote by $[\alpha,\beta]$ this interval. Thanks to the knowledge of $\mu_i([s_j;t_j])$ and $\mu_i([x,t_{\mathsf{f}(x)}])$ we can also get $a'=eval_i^{\mathcal{X}}(x,\alpha)$. Then we have:\
$cut_i^{\mathcal{X}}(x,a)=cut_i(\alpha,a-a')$.\
Therefore, $cut_i^{\mathcal{X}}$ needs two new queries in the Robertson-Webb model: $eval_i(x,t_{\mathsf{f}(x)})$ and $cut_i(\alpha,a-a')$.\
In conclusion, in Step \[step1:aristo\] the algorithm applied with $\eta$ measures uses $\eta$ $cut_i^{\mathcal{X}}$ queries, thus these queries can be computed with $2 \eta$ queries in the Robertson-Webb model. In Step \[step2:aristo\], it uses $\eta(\eta-1)$ $eval_i^{\mathcal{X}}$ queries. These queries can be computed with $\eta(\eta-1)$ queries in the Robertson-Webb model. Therefore, each call of the algorithm applied with $\eta$ measures uses $\eta(\eta+1)$ queries in the Robertson-Webb model of computation. Furthermore, in the worst case, at each call of the algorithm only one player get a part of the cake. Thus we use at most $$n^2+\sum_{\eta=1}^{n-1} \eta(\eta+1)=n^2+ \sum_{\eta=1}^{n-1}\eta^2 +\sum_{\eta=1}^{n-1} \eta=n^2+ \dfrac{n(n-1)(2n-1)}{6}+\dfrac{n(n-1)}{2} \in \bigO(n^3)$$ queries in the Robertson-Webb model.
From the previous propositions we get:
There exists an aristotelian proportional fair division algorithm which uses at most $\bigO(n^3)$ queries in the Robertson-Webb model of computation.
As already mentioned in the introduction, this theorem says that if we just want an aristotelian proportional fair division it is not necessary to use an envy-free algorithm which uses an exponential number of queries.
A symmetric and proportional cake cutting algorithm
---------------------------------------------------
In Section \[sec:symenvyfree\], we have proposed a symmetric and envy-free protocol, this gives then a proportional and symmetric protocol. With this approach we need to compute $n!$ envy-free fair divisions. This raises the following question: Do we need to compute a proportional fair division for all the possible permutations to get a proportional and symmetric division algorithm?\
In this subsection we are going to show that there exists a symmetric and proportional algorithm which uses $\bigO(n^3)$ queries in the Robertson-Webb model.\
The idea of the algorithm is a kind of improvement of the aristotelian algorithm. Indeed, in the aristotelian algorithm if two players get a portion at the same stage of the algorithm then they will evaluate their portion in the same way. Here, we construct an algorithm in order to have also the following property: a player will always receive a portion at the same stage of the algorithm whatever his or her position in the input list $\underline{\mu}$ is.\
Our algorithm works as follows: Instead of asking to the first player to divide the cake in $n$ equal parts, we are going to ask to all players to cut the cake in $n$ equal parts for their own measures. Then, we will select the “smallest partition" relatively to the graded order. Thus, we obtain a partition $X=\sqcup_{j}A_j$ independent of the order of the measures.\
Next, we compute all maximal allocations $\mathcal{A}$ relatively to this partition. For all of these allocations we consider the set $\mathcal{E}_{\mathcal{A}}$, constructed as follows: $i \in \mathcal{E}_{\mathcal{A}}$ if and only if $i$ belongs to the maximal allocation $\mathcal{A}$ and for all $j$, we have: $\mu_i(A_j)=\mu_i(X)/n$. These sets will play the same role as the set $\mathcal{E}$ in the algorithm `AristoProp`. However, here we have several sets $\mathcal{E}_{\mathcal{A}}$ and then we have to choose one of them. We select then a maximal allocation $\mathcal{A}$ where the portions associated to the players in $\mathcal{E}_{\mathcal{A}}$ appear in the leftmost part of the cake. Thus this choice is still independent of the order or the players. At last, we give these portions to their associated players. If a portion $A_j$ belongs to the selected maximal allocation is not given to a player then this portion is used to construct a subcake $\mathcal{X}_1$ as in the aristotelian case. If a portion $A_j$ is not in the selected allocation then this portion is used to construct the subcake $\mathcal{X}_2$. Our algorithm is then constructed in a way such that we always associate the same players to the same subcake $\mathcal{X}_1$ or $\mathcal{X}_2$. As we repeat our strategy on $\mathcal{X}_1$ and $\mathcal{X}_2$ we get a symmetric algorithm.\
`SymProp`\
$\underline{\mu}=[\mu_1,\ldots,\mu_{n}]$, $\mathcal{X} \subset [0,1]$.\
$\mathcal{X}=\mathcal{F}(\mathcal{X},\underline{\mu},1) \sqcup \ldots \sqcup \mathcal{F}(\mathcal{X},\underline{\mu},n)$, where $\mathcal{F}(\mathcal{X},\underline{\mu},i)$ is a finite union of disjoint intervals and $\mathcal{F}(\mathcal{X},\underline{\mu},i)$ is given to the $i$-th player.
1. \[step1:sym\]%*Ask all players to cut the cake in $n$ parts with values $\mu_i(\mathcal{X})/n$.* %\
For $i$ from $1$ to $n$ do\
$x_{i,0}:=\min_{x \in \mathcal{X}}(x)$\
For $j$ from 1 to $n$ do\
$x_{i,j}:=cut_i^{\mathcal{X}}(x_{i,j-1},\mu_i(\mathcal{X})/n)$.\
2. \[step2:sym\]% *Find the smallest partition $\mathcal{X}=\sqcup_{j=1}^n A_j$ for the graded order.* %\
Compute $(x_{0,0}, \ldots, x_{0,n}):=\min_{\succ_{gr}} \{ (x_{i,0},\ldots,x_{i,n}) \, | \, i=1,\ldots,n \}$.\
For $j$ from $1$ to $n$ do\
Set $A_j:=[x_{0,j-1};x_{0,j}] \cap \mathcal{X}$.\
3. \[step3:sym\] % *Ask each player to evaluate each $A_j$.*%\
For $i$ from $1$ to $n$ do\
For $j$ from $1$ to $n$ do\
$eval_i^{\mathcal{X}}(x_{0,j-1},x_{0,j})$.\
4. \[step4:sym\] Compute the set $S$ of all maximal allocations $\mathcal{A}:=\{ (\mu_{i_1},A_{j_1}), \ldots, (\mu_{i_l},A_{j_l})\}$ relatively to the partition $\mathcal{X}=\sqcup_{j=1}^n A_j$.\
5. \[step5:sym\] % *The set $\mathcal{E}_{\mathcal{A}}$ is the set of indices $i \in \{i_1, \ldots,i_l\}$ appearing in the allocation $\mathcal{A}$ such that for $j=1, \ldots, n$, we have $\mu_i(A_j)=\mu_i(\mathcal{X})/n$.*%\
For all $\mathcal{A}=\{ (\mu_{i_{1}},A_{j_{1}}), \ldots, (\mu_{i_{l}},A_{j_{l}}) \}$ in $S$ do\
Set $\mathcal{E}_{\mathcal{A}}:=\emptyset$.\
For $k$ from 1 to $l$ do\
If the vector $(x_{i_k,0}, \ldots,x_{i_k,n})$ associated to $\mu_{i_{k}}$ satisfied\
$(x_{i_k,0}, \ldots,x_{i_k,n})=(x_{0,0},\ldots,x_{0,\eta})$\
Then $\mathcal{E}_{\mathcal{A}}:=\mathcal{E}_{\mathcal{A}} \cup \{i_k\}$.\
6. \[step6:sym\] %*Find an allocation $\hat{\mathcal{A}} \in S$ such that the portions associated to the measures $\mu_{i}$ with $i \in \mathcal{E}_{\hat{\mathcal{A}}}$ are on the leftmost part of $\mathcal{X}$.*%\
For all allocations $\mathcal{A}=\{ (\mu_{i_{1}},A_{j_{1}}), \ldots, (\mu_{i_{l}},A_{j_{l}}) \}$ in $S$ do\
$N_{\mathcal{A}}:=0$;\
For $k$ from 1 to $l$ do\
If $i_k \in \mathcal{E}_{\mathcal{A}}$ Then $N_{\mathcal{A}}:=N_{\mathcal{A}}+2^{j_{k}}$.\
Find an allocation $\hat{\mathcal{A}}\in S$ such that $N_{\hat{\mathcal{A}}}$ is minimal.\
7. \[step7:sym\] % *We consider the allocation $\hat{\mathcal{A}}$. If $i_k \in \mathcal{E}_{\hat{\mathcal{A}}}$ then we give the portion $A_{j_k}$ to the $i_k$-th player else we use $A_{j_k}$ to construct the subcake $\mathcal{X}_1$.* %\
Set $\hat{\mathcal{A}}:=\{(\mu_{i_1},A_{j_1}),\ldots,(\mu_{i_l},A_{j_l})\}$.\
Set $N_{\hat{\mathcal{A}}}:=\sum_{j \in J} 2^j$,\
Set $\mathcal{E}_1:=\emptyset$, $\mathcal{L}_1:=\emptyset$, $\mathcal{X}_1:=\emptyset$.\
For $j_k$ in $\{ j_1, \ldots, j_l\}$ do\
If $j_k \in J$ Then $\mathcal{F}(X,\underline{\mu},i_k):=A_{j_k}$\
Else $\mathcal{E}_1:=\mathcal{E}_1 \cup \{i_k\}$, $\mathcal{L}_1:=\mathcal{L}_1 \cup \{j_k \}$, $\mathcal{X}_1:=\mathcal{X}_1 \cup A_{j_k}$.\
8. \[step8:sym\] Construct a partition $\mathcal{E}_1=\sqcup_{m=1}^d \mathcal{E}_{1,m}$ and a partition $\mathcal{L}_1:=\sqcup_{m=1}^d \mathcal{L}_{1,m}$ sastisfying $(\star)$.\
Set $\underline{\mu}_{1,m}$ as the list of measures associated to players with index in $\mathcal{E}_{1,m}$.\
Set $\mathcal{X}_{1,m}:=\sqcup_{j \in \mathcal{L}_{1,m}} A_{j}$.\
Set $\mathcal{E}_2:=\{1,\ldots,n\} \setminus \{i_1, \ldots,i_l\}$, $\mathcal{X}_2:=\mathcal{X} \setminus \big(\sqcup_{k=1}^l A_{j_k}\big)$.\
Set $\underline{\mu}_2$ as the list of measures associated to players with index in $\mathcal{E}_2$.\
9. \[step9:sym\] Return($\sqcup_{i \in \mathcal{E}} \mathcal{F}(\mathcal{X},\underline{\mu},i) \,\sqcup_{m=1}^d$ `SymProp` $(\underline{\mu}_{1,m},\mathcal{X}_{1,m})\, \sqcup$ `SymProp` $(\underline{\mu}_2,\mathcal{X}_2)\big)$.
\[prop:sym\_prop\] The algorithm `SymProp` applied to $\underline{\mu}=[\mu_1,\ldots,\mu_n]$ and $\mathcal{X}=[0,1]$ terminates, is symmetric and gives a proportional fair division of $[0,1]$.
The algorithm terminates since after one call of the algorithm the number of player decreases strictly since at least one player get a part of the cake.\
Now, we have to prove that this algorithm is symmetric.\
First, we remark that in Step \[step2:sym\] the partition $\mathcal{X}=\sqcup_j A_j$ is independent of the order of the players given in the input. Indeed, this partition is chosen thanks to the graded order.\
Second, we define the set $\mathbb{A}$ as the set of portions $A_{j_k}$ given in Step \[step7:sym\] and the set $\mathbb{M}$ as the set of measures receiving a portion in Step \[step7:sym\].\
We are going to show that these two sets are independent of the order of the players and also independent of the choice of $\hat{\mathcal{A}}$ in Step \[step6:sym\].\
Indeed, we give $A_{j_k}$ if $j_k \in J$, where $J$ is the set defined by the property $N_{\hat{\mathcal{A}}}=\sum_{j \in J} 2^j$. As the binary expansion of $N_{\hat{\mathcal{A}}}$ is unique, the set $J$ is independent of the order of the players and also independent of the allocation $\hat{\mathcal{A}}$ chosen in Step \[step6:sym\]. Thus $\mathbb{A}$ is independent of the order of the players and is also independent of the choice of $\hat{\mathcal{A}}$.\
The set $\mathbb{M}$ is independent of the order of the players and of the choice of $\hat{\mathcal{A}}$ in Step \[step6:sym\].\
Indeed, if $\mu_{i_k} \in \mathbb{M}$ then $(\mu_{i_k},A_{j_k}) \in \hat{\mathcal{A}}$ and $j_k \in J$, where $N_{\hat{\mathcal{A}}}=\sum_{j \in J}2^j$. Therefore as before $\mathbb{M}$ is independent of the choice of $\hat{\mathcal{A}}$ in Step \[step6:sym\]. Furthermore, by contruction in Step \[step5:sym\] and Step \[step6:sym\], $i_k$ belongs to $\mathcal{E}_{\hat{\mathcal{A}}}$. Thus, $\mu_{i_k} \in \mathbb{M}$ means that the associated vector $(x_{i_k,0}, \ldots,x_{i_k,n})$ satisfies the equality $(x_{i_k,0}, \ldots,x_{i_k,n})=(x_{0,0},\ldots,x_{0,n})$. As the choice of the partition $\mathcal{X}=\sqcup_j A_j$ in Step \[step2:sym\] is independent of the order of the players, that is to say the choice of $(x_{0,0},\ldots,x_{0,n})$ is independent of the order of the players, we deduce that $\mathbb{M}$ is indepedent of the order of the players.\
Now, we remark that, for all $\mu \in \mathbb{M}$ and all $A \in \mathbb{A}$ we have: $\mu(A)=\mu(\mathcal{X})/n$.\
Indeed, all players $i_k$ associated to a measure $\mu_{i_k} \in \mathbb{M}$ belongs to $\mathcal{E}_{\hat{A}}$ by construction in Step \[step5:sym\] and Step \[step6:sym\]. Thus $\mu(A)$ is independent of the choice of $\hat{\mathcal{A}}$ in Step \[step6:sym\].\
Therefore, in Step \[step7:sym\] the sets $\mathbb{A}$ and $\mathbb{M}$ and the value $\mu(A)$ for $\mu \in \mathbb{M}$ and $A \in \mathbb{A}$ are independent of the order of the players and independent of the choice of $\hat{\mathcal{A}}$ with minimal $N_{\hat{\mathcal{A}}}$ in Step \[step6:sym\]. Furthermore, we can deduce that:
- $\mathcal{X}_1$ and $\mathcal{E}_1$ and then $\mathcal{X}_{1,m}$ and $\mathcal{E}_{1,m}$,
- $\mathcal{X}_2$ and $\mathcal{E}_2$
are also independent of the order of the players and of the choice of $\hat{A}$ in Step \[step6:sym\]. It follows then that the algorithm is symmetric.\
The algorithm is proportional.\
Indeed, the sets $\mathcal{X}_1$ and $\mathcal{X}_2$ are constructed as in `AristoProp`. The strategy used by `SymProp` is the same than the one used in the algorithm `AristoProp`. Thus with the same approach as the one used in Proposition \[prop:aristo\_prop\] we can deduce that the algorithm `SymProp` is proportional.
The algorithm `SymProp` uses at most $\bigO(n^3)$ queries in the Robertson-Webb model.
In Step \[step1:sym\] we use $n^2$ $cut_i^{\mathcal{X}}$ queries, in Step \[step3:sym\] we use $n(n-1)$ $eval_i^{\mathcal{X}}$ queries. Thus as shown in Proposition \[prop:complexityaristot\] we use at most $\bigO(n^3)$ queries in the Robertson-Webb model.
Suppose that all the measures $\mu_i$ are equal to the Lebesgue measure on $[0,1]$. Then in Step \[step2:sym\] of the algorithm we have $$(x_{0,0},\ldots,x_{0,n})=\big(0,1/n,2/n,\ldots,(n-1)/n,1\big),$$ and then $S$ contains $n!$ allocations.\
Thus in Step \[step6:sym\] we compute $n!$ times the number $N_{\mathcal{A}}=2+2^2+\cdots+2^n$.\
Therefore, there exists a situation where the algorithm computes at least $n!$ sums.\
This is not the only situation where we need to perform an exponential number of arithmetic operations. Another example is the following: Consider $2n+1$ players, suppose that the measure associated to the first $n$ players is the Lebesgue measure on $[0;1]$ and the measure associated to the other players is concentrated on $[\frac{2n}{2n+1},1]$. Then in Step \[step2:sym\] of `SymProp` we have $$(x_{0,0},\ldots,x_{0,n})=\Big(0,\frac{1}{2n+1},\ldots,\frac{2n}{2n+1},1\Big),$$ and $S$ contains $\binom{2n}{n}$ allocations. Indeed, in order to get a maximal allocation we have to associated $n$ intervals among the $2n$ first intervals to the $n$ players with the Lebesgue measure on $[0,1]$. As $\binom{2n}{n} \approx \dfrac{4^n}{\sqrt{\pi n}}$ in this situation we also perform an exponential number of operations or inequality tests.\
These examples show that the algorithm `SymProp` needs a polynomial number of queries in the Robertson-Webb model but needs an exponential number of elementary operations. The combinatorial nature of the problem is processed with classical arithmetic operations and inequality tests.
Conclusion {#sec:conclusion}
==========
In this article we have given an algorithm for computing symmetric and envy-free fair division.\
The complexity in the Robertson-Webb model of this algorithm increases the complexity of an envy-free fair division by a factor $n!$. This raises the following question: Can we avoid or reduce this factor?\
Furthermore, we know a lower bound for the number of queries for an envy-free division. This lower bound is $\Omega(n^2)$, see [@Procaccia-lowerbound]. What is the lower bound for the symmetric and envy-free problem? Do we have necessarily a factorial number of queries? In other words, does the lower bound for the symmetric and envy-free fair division belongs to $\Omega(n!)$? Can we get a lower bound for a symmetric and envy-free fair division?\
In the approximate setting we get an $\varepsilon$-symmetric and $\varepsilon$-envy free fair division algorithm thanks to the $\varepsilon$-perfect division proposed in [@Branzei]. In this case the number of queries is in $\bigO(n^2/\varepsilon)$. However, this algorithm uses an exponential number in $n$ of arithmetic operations and inequality tests.\
This problem appears also in our last algorithm which computes a symmetric and proportional fair division with $\bigO(n^3)$ queries in the Robertson-Webb model. In this algorithm we solve a sub-problem (the computation of the set $S$) with an exponential number (in $n$) arithmetic operations and inequality tests.\
Thus in these kinds of situations (Symmetric and Proportionnal or $\epsilon$-perfect fair division) an algorithm with a polynomial number of queries cannot be considered as a fast algorithm if it uses an exponential number in $n$ of elementary operations.\
A new model of computation has been suggested in [@ChezeBSSRW]. In this model the number of elementary operations must be taken into account. Then, in this model, the algorithm `SymProp` has not a polynomial complexity and cannot be considered as a fast algorithm. However, in a recent work based on a preprint of this article, Aigner-Horev and Segal-Halevi have shown how to modify `SymProp` in order to get an algorithm with a polynomial complexity even if we take into account the number of arithmetic operations, see [@Segal-Halevi-bipartite].\
We have constructed in this article an algorithm giving an aristotelian and proportional algorithm. This algorithm uses less arithmetic operations than our symmetric and proportional algorithm but these two algorithms use the same number of queries. Is it necessary?\
At last, the aristotelian notion comes from the Nichomachean Ethics by Aristotle and one of the contributions of this article is to prove that we can compute an aristotelian and proportional fair division efficiently (with a polynomial number of queries). This result is interesting since until now all aristotelian proportional fair division algorithms were envy-free algorithms and thus have a huge complexity in the Roberston-Webb model. However, another philosopher, Seneca, would have given a sever conclusion about this work:
*“The mathematician teaches me how to lay out the dimensions of my estates; but I should rather be taught how to lay out what is enough for a man to own.\[…\] What good is there for me in knowing how to parcel out a piece of land, if I know not how to share it with my brother? \[…\] The mathematician teaches me how I may lose none of my boundaries; I, however, seek to learn how to lose them all with a light heart."*\
Letters Lucilius/Letter 88; Seneca.\
**Acknowledgement**: The author thanks Erel Segal-Halevi for pointing to him an imprecision at the end of the algorithm `SymProp` in a previous version of this work.\
At last, the author thanks Émèlie, Éloïse and Timothé for having implicitly suggested to study this problem.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We will discuss some results of the paper “Asymptotic estimates on the Von Neumann Inequality for homogeneous polynomials”, of Galicer D., Muro S. and Sevilla-Peris P. [@Ga]. Also, we will see how to extend some of these results using the same techniques in such paper.'
address: 'Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA'
author:
- 'Oscar Zatarain-Vera'
title: Some asymptotic estimates on the Von Neumann Inequality for homogeneous polynomials
---
First, let’s recall some definitions and notation. If $(a_n)$ and $(b_n)$ are two sequences of real numbers, we write $a_n\ll b_n$ if there is a positive constant $C$ (independent of $n$) such that $a_n\leq Cb_n$ for every $n$. We also write $a_n\sim b_n$ if $a_n\ll b_n$ and $b_n\ll a_n$. Given a set $A$ its cardinality is denoted by $|A|$.
A $k-$homogeneous polynomial in $n$ variables is a function $p:\mathbb{C}^n\to \mathbb{C}$ of the form $$p(z_1,...,z_n)=\sum_{\substack{\alpha\in \mathbb{N}^n_0 \\ |\alpha|=k}}a_{\alpha}z_1^{\alpha_1}\cdots z_n^{\alpha_n}=\sum_{\substack{J=(j_1,...,j_k) \\ 1\leq j_1\leq ...\leq j_k\leq n}}c_Jz_{j_1}\cdots z_{j_k},$$ where $a_{\alpha}\in \mathbb{C}$ and $|\alpha|=\alpha_1+\cdots+\alpha_n$. Given $\alpha$ we have $a_{\alpha}=c_J$ where $J=(1,\overset{\alpha_1}{...},1,...,n,\overset{\alpha_k}{...},n)$. We will write $z_1^{\alpha_1}\cdots z_n^{\alpha_n}=z^{\alpha}$ and $z_{j_1}\cdots z_{j_k}=z_J$.\
For $1\leq q\leq \infty$ we denote by $\mathcal{P}(^k\ell_q^n)$ the Banach space of all $k-$homogeneous polynomial on $n$ variables with the norm $$\|p\|_{\mathcal{P}(^k\ell_q^n)}=\sup\{|p(z_1,...,z_n)|:\|(z_1,...,z_n)\|_q\leq 1\}.$$ It is well known that for every $k-$homogeneous polynomial there is a unique symmetric $k-$linear form $L$ on $\mathbb{C}^n$ such that $p(z)=L(z,...,z)$ for all $z\in\mathbb{C}^n$ [@Di Chapter 1]. Also for each $1\leq q\leq \infty$ and $k\geq 2$ there exists a constant $\lambda(k,q)>0$ such that $$\label{firstineq}
\|p\|_{\mathcal{P}(^k\ell_q^n)}\leq\sup\{L(z^{(1)},...,z^{(k)}):\|z^{(j)}\|_q\leq 1, j=1,...,k\}\leq \lambda(k,q)\|p\|_{\mathcal{P}(^k\ell_q^n)}.$$ In general $\lambda(k,q)\leq \frac{k^k}{k!}$, it is worth mentioning that some improvements are know for particular cases: $\lambda(k,2)=1$ and $\lambda(k,\infty)\leq \frac{k^{\frac{k}{2}}(k+1)^{\frac{k+1}{2}}}{2^kk!}$.\
In [@MaTo], the authors considered the next “von Neumann’s inequality type problem”: for each $1\leq q<\infty$, let $C_{k,q}(n)$ be the smallest constant such that $$\|p(T_1,...,T_n)\|_{\mathcal{L}(\mathcal{H})}\leq C_{k,q}(n)\sup\{|p(z_1,...,z_n)|:\sum_{j=1}^n|z_j|^q\leq 1\},$$ for every $k-$homogeneous polynomial $p$ in $n$ variables and every $n-$tuple of commuting contractions $(T_1,...,T_n)$ with $\sum_{j=1}^n\|T_i\|_{\mathcal{\mathcal{L}(\mathcal{H})}}^q\leq 1$. The lower and upper estimates for the growth of $C_{k,q}(n)$ [@MaTo Propositions 11 and 17] (here $q'$ denotes the conjugate of $q$) that they obtained are: $$\begin{aligned}
n^{\frac{k-1}{q'}-\frac{1}{2}\left[\frac{k}{2}\right]}\ll C_{k,q}(n)&\ll n^{\frac{k-2}{q'}} \hspace{0.2cm}\text{for}\hspace{0.2cm} 1\leq q\leq 2,\\
n^{\frac{k}{2}-\frac{1}{2}\left(\left[\frac{k}{2}\right]+1\right)}\ll C_{k,q}(n)&\ll n^{\frac{k-2}{2}} \hspace{0.2cm}\text{for}\hspace{0.2cm} 2\leq q< \infty.\label{upperbound}
\end{aligned}$$ Furthermore, the upper bounds here hold for every $n-$tuple $(T_1,...,T_n)$ satisfying $\sum_{j=1}^n\|T_i\|_{\mathcal{\mathcal{L}(\mathcal{H})}}^q\leq 1$ (and even a weaker condition), not necessarily commuting. If we do not ask the contractions to commute, this bound is shown to be optimal in [@MaTo Proposition 15].
In [@Ga], the lower estimates were improved. Specifically, Galicer et al. showed:
\[maintheo\] For $k\geq 3$ and $1\leq q\leq\infty$, let $C_{k,q}(n)$ be the smallest constant such that $$\|p(T_1,...,T_n)\|_{\mathcal{L}(\mathcal{H})}\leq C_{k,q}(n)\sup\{|p(z_1,...,z_n)|:\|(z_j)_j\|_q\leq 1\},$$ for every $k-$homogeneous polynomial $p$ in $n$ variables and every $n-$tuple of commuting contractions $(T_1,...,T_n)$ with $\sum_{j=1}^n\|T_i\|_{\mathcal{\mathcal{L}(\mathcal{H})}}^q\leq 1$. Then
1. $C_{k,\infty}(n)\sim n^{\frac{k-2}{2}}$
2. for $2\leq q<\infty$ we have $$\log^{-3/q}(n)n^{\frac{k-2}{2}}\ll C_{k,q}(n) \ll n^{\frac{k-2}{2}}.$$ In particular, $n^{\frac{k-2}{2}-\varepsilon} \ll C_{k,q}(n) \ll n^{\frac{k-2}{2}}$ for every $\varepsilon >0$.
Let $n\in\mathbb{N}$ and $1\leq t \leq k\leq n$. An $S_p(t,k,n)$ partial Steiner system is a collection of subsets of $\{1,...,n\}$ of cardinality $k$, called blocks such that every subset of $\{1,...,n\}$ of size $t$ is contained in at most one block of the system.\
A $k-$homogeneous polynomial of $n$ variables is a Steiner unimodular polynomial if there exists an $S_p(t,k,n)$ partial Steiner system $\mathcal{J}$ such that $p(z_1,...,z_n)=\sum_{J\in\mathcal{J}}c_Jz_J$ and $c_J=\pm1$.
Let’s recall the construction of commuting contractions which appeared in the proof of Theorem \[maintheo\] [@Ga Theorem 1.1]. Fix $k, n \in\mathbb{N}$. Let $\mathcal{H}$ be a finite dimensional Hilbert space with the following orthonormal basis: $$\begin{cases}
e;\\
e(j_1,...,j_m) \hspace{0.3cm} 0\leq m\leq k-2 \hspace{0.3cm}\text{and}\hspace{0.3cm} 1\leq j_1\leq j_2\leq \cdots \leq j_m\leq n;\\
f_i \hspace{1.75cm}1\leq i\leq n;\\
g.
\end{cases}$$ Given any subset $\{i_1,...,i_r\}\subset\{1,...,n\}$ we denote by $[i_1,...,i_r]$ its nondecreasing reordering. For $1\leq l\leq n$, we define the operator $T_l\in B(\mathcal{H})$ as $$\begin{cases}
T_le=e(l);\\
T_le(j_1,...,j_m)=e[l,j_1,...,j_m] \hspace{0.3cm} \text{if} \hspace{0.3cm} 0\leq m <k-2;\\
T_le(j_1,...,j_{k-2})=\sum_{i}\gamma_{\{i,l,j_1,...,j_{k-2}\}}f_i;\\
T_lf_i=\delta_{li}g;\\
T_lg=0,
\end{cases}$$ where $\gamma$ is defined as $$\gamma_{\{i_1,...,i_k\}}:=\begin{cases} c_{\{i_1,...,i_k\}} \hspace{0.3cm} \text{if} \hspace{0.1cm} \{i_1,...,i_k\}\in \mathcal{J} \\ 0 \hspace{1.5cm}\text{otherwise} \end{cases}$$ For $t=k-1$, it has been proved that $(T_1,...,T_n)$ is a commuting tuple of contractions on $\mathcal{H}$ and that $\|T_l\|=1$ for $l=1,...,n$. Following [@Ra], we call such a commuting tuple of contractions to be a Dixon’s $n-$tuple.
In [@Ga Theorem 2.5] the authors proved the following:
\[secondtheorem\] Let $k\geq 2$ and $\mathcal{J}$ be an $S_p(k-1,k,n)$ partial Steiner system. Then there exist signs $(c_J)_{J\in\mathcal{J}}$ and a constant $A_{k,q}>0$ independent of $n$ such that the $k-$homogeneous polynomial $p=\sum_{J\in\mathcal{J}}c_Jz_J$ satisfies $$\|p\|_{\mathcal{P}(^k\ell^n_q)}\leq A_{k,q}\times\begin{cases}
\log^{\frac{3}{q}}(n)n^{\frac{k}{2}(\frac{q-2}{q})} \text{ for } 2\leq q <\infty\\
\log^{\frac{3q-3}{q}}(n) \hspace{0.75cm}\text{ for } 1\leq q\leq 2.
\end{cases}$$ Moreover, the constant $A_{k,q}$ may be taken independent of $k$ for $q\neq 2$.
Galicer et al. also obtained the following corollary as an observation about the cardinality of partial Steiner systems.\
It is well known that any partial Steiner system $S_p(t,k,n)$ has cardinality less than or equal to ${n \choose t}/{k \choose t}$. In [@Noga], it is proved that there exists a constant $c>0$ such that there exist partial Steiner systems $S_p(k-1,k,n)$ of cardinality at least $$\label{cardsystem}
\psi(k,n):=\begin{cases}
\frac{{n \choose k-1}}{k}\left(1-\frac{c}{n^{\frac{1}{k-1}}}\right) \text{ for } k>3, \\
\frac{{n \choose k-1}}{k}\left(1-\frac{c\log^{3/2}n}{n^{\frac{1}{k-1}}}\right) \text{ for } k=3.
\end{cases}$$ As a consequence:
\[cor\] Let $k\geq 3$. Then there exists a $k-$homogeneous Steiner unimodular polynomial $p$ of $n$ complex variables with at least $\psi(k,n)$ coefficients satisfying the estimates in Theorem \[secondtheorem\]. Note that in this case $\psi(k,n)>>n^{k-1}$.
Mantero and Tonge [@MaTo] also studied another multivariable extension of von Neumann inequality by considering polynomials on commuting operators $T_1,...,T_n$ satisfying that for any pair $h,g$ of norm one vectors in the Hilbert space $$\label{cond}
\sum_{j=1}^{n}|\langle T_jh,g\rangle|^q\leq 1,$$ or, equivalently, that for any vector $\alpha\in \mathbb{C}^n$ such that $\|\alpha\|_{\ell^n_{q'}}=1$ we have $$\|\sum_{j=1}^{n}\alpha_jT_j\|\leq 1.$$ Let $D_{k,q}(n)$ be the smallest constant such that $$\|p(T_1,...,T_n)\|_{\mathcal{L}(\mathcal{H})}\leq D_{k,q}(n)\sup\{|p(z_1,...,z_n)|:\sum_{j=1}^n|z_j|^q\leq 1\},$$ for every $k-$homogeneus polynomial $p$ in $n$ variables and every $n-$tuple of contractions satisfying (\[cond\]). The upper bound for $D_{k,q}(n)$ found in [@MaTo Proposition 20] is $$D_{k,q}(n)\begin{cases} n^{(k-1)(\frac{1}{2}+\frac{1}{q})} , \hspace{0.2cm}\text{for}\hspace{0.2cm} q\geq 2,\\ n^{(k-1)(\frac{1}{2}+\frac{1}{q'})} , \hspace{0.2cm}\text{for}\hspace{0.2cm} q\leq 2.\end{cases}$$ For $k=3$ and $q=2$, Galicer et al. showed that this is an optimal bound up to a logarithmic factor. Specifically they showed:
We have the following asymptotic behavior: $$\frac{n^2}{\log^{15/4}n}\ll D_{3,2}(n)\ll n^2.$$
Following the same techniques of [@Ga] we obtain:
We have the following asymptotic behavior for $D_{k,2}(n)$: $$\frac{n^{k-1}}{\log^{\frac{3}{4}(k+2)}n}\ll D_{k,2}(n) \ll n^{k-1}.$$
Let $p(z)=\sum_{J\in \mathcal{J}}c_Jz_J$ be a $k-$homogeneous Steiner unimodular polynomial as in Theorem \[secondtheorem\]. Let $(T_1,...,T_n)$ be a Dixon’s $n-$tuple and $h\in \mathcal{H}$. First, we prove that $\displaystyle\frac{T_1}{(1+\|p\|_{\mathcal{P}(^k\ell_2^n)})^{1/2}},...,\displaystyle\frac{T_n}{(1+\|p\|_{\mathcal{P}(^k\ell_2^n)})^{1/2}}$ satisfy (\[cond\]) to get the upper bound. We have for $(1\leq m \leq k-2)$: $$\begin{aligned}
\sum_j\alpha_jT_jh&=\sum_j\alpha_j\left[\langle h,e \rangle T_je
+\sum_{j_m}\cdots\sum_{j_1}\langle h,e(j_m,...,j_1) \rangle T_je(j_m,...,j_1)
+\sum_{i}\langle h,f_i \rangle T_jf_i
+\langle h,g\rangle T_jg \right]\\
\\
&=\sum_j\alpha_j\langle h,e \rangle T_je\\
&+\sum_j\sum_{j_m}\cdots\sum_{j_1}\alpha_j\langle h,e(j_m,...,j_1) \rangle T_je(j_m,...,j_1)\\
&+\sum_j\sum_{j_{k-2}}\cdots\sum_{j_1}\alpha_j\langle h,e(j_{k-2},...,j_1) \rangle T_je(j_{k-2},...,j_1)\\
&+\sum_j\sum_{i}\alpha_j\langle h,f_i \rangle T_jf_i\\
&+\sum_j\alpha_j\langle h,g\rangle T_jg\\
\\
&=\sum_j\alpha_j\langle h,e \rangle e(j)\\
&+\sum_j\sum_{j_m}\cdots\sum_{j_1}\alpha_j\langle h,e(j_m,...,j_1) \rangle e[j,j_m,...,j_1]\\
&+\sum_j\sum_{j_{k-2}}\cdots\sum_{j_1}\alpha_j\langle h,e(j_{k-2},...,j_1) \rangle \sum_l \gamma_{\{l,j,j_{k-2},...,j_1\}}f_l\\
&+\sum_j\sum_{i}\alpha_j\langle h,f_i \rangle \delta_{ji}g\\
\end{aligned}$$ $$\begin{aligned}
&=\sum_j\alpha_j\langle h,e \rangle e(j)\\
&+\sum_j\sum_{j_1}\cdots\sum_{j_m}\alpha_j\langle h,e(j_1,...,j_m) \rangle e[j,j_1,...,j_m]\\
&+\sum_j\sum_l\sum_{j_1}\cdots\sum_{j_{k-2}}\alpha_j\langle h,e(j_1,...,j_{k-2}) \rangle a_{\{l,j,j_1,...,j_{k-2}\}}f_l\\
&+\sum_j\alpha_j\langle h,f_j \rangle g.
\end{aligned}$$ In this way (below $\beta$ is some vector in the unit ball of $\ell_2^n$) $$\begin{aligned}
\|\sum_j\alpha_jT_jh\|^2&=\sum_j|\alpha_j\langle h,e \rangle|^2\\
&+\sum_j\sum_{j_1,...,j_m}|\alpha_j\langle h,e(j_1,...,j_m) \rangle|^2\\
&+\sum_j\left|\sum_{l,j_1,...,j_{k-2}}\alpha_j\langle h,e(j_1,...,j_{k-2}) \rangle a_{\{j,l,j_1,...,j_{k-2}\}}\right|^2\\
&+\left|\sum_j\alpha_j \langle h,f_j\rangle\right|^2\\
&\leq \|\alpha\|^2_{\ell_2^n}|\langle h,e \rangle|^2\\
&+\|\alpha\|^2_{\ell_2^n}\|(\langle h,e(j_1,...,j_m) \rangle)_{j_1,...,j_m}\|^2_{\ell_2^n}\\
&+\left(\sum_j\beta_j \sum_{l,j_1,...,j_{k-2}}\alpha_j\langle h,e(j_1,...,j_{k-2}) \rangle a_{\{j,l,j_1,...,j_{k-2}\}}\right)^2\\
&+\|\alpha\|^2_{\ell_2^n}\| \|(\langle h,f_j \rangle)_j\|^2_{\ell_2^n}\\
\\
&\leq \|\alpha\|^2_{\ell_2^n}|\langle h,e \rangle|^2\\
&+\|\alpha\|^2_{\ell_2^n}\|(\langle h,e(j_1,...,j_m) \rangle)_{j_1,...,j_m}\|^2_{\ell_2^n}\\
&+\|\alpha\|^2_{\ell_2^n} \|p\|_{\mathcal{P}(^k\ell_2^n)} \|(\langle h,e(j_1,...,j_{k-2}) \rangle)_{j_1,...,j_{k-2}}\|^2_{\ell_2^n}\\
&+\|\alpha\|^2_{\ell_2^n} \|(\langle h,f_j \rangle)_j\|^2_{\ell_2^n}\\
\\
&=\|\alpha\|^2_{\ell_2^n}(|\langle h,e \rangle|^2 + \|(\langle h,e(j_1,...,j_m) \rangle)_{j_1,...,j_m}\|^2_{\ell_2^n}
+\|(\langle h,f_j \rangle)_j\|^2_{\ell_2^n}\\
&+ \|p\|_{\mathcal{P}(^k\ell_2^n)} \|(\langle h,e(j_1,...,j_{k-2}) \rangle)_{j_1,...,j_{k-2}}\|^2_{\ell_2^n})\\
\\
&\leq \|\alpha\|^2_{\ell_2^n} \left(\|h\|^2_{\mathcal{H}}+\|p\|_{\mathcal{P}(^k\ell_2^n)}\|h\|^2_{\mathcal{H}}\right)\\
\\
&=\|\alpha\|^2_{\ell_2^n} \|h\|^2_{\mathcal{H}}\left(1+\|p\|_{\mathcal{P}(^k\ell_2^n)}\right)
\end{aligned}$$.
Second, for the lower bound we use corollary \[cor\], $$\begin{aligned}
\left\|p\left(\frac{T_1}{(1+\|p\|_{\mathcal{P}(^k\ell_2^n)})^{1/2}},...,\frac{T_n}{(1+\|p\|_{\mathcal{P}(^k\ell_2^n)})^{1/2}}\right)\right\|_{\mathcal{L}(\mathcal{H})}
&\geq (1+\|p\|_{\mathcal{P}(^k\ell_2^n)})^{-k/2} \|p(T_1,...,T_n)e\|_{\mathcal{H}}\\
&= (1+\|p\|_{\mathcal{P}(^k\ell_2^n)})^{-k/2} |\mathcal{J}|\\
&= \frac{1}{(1+\|p\|_{\mathcal{P}(^k\ell_2^n)})^{k/2}} \frac{\|p\|_{\mathcal{P}(^k\ell_2^n)}}{\|p\|_{\mathcal{P}(^k\ell_2^n)}} |\mathcal{J}|\\
&\geq \frac{1}{C'(\log^{3/2}n)^{k/2}} \frac{\|p\|_{\mathcal{P}(^k\ell_2^n)}}{A_{k,2}\log^{3/2}n} n^{k-1}\\
&= C \frac{\|p\|_{\mathcal{P}(^k\ell^n_2)}}{(\log^{3/2}n)^{\frac{k}{2}+1}} n^{k-1}\\
&= C \frac{\|p\|_{\mathcal{P}(^k\ell^n_2)}}{\log^{\frac{3}{4}(k+2)}n} n^{k-1}\\
&\gg \|p\|_{\mathcal{P}(^k\ell^n_2)}\frac{n^{k-1}}{\log^{\frac{3}{4}(k+2)}n}.
\end{aligned}$$ where we have been used that for large enough $n$ we have $1\leq \log^{3/2}n$, and by Theorem \[secondtheorem\] also $\|p\|_{\mathcal{P}(^k\ell^n_2)}\leq A_{k,2}\log^{3/2}n$, and consequently $(1+\|p\|_{\mathcal{P}(^k\ell^n_2)})\leq C'\log^{3/2}n$, where $C'$ is a constant independent of $n$.
Again, following the same path than Galicer et al. [@Ga], we are able to show a variant of Theorem \[secondtheorem\]. To see this, let’s recall some definitions and facts about probability.
A Young function $\psi$ is an increasing convex function defined on $(0,\infty)$ such that $\lim_{t\to\infty}\psi(t)=\infty$ and $\psi(0)=0$. For a probability space $(\Omega,\Sigma,\mathbb{P})$, the Orlicz space $L_{\psi}=L_{\psi}(\Omega,\Sigma,\mathbb{P})$ is defined as the space of all real-valued random variables $Z$ for which there exists $c>0$ such that $\mathbb{E}(\psi(|Z|/c))<\infty$. It is a Banach space with the norm $\|Z\|_{L_{\psi}}=\inf\{c>0:\mathbb{E}(\psi(|Z|/c))\leq 1\}$. For more information about these and random process see [@LeTa].\
Let $k\geq 2$ and let $\mathcal{J}$ be a $S_p(k-1,k,n)$ partial Steiner system. Consider a family of independent Bernoulli variables $(\varepsilon_J)_{J\in\mathcal{J}}$ on $(\Omega,\Sigma,\mathbb{P})$. For $z\in B_{\ell_{2}^{n}}$ we define the following Rademacher process indexed by $B_{\ell_{2}^{n}}$ as $$\label{process}
Y_z=\frac{1}{k}\sum_{J\in\mathcal{J}} \varepsilon_Ja_Jz_J,$$ where the $(a_J)_{J\in\mathcal{J}}$ are complex coefficients with modulus bounded by 1. We view it as a random process in the Orlicz space defined by the Young function $\psi_2(t)=e^{t^2}-1$, recall $(\varepsilon_i)$ still spans a subspace isomorphic to $\ell_2$ in $\psi_2$.
\[Lipcond\] The Rademacher process defined in (\[process\]) fulfills the following Lipschitz condition $$\|Y_z-Y_{z'}\|_{L_{\psi_2}}\leq C\left(\max_{J\in\mathcal{J}}|a_J|\right)\|z-z'\|_{\infty}$$ for some universal constant $C\geq 1$ and every $z,z'\in B_{\ell_{2}^{n}}$.
By the Khintchine’s inequality the $\psi_2-$norm of a Rademacher process is comparable to its $L_2-$norm. So, $$\begin{aligned}
\|Y_z-Y_{z'}\|_{L_2}&=\displaystyle\frac{1}{k}\left(\int_{\Omega}\left|\displaystyle\sum_{J\in\mathcal{J}}\varepsilon_J(\omega)a_J(z_J-z'_{J})\right|^2\,d\mathbb{P}(\omega)\right)^{1/2}
=\frac{1}{k}\left(\displaystyle\sum_{J\in\mathcal{J}}|a_J|^2|z_J-z'_{J}|^2\right)^{1/2}\\
&=\displaystyle\frac{1}{k}\left(\displaystyle\sum_{J\in\mathcal{J}}|a_J|^2\left|\sum_{u=1}^{k}z_{j_1}\cdots z_{j_{u-1}}(z_{j_u}-z'_{j_u})z'_{j_{u+1}}\cdots z'_{j_k}\right|^2\right)^{1/2}\\
&\leq \displaystyle\frac{1}{k}\sum_{u=1}^{k}\left(\displaystyle\sum_{J\in\mathcal{J}}|a_J|^2|z_{j_1}\cdots z_{j_{u-1}}(z_{j_u}-z'_{j_u})z'_{j_{u+1}}\cdots z'_{j_k}|^2\right)^{1/2}\\
&\leq \displaystyle\frac{1}{k}\sum_{u=1}^{k}\|z-z'\|_{\infty}\left(\displaystyle\sum_{J\in\mathcal{J}}|a_J|^2|z_{j_1}\cdots z_{j_{u-1}}z'_{j_{u+1}}\cdots z'_{j_k}|^2\right)^{1/2}\\
&\leq \displaystyle\frac{1}{k}\sum_{u=1}^{k}\|z-z'\|_{\infty}\left(\max_{J\in\mathcal{J}}|a_J|^2\right)^{1/2}\left(\displaystyle\sum_{J\in\mathcal{J}}|z_{j_1}\cdots z_{j_{u-1}}z'_{j_{u+1}}\cdots z'_{j_k}|^2\right)^{1/2}\\
&\leq\left(\displaystyle\max_{J\in\mathcal{J}}|a_J|\right)\|z-z'\|_{\infty},
\end{aligned}$$ where the last inequality is true for the following: Since $\mathcal{J}$ is an $S_p(k-1,k,n)$ partial Steiner system, given $z_{j_1}\cdots z_{j_{u-1}}z'_{j_{u+1}}\cdots z'_{j_k}$ for a fixed $u$, there is at most one index $j_u$ such that $(j_1,...,j_k)$ belongs to $\mathcal{J}$. Therefore the sum $\sum_{J\in\mathcal{J}}|z_{j_1}\cdots z_{j_{u-1}}z'_{j_{u+1}}\cdots z'_{j_k}|^2$ can be bounded by $$(\sum_{l_1=1}^{n}|z_{l_1}|^2)\cdots(\sum_{l_{u-1}=1}^{n}|z_{l_{u-1}}|^2)(\sum_{l_{u+1}=1}^{n}|z'_{l_{u+1}}|^2)\cdots(\sum_{l_k=1}^{n}|z'_{l_k}|^2),$$ which is less or equal than one since $z,z'$ are in the unit ball of $\ell_2^n$.
For a metric space $(X,d)$, given $\varepsilon>0$, the entropy number $N(X,d;\varepsilon)$ is defined as the smallest number of open balls of radius $\varepsilon$ in the metric $d$, which form a covering of the metric space $X$. Then, the entropy integral of $(X,d)$ with respect to $\psi$ (Young function) is given by $$J_{\psi}(X,d):=\int_{0}^{{\operatorname{diam}}X}\psi^{-1}(N(X,d;\varepsilon))\,d\varepsilon.$$
The next theorem due to Pisier ([@Pi]) bounds the expectation of a random process with the entropy integral, provided the process satisfy a contraction condition.
\[Pisier\] Let $Z=(Z_x)_{x\in X}$ be a random process indexed by $(X,d)$ in $L_{\psi}$ such that, for every $x,x'\in X$, $$\|Z_x-Z_{x'}\|_{L_{\psi}}\leq d(x,x').$$ Then if $J_{\psi}(X,d)$ is finite, $Z$ is almost surely bounded and $$\mathbb{E}\left(\sup_{x,x'\in X}|Z_x-Z_{x'}|\right)\leq 8J_{\psi}(X,d).$$
So, by Lemma \[Lipcond\], we can use the previous theorem with $L_{\psi_2}, X=B_{\ell_2^n}$ and $d=\|\cdot\|_{\infty}$, to bound the expectation of the supremum of the random process, i.e. we need to estimate the integral $J_{\psi_2}(B_{\ell_2^n},\|\cdot\|_{\infty})$. Note that $\psi_{2}^{-1}(t)=\log^{1/2}(t+1)$but we can use instead $\log^{1/2}(t)$ since it does not change the computation in the integral.
\[entropy\] There exists $C>0$ such that for every $n\geq 2$ we have $J_{\psi_2}(B_{\ell_2^n},\|\cdot\|_{\infty})\leq C\log^{3/2}(n)$.
See [@Ga].
Now, following the exact argument in [@Ga Theorem 2.5] we obtain the next result:
Let $k\geq 2$ and $\mathcal{J}$ be an $S_p(k-1,k,n)$ partial Steiner system. Then there exist signs $(c_J)_{J\in\mathcal{J}}$, coefficients $(a_J)_{J\in\mathcal{J}}$ of modulus less than or equal to 1, and a constant $A_{k,q}>0$ independent of $n$, such that the $k-$homogeneous polynomial $p=\sum_{J\in\mathcal{J}}c_Ja_Jz_J$ satisfies $$\|p\|_{\mathcal{P}(^k\ell^n_q)}\leq A_{k,q}\times\begin{cases}
\|a_J\|_{\infty}^{\frac{q-2}{q}}\log^{\frac{3}{q}}(n)n^{\frac{k}{2}\frac{(q-2)}{q}}, \text{ for } 2\leq q <\infty\\
\|a_J\|_{\infty}^{\frac{2-q}{2}}\log^{\frac{3q-3}{q}}(n), \text{ for } 1\leq q\leq 2.
\end{cases}$$ Moreover, the constant $A_{k,q}$ may be taken independent of $k$ for $q\neq 2$.
Any $S_p(k-1,k,n)$ partial Steiner system $\mathcal{J}$ satisfies $|\mathcal{J}|\leq\frac{1}{k}{n \choose k-1}$, now we use $\mathcal{J}$ to define a Rademacher process $(Y_z)_{z\in B_{\ell^n_2}}$ as in (\[process\]). By Lemmas \[Lipcond\], \[entropy\] and Theorem \[Pisier\] there exists a constant $K>0$ such that $\mathbb{E}(\sup_{z\in B_{\ell_{2}^{n}}}|Y_z|)\leq K\log^{3/2}(n)$. By Markov’s inequality we have $$\mathbb{P}\{\omega\in\Omega:\|\sum_{J\in\mathcal{J}}\varepsilon_J(\omega)a_Jz_J\|_{\mathcal{P}(^k\ell_2^n)}\geq MkK\log^{3/2}(n)\}\leq \frac{1}{M},$$ where $M$ is a constant to be determined. Now, recall that by [@Ka] we have $$\mathbb{P}\{\omega\in\Omega:\|\sum_{J\in\mathcal{J}}\varepsilon_J(\omega)a_Jz_J\|_{\mathcal{P}(^k\ell_{\infty}^n)}\geq D(n|\mathcal{J}|\|a_J\|^2_{\infty}\log(k))^{1/2}\}\leq \frac{1}{k^2e^n}.$$ Therefore if $M>1+\frac{1}{k^2e^n-1}$, for $\omega$ in a set of positive measure, we have the following $$\label{ineq}\begin{cases}
\|\sum_{J\in\mathcal{J}}\varepsilon_J(\omega)a_Jz_J\|_{\mathcal{P}(^k\ell_2^n)}\leq MkK\log^{3/2}(n),\\
\|\sum_{J\in\mathcal{J}}\varepsilon_J(\omega)a_Jz_J\|_{\mathcal{P}(^k\ell_{\infty}^n)} \leq D(\frac{\log(k)}{k}{n\choose k-1}n\|a_J\|^2_{\infty})^{1/2}
\leq D(\frac{\log(k)}{k!}n^k\|a_J\|^2_{\infty})^{1/2}.
\end{cases}$$ There is a choice of signs $(C_J)_{J\in\mathcal{J}}$ and coefficients $(a_J)_{J\in\mathcal{J}}$ such that $p(z):=\sum_{J\in{\mathcal{J}}}c_ja_Jz_J$ satisfies the inequalities in (\[ineq\]). We shall use an interpolation argument to get a bound for the norm of $p$ in $\mathcal{P}(^k\ell_2^n)$ for $2<q<\infty$. Consider the $k-$linear form associated to $p$, then by interpolation, and inequalities (\[firstineq\]) and (\[ineq\]) we obtain $$\begin{aligned}
\|p\|_{\mathcal{P}(^k\ell_2^n)}&\leq (MkK)^{\frac{2}{q}}\log^{\frac{3}{q}}(n)(D\lambda(k,\infty)\frac{\log^{\frac{1}{2}}(k)}{\sqrt{k!}}\|a_J\|_{\infty})^{\frac{q-2}{q}}n^{\frac{k}{2}\frac{q-2}{q}}\\
&\leq \underbrace{\max\{M,K,D\}\left(\frac{k^{\frac{k}{2}}(k+1)^{\frac{k+1}{k}}\sqrt{\log(k)}}{2^kk!\sqrt{k!}}\right)^{\frac{q-2}{q}}k^{\frac{2}{q}}}_{A_{k,q}}\|a_J\|_{\infty}^{\frac{q-2}{q}}\log^{\frac{3}{q}}(n)n^{\frac{k}{2}(\frac{q-2}{q})}.
\end{aligned}$$ Note that for $q>2, A_{q,k}\to 0$ as $k\to\infty$ and thereby we can take a constant independent of $k$ in this case.\
For $q=1$, let $P(z)=\sum_{|\alpha|=k}a_{\alpha}z^{\alpha}$ be any $k-$homogeneous polynomial, then $$|P(z)|\leq \sum_{|\alpha|=k}|a_{\alpha}z^{\alpha}|\leq \sup_{|\alpha|=k}\left\{|a_{\alpha}\frac{\alpha!}{k!}|\right\}\sum_{|\alpha|=k}|\frac{k!}{\alpha!}z^{\alpha}|=
\sup_{|\alpha|=k}\left\{|a_{\alpha}\frac{\alpha!}{k!}|\right\}\left(\sum_{j=1}^{n}|z_j|\right)^k.$$ In particular the polynomial $p$ considered above satisfies $\|p\|_{\mathcal{P}(^k\ell_1^n)}\leq\frac{\|a_J\|_{\infty}}{k!}$. So, finally proceeding again by interpolation between the $\ell_1^n$ and $\ell_2^n$ cases we have for $1<q<2$ the next estimate $$\|p\|_{\mathcal{P}(^k\ell_q^n)}\leq \left(\|a_J\|_{\infty}\frac{k^k}{(k!)^2}\right)^{\frac{2-q}{q}}(MkK\log^{3/2}(n))^{\frac{2q-2}{q}}=A_{k,q}\|a_J\|_{\infty}^{\frac{2-q}{2}}\log^{\frac{3q-3}{q}}(n).$$ Also, in this case, for every $1\leq q<2$ we have $A_{k,q}\to 0$ as $k\to \infty$.
[8]{}
Noga Alon, Jeong-Han Kim, and Joel Spencer. [*Nearly perfect matchings in regular simple hypergraphs*]{}. Israel J. Math., 100:171-187, 1997. Seán Dineen. [*Complex analysis on infinite dimensional spaces*]{}. Springer Monographs in Mathematics. London: Springer, 1999. Daniel Galicer, Santiago Muro and Pablo Sevilla-Peris. (2016). [*Asymptotic estimates on the von Neumann inequality for homogeneous polynomials*]{}. Journal für die reine und angewandte Mathematik (Crelles Journal), 2018(743), pp. 213-227. Jean-Pierre Kahane. [*Some random series of functions*]{}. D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. Michel Ledoux and Michel Talagrand. [*Probability in Banach spaces*]{}, volume 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) \[Results inMathematics and Related Areas (3)\]. Springer-Verlag, Berlin, 1991. Isoperimetry and processes. Anna Maria Mantero and Andrew Tonge. [*Banach algebras and von Neumann’s inequality*]{}. Proc. London Math. Soc. (3), 38(2):309-334, 1979. Gilles Pisier. [*Some applications of the metric entropy condition to harmonic analysis*]{}. In Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981), volume 995 of Lecture Notes in Math., pages 123-154. Springer, Berlin, 1983. Samia Kumar Ray. [*On Multivariate Matsaev’s Conjecture*]{}. arXiv:1703.00733.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Gao-Min Tang'
- Fuming Xu
- Jian Wang
title: 'Supplemental material: Waiting Time Distribution of Quantum Electronic Transport in Transient Regime'
---
In this supplemental material we present details on how to calculate Green’s function and self-energy in the time domain in the presence of an upward pulse.[@35] The equilibrium self-energies are chosen to be energy dependent with a finite band width $W_0$ $$\label{eq1}
\tilde{\Sigma}_{\alpha}^r(\omega)=\frac{\Gamma_{\alpha} W_0}{2(\omega+iW_0)} \tag{1}$$ so that the linewidth function is the following Lorentzian form $$\label{eq2}
{\bf \Gamma}_{\alpha}(\epsilon)=\frac{\Gamma_{\alpha} W_0^2}{\epsilon^2+W_0^2} \tag{2}$$ where $\Gamma_{\alpha}$ is the linewidth amplitude. The self-energy in the time domain is defined as $$\label{eq3}
\Sigma_{\beta}^{r,<}(\tau_1,\tau_2)=\int \frac{d\omega}{2\pi} e^{-i\omega(\tau_1-\tau_2)}\tilde{\Sigma}_{\beta}^{r,<}(\omega) e^{ -i\int_{\tau_2}^{\tau_1}\Delta_L(t) dt } \tag{3}$$ where $\tilde{\Sigma}_{\beta}^{r,<}$ is the equilibrium self-energy in the energy domain. Using Eq.(\[eq3\]), we find the retarded self-energy of the left lead: $$\label{eq4}
\Sigma_L^r(\tau_1,\tau_2)=-\frac{i}{4}
\theta(\tau_1-\tau_2)\Gamma W_0 e^{-(i\Delta_L+W_0)(\tau_1-\tau_2)} \tag{4}$$ where we have assumed $\Gamma_L=\Gamma_R=\Gamma/2$. For the lesser self-energy $$\label{eq5}
\Sigma_L^<(\tau_1,\tau_2)=i\int\frac{d\omega}{2\pi} e^{-i\omega(\tau_1-\tau_2)}
e^{-i\Delta_L(\tau_1 -\tau_2)}f(\omega){\bf \Gamma}_L(\omega) \tag{5}$$ with $f(\omega)=1/\left[e^{\beta(\omega-E_F)}+1\right]$.
At zero temperature, we have $$\begin{aligned}
\label{eq6}
\Sigma_L^<(\tau_1,\tau_2)
&=i e^{-i\Delta_L(\tau_1 -\tau_2)} \int_{-\infty}^{0}\frac{d\omega}{2\pi} e^{-i\omega(\tau_1-\tau_2)}\frac{\Gamma_L W_0^2}{\omega^2+W_0^2} \tag{6}\end{aligned}$$ where we have set $E_F=0$.\
1. If $\tau_1=\tau_2$ $$\label{eq7}
\Sigma_L^<(\tau_1,\tau_2)=\frac{i}{8}\Gamma W_0 \tag{7}$$ 2. If $\tau_1>\tau_2$, let $\tau=\tau_1-\tau_2$ $$\begin{aligned}
\label{eq8}
\Sigma_L^<(\tau_1,\tau_2) &=\frac{i}{8}\Gamma W_0
\left\{\frac{i}{\pi} e^{(W_0-i\Delta_L)\tau} E1(W_0\tau) \right. \notag \\
&\left. +e^{-(W_0+i\Delta_L)\tau} \left[2-\frac{i}{\pi}E1(-W_0\tau) \right]\right\}
\tag{8}\end{aligned}$$ where $E1(x)=\int_x^{\infty}\frac{e^{-t}}{t}dt$.
At non-zero temperature, we have
1\. if $\tau_1=\tau_2$, the integral is actually Hilbert transformation of the Fermi distribution function.[@36] $$\label{eq9}
\Sigma_L^<(\tau_1,\tau_2)=\frac{i\Gamma W_0}{8} \tag{9}$$ 2. if $\tau_1>\tau_2$, it has poles $\frac{-i(2n+1)\pi}{\beta}$ and $-iW_0$, where $n=0,1,2,3...$ $$\begin{aligned}
\label{eq10}
&\Sigma_L^<(\tau_1,\tau_2)=
\frac{i\Gamma_L W_0\exp[-(W_0+i\Delta_L)(\tau_1-\tau_2)]}{2\exp(-i\beta W_0)+2}
-\frac{1}{\beta} \times \notag \\
&\sum_{n=0}^{+\infty}\exp\left\{-[\frac{(2n+1)\pi}{\beta}+i\Delta_L](\tau_1-\tau_2) \right\}\frac{\Gamma_L W_0^2}{W_0^2-[\frac{(2n+1)\pi}{\beta}]^2} \tag{10}\end{aligned}$$ Using the relation $\Sigma_L^<(\tau_1,\tau_2)\big|_{\tau_1<\tau_2}=-[\Sigma_L^<(\tau_1,\tau_2)\big|_{\tau_1>\tau_2}]^*$, we obtain the full expression of $\Sigma_L^<(\tau_1,\tau_2)$. The expression of $\Sigma_R^<(\tau_1,\tau_2)$ can be obtained similarly.
Taking Fourier transform of the retarded self-energy $\Sigma^r(\tau_1,\tau_2)=\Sigma_L^r(\tau_1,\tau_2)+\Sigma_R^r(\tau_1,\tau_2)$ to energy domain, we find $$\label{eq11}
\Sigma^r(\omega)=\frac{\Gamma W_0}{4}[\frac{1}{\omega+iW_0}+\frac{1}{\omega-\Delta_L+iW_0}] \tag{11}$$ The retarded Green’s function in time domain is given by $$\begin{aligned}
\label{eq12}
G^r(\tau_1,\tau_2)&=\int\frac{d\omega}{2\pi}e^{-i\omega(\tau_1-\tau_2)}\frac{1}{\omega-\epsilon_0-\Sigma^r(\omega)} \notag \\
&=\int\frac{d\omega}{2\pi}e^{-i\omega(\tau_1-\tau_2)}\frac{(\omega+iW_0)(\omega-\Delta_L+iW_0)}{(\omega-\omega_1)(\omega-\omega_2)(\omega-\omega_3)}
\tag{12}\end{aligned}$$ where $\omega_1, \ \omega_2, \ \omega_3$ are poles of equilibrium retarded Green’s function satisfying the following equation $$\label{eq13}
\omega^3 + a \omega^2 +b\omega +c=0 \tag{13}$$ with $$\begin{aligned}
\label{eq14}
a &=(2iW_0-\Delta_L-\epsilon_0) \notag \\
b &=iW_0(iW_0+\frac{i\Gamma}{2}-2\epsilon_0-\Delta_L)+\epsilon_0\Delta_L \notag \\
c &=\frac{\Gamma W_0}{4} (\Delta_L-2iW_0)+iW_0\epsilon_0(\Delta_L-iW_0) \tag{14}\end{aligned}$$ This integral can be calculated using the theorem of residue. $$\label{eq15}
G^r(\tau_1,\tau_2)=-i\sum_i^3\frac{(\omega_i+iW_0)(\omega_i-\Delta_L+iW_0)e^{-i\omega_i(\tau_1-\tau_2)}}{\sum_{j,k=1}^3|\epsilon_{ijk}|(\omega_i-\omega_j)(\omega_i-\omega_k)} \tag{15}$$ where $\epsilon_{ijk}$ is the usual Levi-Civita symbol. Here $G^<$ can be calculated through the Keldysh equation: $$\label{eq16}
G^<=(1+G^r\Sigma^r)g^<(1+\Sigma^aG^a)+G^r\Sigma^<G^a \tag{16}$$ The first term is zero if $(g^r)^{-1}g^< (g^a)^{-1}=0$ which is the case for steady states. In our case we should keep it in the transient regime calculation. In the calculation, we have chosen $g^<=0.5i$ at $t=0$ which makes sure that $I_L+I_R=0$ and $<(\Delta I_L)^2>=<(\Delta I_R)^2>$.
[22]{} Y. Zhu, J. Maciejko, T. Ji, and H. Guo, Phys. Rev. B **71**, 075317 (2005). G. Bevilacqua, arXiv:1303.6206 \[math-ph\]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We extend classical results on simple varieties of trees (asymptotic enumeration, average behavior of tree parameters) to trees counted by their number of leaves. Motivated by genome comparison of related species, we then apply these results to strong interval trees with a restriction on the arity of prime nodes. Doing so, we describe a filtration of the set of permutations based on their strong interval trees. This filtration is also studied from a purely analytical point of view, thus illustrating the convergence of analytic series towards a non-analytic limit at the level of the asymptotic behavior of their coefficients.'
author:
- 'Mathilde Bouvel[^1], Marni Mishna[^2], Cyril Nicaud[^3]'
title: 'Some families of trees arising in permutation analysis[^4]'
---
#### Keywords: {#keywords .unnumbered}
permutations, simple varieties of trees, random generation, tree parameters, asymptotic formulas.
Introduction {#sec:Introduction}
============
The idea of viewing permutations as enriched trees has been around for several decades in different research communities. For example, the recent enumerative study [@AlAt05] of pattern avoiding permutations, in which *(substitution) decomposition trees* play a crucial role. Also, the analysis of some sorting algorithms is very linked to tree representations of permutations: *PQ trees* [@BoLu76] appear in the context of graph algorithms and *strong interval trees* arise in comparative genomics [@BoChMiRo11 and references therein, for instance].
The focus of this article is the study of strong interval trees (a.k.a. decomposition trees). Their typical shape (under the uniform distribution) has been described in [@BoChMiRo11], showing that they have a very flat, somehow degenerate, shape. Strong interval trees are an essential tool to model and study genome rearrangements. But contrary to what this average shape shows, the trees associated to permutations that arise in the comparison of mammalian genomes seem to have a richer, deeper structure. This suggest that trees coming from permutations under the uniform distribution do not adequately represent the trees that arise in genomic comparisons.
In [@BoChMiRo11], Bouvel *et al.* considered a subclass of strong interval trees –selected because they represent what is known as *commuting scenarios* [@BeBeChPa07]– that correspond to the class of *separable permutations*. This is a first step towards a more relevant model of permutations which arise in genome comparison. By studying asymptotic enumeration and parameter formulas for separable permutations, they proved that the complexity of the algorithm of [@BeBeChPa07] solving the *perfect sorting by reversals* problem is polynomial time on separable permutations, whereas this problem is NP-complete in general. Furthermore they were also able to describe some average-case properties of the perfect sorting scenarios for separable permutations.
Ultimately, a clear understanding of the properties possessed by the strong interval trees that represent the comparison of actual genomes might tell us something about the evolutionary process. Bouvel *et al.* [@BoChMiRo11] conclude their study on separable permutations with a suggestion for the next step: strong interval trees with degree restrictions on certain internal nodes. It is a very controlled way to introduce bias in the distribution of strong interval trees. This is precisely what we do in this work; namely, we study strong interval trees where the prime nodes have a bounded number of children. In this work, we focus on the combinatorial analysis of these restricted sets of trees. They can be completely understood combinatorially and analytically, and so we have access to enumeration and analysis of some tree parameters that are ultimately related to the complexity of computing perfect sorting scenarios, or to properties of these scenarios.
Although our initial motivation is the application of combinatorial analysis to a better understanding of models for genome rearrangements, we believe our study has ramifications of independent interest in analytic combinatorics. Indeed, our work reveals a very lush substructure of permutations whose study from an analytical point of view allows us to formulate new questions on the convergence of sequences of combinatorial series.
Specifically, we define nested families of trees (which are almost simple varieties of trees) whose limit is the set of all strong interval trees. The components are then families of trees to which we are able to apply a very complete set of tools: asymptotic analysis, parameter analysis, random generation. But the complete set of strong interval trees is not even close to be a simple variety of trees, so that these tools are inaccessible to the full class without working through the size preserving bijection between strong interval trees and permutations. The question that we ask is then to understand at the analytical level the convergence of algebraic subclasses of a non-analytic class towards the full class. As we explain in details in our work, this question is naturally asked for strong interval trees, but it could also be considered for other classes such as $k$-regular graphs [@Gess90] or $\lambda$-terms of bounded unary height [@BGG11].
The organization of this article is as follows. First, in Section \[sec:General\] we present some very general theorems for asymptotic enumeration and parameter analysis in families of trees counted by leaves, that are widely applicable. Then in Section \[sec:CIT\] we describe strong interval trees, a decomposable combinatorial class of trees counted by leaves, in bijection with permutations. Next, in Section \[sec:restrictedCIT\] we introduce a filtration of strong interval trees, where we bound the arity of so-called *prime* nodes. As discussed in Section \[sec:restrictedCIT\], this filtration has applications in bio-informatics for the study of genome rearrangements. But it also witnesses an intriguing analytic phenomenon: the convergence of a sequence of (well-behaved and algebraic) families of trees towards the (transcendental and non-analytic) class of permutations. Section \[sec:filtration\] establishes some first results in the exploration of this phenomenon.
When the size of a tree is the number of leaves {#sec:General}
===============================================
There are many works which study the average case behavior of tree parameters, where the size of a tree is the number of internal nodes or of both internal nodes and leaves. The generating functions of these trees satisfy a functional equation of the form $T(z)=z\cdot\Phi(T(z))$, and when $\Phi$ satisfies certain conditions, such as analyticity, then there are formulas for inversion, resulting in explicit enumerative results. A class of trees amenable to this treatment is said to be a *simple variety of trees*. The subject is exhaustively treated in Section VII.3 of [@FlSe09].
If, instead, we define the size of a tree as the number of leaves, the generating function satisfies a relation of the form $T(z)=z+\Lambda(T(z))$. The same general theorems on inversion still work, and it suffices to apply them and unravel the results. Even though they are less frequent, these have also been studied in the literature, and the applicability of the inversion lemmas is noted in Example VII.13 of [@FlSe09]. In this section we do this explicitly. In this work, when refering to simple varieties of trees, we mean a family of trees where the size is defined as the number of leaves, and whose study falls into the scope of the results of the present section.
Table \[tab:summary\] summarizes the results of this section. We determine asymptotic formulas for number of trees, and several key parameters. The shape of the formulas are, unsurprisingly, not unlike those that arise in the study of trees counted by internal nodes.
-------------------------------------------------------------------------------- -------------------------------------------------------------------------- --
Asymptotic number of trees with $n$ leaves $\sqrt{\frac{\rho}{2\pi\Lambda''(\tau)}}\cdot \frac{\rho^{-n}}{n^{3/2}}$
\[4mm\] The average number of nodes of arity $\kappa$ in trees with $n$ leaves $\frac{\lambda_\kappa\tau^\kappa}{\rho} \cdot n$
\[4mm\] The average number of internal nodes in trees with $n$ leaves $\frac{\Lambda(\tau)}{\rho}\cdot n=\frac{\tau-\rho}{\rho}\cdot
n$
\[4mm\] The average subtree size sum in trees with $n$ leaves $\sqrt{\frac{\pi}{2\rho\Lambda''(\tau)}}\cdot n^{3/2}$
\[2mm\]
-------------------------------------------------------------------------------- -------------------------------------------------------------------------- --
: A summary of parameters of trees given by $T = z +
\Lambda(T)$. The value $\tau$ is the unique solution to $\Lambda'(\tau)=1$ between $0$ and $R_{\Lambda}<1$, and $\rho=\tau-\Lambda(\tau)$. []{data-label="tab:summary"}
Asymptotic number of trees
--------------------------
Our entire analysis is roughly a consequence of the Analytic Inversion Lemma and Transfer Theorems. The version to which we appeal is given and proved in [@FlSe09]. Citations to original sources may be found therein. The following theorem is a slight adaptation of Proposition IV.5 and Theorem VI.6 of [@FlSe09] to combinatorial equations of the form ${\mathcal{T}}=\mathcal{Z}+\Lambda({\mathcal{T}})$ instead of ${\mathcal{T}}=\mathcal{Z}\cdot\Phi({\mathcal{T}})$.
\[thm:main\] Let $\Lambda$ be a function analytic at $0$, with non-negative Taylor coefficients, and such that, near $0$, $$\Lambda(z) = \sum_{n\geq 2} \lambda_n z^n.$$ Let $R_\Lambda$ be the radius of convergence of this series. Under the condition , there exists a unique solution $\tau \in (0,R_\Lambda)$ of the equation $\Lambda'(\tau) = 1$.
Then, the formal solution $T(z)$ of the equation $$T(z) = z+\Lambda(T(z)) \label{eq:main}$$ is analytic at $0$, its unique dominant singularity is at $\rho = \tau - \Lambda(\tau)$ and its expansion near $\rho$ is $$T(z) = \tau - \sqrt{\frac{2\rho}{\Lambda''(\tau)}} \sqrt{1-\frac{z}{\rho}} + \mathcal{O}\left(1-\frac{z}{\rho}\right).$$ Moreover, if $T$ is aperiodic, then one has $$[z^n]T(z) \sim \sqrt{\frac{\rho}{2\pi\Lambda''(\tau)}}\cdot \frac{\rho^{-n}}{n^{3/2}}.$$
The conditions on $\Lambda$ imply that both $\Lambda(x)$ and $\Lambda'(x)$ are increasing functions for $x$ in the real interval $(0,R_\Lambda)$. Since $\Lambda'(0) = 0$ and since , there exists $R'\in (0,R_\Lambda)$ such that $\Lambda'(R')>1$. Hence there exists a unique $\tau\in(0,R')$, and thus on $(0,R_{\Lambda})$, such that $\Lambda'(\tau)=1$.
Now observe that Equation \[eq:main\] admits a unique formal power series solution $T(z)$, which has non-negative coefficients, by bootstrapping the coefficients. By Analytic Inversion [@FlSe09 Lemma IV.2], this solution is analytic at $z=0$ and with $T(0)=0$: Equation \[eq:main\] can be rewritten $\Psi(T(z)) = z$, with $\Psi(x) = x-\Lambda(x)$, and $\Psi'(0) \neq 0$.
By Pringsheim’s Theorem, a dominant singularity of $T(z)$, if any, lies on the positive real axis. Let $r\in(0,+\infty]$ be the radius of convergence of $T(z)$ at $0$, and set $T(r)\in (0,+\infty]$ be defined by $T(r) = \lim_{x\rightarrow r^-}T(x)$. Following almost exactly the proof of Proposition IV.5 in [@FlSe09 p. 278], we get that $T(r)=\tau$. Since $T$ and $\Psi$ are inverse functions, we get by continuity that $T(z)$ has a unique dominant singularity at $\rho = \tau-\Lambda(\tau)$.
The remainder of the proof follows almost readily the one of Theorem VI.6 in [@FlSe09 p. 405], using our specific equations.
Parameter Analysis
------------------
In the case of trees counted by internal nodes, the study of recursively defined parameters is very straightforward, starting from generating function equations. We can describe analogous versions for trees counted by leaves. In particular, we consider additive parameters, and describe a Modified Iteration Lemma, adapted to our notion of size. We illustrate the lemma on number of internal nodes, subtree size sum and number of nodes of a given arity.
### General additive parameters
Our focus is on tree parameters that can be computed additively by parameters of subtrees. More precisely, we consider a parameter $\xi(t)$ for trees $t\in{\mathcal{T}}$ which satisfy the relation $$\xi(t) = \eta(t) + \sum_{j=1}^{\operatorname{deg}(t)}\sigma(t_j),$$ where $\operatorname{deg}(t)$ is the arity of the root, $t_j$ are its children, $\eta$ is a simpler tree parameter, and $\sigma$ is either $\xi$ or a simpler tree parameter. Let $\Xi(z)$, $H(z)$ and $\Sigma(z)$ be the associated cumulative functions of $\xi$, $\eta$ and $\sigma$. That is, $$\Xi(z)=\sum\limits_{t\in{\mathcal{T}}}\xi(t) z^{|t|}, \quad
H(z)=\sum\limits_{t\in{\mathcal{T}}}\eta(t) z^{|t|} \textrm{\quad and \quad }
\Sigma(z)=\sum\limits_{t\in{\mathcal{T}}}\sigma(t) z^{|t|}.$$
Lemma VII.1 in [@FlSe09] has an analogue for trees counted by their leaves, and it is proved in a very similar way.
\[lm:inv\] Let ${\mathcal{T}}$ be a class of trees satisfying ${\mathcal{T}}=\mathcal{Z}+\Lambda({\mathcal{T}})$. The cumulative generating functions are related by $$\Xi(z) = H(z) + \Lambda'(T(z))\,\Sigma(z).$$ In particular, if $\sigma\equiv\xi$, one has $
\Xi(z) = \frac{H(z)}{1-\Lambda'(T(z))} = H(z)\cdot T'(z).
$
Unraveling the definition of $ \xi(t)$, we have $$\Xi(z) = H(z) + \widetilde{\Xi}(z) \textrm{ \ with \ } \widetilde{\Xi}(z) = \sum_{t \in {\mathcal{T}}} z^{|t|} \sum_{j=1}^{\operatorname{deg}(t)} \sigma(t_j).$$ Splitting the sum defining $\widetilde{\Xi}(z)$ according to the value $r$ of the degree of the root of $t$, we get: $$\begin{aligned}
\widetilde{\Xi}(z) &= \sum_{r \geq 1} \lambda_r z^{|t_1| + \ldots + |t_r|}(\sigma(t_1) + \ldots + \sigma(t_r)) \\ &=
\sum_{r \geq 1} \lambda_r \left( \sigma(t_1)z^{|t_1|}z^{|t_2| + \ldots + |t_r|} + \ldots + \sigma(t_r) z^{|t_r|}z^{|t_1| + \ldots + |t_{r-1}|} \right)\\ &=
\sum_{r \geq 1} \lambda_r \times r \times \Sigma(z) T(z)^{r-1}=\Lambda'(T(z)) \, \Sigma(z).\end{aligned}$$
In the case $\sigma\equiv\xi$, $\Xi(z) = \frac{H(z)}{1-\Lambda'(T(z))}$ is derived immediately. The last equality is a consequence of $T'(z)(1-\Lambda'(T(z))) = 1$, which is obtained by differentiating $T(z) = z + \Lambda(T(z))$ with respect to $z$.
Note that if $\sigma\equiv\xi$, the parameter is said to be *recursive*. Most basic parameters are recursive, and in what follows we shall use this case only.
Note also that when analytic treatment applies, $T(z)$ has a square-root singularity (see Theorem \[thm:main\]), so that $T'(z)$ has an inverse square-root singularity (by analytic derivation). Therefore, whenever $H(z)$ tends to a positive real when $z\rightarrow\rho$ (under some analytic conditions), then the Transfer Theorem yields an asymptotic equivalent of the mean value of the parameter of the form $c\cdot n$. This is for instance the case for the number of nodes of fixed arity and the number of internal nodes, as shown below.
### Three applications
#### Number of nodes with exactly $\kappa$ children.
We “mark” nodes of arity $\kappa$ by setting $$\eta(t) = \begin{cases}
1 & \text{if the root of $t$ is of arity }\kappa,\\
0 & \text{otherwise}.
\end{cases}$$ Hence if $\kappa\geq 2$, $
H(z) = \sum\limits_{t\in {\mathcal{T}}} \eta(t)z^{|t|} =
\sum\limits_{t_1,\ldots,t_\kappa\in{\mathcal{T}}}\lambda_\kappa
z^{|t_1|+|t_2|+\ldots+|t_\kappa|} $ so that $H(z) = \lambda_\kappa\,T(z)^\kappa$. Now,[^5] if $\kappa=0$, then $H(z) = z$ which is not interesting since it is simply counting the number of leaves *i.e.* the size of the tree.
By Lemma \[lm:inv\], for any $\kappa\geq 2$ one has $\Xi(z) =
\lambda_\kappa T(z)^{\kappa}\cdot T'(z)$. Since the singular expansion of $T(z)$ near $\rho$ is $$\label{eq:gamma}
T(z) = \tau - \gamma \sqrt{1-z/\rho} +o\left(\sqrt{1-z/\rho}\right), \text{with }\gamma=\sqrt{\frac{2\rho}{\Lambda''(\tau)}}$$ then near $\rho$, one has $T(z)^\kappa = \tau^\kappa + \mathcal{O}\left(\sqrt{1-z/\rho}\right).$ Using the Singular Differentiation Theorem we have $$T'(z) = \frac{\gamma}{2\rho \sqrt{1-z/\rho}} +o\left(\frac1{\sqrt{1-z/\rho}}\right),
\text{ so that }
\Xi(z) = \frac{\lambda_\kappa \gamma\tau^\kappa}{2\rho \sqrt{1-z/\rho}} + o\left(\frac1{\sqrt{1-z/\rho}}\right),$$ from which we get the asymptotics of the cumulative generating function $$[z^n]\Xi(z)\sim\frac{\lambda_\kappa \gamma\tau^\kappa\rho^{-n-1}}{2\sqrt{\pi n}}.$$ The asymptotics of the average value across all trees of size $n$ is then $$\frac{[z^n]\Xi(z)}{[z^n]T(z)} \sim
\frac{\lambda_\kappa \gamma\tau^\kappa\rho^{-n-1}}{2\sqrt{\pi n}} \cdot \sqrt{\frac{2\pi\Lambda''(\tau)}{\rho}} \frac{n^{3/2}}{\rho^{-n}}
\sim \frac{\lambda_\kappa\tau^\kappa}{\rho} \cdot n ,$$ as reported in Table \[tab:summary\].
#### Number of internal nodes.
For this parameter, just take the following definition for $\eta$: $$\eta(t) = \begin{cases}
0 & \text{if }t\text{ is just one leaf,} \\
1 & \text{otherwise}.
\end{cases}$$ One has $H(z) = \sum\limits_{t\in{\mathcal{T}}} \eta(t)z^{|t|} = T(z)-z$, and therefore (with the $\gamma$ of Equation ) $$\Xi(z) = \left(T(z)-z\right)\,T'(z) = \frac{\gamma(\tau-\rho)}{2\rho \sqrt{1-z/\rho}} +o\left(\frac1{\sqrt{1-z/\rho}}\right).$$ It follows that $$[z^n]\Xi(z)\sim\frac{\gamma(\tau-\rho)\rho^{-n-1}}{2\sqrt{\pi n}} \textrm{\quad and \quad }
\frac{[z^n]\Xi(z)}{[z^n]T(z)} \sim \frac{\tau-\rho}{\rho} \cdot n.$$
#### Subtree size sum.
We are interested in the subtree size sum parameter, defined by $\eta(t) = |t|$. This implies that $H(z)=zT'(z)$, so that $$\Xi(z) = z T'(z)^2 = \frac{\gamma^2}{4\rho(1-z/\rho)} +o\left(\frac{1}{1-z/\rho}\right)
\text{\quad and \quad}
[z^n]\Xi(z)\sim\frac{\gamma^2}{4\rho} \cdot \rho^{-n}.$$ Unlike the two previous examples, this is not an inverse of square-root singularity. In this case, for the average value of the subtree size sum, we find $$\frac{[z^n]\Xi(z)}{[z^n]T(z)} \sim \frac{\gamma^2}{4\rho} \rho^{-n} \cdot \sqrt{\frac{2\pi\Lambda''(\tau)}{\rho}} \frac{n^{3/2}}{\rho^{-n}}
\sim \sqrt{\frac{\pi}{2\rho\Lambda''(\tau)}}\cdot n^{3/2},$$ that is, an asymptotic equivalent in $n^{\frac32}$. This behavior is typical for such path length related parameters.
There are many other tree parameters that we could consider in a similar fashion.
Strong Interval Trees {#sec:CIT}
=====================
Our interest in trees counted by leaves is spawned by *strong interval trees*. Strong interval trees are in a size preserving bijection with permutations. They have been introduced in the early 2000’s in a bio-informatics context [@HeSt01; @BeChdeMRa05]: they are a very effective data structure for algorithms in reconstruction of genome evolution scenarios, as we briefly mentioned in Section \[sec:Introduction\]. Under a different name, and roughly at the same time, these objects also made their appearance in combinatorics, in the study of permutation patterns: strong interval trees (rather called (substitution) decomposition trees) are a tree representation of the block decomposition of permutations described by Albert and Atkinson [@AlAt05]. Although the proper definition of strong interval trees is relatively recent, it can be traced to older notions of decomposition (of graphs in particular): it is a close relative of the modular decomposition trees of permutation graphs [@BeChdeMRa05] and even has origins in the PQ-trees of Booth and Lueker [@BoLu76].
In this section, we review the definition of strong interval trees and the bijection with permutations. Then, we turn to a presentation of these objects as a constructible combinatorial class, in the flavor of what is done in Section \[sec:General\].
Definition and bijection with permutations
------------------------------------------
Strong interval trees are most often defined via the bijection that relates them to permutations. Different presentations of this bijection can be found for instance in [@AlAt05; @BeChdeMRa05; @BoChMiRo11]. For the reader who is not familiar with these objects, we review the definition of strong interval trees, and the correspondence with permutations below.
In the context of our work, a permutation of size $n$ is a word containing exactly once each symbol in $\{1,2,\ldots, n\}$.
An *interval* of a permutation $\sigma$ is a factor of $\sigma$, such that the underlying set of symbols is an interval of integers. For instance, $7\,\,9\,\,10\,\,11\,\,13\,\,8\,\,12$ and $3\,\,1\,\,5\,\,4\,\,2$ are intervals of the permutation $6\,\,7\,\,9\,\,10\,\,11\,\,13\,\,8\,\,12\,\,3\,\,1\,\,5\,\,4\,\,2$, but $10\,\,11\,\,13$ is not ($12$ is missing). For every permutation $\sigma$ of size $n$, the singletons $i$ (for $1\leq i \leq n$) and $\sigma$ itself are intervals of $\sigma$. They are called *trivial* intervals of $\sigma$.
A permutation is said to be *simple* when its only intervals are the trivial ones. Note that our convention will be that $1$, $1\,\,2$ and $2\,\,1$ are not simple permutations, although they satisfy the above definition. It is immediate to check that there is no simple permutation of size $3$ (each permutation of size $3$ containing an interval of size $2$), and that there are $2$ simple permutations of size $4$, namely $2\,\,4\,\,1\,\,3$ and $3\,\,1\,\,4\,\,2$. A larger simple permutation is for instance $3\,\,5\,\,7\,\,1\,\,4\,\,2\,\,6$. We will go back to the enumeration of simple permutations in the next subsection.
Two intervals of $\sigma$ *overlap* when their intersection is neither empty nor equal to one of them. Returning to our example of $6\,\,7\,\,9\,\,10\,\,11\,\,13\,\,8\,\,12\,\,3\,\,1\,\,5\,\,4\,\,2$, the intervals $6\,\,7$ and $7\,\,9\,\,10\,\,11\,\,13\,\,8\,\,12$ overlap (their intersection is $7$), but $10\,\,11$ and $5\,\,4$ don’t. A *strong* interval of $\sigma$ is an interval that does not overlap any other interval of $\sigma$. The trivial intervals are obviously strong. On our running example, the non-trivial strong intervals are $$5\,\,4 \ ; \quad
3\,\,1\,\,5\,\,4\,\,2 \ ;\quad
9\,\,10\,\,11 \ ;\quad
9\,\,10\,\,11\,\,13\,\,8\,\,12 \quad \textrm{ and } \quad
6\,\,7\,\,9\,\,10\,\,11\,\,13\,\,8\,\,12.$$
From their definition, it follows immediately that the inclusion order on the set of strong intervals of a permutation $\sigma$ induces a tree structure, where the leaves are the singletons, and the root is the $\sigma$ itself. This is the *strong interval tree* of $\sigma$.
From there, and depending on the context, the definition of the strong interval tree may vary. For us, these trees are *embedded in the plane*, by imposing the order of the leaves. Namely, from left to right, the leaves (corresponding to singletons of $\sigma$) are required to appear in the same order as in $\sigma$. The strong interval tree of our running example would then be:
child[ node[$6\,\,7\,\,9\,\,10\,\,11\,\,13\,\,8\,\,12$]{} child [ node[$6$]{} ]{} child [ node[$7$]{} ]{} child\[missing\] child [ node[$9\,\,10\,\,11\,\,13\,\,8\,\,12$]{} child[ node[$9\,\,10\,\,11$]{} child[ node[$9$]{} ]{} child[ node[$10$]{} ]{} child[ node[$11$]{} ]{} ]{} child[ node[$13$]{} ]{} child[ node[$8$]{} ]{} child[ node[$12$]{} ]{} ]{} ]{} child\[missing\] child\[missing\] child\[missing\] child\[missing\] child[ node[$3\,\,1\,\,5\,\,4\,\,2$]{} child[ node[$3$]{} ]{} child[ node[$1$]{} ]{} child[ node[$5\,\,4$]{} child [node[$5$]{} ]{} child [node[$4$]{} ]{} ]{} child[ node[$2$]{} ]{} ]{};
From this tree, there is a last step that we perform before obtaining what we refer to as the strong interval tree in our work. It relies on an important remark, proved for instance in [@BXHaPa05]. To state it, we first need to describe how to associate a permutation of size $k$ to each node of the strong interval tree with $k$ children.
Note first that given several disjoint strong intervals, the natural order on integers induces an order among them: the smaller the elements contained in the interval, the smaller the interval itself. On our running example, we have for instance that $5\,\,4$ is smaller than $8$ which is itself smaller than $9\,\,10\,\,11$. Consider now a non-singleton strong interval, corresponding to an internal node of the strong interval tree. Its children (assume there are $k$ of them) are also strong intervals, and they are disjoint. So they can be ordered as described above. To this node of the tree, we associate a permutation $\tau$ of size $k$ built as follows: $\tau_i = j$ if the $i$-th child from the left is the $j$-th smallest one. For instance, the permutation $\tau$ associated with the node labeled $9\,\,10\,\,11\,\,13\,\,8\,\,12$ in our running example is $2\,\, 4 \,\, 1 \,\, 3$ since $8$ is smaller than $9\,\,10\,\,11$, itself smaller than $12$ and $13$.
The permutations labeling the node enjoy a remarkable property (see [@BXHaPa05], or in a somewhat different presentation [@AlAt05]): they are either increasing ($1 \, \, 2 \ldots k$) or decreasing ($k \ldots 2 \, \, 1$) or simple. In the reminder of this article, when speaking about strong interval trees, we mean the plane tree whose structure has been described above, but whose internal nodes are only labeled by $\oplus$, $\ominus$ (corresponding to increasing or decreasing permutations respectively), or by a simple permutation. In particular, the leaves carry no label. Nodes labeled by $\oplus$ or $\ominus$ are called *linear*, whereas those labeled by simple permutations are called *prime*. Figure \[fig:example-CIT\]$(a)$ shows the strong interval tree of our running example. Figure \[fig:example-CIT\]$(b)$ represents this tree for a simple permutation. What can be observed on this example is true in general: the trees corresponding to simple permutations consist of a single prime node labeled by the permutation itself, with pending leaves.
In a strong interval tree, it is impossible for a node labeled by $\oplus$ (resp. $\ominus$) to have a child carrying the same label. This property appears (although in disguise) in [@AlAt05], but is simply proved by contradiction: assuming that a parent and a child both carry the label $\oplus$ (resp. $\ominus$) contradicts that the child is a strong interval (indeed, it overlaps an interval resulting from the union of one of its own children with one of its siblings). With this in mind, strong interval trees are now just plane trees, where internal nodes are of arity at least $2$ and carry labels $\oplus$, $\ominus$ or $\tau$ for any simple permutation $\tau$, with the additional conditions that a node labeled by $\oplus$ (resp. $\ominus$) does not have a child carrying the same label, and that the number of children of a node labeled by a simple permutation $\tau$ is exactly the size of $\tau$.
It turns out (and the proof follows immediately from [@AlAt05]) that any such tree is the strong interval tree of a permutation. Moreover, the above construction provides a bijection between permutations and strong interval trees. Note that the bijection is completely constructive, and that it can be computed in linear time, although this is quite difficult to achieve, see [@BeChdeMRa05].
Strong interval trees as a constructible class
----------------------------------------------
From now on, we denote $\mathcal{P}$ the class of strong interval trees. As we have seen above, this class is a set of trees where some internal nodes are signed and others are enriched with a simple permutation. More precisely, the characterization of strong interval trees given above can be rephrased as show in the following theorem.
The class of permutations is in a size-preserving bijection with the combinatorial class $\mathcal{P}$ of strong interval trees. These are enriched trees defined by the following relations, where size is given by the number of leaves: $$\label{eq:initialsystem}
\begin{aligned}
\mathcal{P} &= \mathcal{Z}_{\Box} +\, \mathcal{N}_{\oplus}\cdot
\operatorname{Seq}_{\geq 2}{\mathcal{U_{\oplus}}} +\, \mathcal{N}_{\ominus}\cdot
\operatorname{Seq}_{\geq 2}{\mathcal{U_{\ominus}}}+\,
\mathcal{N}_{\bullet}\cdot\, S(\mathcal{P}),\\
\mathcal{U_{\oplus}} &= \mathcal{Z}_{\Box} +\,
\mathcal{N}_{\ominus}\cdot\operatorname{Seq}_{\geq 2}{\mathcal{U}_{\ominus}} +\,
\mathcal{N}_{\bullet}\cdot\, S(\mathcal{P}),\\
\mathcal{U_{\ominus}} &= \mathcal{Z}_{\Box} +\,
\mathcal{N}_{\oplus}\cdot\operatorname{Seq}_{\geq 2}{\mathcal{U}_{\oplus}} +\,
\mathcal{N}_{\bullet}\cdot\, S(\mathcal{P}).
\end{aligned}$$ Above, the class $\mathcal{Z}$ is an atomic class with a single element of size $1$, the $\mathcal{N}$ classes are all epsilon classes containing a single element of size $0$, marking internal nodes, and the function $S(z) = \sum_{j \geq 4} s_j z^j$ is the generating function for simple permutations. \[thm:AA05\]
Notice that $\mathcal{U}_{\oplus}$ and $\mathcal{U}_{\ominus}$ define combinatorial classes which are in size-preserving bijection. In the following, in order to deal with one class instead of two, we replace them by the equivalent class $\mathcal{U} = \mathcal{Z}_{\Box} +\, \mathcal{N}_{\circ}\cdot\operatorname{Seq}_{\geq 2}{\mathcal{U}} +\,\mathcal{N}_{\bullet}\cdot\, S(\mathcal{P})$. Doing so, we change the labels of the linear nodes having a linear parent (replacing them by $\circ$). This does not affect the enumeration of the class. Indeed, these labels are determined since a linear node and its linear parent have different labels.
It is not hard to view $\mathcal{U}$ as a family of trees not unlike those of studied in Section \[sec:General\]:
The following combinatorial equivalences are true:
[ $$\mathcal{P}\equiv \operatorname{Seq}_{\geq 1}{\mathcal{U}} \qquad \text{and} \qquad
\mathcal{U}\equiv \mathcal{Z} + \operatorname{Seq}_{\geq
2}{\mathcal{U}} + S(\operatorname{Seq}_{\geq 1}{\mathcal{U}}).$$ ]{} Consequently, $\mathcal{U}$ is in bijection with a class of $\Lambda$-trees, or in other words its generating function $U(z)$ satisfies $U(z) = z + \Lambda(U(z))$, for $\Lambda(x)= \frac{x^2}{1-x} +\sum_{j\geq 4} s_j \left(\frac{x}{1-x}\right)^j$, where $s_j$ is the number of simple permutations of size $j$.
This equivalence is derived from Equation , the fact that $\mathcal{U} \equiv \mathcal{U}_{\oplus} \equiv \mathcal{U}_{\ominus}$, and the intermediary equivalence $\mathcal{P} \equiv \mathcal{U}+
\operatorname{Seq}_{\geq 2}{\mathcal{U}}$.
There is however an important difference between $\mathcal{U}$ and the set of classes that are covered by Theorem \[thm:main\]: the function $\Lambda$ defined by $\Lambda(x)= \frac{x^2}{1-x} +\sum_{j\geq 4} s_j \left(\frac{x}{1-x}\right)^j$ is not analytic at the origin. This is due to $S(z)$, the generating function for simple permutation, not being analytic at the origin. This somewhat undesirable property follows from an enumerative study of simple permutations done by Albert *et al.* [@AlAtKl03]. We can, however, make use of their asymptotic enumeration formulas, which we recall below.
Recall that $s_n$ denotes the number of simple permutations of size $n$. The sequence $(s_n)$ has label A111111 in the On-Line Encyclopedia of Integer Sequences [@OEIS-simple]. This sequence is not P-recursive, but it does satisfy a simple functional inversion formula (see [@AlAtKl03]), and we have calculated exact values of $s_n$ for $n<800$. Albert *et al.* [@AlAtKl03] determined the following bounds: $$\label{eq:bounds}
\frac{n!}{e^2}\left(1-\frac{4}{n}\right) \leq s_n\leq \frac{n!}{e^2}\left(1-\frac{4}{n}+\frac{2}{n(n-1)}\right).$$ Here are the first few terms in the generating function for simple permutations: $$S(z)=2z^4+6z^5+46z^6+338z^7+ 2926z^8+ 28146z^9+ 298526z^{10}+ 3454434z^{11}+\dots$$
Because $S(z)$, and hence $\Lambda(x)$, are not analytic at the origin, neither $\mathcal{P}$ nor $\mathcal{U}$ are simple varieties of trees whose analysis is covered by Section \[sec:General\]. Of course, $\mathcal{P}$ being in bijection with permutations, this gives access to a very good understanding of $\mathcal{P}$ (in particular, enumeration results and random generation tools).
However, very shortly we propose a different strategy for studying $\mathcal{P}$: we describe a filtration $(\mathcal{P}^{(k)})_{k \geq 4}$ such that each $\mathcal{P}^{(k)}$ is built in a straightforward manner from a simple variety of trees $\mathcal{U}^{(k)}$. This makes these subclasses $\mathcal{P}^{(k)}$ easy to analyze, particularly given the generic analysis we have completed in Section \[sec:General\]. This is done in Section \[sec:restrictedCIT\]. Next, we view the entire class $\mathcal{P}$ as the (combinatorial) limit of the filtration. As explained in further details in Section \[sec:filtration\], one of our goals is to understand how much this strategy can yield concerning the asymptotic behavior of $\mathcal{P}$, from that of the subclasses $\mathcal{P}^{(k)}$.
Prime-Degree Restricted Strong Interval Trees {#sec:restrictedCIT}
=============================================
The filtration of the class of trees $\mathcal{P}$ that we consider consists in bounding the maximal arity of prime nodes. As we shall see, the results of Section \[sec:General\] are applicable to the subclasses of $\mathcal{P}$ where the arity of prime nodes is bounded. Our motivation for studying this restriction of strong interval trees is twofold.
First, as indicated above, this gives an example of a non-analytic class which is the limit of a sequence of families of trees which are almost simple varieties of trees: we believe this example is instructive and opens the way to adapting this strategy to study other non-analytic classes, as we discuss in Section \[sec:filtration\].
Our second motivation comes from the study of genome rearrangements, specifically in the model of perfect sorting by reversals. Indeed, as shown in [@BeBeChPa07; @BoChMiRo11], the algorithmic complexity of finding an evolutionary scenario in this model depends heavily on the maximal arity of prime nodes in the strong interval trees of the permutations that encodes the genomes (recording the order of the genes): the smaller this maximal arity, the more efficient the algorithm. Based on biological data for mammalian genomes [@ChMcMi11], it appears that prime nodes occur relatively rarely, and are of small arity. In [@BoChMiRo11], the combinatorics of strong interval trees without any prime nodes was investigated, resulting in a better understanding of the so-called *commuting scenarios*. Now allowing some prime nodes to occur, but with a bounded arity, we are going a step further in this analysis, while focusing on subclasses of strong interval trees that seem to represent well the biological data.
The filtration for permutations
-------------------------------
We define the class $\mathcal{P}^{(k)}$ as follows, where $S^{\leq k}(z)=\sum_{j=4}^ks_jz^j$: $$\mathcal{P}^{(k)} = \mathcal{Z} + 2\operatorname{Seq}_{\geq 2}{\mathcal{U}^{(k)}} + S^{\leq k}(\mathcal{P}^{(k)})\qquad
\text{and} \qquad
\mathcal{U}^{(k)} = \mathcal{Z} + \operatorname{Seq}_{\geq 2}{\mathcal{U}^{(k)}} + S^{\leq k}(\mathcal{P}^{(k)}).$$
That is, only prime nodes of arity at most $k$ are allowed. We refer to the classes denoted by $\mathcal{P}^{(k)}$, as *prime-degree restricted* strong interval trees.
The containment $\mathcal{P}^{(k)}\subset\mathcal{P}^{(k+1)}$ is straightforward, and since $\mathcal{P}^{(k)}_n = \mathcal{P}_n$ when $k\geq n$, we can derive the limit of combinatorial classes $\lim_{k\rightarrow \infty}
\mathcal{P}^{(k)}=\mathcal{P}$.
Furthermore, by the same manipulations as for the full class, we derive: $$\label{Pk-modified}
\mathcal{P}^{(k)} \equiv \operatorname{Seq}_{\geq 1}{\mathcal{U}^{(k)}}\qquad \text{and} \qquad
\mathcal{U}^{(k)} \equiv \mathcal{Z} + \operatorname{Seq}_{\geq
2}{\mathcal{U}^{(k)}} + S^{\leq k}(\operatorname{Seq}_{\geq
1}{\mathcal{U}^{(k)}}).$$
Again similarly to the case of the full class, remark that $\mathcal{U}^{(k)}$ is isomorphic to a $\Lambda_k$-tree with $\Lambda_k(x)=\frac{x^2}{1-x}+ \sum_{j=4}^k s_j \left(\frac{x}{1-x}\right)^j$. This class is certainly algebraic and is a simple variety of trees. The enumerative analysis of Section \[sec:General\] applies directly to these families of trees $\mathcal{U}^{(k)}$, then giving access to enumeration and parameter average behavior for $\mathcal{P}^{(k)}$ also, even if $\mathcal{P}^{(k)}$ is not itself a simple variety of trees. This is done in the remaining part of this section, focusing on applications to the study of genome rearrangements. Also, keeping in mind our next goal of letting $k$ go to infinity to recover the class $\mathcal{P}$, we would like to preserve $k$ as much as possible in the formulas.
Asymptotic enumeration
----------------------
The equations allow us to directly apply Theorem \[thm:main\] to determine asymptotic formulas for the coefficients of the generating functions $P^{(k)}$ of the classes $\mathcal{P}^{(k)}$.
For fixed $k$, the number of prime-degree restricted strong interval trees of size $n$, denoted $P^{(k)}_n$, grows asymptotically like
$$\label{eq:pkn-sim}
P^{(k)}_n \sim\frac{\gamma_k}{(1 - \tau_k)^2} \rho_k^{-n}n^{-3/2}\quad\text{ as }n\rightarrow \infty, \quad \text{where} \quad \gamma_k=\sqrt{\frac{\rho_k}{2\pi\Lambda_k''(\tau_k)}}.$$
Here, $\Lambda_k(x)=\frac{x^2}{1-x}+\sum_{j=4}^k s_j
(\frac{x}{1-x})^j$, $\tau_k$ satisfies $1-\Lambda_k'(\tau_k)=0$ and $\rho_k=\tau_k-\Lambda_k(\tau_k)$.
\[thm:asym\_pnk\]
First, we note that since $\sum_{j=4}^k s_j
(\frac{x}{1-x})^j$ is a polynomial in $\frac{x}{1-x}$, $\Lambda_k$ is certainly analytic at $0$. The radius of convergence of $\Lambda_k$ is easily seen to be $1$, and $\lim_{x \rightarrow 1^-} \Lambda_k'(x) = +\infty$. Hence, Theorem \[thm:main\] gives that at $\rho_k$, it holds that: $$U^{(k)}(z) = \tau_k - \beta_k \sqrt{1-\frac{z}{\rho_k}} + \mathcal{O}(1-z/\rho_k) \quad \textrm{ with } \beta_k = \sqrt{\frac{2 \rho_k}{\Lambda''(\tau_k)}}.$$
Note that the enumerative formulas of the first section also yield the asymptotic estimate $U^{(k)}_n\sim \gamma_k \rho_k^{-n}n^{-3/2}$ where $\gamma_k=\frac{\beta_k}{2\sqrt{\pi}} = \sqrt{\frac{\rho_k}{2\pi\Lambda''(\tau_k)}}$.
Next, we note that by the first relation in Equation , $P^{(k)}(z)=\frac{U^{(k)}(z)}{1-U^{(k)}(z)}$. By Theorem \[thm:main\], the value of $U^{(k)}(z)$ at its dominant singularity $\rho_k$ is $\tau_k$. Moreover, $\tau_k$ is less than the radius of convergence of $\Lambda_k$, *i.e.*, $\tau_k <1$. So the composition $P^{(k)}(z)=\frac{U^{(k)}(z)}{1-U^{(k)}(z)}$ is subcritical (see [@FlSe09 paragraph VI.9]): this implies that the dominant singularity of $P^{(k)}(z)$ is also $\rho_k$, and that at $\rho_k$, we have: $$\begin{aligned}
P^{(k)}(z) & = \frac{\tau_k - \beta_k \sqrt{1-\frac{z}{\rho_k}} + \mathcal{O}(1-z/\rho_k)}{1 - \tau_k + \beta_k \sqrt{1-\frac{z}{\rho_k}} + \mathcal{O}(1-z/\rho_k)} \\
& = \frac{\tau_k}{1 - \tau_k} \left( 1 - \frac{\beta_k}{\tau_k} \sqrt{1-\frac{z}{\rho_k}}\right) \left( 1 - \frac{\beta_k}{1- \tau_k} \sqrt{1-\frac{z}{\rho_k}}\right)+ \mathcal{O}(1-z/\rho_k)\\
& = \frac{\tau_k}{1 - \tau_k} - \frac{\beta_k}{(1 - \tau_k)^2} \sqrt{1-\frac{z}{\rho_k}} + \mathcal{O}(1-z/\rho_k).\end{aligned}$$ The classic Transfer Theorem of asymptotic enumeration then gives $$P^{(k)}_n \sim \frac{\beta_k}{(1 - \tau_k)^2} \frac{1}{2\sqrt{\pi n^3}} \rho_k^{-n}
= \frac{\gamma_k}{(1-\tau_k)^2}\rho_k^{-n} n^{-3/2} \textrm{ \quad as claimed.} \qedhere$$
Note that along the proof of Theorem \[thm:asym\_pnk\], we have seen that $\tau_k <1$, an inequality that will be useful in Section \[sec:filtration\] to bound the asymptotic estimate of Equation .
Table \[tab:rho-tau\] contains numeric approximations for $\tau_k$ and $\rho_k$ in the range $k=4\dots 13$. Using these estimates gives good asymptotic approximations and the enumerative formulas given in Equation converge quickly for fixed $k$. For example, when $k=8$, our asymptotic formula is within 2% of the correct value at $n=10$.[^6]
$k$ $\tau_k$ $\rho_k$ $k$ $\tau_k$ $\rho_k$
----- ---------------- ---------------- ------ ---------------- -----------------
$4$ $0.2258458016$ $0.1454726242$ $9$ $0.1463252500$ $0.1102193554$
$5$ $0.2043553556$ $0.1364583031$ $10$ $0.1375961304$ $0.1057725121$
$6$ $0.1841224072$ $0.1277948168$ $11$ $0.1300393555$ $0.1017629085$
$7$ $0.1689470150$ $0.1210046262$ $12$ $0.1234001218$ $0.09810173382$
$8$ $0.1565912704$ $0.1152312243$ $13$ $0.1174959122$ $0.09472586497$
: Computed approximate values for $\rho_k$ and $\tau_k$ for small values of $k$. []{data-label="tab:rho-tau"}
Parameter analysis
------------------
The average shape of general strong interval trees was described in [@BoChMiRo11]. This study is essentially based on Equation , which shows that simple permutations make up about $1/9$ of all permutations. As a consequence, general strong interval trees have a very flat shape with probability tending to $1$, and this shape governs the average case behavior of any tree parameter. However, the prime-degree restricted trees are much more rich in this regards, and parameter analysis follows from Section \[sec:General\].
We focus here on some parameters which are to some extent linked to the perfect sorting scenarios for $\sigma$, that is, to parsimonious evolutionary scenarios in the model of perfect sorting by reversals (see [@BoChMiRo11] for a detailed explanation of this connection). We will be specifically interested in the number of internal nodes (which is related to the number of reversals in a scenario), the number of prime nodes (since the complexity of computing a parsimonious scenario depends on it) and the average subtree size (which has a tight connection to the average reversal size). These parameters give important insight into the average case analysis of perfect sorting by reversals.
We have seen above that the generating function of $\mathcal{U}^{(k)}$ satisfies $U^{(k)}(z) = z + \Lambda_k(U^{(k)}(z))$ with $\Lambda_k(x)=\frac{x^2}{1-x}+ \sum_{j=4}^k s_j \left(\frac{x}{1-x}\right)^j$. Consequently, $\mathcal{U}^{(k)}$ is a simple variety of trees, and this allows to apply directly the results of Section \[sec:General\] for the average number of internal nodes or the average subtree size sum in $\mathcal{U}^{(k)}$ trees. The average number of prime nodes in $\mathcal{U}^{(k)}$ trees can also be derived using the general framework developed in Section \[sec:General\]. Then, the behaviour of these parameters in $\mathcal{P}^{(k)}$ trees is deduced from the already observed identity $$\mathcal{P}^{(k)} = \mathcal{U}^{(k)} + \operatorname{Seq}_{\geq 2}{\mathcal{U}^{(k)}}. \label{eq:fromUtoP}$$ Note that even though $\mathcal{P}^{(k)}$ is not a simple variety of trees, the behaviour of the studied parameters are of the same order as in such families of trees.
The results proved in this section are summarized in Table \[tab:summary\_Pk\].
------------------------------------------- -- ------------------------------------------------------- --
The average number of internal nodes $\frac{\tau_{k}-\rho_{k}}{\rho_{k}}\,n$
\[4mm\] The average number of prime nodes $\frac{S^{\leq k}(\tau_{k})}{\rho_{k}}\, n$
\[4mm\] The average subtree size sum $\frac{\beta_{k}^{2}}{4\rho_{k}\gamma_{k}}\, n^{3/2}$
\[2mm\]
------------------------------------------- -- ------------------------------------------------------- --
: A summary of asymptotic behavior of parameters for trees in $\mathcal{P}^{(k)}$.[]{data-label="tab:summary_Pk"}
### Number of internal nodes
Let $U^{(k)}(z,y)$ (resp. $P^{(k)}(z,y)$) be the bivariate generating function of $\mathcal{U}^{(k)}$ trees (resp. $\mathcal{P}^{(k)}$ trees), where $z$ counts the size (*i.e.*, the number of leaves) and $y$ counts the number of internal nodes. It follows from Eq. that $$P^{(k)}(z,y) = U^{(k)}(z,y) + y \cdot \frac{U^{(k)}(z,y)^2}{1-U^{(k)}(z,y)}.$$ Consequently, we have $$\begin{aligned}
\label{eq:Pderivate_internalNodes}
\frac{\partial}{\partial y} P^{(k)}(z,y) \Big|_{y=1} = \quad & \frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1}
+ \frac{2U^{(k)}(z,1)}{1-U^{(k)}(z,1)} \frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1} \nonumber \\
& + \frac{U^{(k)}(z,1)^2}{1-U^{(k)}(z,1)}
+ \frac{U^{(k)}(z,1)^2}{(1-U^{(k)}(z,1))^2}\frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1}. \end{aligned}$$ Near $\rho_k$, we know that $
U^{(k)}(z,1) = U^{(k)}(z) = \tau_k - \beta_k \sqrt{1-z/\rho_k} + \mathcal{O}(1-z/\rho_k)
$. The weaker estimate $U^{(k)}(z) = \tau_k + \mathcal{O}(\sqrt{1-z/\rho_k})$ gives $\frac{1}{1-U^{(k)}(z)} = \frac{1}{1-\tau_k} + \mathcal{O}(\sqrt{1-z/\rho_k})$, and these are enough to estimate all rational fractions in $U^{(k)}(z)$ that appear in Eq.. Moreover, the generating function $\frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1} $ counts $\mathcal{U}^{(k)}$ trees weighted by their number of internal nodes. As seen in Section \[sec:General\], it then follows from Lemma \[lm:inv\] that, near $\rho_k$, $$\frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1} = \frac{\beta_k (\tau_k - \rho_k)}{2\rho_k (1-z/\rho_k)^{1/2}} + o\left(\frac{1}{(1-z/\rho_k)^{1/2}}\right).$$ Combining these asymptotic estimates gives, near $\rho_k$, $$\frac{\partial}{\partial y} P^{(k)}(z,y) \Big|_{y=1} = \frac{\beta_k (\tau_k - \rho_k)}{2\rho_k (1-\tau_k)^{2}}\frac{1}{(1-z/\rho_k)^{1/2}} + o\left(\frac{1}{(1-z/\rho_k)^{1/2}}\right).$$ Recalling the identity $\gamma_k=\frac{\beta_k}{2\sqrt{\pi}}$ and the asymptotic behaviour of $[z^n]P^{(k)}(z,1) = P^{(k)}_n$ given in Theorem \[thm:asym\_pnk\], we deduce that the average number of internal nodes in $\mathcal{P}^{(k)}$ trees is $$\frac{[z^n]\frac{\partial}{\partial y} P^{(k)}(z,y) \Big|_{y=1}}{[z^n]P^{(k)}(z,1)} \sim_{n \to \infty}
\frac{\beta_k(\tau_k - \rho_k)}{2\rho_k (1-\tau_k)^{2} \sqrt{\pi n}} \rho_k^{-n} \cdot \frac{(1-\tau_k)^2}{\gamma_k\rho_k^{-n}} n^{3/2} = \frac{(\tau_k - \rho_k)}{\rho_k} \cdot n.$$
### Number of prime nodes
Like before, let us denote by $U^{(k)}(z,y)$ (resp. $P^{(k)}(z,y)$) the bivariate generating function of $\mathcal{U}^{(k)}$ trees (resp. $\mathcal{P}^{(k)}$ trees), counted by size (for $z$) and number of prime nodes (for $y$). We know an asymptotic estimates of $U^{(k)}(z,1) = U^{(k)}(z)$ near $\rho_k$, and we now apply the method of Section \[sec:General\] to compute one for $\frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1}$.
For any $\mathcal{U}^{(k)}$ tree $t$, let $\sigma(t) = \xi(t)$ denote the number of prime nodes in $t$ and let $\eta(t)$ be $1$ if the root of $t$ is a prime node, $0$ otherwise. With the notations of Lemma \[lm:inv\], we have $$\begin{aligned}
H(z) & = \sum_{t \in \mathcal{U}^{(k)}} \eta(t) z^{|t|} = S^{\leq k}(U^{(k)}(z)) \text{ \quad where } S^{\leq k}(u) = \sum_{j=4}^{k}s_{j}u^{j} \\
\text{and \quad} \frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1} & = \Xi(z) = H(z)\cdot \frac{\partial}{\partial z}U^{(k)}(z)
= S^{\leq k}(U^{(k)}(z)) \cdot \frac{\partial}{\partial z}U^{(k)}(z).\end{aligned}$$ The asymptotic estimate of $U^{(k)}(z)$ near $\rho_k$ is $U^{(k)}(z) = \tau_k - \beta_k \sqrt{1-z/\rho_k} + \mathcal{O}(1-z/\rho_k)$, from which we deduce $U^{(k)}(z)^j = \tau_k^j + o(1)$. Moreover, Singular Differentiation gives, near $\rho_k$, $$\frac{\partial}{\partial z}U^{(k)}(z) = \frac{\beta_k}{2\rho_k(1-z/\rho_k)^{1/2}} + o\left(\frac{1}{(1-z/\rho_k)^{1/2}}\right).$$ Consequently, we obtain that near $\rho_k$, $$\frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1} =
S^{\leq k}(\tau_k) \cdot \frac{\beta_k}{2\rho_k} \cdot \frac{1}{(1-z/\rho_k)^{1/2}} + o\left(\frac{1}{(1-z/\rho_k)^{1/2}}\right).$$
Now turning to $\mathcal{P}^{(k)}$ trees, Eq. implies that $$P^{(k)}(z,y) = U^{(k)}(z,y) + \frac{U^{(k)}(z,y)^2}{1-U^{(k)}(z,y)}.$$ Differentiation gives $$\frac{\partial}{\partial y} P^{(k)}(z,y) \Big|_{y=1} = \left( 1 + \frac{2U^{(k)}(z,1)}{1-U^{(k)}(z,1)} + \frac{U^{(k)}(z,1)^2}{(1-U^{(k)}(z,1))^2} \right)
\frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1}.$$ The asymptotic estimates obtained above give that, near $\rho_k$, $$\frac{\partial}{\partial y} P^{(k)}(z,y) \Big|_{y=1} =
S^{\leq k}(\tau_k) \cdot \frac{\beta_k}{2\rho_k (1-\tau_k)^2} \cdot \frac{1}{(1-z/\rho_k)^{1/2}} + o\left(\frac{1}{(1-z/\rho_k)^{1/2}}\right).$$ Finally, we deduce that the average number of prime nodes in $\mathcal{P}^{(k)}$ trees is $$\frac{[z^n]\frac{\partial}{\partial y} P^{(k)}(z,y) \Big|_{y=1}}{[z^n]P^{(k)}(z,1)} \sim_{n \to \infty}
\frac{S^{\leq k}(\tau_k) \cdot \beta_k }{2\rho_k (1-\tau_k)^2 \sqrt{\pi n}} \rho_k^{-n} \cdot \frac{(1-\tau_k)^2}{\gamma_k\rho_k^{-n}} n^{3/2}
= \frac{S^{\leq k}(\tau_k)}{\rho_k} \cdot n.$$
### Subtree size sum
Again, we denote by $U^{(k)}(z,y)$ (resp. $P^{(k)}(z,y)$) the bivariate generating function of $\mathcal{U}^{(k)}$ trees (resp. $\mathcal{P}^{(k)}$ trees), counted by size (for $z$) and number of prime nodes (for $y$). In this case, Eq. gives $$P^{(k)}(z,y) = U^{(k)}(z,y) + \frac{U^{(k)}(zy,y)^2}{1-U^{(k)}(zy,y)}.$$ As before, we have $U^{(k)}(z,1) = U^{(k)}(z)$. Note also that $ \frac{\partial}{\partial z} U^{(k)}(z,y) \Big|_{y=1} = \frac{\partial}{\partial z}U^{(k)}(z)$. It follows that $$\begin{aligned}
\frac{\partial}{\partial y} P^{(k)}(z,y) \Big|_{y=1} = &
\frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1} \\
+ &
\left(\frac{2U^{(k)}(z,1)}{1-U^{(k)}(z,1)} + \frac{U^{(k)}(z,1)^2}{(1-U^{(k)}(z,1))^2} \right)
\left( z \frac{\partial}{\partial z}U^{(k)}(z) + \frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1} \right)\end{aligned}$$ and we proceed like in the previous cases. Near $\rho_k$, the asymptotic estimate of $\frac{\partial}{\partial z}U^{(k)}(z)$ is $$\frac{\partial}{\partial z}U^{(k)}(z) = \frac{\beta_k}{2\rho_k(1-z/\rho_k)^{1/2}} + o\left(\frac{1}{(1-z/\rho_k)^{1/2}}\right),$$ and we have seen in Section \[sec:General\] that $$\frac{\partial}{\partial y} U^{(k)}(z,y) \Big|_{y=1} = \frac{\beta_k^2}{4\rho_k(1-z/\rho_k)} + o\left(\frac{1}{1-z/\rho_k}\right),$$ since this function counts $\mathcal{U}^{(k)}$ trees weighted by their substree size sum. Consequently, the asymptotic estimate of $\frac{\partial}{\partial y} P^{(k)}(z,y) \Big|_{y=1}$ near $\rho_k$ is $$\frac{\partial}{\partial y} P^{(k)}(z,y) \Big|_{y=1} = \frac{\beta_k^2}{4\rho_k (1-\tau_k)^2(1-z/\rho_k)} + o\left(\frac{1}{1-z/\rho_k}\right).$$ We conclude that the average value of the subtree size sum in $\mathcal{P}^{(k)}$ trees is $$\frac{[z^n]\frac{\partial}{\partial y} P^{(k)}(z,y) \Big|_{y=1}}{[z^n]P^{(k)}(z,1)} \sim_{n \to \infty}
\frac{\beta_k^2}{4\rho_k (1-\tau_k)^2} \rho_k^{-n} \cdot \frac{(1-\tau_k)^2}{\gamma_k\rho_k^{-n}} n^{3/2}
= \frac{\beta_k^2}{4\rho_k\gamma_k} \cdot n^{3/2}.$$
Random generation {#sec:GRT}
-----------------
Equation gives immediate access to random sampling of trees in $\mathcal{P}^{(k)}$. Thinking of the classes $\mathcal{P}^{(k)}$ as possible models for the biological data collected in [@ChMcMi11], it is interesting to generate random trees in $\mathcal{P}^{(k)}$, to compare them with the trees obtained from the data. In this context, our interest is the global shape of the trees, and not the particulars of the internal nodes. We have produced a Boltzmann generator which generates trees in $\mathcal{P}^{(k)}$ of size approximately $10000$ for $k$ up to $800$ without generating the simple permutation labels (prime and linear nodes are however distinguished). Figure \[fig:random-k\] illustrates a randomly generated tree from $\mathcal{P}^{(7)}$ with approximately $1000$ leaves. Remark that the structure is dominated by prime nodes of arity $7$.
![A tree from $\mathcal{P}^{(7)}$ generated uniformly at random[]{data-label="fig:random-k"}](random-k=7.pdf){width="14cm"}
The results of this random generation experiment are somehow disappointing, because the trees generated do not look like the trees obtained from the biological data. These have for instance significantly fewer prime nodes and are flatter. A careful statistical analysis would however be necessary to properly invalidate our proposed model. It would also set a solid basis for the comparison of other proposed models with the data, since this statistical analysis could then be reproduced for other subclasses of strong interval trees defined by finer restrictions.
One of our long term goals on the biological side is to identify the very specific traits which arise in permutations which encode mammalian genome comparisons, and to provide more adequate models. Chauve, McCloskey and Mishna [@ChMcMi11] have taken some preliminary steps in this direction, and a reasonable model might use $\mathcal{P}^{(k)}$ trees as components to build realistic trees. Note that permutations are very often used to model real-world problems, not only in biology: for instance also in sorting problems in algorithmic. We can therefore view this goal as part of a larger project: that of defining subclasses of permutations modeling “well” the permutations occurring in these real-world contexts (using for instance strong interval trees), and of proving statistically that these models indeed represent well the data.
Studying a combinatorial class *via* its filtration {#sec:filtration}
===================================================
We now turn to the second aspect of the study of the classes $\mathcal{P}^{(k)}$: understanding the convergence towards the full class $\mathcal{P}$, in particular at the level of the asymptotic number of trees in the class. The example of $\mathcal{P}$ we have in hand is a particularly instructive one, since the limit is known by other means, namely from the correspondence between $\mathcal{P}$ and permutations.
This example is also meant to illustrate a more general goal, discussed further in Subsection \[subsec:analytic\]: that of analyzing classes of trees $\mathcal{C}$ via a filtration $\mathcal{C}^{(k)}$ similar to $\mathcal{P}^{(k)}$, especially when $\mathcal{C}$ fail to be a simple variety of trees because the series governing the number of children available (our $\Lambda$) is not analytic.
From the asymptotic estimate of $P^{(k)}_n$ to the factorial {#subsec:asymptotic}
------------------------------------------------------------
Recall that $P^{(k)}_n$ denotes the number of strong interval trees with $n$ leaves and arity of prime nodes at most $k$. For any fixed $k$, the asymptotic behavior of $P^{(k)}_n$ is given in Equation , and is of the form $\gamma \cdot \rho^{-n} \cdot n^{-3/2}$, which is typical for trees. However, taking the limit as $k$ tends to infinity, we recover the class of all strong interval trees, or equivalently of permutations, and hence an asymptotic behavior in $n! \sim \sqrt{2\pi} e^{-n} n^{n+1/2}$. Our goal is to reconcile those estimates.
Most of this section aims at producing an upper bound for the asymptotic estimate of $P^{(k)}_n$ given in Equation . This is obtained by bounding $\rho_k$ and $\Lambda_k''(\tau_k)$. The first ingredient is a more explicit bound for $s_n$, the number of simple permutations of size $n$.
\[lm:sn bound\] For every $n\geq 4$, $s_n \leq \sqrt{2\pi}\,n^{n+1/2}\,e^{-n-2}$.
This inequality follows from Equation , stating that $s_n \leq \frac{n!}{e^2}\left(1-\frac{4}{n}+\frac{2}{n(n-1)}\right)$, and the following upper bound on $n!$: $n! \leq \sqrt{2\pi}n^{n+\frac12}e^{-n}\,e^{\frac1{12n}}$. Combining these two, our claim will follow if we prove that $(1-\frac4n+\frac2{n(n-1)})e^{\frac1{12n}} \leq 1$ for $n\geq 4$. This is equivalent to $(1-\frac4n+\frac2{n(n-1)})\leq e^{-\frac1{12n}}$. And since $1-x\leq e^{-x}$, it is sufficient to prove that $1-\frac4n+\frac2{n(n-1)} \leq 1 -\frac1{12n} $, *i.e.*, that $4-\frac2{n-1} \geq \frac1{12} $. This obviously holds for $n \geq 4$, concluding the proof.
From this estimate, the derivations of the bounds on $\tau_k$ and $\rho_k$ are relatively straightforward, but technical. Working with the value ${\tilde{\tau}_k}=\frac{\tau_k}{1-\tau_k}$ simplifies the expressions. To derive those bounds, it is essential to keep in mind this sequence of inequalities, which follow from $\tau_k <1$ and $\rho_k = \tau_k - \Lambda_k(\tau_k)$: $$0<\rho_k<\tau_k<{\tilde{\tau}_k}<1.$$
\[prop:bounds\_tautilde\] For any $\alpha<\frac{e-2}{e-1}$, there exists $k(\alpha)$ such that for $k>k(\alpha)$ $$\left(\frac{\alpha}{ks_k}\right)^{\frac{1}{k-1}}<{\tilde{\tau}_k}<\left(\frac{1}{ks_k}\right)^{\frac{1}{k-1}}.$$ Consequently, $$\frac{e}{k}\left(\frac{\alpha e^{3}}{\sqrt{2\pi}\,k^{5/2}}\right)^{\frac1{k-1}}<{\tilde{\tau}_k}<\frac{e}{k} \left(\frac{e^3}{\sqrt{2\pi}\,k^{3/2}(k-4)} \right)^{\frac{1}{k-1}} <\frac{e}{k}.$$
Computational evidence suggests that $k(\alpha)=4$, for all $\alpha$ near $\frac{e-2}{e-1}$.
The starting point is the equation $\Lambda_k'(x)=1$, satisfied by $\tau_k$. Because $\Lambda_k(x) = \frac{x^2}{1-x}+\sum_{j=4}^k s_j (\frac{x}{1-x})^j$, it is convenient to consider this equation under the change of variables $y=\frac{x}{1-x}$, *i.e.*, $x=\frac{y}{1+y}$. Notice that it implies $\frac{1}{(1-x)^2} = (1+y)^2$.
Derivation gives $\Lambda_k'(x) = \frac{1}{(1-x)^2} -1 + \frac{1}{(1-x)^2} \sum_{j=4}^k j s_j(\frac{x}{1-x})^{j-1}$, so that the equation $\Lambda_k'(x)=1$ can be rewritten as $$\label{eq:lambda}
(1+y)^2 -1 + (1+y)^2\sum_{j=4}^k j s_j y^{j-1} = 1\quad \text{ which implies } \quad
\frac{2-(1+y)^2}{(1+y)^2} = \sum_{j=4}^k j s_j y^{j-1}.$$ The next step towards proving the stated inequalities is the fact that for $0<y<1$, $1-5y < \frac{2-(1+y)^2}{(1+y)^2} < 1$, which is immediately proved by simple manipulations of inequalities. Indeed, we now observe that (by definition of ${\tilde{\tau}_k}$), Equation is satisfied at $y={\tilde{\tau}_k}$. Consequently, these inequalities yield an upper and a lower bound for $\sum_{j=4}^k j s_j {\tilde{\tau}_k}^{j-1}$: $$1-5{\tilde{\tau}_k}< \sum_{j=4}^k j s_j {\tilde{\tau}_k}^{j-1} < 1.
\label{eqn:ttk}$$
From Equation , we get $k s_k {\tilde{\tau}_k}^{k-1} \leq \sum_{j=4}^k j s_j {\tilde{\tau}_k}^{j-1} < 1$, from which the upper bound ${\tilde{\tau}_k}< \left(\frac{1}{ks_k}\right)^{\frac{1}{k-1}}$ follows. From there, deriving ${\tilde{\tau}_k}<\frac{e}{k} \left(\frac{e^3}{\sqrt{2\pi}\,k^{3/2}(k-4)} \right)^{\frac{1}{k-1}}$ is then a routine exercise using $\frac{k!}{e^2}\left(1-\frac{4}{k}\right) \leq s_k$ (see Equation ) and Stirling’s inequality $\left(\frac{k}{e}\right)^k \sqrt{2\pi k}\leq k!$. This quantity is no larger than $\frac{e}{k}$ as soon as $k\geq 5$. This concludes the part of the proof about upper bounds.
For the lower bounds, we start again from Equation above. We use the inequality $ 1-5{\tilde{\tau}_k}- \sum_{j=4}^{k-1} j s_j {\tilde{\tau}_k}^{j-1} < k s_k {\tilde{\tau}_k}^{k-1}$, and combine it with the bound $0 <{\tilde{\tau}_k}\leq e/k = o(1)$, and an upper bound on $\sum_{j=4}^{k-1} j s_j {\tilde{\tau}_k}^{j-1}$ obtained below. We split this sum as $$\sum_{j=4}^{k-1} j\,s_j\,{\tilde{\tau}_k}^{j-1} =
\underbrace{\sum_{j=4}^{k-\iota_k-1} j\,s_j\,{\tilde{\tau}_k}^{j-1}}_{A(k)} + \underbrace{\sum_{j=k-\iota_k}^{k-1} j\,s_j\,{\tilde{\tau}_k}^{j-1}.}_{B(k)}$$ where $\iota_k = \lfloor k^{\frac13}\rfloor$. Note that $\iota_k$ an non-decreasing integer function of $k$ that tends to infinity and such that $\iota_k=o(\sqrt{k})$. Lemmas \[lem:A\] and \[lem:B\] below prove that $$A(k) =
\sum_{j=4}^{k-\iota_k-1}
j s_j {\tilde{\tau}_k}^{j-1}=\mathcal{O}\left(\frac{1}{k^3}\right)
\qquad\text{ and that }\qquad
B(k) = \sum_{k-\iota_k}^{k-1}
j s_j {\tilde{\tau}_k}^{j-1}=\frac{1}{e-1} + o(1).$$ It follows that $$k s_k {\tilde{\tau}_k}^{k-1} > 1 - 5 {\tilde{\tau}_k}- \sum_{j=4}^{k-1} j s_j {\tilde{\tau}_k}^{j-1} = 1 - \frac1{e-1} +o(1) = \frac{e-2}{e-1} + o(1).$$ Hence for any $\alpha < \frac{e-2}{e-1}$, there exists $k(\alpha)$ such that for any $k\geq k(\alpha)$, we have $k\,s_k\,{\tilde{\tau}_k}^{k-1} > \alpha$, and therefore ${\tilde{\tau}_k}> \left(\frac{\alpha}{k\,s_k}\right)^{\frac1{k-1}}$. To conclude the proof, we plug in the upper bound on $s_k$ from Lemma \[lm:sn bound\].
The quantity $A(k) = \sum_{j=4}^{k-\iota_k-1} j s_j {\tilde{\tau}_k}^{j-1}$ defined in the proof of Proposition \[prop:bounds\_tautilde\] satisfies $A(k)=\mathcal{O}\left(\frac{1}{k^3}\right)$. \[lem:A\]
It is convenient to define $b_j = j\,s_j$. For any $k$, since ${\tilde{\tau}_k}< e/k$, $a_j = b_j \,e^{j-1}\,k^{1-j}$ is an upper bound on $j \,s_j \,{\tilde{\tau}_k}^{j-1}$, so that $A(k) \leq \sum_{j=4}^{k-\iota_k-1} a_j$. In what follows, we prove that $\sum_{j=4}^{k-\iota_k-1} a_j =\mathcal{O}\left(\frac{1}{k^3}\right)$ (which is enough to conclude, since $A(k)>0$).
We claim that for some integer $j_0$, the sequence $(b_j)_{j\geq j_0}$ is log-convex. Indeed, for any $j\geq 6$, $ \frac{b_j^2}{b_{j-1}b_{j+1}} = \frac{j^2}{(j-1)(j+1)}\frac{s_j^2}{s_{j+1}s_{j-1}}$, and Equation then gives $$\begin{aligned}
\frac{b_j^2}{b_{j-1}b_{j+1}}
& \leq \frac{j^2}{(j-1)(j+1)} \frac{j!^2\left(1-\frac{4}{j}+\frac2{j(j-1)}\right)^2}{(j-1)!(j+1)!\left(1-\frac{4}{j+1}\right)\left(1-\frac{4}{j-1}\right)}
= 1-\frac1j + \mathcal{O}\left(\frac1{j^2}\right). \end{aligned}$$ In particular, there exists an integer $j_0\geq 6$ such that for any $j\geq j_0$, $\frac{b_j^2}{b_{j-1}b_{j+1}}<1$ and therefore the sequence $(b_j)_{j\geq j_0}$ is log-convex.
Now, note that for any $k$, $(a_j)_{j\geq j_0}$ is also log-convex, since $\frac{a_j^2}{a_{j-1}a_{j+1}} = \frac{b_j^2}{b_{j-1}b_{j+1}}$ for all $j$. The reason for considering the sequence $(b_j)$ instead of $(a_j)$ in the first place is to ensure that $j_0$ does not depend on $k$, although the definition of $a_j = j\,s_j\,e^{j-1}\,k^{1-j}$ depends on $k$.
Log-convex sequences are decreasing down to a given minimum then increasing, and therefore are bounded from above by the values reached at the extremities. Thus for all $ j\in \{j_0,\ldots,k-\iota_k-1\}, a_j \leq \max\{a_{j_0}, a_{k-\iota_k-1}\} \leq a_{j_0} + a_{k-\iota_k-1}$. Consequently, $$\sum_{j=4}^{k-\iota_k-1} a_j = \sum_{j=4}^{j_0-1} a_j + \sum_{j=j_0}^{k-\iota_k-1} a_j \leq \sum_{j=4}^{j_0-1} a_j + k\,a_{j_0} + k\,a_{k-\iota_k-1},$$ and the result will follow if we find adequate upper bounds on each on these three terms, which we now do.
For any $j\in\{4,\ldots j_0-1\}$, we have $ a_j = j\,s_j\,e^{j-1}\,k^{1-j} \leq j j! \,e^{j-1}\,k^{1-j}\leq j_0 j_0!\,e^{j_0-1}\,k^{-3}
$ so that $\sum_{j=4}^{j_0-1} a_j \leq j_0^2 j_0!\,e^{j_0-1}\,k^{-3} = \mathcal{O}(k^{-3})$. For the term $k\,a_{j_0}$, we have $k\,a_{j_0} = j_0\,s_{j_0}\,e^{j_0-1}\,k^{2-j_0} =\mathcal{O}(k^{-3})$ since $j_0 \geq 6$. Using Lemma \[lm:sn bound\] and the fact that for all $x\in (0,1)$, $\log(1-x)<-x$, we obtain the bound for the last term. More precisely, we have: $$\begin{aligned}
k\,a_{k-\iota_k-1} & \leq k^2\cdot s_{k-\iota_k-1}\cdot e^{k-\iota_k-2}\cdot k^{2-k+\iota_k} \\
& \leq k^2\cdot \sqrt{2\pi}\cdot (k-\iota_k-1)^{k-\iota_k-1/2}\cdot e^{-k+\iota_k-1}\cdot e^{k-\iota_k-2}\cdot k^{2-k+\iota_k} \\
& \leq \frac{\sqrt{2\pi}}{e^3}k^{7/2}\cdot\left(1-\frac{\iota_k+1}{k}\right)^{k-\iota_k-1/2} \\
& \leq \frac{\sqrt{2\pi}}{e^3}k^{7/2}\cdot\exp\left((k-\iota_k-1/2)\log\left(1-\frac{\iota_k+1}{k}\right)\right)\\
& \leq \frac{\sqrt{2\pi}}{e^3}k^{7/2}\cdot\exp\left(-\frac{(k-\iota_k-1/2)(\iota_k+1)}k\right).\end{aligned}$$ The quantity in the exponential is asymptotically equivalent to $-\iota_k = -\lfloor k^{\frac13}\rfloor$. Hence $k\,a_{k-\iota_k-1}$ decreases super-polynomially fast toward $0$, and is therefore a $\mathcal{O}(k^{-3})$ too.
The quantity $B(k) = \sum_{k-\iota_k}^{k-1} j s_j {\tilde{\tau}_k}^{j-1}$ defined in the proof of Proposition \[prop:bounds\_tautilde\] satisfies $B(k)=\frac{1}{e-1} +o(1)$. \[lem:B\]
With the change of variable $i=k-j$, we can write $B(k) = \sum_{i=1}^{\iota_k} (k-i)\,s_{k-i}\,{\tilde{\tau}_k}^{k-i-1}$. By Lemma \[lm:sn bound\] and the upper bound on ${\tilde{\tau}_k}$ proved in Proposition \[prop:bounds\_tautilde\], we have $$\begin{aligned}
(k-i)\,s_{k-i} & \leq \frac{\sqrt{2\pi}(k-i)^{k-i+3/2}}{e^{k-i+2}},\\
{\tilde{\tau}_k}^{k-i-1} & \leq \frac{e^{k-i-1}}{k^{k-i-1}}\left(
\frac{e^{3}}{\sqrt{2\pi}\,k^{3/2}\left(k-4\right)}\right)\cdot\left(
\frac{e^{3}}{\sqrt{2\pi}\,k^{3/2}\left(k-4\right)}\right)^{\frac{-i}{k-1}}.\end{aligned}$$ Therefore $
(k-i)\,s_{k-i}\,{\tilde{\tau}_k}^{k-i-1} \leq \left(1-\frac{i}{k}\right)^{k-i+3/2}\left(
e^{-3}\sqrt{2\pi}\,k^{3/2}\left(k-4\right)\right)^{\frac{i}{k-1}}\cdot\frac{1}{1-\frac4{k}}
$. Since $i \leq \iota_k$ and $e^{-3}\sqrt{2\pi}\,k^{3/2}\left(k-4\right) \geq 1$ as soon as $k\geq 5$, we obtain that for $k\geq 5$, $$(k-i)\,s_{k-i}\,{\tilde{\tau}_k}^{k-i-1} \leq \left(1-\frac{i}{k}\right)^{k-i+3/2}\left(
e^{-3}\sqrt{2\pi}\,k^{3/2}\left(k-4\right)\right)^{\frac{\iota_k}{k-1}}\cdot\underbrace{\frac{1}{1-\frac4{k}}}_{1+\mathcal{O}(\frac1k)}.$$ Using again that $\log(1-x)<-x$ for $x\in (0,1)$, we have $$\left(1-i/k\right)^{k-i+3/2} = e^{(k-i+3/2)\log\left(1-\frac{i}{k}\right)} \leq e^{ - \frac{(k-i+3/2)i}{k}} = e^{ - i +\frac{i^2}k -\frac{3i}{2k}}.$$ Recalling that $i \leq \iota_k = \lfloor k^{\frac13}\rfloor $, this gives $\left(1-i/k\right)^{k-i+3/2} \leq e^{-i} \exp(k^{-1/3})= e^{-i}\left(1+o(1)\right)$. Proceeding similarly, the middle term satisfies $$\left(e^{-3}\sqrt{2\pi}\,k^{3/2}\left(k-4\right)\right)^{\frac{\iota_k}{k-1}} = 1 + o(1).$$ Therefore, we obtain $(k-i)\,s_{k-i}\,{\tilde{\tau}_k}^{k-i-1} \leq e^{-i}\left(1+o(1)\right)$, where the function hidden in the $o(1)$ notation depends on $k$ but not on $i$. Consequently, summing over $i$, we obtain $$B(k) \leq \left(\sum_{i=1}^{\lambda_k}e^{-i}\right) \left(1+o(1)\right) \leq \left(\sum_{i=1}^{\infty}e^{-i}\right) \left(1+o(1)\right)
= \frac{1+o(1)}{e-1} \textrm{ as claimed.} \qedhere$$
\[thm:bounds\_rho\] There exists a constant $\beta$ such that for any $\alpha < \frac{e-2}{e-1}$, there exist $k(\alpha,\beta)$ such that for any $k \geq k(\alpha,\beta)$, $$\frac{e}{k} \left(\frac{\alpha e^3}{\sqrt{2\pi}\,k^{5/2}} \right)^{\frac{1}{k-1}}
\left(1-\frac{\beta}{k}\right)
< \rho_k <
\frac{e}{k} \left(\frac{e^3}{\sqrt{2\pi}\,k^{3/2}(k-4)} \right)^{\frac{1}{k-1}}.$$ Consequently, $\rho_k=\frac{e}{k}\left(1-\frac{5}{2}\, \frac{\log k}{k} + \mathcal{O}(\frac1k) \right)$.
The upper bound is immediate from the bound $\rho_k<{\tilde{\tau}_k}$ and Proposition \[prop:bounds\_tautilde\].
For the lower bound, we start from $\rho_k = \tau_k-\Lambda_k(\tau_k)$. The definitions of ${\tilde{\tau}_k}$ and $\Lambda_k$ give $\rho_k={\tilde{\tau}_k}\left(1-\frac{2{\tilde{\tau}_k}}{1+{\tilde{\tau}_k}}-\sum_{j=4}^k s_j{\tilde{\tau}_k}^{j-1}\right)$. Our main step is to deduce from this equality that $\rho_k \geq {\tilde{\tau}_k}(1-\beta/k)$ for some constant $\beta$. The lower bound will then follow from Proposition \[prop:bounds\_tautilde\].
As in the proof of Proposition \[prop:bounds\_tautilde\], we leverage upper bounds on ${\tilde{\tau}_k}$ to build a lower bound on $1-\frac{2{\tilde{\tau}_k}}{1+{\tilde{\tau}_k}}-\sum_{j=4}^k s_j{\tilde{\tau}_k}^{j-1}$. In this case, we use $\frac{2{\tilde{\tau}_k}}{1+{\tilde{\tau}_k}}\leq2{\tilde{\tau}_k}\leq
2\frac{e}{k}$, and we will bound the summation by splitting the sum at the same place: $$\sum\limits_{j=4}^{k} s_j\,{\tilde{\tau}_k}^{j-1} =
\sum\limits_{j=4}^{k-\iota_k-1} s_j\,{\tilde{\tau}_k}^{j-1} +
\sum\limits_{j=k-\iota_k}^{k-1} s_j\,{\tilde{\tau}_k}^{j-1} +
s_k\,{\tilde{\tau}_k}^{k-1}.$$ Even though it is not the same summation, we can re-use the bounds from Lemmas \[lem:A\] and \[lem:B\]. Indeed, $$\begin{aligned}
\sum\limits_{j=4}^{k-\iota_k-1} s_j\,{\tilde{\tau}_k}^{j-1} & \leq \sum\limits_{j=4}^{k-\iota_k-1} j \, s_j\,{\tilde{\tau}_k}^{j-1}
= A(k) = \mathcal{O}\left(\frac{1}{k^3}\right) \\
\textrm{and } \sum\limits_{j=k-\iota_k}^{k-1} s_j\,{\tilde{\tau}_k}^{j-1} & \leq \sum\limits_{j=k-\iota_k}^{k-1} \frac{j}{k - \iota_k} s_j\,{\tilde{\tau}_k}^{j-1}
= \frac{B(k)}{k - \iota_k} = \mathcal{O}\left(\frac{1}{k}\right).\end{aligned}$$ Finally, Proposition \[prop:bounds\_tautilde\] ensures that $k\,s_k\,{\tilde{\tau}_k}^{k-1} \leq 1$, and we obtain $\frac{2{\tilde{\tau}_k}}{1+{\tilde{\tau}_k}}+\sum_{j=4}^k s_j{\tilde{\tau}_k}^{j-1} = \mathcal{O}\left(\frac1k\right)$. It follows that for some $\beta$, there exists $k(\beta)$ such that when $k \geq k(\beta)$ we have: $$\frac{2{\tilde{\tau}_k}}{1+{\tilde{\tau}_k}}+\sum_{j=4}^k s_j{\tilde{\tau}_k}^{j-1} \leq \frac{\beta}{k}
\textrm{ \quad and hence \quad } \rho_k \geq {\tilde{\tau}_k}\left( 1-\frac{\beta}{k} \right),$$ which together with Proposition \[prop:bounds\_tautilde\] proves the lower bound.
To obtain the claimed asymptotic estimate of $\rho_k$, it is enough to observe that both the upper and the lower bound behave like $\frac{e}{k}\left(1-\frac{5}{2}\, \frac{\log k}{k} + \mathcal{O}(\frac1k) \right)$. More precisely, $$\begin{aligned}
\left(\frac{\alpha e^3}{\sqrt{2\pi}\,k^{5/2}} \right)^{\frac{1}{k-1}}
& = \exp\left( \frac{\log(k^{-5/2}) + cst}{k-1} \right)
= \exp\left( -\frac{5}{2}\frac{\log k}{k-1}+ \frac{cst}{k-1} \right) \\
& = 1 - \frac{5}{2}\frac{\log k}{k} +\mathcal{O}\left( \frac{1}{k}\right) \\
\textrm{so that } \left(\frac{\alpha e^3}{\sqrt{2\pi}\,k^{5/2}} \right)^{\frac{1}{k-1}} \left(1-\frac{\beta}{k}\right) & =
1 - \frac{5}{2}\frac{\log k}{k} +\mathcal{O}\left( \frac{1}{k}\right) \\
\textrm{and }
\left(\frac{e^3}{\sqrt{2\pi}\,k^{3/2}(k-4)} \right)^{\frac{1}{k-1}} & =
\exp\left( \frac{\log(k^{-5/2}) + \log (\frac{1}{1-4/k}) + cst}{k-1} \right) \\
& = 1 - \frac{5}{2}\frac{\log k}{k} +\mathcal{O}\left( \frac{1}{k}\right). \qedhere \end{aligned}$$
It was known in [@PP2011] that $\rho_k = \frac{e}{k} (1+ o(1))$, but we are able to produce a more precise estimate. We require this precision when we consider the limit as $k\rightarrow \infty.$
Looking at the asymptotic estimate of $P^{(k)}_n$ provided by Theorem \[thm:asym\_pnk\], and aiming at obtaining an upper bound on this estimates, the only missing piece is a lower bound on $\Lambda_k''(\tau_k)$. The definition of $\Lambda_k$ (see Theorem \[thm:asym\_pnk\]) gives $$\begin{aligned}
\Lambda_k''(x) &= \frac{2}{(1-x)^3}\left( 1 + \sum_{j=4}^{k} j s_j \left(\frac{x}{1-x} \right)^{j-1}+
\frac{1}{2(1-x)}\sum_{j=4}^{k} j (j-1) s_j \left(\frac{x}{1-x} \right)^{j-2} \right) \\
& \geq \frac{2}{(1-x)^3} \textrm{ for all } x \in (0,1), \end{aligned}$$ and therefore the series expansion of $(1-x)^{-3}$ ensures that $\Lambda_k''(\tau_k) \geq 2+6{\tilde{\tau}_k}$. We could expand this expression further, and use lower bounds on ${\tilde{\tau}_k}$, but it turns out that for our purposes, the bound $\Lambda_k''(\tau_k) \geq 2$ is sufficient.
Finally, we have all of the elements to find an upper bound for the asymptotic estimate of $P^{(k)}_n$.
For any fixed $k$, as $n$ tends to infinity, $P^{(k)}_n$ behaves like $\frac{\gamma_k}{(1 - \tau_k)^2} \rho_k^{-n}n^{-3/2}$, where where $\gamma_k = \sqrt{\frac{\rho_k}{2\pi\Lambda_k''(\tau_k)}}$. And when $k$ grows to infinity, this estimates is no larger than
[ $$\label{eq:estimate_upper_bound}
\frac{1}{(1-\frac{e}{k})^2}\sqrt{\frac{e}{4k\pi}}\left(\frac{k}{e}\right)^{n}
\left(1+\frac{5}{2}\frac{\log k}{k} + \mathcal{O}\left(\frac{1}{k} \right)\right)^n \, n^{-3/2}$$ ]{}
Theorem \[thm:main\] ensures that $P^{(k)}_n \sim \frac{\gamma_k}{(1 - \tau_k)^2} \rho_k^{-n}n^{-3/2}$ as $n \rightarrow \infty$. From Proposition \[prop:bounds\_tautilde\] and Theorem \[thm:bounds\_rho\], assuming now that $k$ is large enough, we have:
- $\tau_k \leq {\tilde{\tau}_k}\leq \frac{e}{k}$, hence $\frac{1}{(1 - \tau_k)^2} \leq \frac{1}{(1-\frac{e}{k})^2}$;
- $\rho_k \leq \frac{e}{k}$ and $\Lambda_k''(\tau_k) \geq 2$, hence $\gamma_k \leq \sqrt{\frac{e}{4k\pi}}$;
- and $\rho_k^{-n} = \left(\frac{e}{k}\right)^{-n} \left(1-\frac{5}{2}\frac{\log k}{k} + \mathcal{O}\left(\frac{1}{k} \right)\right)^{-n}
= \left(\frac{k}{e}\right)^{n} \left(1+\frac{5}{2}\frac{\log k}{k} + \mathcal{O}\left(\frac{1}{k} \right)\right)^n$.
#### In the limit, Stirling’s approximation.
Our analysis of $\mathcal{P}$ has brought together two classic asymptotic facts. The asymptotic growth of each $\mathcal{P}^{(k)}$ is of the form $P^{(k)}_n \sim \gamma \rho^{-n} n^{-3/2}$ for some real valued $\rho$ and $\gamma$. (Note that although $\mathcal{P}^{(k)}$ is not a simple variety of trees, the asymptotic behaviour of $P^{(k)}_n$ is of the same form as for such families.) But for the full class $\mathcal{P}$, the classical Stirling’s approximation of $n!$ gives $P_n \sim \left(\frac{n}{e}\right)^n\sqrt{2\pi n}$. Subtle analysis is required to reconcile these two estimates, and our upper bound on the asymptotic estimate of $P^{(k)}_n$ allows us to take a first step in this direction.
For any $n$, the strong interval tree of a permutation of size $n$ contains no prime node of arity larger than $n$. Thus, if $k\geq n$, $\mathcal{P}^{(k)}_n$ contains all trees corresponding to permutations of size $n$, and hence $P^{(k)}_n = n!$ for $k\geq n$. Now, forget for a moment that the estimates for $\mathcal{P}^{(k)}_n$ as $n \rightarrow \infty$ is valid only for fixed $k$, and consider the expression in with $k=n$. It simplifies as follows: $$\frac{1}{(1-\frac{e}{n})^2}\sqrt{\frac{e}{4n\pi}}\left(\frac{n}{e}\right)^{n}
\left(1+\frac{5}{2}\frac{\log n}{n} + \mathcal{O}\left(\frac{1}{n} \right)\right)^n \, n^{-3/2}
= \sqrt{\frac{e}{4\pi}} \left(\frac{n}{e} \right)^n \sqrt{n} \cdot (1+o(1)).$$ This is a constant times Stirling’s formula (the constant being $\sqrt{\frac{e}{8 \pi^2}}$). And this is encouraging: indeed, even though setting $k=n$ was not justified, these purely formal manipulations do reconcile the two asymptotics, up to a constant factor.
However, if we follow the same route for $P^{(2n)}_n$, which is also $n!$, the quantity in gains an unwanted factor of $2^n$. This does not contradict the correctness of our asymptotic form for fixed $k$. It rather emphasizes that it is an open problem to develop asymptotic formulas when $k$ is a function of $n$, and they go to infinity together. This will require a very delicate treatment of the bounds, a much stronger understanding of how to take the limit as $k\rightarrow\infty$, and a return to the analytic inversion and transfer theorems to study how the error terms depend on $k$.
A simpler case: when $\Lambda$ is analytic {#subsec:analytic}
------------------------------------------
The example developed above is meant to illustrate a strategy to enumerate classes $\mathcal{C}$ of trees whose generating functions satisfy $C(z) = z + \Lambda(C(z))$, in particular in the case where $\Lambda$ is not analytic.
The method we proposed is to consider a sequence of analytic $\Lambda_k$ (obtained for instance by truncations at order $k$) such that as formal power series, $\lim_{k\rightarrow\infty}\Lambda_k = \Lambda$, and to study first the sets $\mathcal{C}^{(k)}$ of $\Lambda_k$-trees. The next step, which is for the moment not accessible to us, is to obtain results about $\mathcal{C}$ from what is known on the classes $\mathcal{C}^{(k)}$. We view the following questions as particularly interesting in this regard: Can we describe conditions so that the limit of the asymptotics of the subclasses tends to the asymptotics of the whole class? To which extent are the parameter formulas valid under the limit? We hope that our work will help developing techniques to obtain information on $\mathcal{C}$ by letting $k$ go to infinity.
The difficulty here lies in $\Lambda$ being not analytic. Notice however that the same filtration by truncations at order $k$ may also be defined when $\Lambda$ is analytic. Next, we show that in this case, we obtain the correct asymptotic formula when taking the limit as $k$ tends to infinity, *i.e.* that limits in $n$ and $k$ commute.
Consider a series $\Lambda(x) = \sum_{i\geq 2} \lambda_i x^i$ with non-negative coefficients. And for all $k \geq 2$, define $\Lambda_k(x) = \sum_{i = 2}^k \lambda_i x^i$. We denote respectively by $\mathcal{C}$ and $\mathcal{C}^{(k)}$ the classes of trees whose generating functions satisfy $$C(z) = z + \Lambda(C(z)) \textrm{\qquad and \qquad} C^{(k)}(z) = z + \Lambda_k(C^{(k)}(z)).$$ We make the following assumptions: $\Lambda$ is analytic at $0$, and denoting $R$ the radius of convergence of $\Lambda$, there is a unique solution $\tau \in (0,R)$ to the equation $\Lambda'(x) =1$. Then, it follows from Theorem \[thm:main\] that $C(z)$ is analytic at $0$, has a unique dominant singularity $\rho = \tau - \Lambda(\tau)$, and, assuming further that $C(z)$ is aperiodic, that the coefficients of this series behave asymptotically like $[z^n]C(z) \sim \sqrt{\frac{\rho}{2\pi\Lambda''(\tau)}}\cdot \frac{\rho^{-n}}{n^{3/2}}$.
For all $k\geq 2$, there exists a unique $\tau_k \in (0,+\infty)$ such that $\Lambda_k'(\tau_k) =1$. Moreover, the sequence $(\tau_k)_{k\geq 2}$ is decreasing and converges to $\tau$ as $k$ goes to infinity.
Fix some $k\geq 2$. From the definition of $\Lambda_k(x) = \sum_{i= 2}^k \lambda_i x^i$, it follows that $\Lambda'_k(x)$ is a polynomial with non-negative coefficients, increasing from $0$ to $+\infty$ when $x$ varies from $0$ to $+\infty$. Moreover, its derivative $\Lambda''_k$ being nowhere zero on $(0,+\infty)$, $\Lambda'_k$ is strictly increasing. Therefore, there is a unique positive solution to $\Lambda_k'(x) =1$, that we denote $\tau_k$.
The fact that the sequence $(\tau_k)_{k\geq 2}$ is decreasing is immediate from $$1 = \Lambda_k'(\tau_k) = \sum_{i = 2}^k i \lambda_i \tau_k^{i-1} \leq \sum_{i= 2}^{k+1} i \lambda_i \tau_k^{i-1} = \Lambda_{k+1}'(\tau_k)$$ and the fact that $\Lambda_{k+1}'$ is increasing.
The sequence $(\tau_k)_{k\geq 2}$ being decreasing and non-negative, it admits a limit, that we denote $\ell$. We want to prove that $\ell = \tau$, *i.e.*, that $\Lambda'(\ell) =1$. First, for all $k$, $\ell \leq \tau_k$, so that $\Lambda'_k(\ell) \leq 1$. Moreover, the sequence $(\Lambda'_k(\ell))$ is increasing (we keep adding non-negative terms), and thus converges towards a limit that is no larger than $1$. This limit being $\Lambda'(\ell)$, we obtain that $\Lambda'(\ell) \leq 1$. Second, since $(\tau_k)_{k\geq 2}$ is decreasing towards $\tau < R$, we get that for $k$ large enough, $\Lambda'$ is defined in $\tau_k$. For any such large $k$, we have $$1 = \Lambda_k'(\tau_k) = \sum_{i= 2}^k i \lambda_i \tau_k^{i-1} \leq \sum_{i\geq 2} i \lambda_i \tau_k^{i-1} = \Lambda'(\tau_k),$$ and taking the limit in $k$ gives $\Lambda'(\ell) \geq 1$.
For all $k\geq 2$, define $\rho_k = \tau_k - \Lambda_k(\tau_k)$. The sequence $(\rho_k)_{k\geq 2}$ converges to $\rho$ as $k$ goes to infinity.
It is enough to prove that $(\Lambda_k(\tau_k))$ converges to $\Lambda(\tau)$. Like in the previous proof, since $(\tau_k)_{k\geq 2}$ is decreasing towards $\tau < R$, we get that for $k$ large enough, $\Lambda$ is defined in $\tau_k$. For such large $k$, we have $$\sum_{j=2}^k \lambda_k \tau^k \leq \sum_{j=2}^k \lambda_k \tau_k^k \leq \sum_{j\geq 2} \lambda_k \tau_k^k
\textrm{\quad that is to say } \Lambda_k(\tau) \leq \Lambda_k(\tau_k) \leq \Lambda(\tau_k).$$ Because $(\Lambda_k(\tau))$ and $(\Lambda(\tau_k))$ share the same limit $\Lambda(\tau)$ as $k$ goes to infinity, we obtain that $\lim\limits_{k \rightarrow +\infty} \Lambda_k(\tau_k) = \Lambda(\tau)$.
From Theorem \[thm:main\], we obtain $[z^n] C^{(k)}(z) \sim \sqrt{\frac{\rho_k}{2\pi\Lambda_k''(\tau_k)}}\cdot \frac{\rho_k^{-n}}{n^{3/2}}$, and the two lemmas above ensure that taking the limit in $k$ in this estimates give $\sqrt{\frac{\rho}{2\pi\Lambda''(\tau)}}\cdot \frac{\rho^{-n}}{n^{3/2}}$. In addition, $\lim\limits_{k \rightarrow +\infty} C^{(k)}(z) = C(z)$ from which we get $[z^n] \lim\limits_{k \rightarrow +\infty} C^{(k)}(z) \sim \sqrt{\frac{\rho}{2\pi\Lambda''(\tau)}}\cdot \frac{\rho^{-n}}{n^{3/2}}$. In other words, taking the limit in $k$ in the estimate of the number of trees of size $n$ in $\mathcal{C}^{(k)}$ gives the estimates of the number of trees of size $n$ in $\mathcal{C}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by ANR project <span style="font-variant:small-caps;">Magnum</span> (2010-BLAN-0204), NSERC Discovery grant 31-611453, and funding by Université Paris-Est.
We are indebted to Cedric Chauve for his guidance and access to the mammalian genome data set, prepared by Bradley Jones and Rosemary McCloskey. Furthermore, Ms. McCloskey wrote the code for the Boltzmann generator, amongst other extremely useful things. We thank Carine Pivoteau for demonstrating her interest in our project at several stages. MM is particularly grateful to both LIGM and LaBRI for hosting her during the course of this work.
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The On-Line Encyclopedia of Integer Sequences. Published electronically at http://oeis.org.
[^1]: LaBRI/CNRS, Université Bordeaux, and Institut für Mathematik, Universität Zürich.
[^2]: Dept. Mathematics, Simon Fraser University, Burnaby, Canada.
[^3]: Laboratoire d’Informatique Gaspard Monge (LIGM), Université Paris-Est, Marne-la-Vallée.
[^4]: A preliminary version of this work appeared in the extended abstract [@ShortVersion].
[^5]: This is the only other possibility since there can be no unary nodes in a proper specification.
[^6]: Maple code to compute Table \[tab:rho-tau\] is available at https://github.com/marnijulie/strong-interval-trees-maple
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Full Bayesian Significance Test (FBST) for precise hypotheses was presented by Pereira and Stern (1999) as a Bayesian alternative instead of the traditional significance test using *p-value*. The FBST is based on the evidence in favor of the null hypothesis (**H**). An important practical issue for the implementation of the FBST is the determination of how large the evidence must be in order to decide for its rejection. In the Classical significance tests, it is known that *p-value* decreases as sample size increases, so by setting a single significance level, it usually leads **H** rejection. In the FBST procedure, the evidence in favor of **H** exhibits the same behavior as the *p-value* when the sample size increases. This suggests that the cut-off point to define the rejection of **H** in the FBST should be a sample size function. In this work, the scenario of Linear Regression Models with known variance under the Bayesian approach is considered, and a method to find a cut-off value for the evidence in the FBST is presented by minimizing the linear combination of the averaged type I and type II error probabilities for a given sample size and also for a given dimension of the parametric space.'
author:
- Alejandra Estefanía Patiño Hoyos
- Victor Fossaluza
title: Adaptative significance levels in linear regression models with known variance
---
Introduction
============
The main goal of our work is to determine how small the Bayesian evidence in the FBST should be in order to reject the null hypothesis. Therefore, considering the concepts in Pereira (1985), in Oliveira (2014) and the recent work of Pereira [*et al.*]{} (2017) and Gannon [*et al.*]{} (2019) related to the adaptive significance levels (levels that are function of sample size which are obtained from the generalized form of the Neyman-Pearson Lemma ), we propose to establish a cut-off value $k^*$ for the $ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)$ as a function of the sample size $n$ and the dimension of the parametric space $d$, i.e., $ k^*= k^*(n,d)$ with $k^*\in [0,1]$, such that $k^*$ minimizes the linear combination of the averaged type I and type II error probabilities, $a\alpha_{\varphi}+b\beta_{\varphi}$. We will focus on model selection for Linear Regression Models with known variance.
Methodology
===========
Consider de normal linear regression model $$\label{model}
{\boldsymbol{\mathrm{y}}}={\boldsymbol{\mathrm{X}}}\boldsymbol{\theta}+\boldsymbol{\varepsilon}, \quad \boldsymbol{\varepsilon} \sim N_n(\boldsymbol{0},\sigma^2\mathbb{I}_n),$$ where ${\boldsymbol{\mathrm{y}}}=(y_1,\dots, y_n)^{\top}$ is an $n \times 1$ vector of $y_i$ observations, ${\boldsymbol{\mathrm{X}}}=(\boldsymbol{x}_1,\dots, \boldsymbol{x}_n)^{\top}$ is an $n \times p$ matrix of known coefficients with $\boldsymbol{x}_i=(1,x_{i1},\dots, x_{ip-1})^{\top}$, $\boldsymbol{\theta}=({\boldsymbol{\theta}}_{1}^{\top},{\boldsymbol{\theta}}_{2}^{\top})^{\top}$ is a $p \times 1$ vector of parameters, and $\boldsymbol{\varepsilon}=(\varepsilon_1,\dots, \varepsilon_n)^{\top}$ an $n\times1$ vector of random errors. Suppose that the residual error variance $\sigma^2$ is known, then $f({\boldsymbol{\mathrm{y}}}\vert \boldsymbol{\theta}) \sim N_n({\boldsymbol{\mathrm{X}}}\boldsymbol{\theta},\sigma^2\mathbb{I}_n)$. The natural conjugate prior family is the family of normal distributions. Suppose therefore that ${\boldsymbol{\theta}}$ has the $N_p({\boldsymbol{\mathrm{m}_0}},{\boldsymbol{\mathrm{W}_0}})$ prior distribution $$\label{likelkv2}
g(\boldsymbol{\theta}) \propto \exp\left\lbrace -\frac{(\boldsymbol{\theta}-{\boldsymbol{\mathrm{m}_0}})^{\top}{\boldsymbol{\mathrm{W}_0}}^{-1} (\boldsymbol{\theta}-{\boldsymbol{\mathrm{m}_0}})}{2} \right\rbrace.$$\
Then, the posterior distribution of ${\boldsymbol{\theta}}$ is ${\boldsymbol{\theta}}\vert {\boldsymbol{\mathrm{y}}}\sim N_p({\boldsymbol{\mathrm{m}^*}},{\boldsymbol{\mathrm{W}^*}})$, with $$\begin{aligned}
{\boldsymbol{\mathrm{m}^*}}&= ({\boldsymbol{\mathrm{W}_0}}^{-1}+\sigma^{-2}{\boldsymbol{\mathrm{X}}}^{\top}{\boldsymbol{\mathrm{X}}})^{-1}({\boldsymbol{\mathrm{W}_0}}^{-1}{\boldsymbol{\mathrm{m}_0}}+\sigma^{-2}{\boldsymbol{\mathrm{X}}}^{\top}{\boldsymbol{\mathrm{y}}}),\\
{\boldsymbol{\mathrm{W}^*}}&=({\boldsymbol{\mathrm{W}_0}}^{-1}+\sigma^{-2}{\boldsymbol{\mathrm{X}}}^{\top}{\boldsymbol{\mathrm{X}}})^{-1}\end{aligned}$$ If ${\boldsymbol{\theta}}_1$ has $s$ elements and ${\boldsymbol{\theta}}_2$ has $r$ elements write $${\boldsymbol{\mathrm{m}_0}}= \left(\begin{array}{c}
{\boldsymbol{\mathrm{m}_0}}_1\\
{\boldsymbol{\mathrm{m}_0}}_2\\
\end{array}\right), ~~~~~~
{\boldsymbol{\mathrm{W}_0}}= \left(\begin{array}
{cc}
{\boldsymbol{\mathrm{W}_0}}_{11} & {\boldsymbol{\mathrm{W}_0}}_{12} \\
{\boldsymbol{\mathrm{W}_0}}_{21}& {\boldsymbol{\mathrm{W}_0}}_{22}
\end{array}\right),$$
where ${\boldsymbol{\mathrm{m}_0}}_1$ is $s\times 1$, ${\boldsymbol{\mathrm{W}_0}}_{11}$ is $s\times s$, ${\boldsymbol{\mathrm{m}_0}}_2$ is $r\times 1$, ${\boldsymbol{\mathrm{W}_0}}_{22}$ is $r\times r$. So, $$\begin{aligned}
{\boldsymbol{\theta}}_1 \sim N_s\left( {\boldsymbol{\mathrm{m}_0}}_1,{\boldsymbol{\mathrm{W}_0}}_{11}\right),\qquad
{\boldsymbol{\theta}}_2 \sim N_r\left( {\boldsymbol{\mathrm{m}_0}}_2,{\boldsymbol{\mathrm{W}_0}}_{22}\right),\end{aligned}$$
Using general results on multivariate normal distributions, $$\begin{aligned}
{\boldsymbol{\theta}}_1 \vert {\boldsymbol{\theta}}_2 &\sim & N_s({\boldsymbol{\mathrm{m}_0}}_{1.2}({\boldsymbol{\theta}}_2),{\boldsymbol{\mathrm{W}_0}}_{11.2}),\end{aligned}$$
where ${\boldsymbol{\mathrm{m}_0}}_{1.2}({\boldsymbol{\theta}}_2)={\boldsymbol{\mathrm{m}_0}}_1+{\boldsymbol{\mathrm{W}_0}}_{12}{\boldsymbol{\mathrm{W}_0}}_{22}^{-1}({\boldsymbol{\theta}}_2-{\boldsymbol{\mathrm{m}_0}}_2)$ and ${\boldsymbol{\mathrm{W}_0}}_{11.2}={\boldsymbol{\mathrm{W}_0}}_{11}-{\boldsymbol{\mathrm{W}_0}}_{12}{\boldsymbol{\mathrm{W}_0}}_{22}^{-1}{\boldsymbol{\mathrm{W}_0}}_{21}$. A corresponding distribution result if we change ${\boldsymbol{\mathrm{m}_0}}$ to ${\boldsymbol{\mathrm{m}^*}}$ and ${\boldsymbol{\mathrm{W}_0}}$ to ${\boldsymbol{\mathrm{W}^*}}$.
Let $f({\boldsymbol{\theta}}\vert {\boldsymbol{\mathrm{y}}})$ be the posterior density of ${\boldsymbol{\theta}}$ given the observed sample. Consider a sharp hypothesis $\text{\textbf{H}}: \boldsymbol{\theta} \in \boldsymbol{\Theta}_\text{\textbf{H}}$ and let ${T_{{\boldsymbol{\mathrm{y}}}}=\left\lbrace \boldsymbol{\theta} \in \boldsymbol{\Theta}: f(\boldsymbol{\theta} \vert {\boldsymbol{\mathrm{y}}})>\text{sup}_\text{\textbf{H}}f(\boldsymbol{\theta} \vert {\boldsymbol{\mathrm{y}}}) \right\rbrace}$ be the set tangential to **H**. The measure of evidence in favor **H** is defined as ${ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)=1-P(\boldsymbol{\theta} \in T_{{\boldsymbol{\mathrm{y}}}}\vert {\boldsymbol{\mathrm{y}}})}$. The **FBST** is the procedure that rejects **H** whenever $ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)$ is small (Pereira et al., 2008).
Suppose that we want to test the hypotheses $$\begin{aligned}
\text{\textbf{H}}&: {\boldsymbol{\theta}}_2=\boldsymbol{0}\nonumber\\
\text{\textbf{A}}&: {\boldsymbol{\theta}}_2\neq \boldsymbol{0}\end{aligned}$$
The tangential set to the null hypothesis is $$T_{{\boldsymbol{\mathrm{y}}}}=\left\lbrace ({\boldsymbol{\theta}}_1,{\boldsymbol{\theta}}_2)\in \boldsymbol{\Theta}: f({\boldsymbol{\theta}}_1,{\boldsymbol{\theta}}_2 \vert {\boldsymbol{\mathrm{y}}})>\underset{\text{\textbf{H}}}{\operatorname{sup}}f({\boldsymbol{\theta}}_1,{\boldsymbol{\theta}}_2\vert {\boldsymbol{\mathrm{y}}}) \right\rbrace, $$ and, since $(\boldsymbol{\theta}-{\boldsymbol{\mathrm{m}^*}})^{\top}{\boldsymbol{\mathrm{W}^*}}^{-1} (\boldsymbol{\theta}-{\boldsymbol{\mathrm{m}^*}}) \sim \chi^2_p$, the evidence in favor of is $$\small
ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)=1-P\left(\chi^2_p<-2\log \left\lbrace \left[\underset{\text{\textbf{H}}}{\operatorname{sup}}f({\boldsymbol{\theta}}_1,{\boldsymbol{\theta}}_2\vert {\boldsymbol{\mathrm{y}}})\right] \left|{\boldsymbol{\mathrm{W}^*}}\right| ^{1/2}\,(2\pi)^{p/2}\right\rbrace\right),$$
where, $\underset{\text{\textbf{H}}}{\operatorname{sup}}f({\boldsymbol{\theta}}_1,{\boldsymbol{\theta}}_2\vert {\boldsymbol{\mathrm{y}}})=f({\boldsymbol{\mathrm{m}^*}}_{1.2}({\boldsymbol{\theta}}_2=\boldsymbol{0}),\, \boldsymbol{0}\vert {\boldsymbol{\mathrm{y}}})$.\
Consider $\varphi({\boldsymbol{\mathrm{y}}})$ as the test such that
$$\varphi({\boldsymbol{\mathrm{y}}})= \left\{ \begin{array}{l} 0 \quad if \quad ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)> k\\ 1 \quad if \quad ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)\leq
k. \end{array} \right.\;$$
Thus, define the set $$\Psi= \left\lbrace {\boldsymbol{\mathrm{y}}}\in \boldsymbol{\Omega}: ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)\leq
k\right\rbrace. $$
The averaged error probabilities can be expressed in terms of the Bayesian prior predictive densities under the respective hypotheses as follows $$\begin{aligned}
\alpha_{\varphi}&= P(\varphi({\boldsymbol{\mathrm{y}}})=1\vert \text{\textbf{H}}) \nonumber\\
&= \int_{{\boldsymbol{\mathrm{y}}}\in {\Psi}}f_\text{\textbf{H}}({\boldsymbol{\mathrm{y}}})\,d{\boldsymbol{\mathrm{y}}}\nonumber\\
&=\int_{{\boldsymbol{\mathrm{y}}}\in {\Psi}}\,\int_{\textbf{H}} f({\boldsymbol{\mathrm{y}}}\vert {\boldsymbol{\theta}}_1,{\boldsymbol{\theta}}_2) \, g_\textbf{H}({\boldsymbol{\theta}}_1,{\boldsymbol{\theta}}_2)\, d{\boldsymbol{\theta}}_1\,d{\boldsymbol{\theta}}_2\nonumber\\
&=\int_{{\boldsymbol{\mathrm{y}}}\in {\Psi}}\,\int_{\textbf{H}} f({\boldsymbol{\mathrm{y}}}\vert {\boldsymbol{\theta}}_1,{\boldsymbol{\theta}}_2) \, g({\boldsymbol{\theta}}_1\vert {\boldsymbol{\theta}}_2=\boldsymbol{0})\, d{\boldsymbol{\theta}}_1\,d{\boldsymbol{\theta}}_2\\[5pt]
&=\int_{{\boldsymbol{\mathrm{y}}}\in {\Psi}}\,\int_{{\boldsymbol{\theta}}_1 \in \mathbb{R}^s} f({\boldsymbol{\mathrm{y}}}\vert {\boldsymbol{\theta}}_1,{\boldsymbol{\theta}}_2=\boldsymbol{0}) \, g({\boldsymbol{\theta}}_1\vert {\boldsymbol{\theta}}_2=\boldsymbol{0})\, d{\boldsymbol{\theta}}_1\nonumber\\[5pt]
&=\int_{{\boldsymbol{\mathrm{y}}}\in {\Psi}}{\small N_n\left( {\boldsymbol{\mathrm{X}}}{\boldsymbol{\mathrm{C}}}{\boldsymbol{\mathrm{m}_0}}_{1.2}({\boldsymbol{\theta}}_2=\boldsymbol{0}),\,\left(\sigma^2\mathbb{I}_n+({\boldsymbol{\mathrm{X}}}{\boldsymbol{\mathrm{C}}}){\boldsymbol{\mathrm{W}_0}}_{11.2}({\boldsymbol{\mathrm{X}}}{\boldsymbol{\mathrm{C}}})^{\top}\right)\right),}\end{aligned}$$
where ${\boldsymbol{\mathrm{C}}}_{(s+r) \times s}=[\mathbb{I}_s,\boldsymbol{0}_{s\times r}]^{\top}$. $$\begin{aligned}
\beta_{\varphi}&=P(\varphi({\boldsymbol{\mathrm{y}}})=0\vert \text{\textbf{A}})\nonumber\\
&= \int_{{\boldsymbol{\mathrm{y}}}\notin {\Psi}} f_\text{\textbf{A}}({\boldsymbol{\mathrm{y}}})\,d{\boldsymbol{\mathrm{y}}}\nonumber\\
&=\int_{{\boldsymbol{\mathrm{y}}}\notin {\Psi}} \int_{\textbf{A}} f({\boldsymbol{\mathrm{y}}}\vert {\boldsymbol{\theta}}) \, g_\textbf{A}({\boldsymbol{\theta}})\, d{\boldsymbol{\theta}}\nonumber\\[3pt]
&=\int_{{\boldsymbol{\mathrm{y}}}\notin {\Psi}} \int_{\textbf{A}} f({\boldsymbol{\mathrm{y}}}\vert {\boldsymbol{\theta}}) \, g({\boldsymbol{\theta}})\, d{\boldsymbol{\theta}}\nonumber\\[3pt]
&=\int_{{\boldsymbol{\mathrm{y}}}\notin {\Psi}} N_n\left( {\boldsymbol{\mathrm{X}}}{\boldsymbol{\mathrm{m}_0}},\left(\sigma^2\mathbb{I}_n+{\boldsymbol{\mathrm{X}}}{\boldsymbol{\mathrm{W}_0}}{\boldsymbol{\mathrm{X}}}^{\top}\right)\right).\end{aligned}$$
So, the adaptive cut-off value $k^{*}$ for $ev\left(\text{\textbf{H}};x\right)$ will be the $k$ that minimizes $a\alpha_{\varphi}+b\beta_{\varphi}$.
Finally, define $\varphi^{*}({\boldsymbol{\mathrm{y}}})$ as the test such that $$\varphi^{*}({\boldsymbol{\mathrm{y}}})= \left\{ \begin{array}{l} 0 \quad if \quad ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)> k^{*}\\1 \quad if \quad ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)\leq
k^{*}. \end{array} \right.\;$$ The optimal averaged error probabilities that depend on the sample size will be $$\alpha_{\varphi^{*}}^{*}= P(\varphi^{*}({\boldsymbol{\mathrm{y}}})=1\vert \text{\textbf{H}}), \quad \beta_{\varphi^{*}}^{*}=P(\varphi^{*}({\boldsymbol{\mathrm{y}}})=0\vert \text{\textbf{A}}).$$
Results
=======
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
 
\(a) ${\boldsymbol{\mathrm{y}}}=\theta_1+\boldsymbol{\varepsilon}, \,\, \text{\textbf{H}}:\theta_1=0,$ \(b) ${\boldsymbol{\mathrm{y}}}=\theta_1+\theta_2 \, x_{i1}+\boldsymbol{\varepsilon}, \,\, \text{\textbf{H}}:\theta_2=0,$
$\mathrm{m}_0=0, \, \mathrm{W}_0=1. $ ${\boldsymbol{\mathrm{m}_0}}=[0,0]^{\top},\,\, {\boldsymbol{\mathrm{W}_0}}=\mathbb{I}_2, $
\[0pt\]
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[2]{}
------ --------- ---------
$k^*$
$n$ $d=1$ $d=2$
10 0.10040 0.35260
50 0.05166 0.11262
100 0.04447 0.10473
150 0.03776 0.09698
200 0.03179 0.08946
250 0.02667 0.08226
300 0.02244 0.07544
350 0.01905 0.06904
400 0.01639 0.06311
450 0.01429 0.05767
500 0.01264 0.05274
1000 0.00649 0.02823
1500 0.00622 0.02954
2000 0.00610 0.03000
------ --------- ---------
: Cut-off values $k^*$ for $ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)$ as function of $n$, with $d=\text{dim}(\boldsymbol{\Theta})$, $a=b=1$.[]{data-label="tabknpriorisNN"}
![ Cut-off values $k^*$ for $ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)$ as function of $n$, with $d=\text{dim}(\boldsymbol{\Theta})$, $a=b=1$.[]{data-label="knpriorisNN"}](ksvC_d1d2.png)
By increasing $n$, $k^*$ shows a decreasing trend, which means that the influence of sample size on the determination of the cut-off for $ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)$ is very relevant.\
On the other hand, it is possible to notice the differences in the results between the two models. Then, the cut-off value for $ev\left(\text{\textbf{H}};{\boldsymbol{\mathrm{y}}}\right)$ will depend not only on the sample size but also on the dimension of the parametric space. More specifically, the $k^*$ value is greater when $d$ is higher.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
 
\(a) ${\boldsymbol{\mathrm{y}}}=\theta_1+\boldsymbol{\varepsilon}, \,\, \text{\textbf{H}}:\theta_1=0,$ \(b) ${\boldsymbol{\mathrm{y}}}=\theta_1+\theta_2 \, x_{i1}+\boldsymbol{\varepsilon}, \,\, \text{\textbf{H}}:\theta_2=0,$
$\mathrm{m}_0=0, \, \mathrm{W}_0=1. $ ${\boldsymbol{\mathrm{m}_0}}=[0,0]^{\top},\,\, {\boldsymbol{\mathrm{W}_0}}=\mathbb{I}_2, $
\[0pt\]
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\[kn11\]
With this procedure, increasing the sample size implies that the probabilities of both kind of errors and their linear combination decrease, when in most cases, setting a single level of significance independent of sample size, only type II error probability decreases.
References {#references .unnumbered}
==========
Gannon, M. A., Pereira, C. A. B. and Polpo, A. (2019). Blending bayesian and classical tools to define optimal sample-size-dependent significance levels.
[[ *The American Statistician*]{}, [**73**]{}(sup1), 213-222]{}
Oliveira, M. C. (2014). [*Definição do nível de significância em função do tamanho amostral*]{}. Dissertação de Mestrado, Universidade de São Paulo, Instituto de Matemática e Estatística. Departamento de Estatística, São Paulo.
Pereira, C. A. B., Nakano, E. Y., Fossaluza, V., Esteves, L. G., Gannon, M. A. and Polpo, A. (2017). Hypothesis tests for bernoulli experiments: Ordering the sample space by bayes factors and using adaptive significance levels for decisions. [*Entropy*]{}, [**19**]{}(12), 696.
Pereira, C. A. B., Stern, J. M. and Wechsler, S. (2008). an a significance test be genuinely bayesian?. [*Bayesian Analysis*]{} [**3**]{}(1), 79-100.
Pereira, C. A. B. (1985). [*Teste de hipóteses definidas em espaços de diferentes dimensões: visão Bayesisana e\
interpretação Clássica*]{}. Tese de Livre Docência, Universidade de São Paulo, Instituto de Matemática e Estatística. Departamento de Estatística, São Paulo.
Pereira, C. A. B. and Stern, J. M. (1999). Evidence and credibility: Full bayesian significance test for precise hypotheses. [*Entropy*]{} [**1**]{}(4), 99-110.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The recent radio observations (Mooley et al, 2018) of a superluminal radio afterglow following GRB 170817A are interpreted in terms of a jet impacting a baryonic cloak, which is presumably the material caught at the front of the jet as the latter emerges from a denser ejected material. Assuming that we the observers are located [at a viewing angle of $\sim 0.2$ radians from the emitting material (perhaps slightly more from jet axis)]{}, we suggest that the Lorentz factor of the jet is $\lesssim 20$ at the time of the prompt emission, and that, as suggested previously, it is accelerated to much higher values before finally decelerating during the afterglow phase. A less extreme example of a short GRB being observed off axis may have been GRB 150101b (Fong, et al., 2016). A feature of GRBs viewed from large offset angles is a large afterglow isotropic equivalent energy as compared to prompt emission, as predicted (Eichler, 2017), and this is born out by the observations of these two GRB.
It is also shown that the prompt emission of GRB 170817A, if seen way off-axis ($\theta \gg 1/\Gamma $), could not be made by internal shocks in the baryonic material that powers the afterglow.
author:
- 'David Eichler$^{*}$'
title: Short GRBs Viewed from Far Off Axis
---
Introduction
=============
Mooley et al (2018) have recently measured apparent superluminal motion in the radio afterglow emission of GRB 170817A, which was associated with a neutron star merger. The apparent motion is of order 4c around the time of radio maximum ([*circa*]{} day 150 after the prompt emission). From this the authors infer that at the epoch of this maximum, the Lorentz factor of the radio emitting region is about (or slightly greater than) 4 and that it is seen from an angle of about 0.25 to 0.35 radian, or about 14 to 20 degrees, from the jet axis. From the sharpness of the peak, they also infer that the angular width of the jet itself cannot be much more than about 0.1 radian Hence, if the jet is axisymmetric, the distance to the closest part of its perimeter cannot be less than about 0.15 radian from our line of sight, and is more likely offset by at least 0.175 or 0.2 radians.
Mooley et al. (2018) also conclude that the energy in the blast that make the afterglow has an isotropic equivalent energy $E_{iso} \equiv 10^{52}E_{52}$ ergs of about $10^{52}$ ergs . If the asymptotic Lorentz factor $\Gamma_a$ of the baryonic material that drives the blast is $10^{2.5}\Gamma_{a,2.5}$, then its isotropic equivalent mass $m_{iso} \equiv 10^{28.5}m_{28.5}$ g is then $m_{iso} \simeq 10^{28.5} E_{52}/\Gamma_{a,2.5}$ g.
The angular extent of the material that last interacted with ( i.e. emitted or scattered) the prompt emission need not be the same as that of the material driving the late time afterglow. In particular, the distribution of the former to may be wider than the latter because the criterion for contributing to the prompt emission is determined by optical depth, which need not require much of the total mass, whereas the contribution to late time afterglow depends on the angular distribution of ejected mass and momentum. Virtually all of the discussion below is qualified by this uncertainty. For the sake of concreteness however, it is assumed below for concreteness that the observer’s line of sight is 0.2 radians from the material that most contributes to the prompt emission and the latter material is taken to be a pencil beam that is somewhat closer to our line of sight than the jet axis.
The relatively large viewing angle inferred for GRB 170817A was anticipated, (e.g. Eichler, 2017) if only because the accompanying gravitational wave signal selected it for unusually close proximity while more distant GRBs are selected for small viewing angles, where relativistic beaming makes them bright enough to be detected. The observed initial rise in afterglow intensity with time, which has been recorded since the discovery, is consistent with an off-axis jet, whose forward shock becomes increasingly visible as it decelerates (e.g. Lazzati et al, 2018; Lyman et al, 2018; Zhang, et al. 2018; Lamb and Kobayashi, 2018;; Ioka, and Nakamura, 2018; Resmi et al. 2018). However, the relatively sharp reversal from rising to declining afterglow in both the radio and X-ray (Alexander et al. 2018; Margutti et al., 2018; Lamb, Mandel, and Resmi, 2018), together with the apparently superluminal motion in the radio, seems to clinch the case for a strongly relativistic off-axis jet over a quasi-spherical, mildly relativistic outflow.
That short GRB [as a class]{} are viewed from a larger viewing angle than long ones has been advocated for many years, \[e.g. Eichler, Guetta, and Manis (2009)\]. [In this particular model, photons (or in any case a baryon poor fireball) overtake baryonic matter in front of them, scattering off it and accelerating it. The baryonic matter so accelerated eventually powers the afterglow. ]{} While a wider viewing angle is kinematically associated with lower Doppler factors and hence longer observed durations, in this particular model, photons seen near the peak of the burst are last scattered (or emitted and then never scattered) from an accelerating surface that is, at the time of the last scattering, moving at a Lorentz factor $\Gamma$ of at least $1/\theta$ relative to the observer’s line of sight. After the peak, the Lorentz factor is larger, causing the observer to see a ”soft tail” (often called “extended emission”) of softening, dimming emission. [ This tail could also be due in part to a light echo off parts of the jet that are oriented further away from our line of sight.]{} The short duration at larger viewing angle is attributed to the shorter acceleration time of the surface of last scattering when $\theta$ is large and $\Gamma_{peak}$, therefore, small. [ The acceleration time (in the frame of the central engine) $d (ln{1-\beta})/dt$ is of order $\Gamma^3 m_{iso} c/L_{iso}$ s (Eichler and Manis, 2007), where $L_{iso}\equiv 10^{53}L_{53}$ is the isotropic equivalent luminosity along the jet axis, which, for $m_{28.5}$, $E_{52}$ and $\Gamma_{a, 2.5}$ all of order unity, is of order $10^{-3.5} \Gamma^3/L_{53}$ s. In observer time, this time interval is compressed by reduction in propagation distance by the factor $ (1-\beta \cos \theta)$, which, for GRB 170817A, is estimated below to be a factor of $(1- \cos\theta ) \simeq 0.02$. Altogether, we might expect that within the first 0.1 seconds of observer time, the baryons are accelerated to a Lorentz factor of $\sim 25 L_{53}^{1/3}$.]{}
In anticipation of the announcement of the discovery of GRB 170817A, and in more detail just afterwards, it was noted (e.g. Eichler, 2017) that this GRB, because it was so close, could be seen at even larger viewing angles, where it would appear softer and fainter than most GRB, yet where it would be within a [broader and therefore]{} more likely range of viewing angles for a random distribution of observers. As the spectral peak - at 180 keV +,- 60 keV - was about 22 times softer than the spectral peak of GRB 090510 (which was taken as the benchmark spectral peak of an on-axis observer), it was conjectured in the above reference that the viewing angle was about $\sqrt{21}/\Gamma \sim 4.5/\Gamma$. This estimate follows from the fact that the observed photon energy $E''$ depends on the viewing angle $\theta$ to the direction of motion of the scatterer $\hat \beta$ as $(1-\beta)E/(1-\beta \cos \theta) \sim E/[1+(\theta \Gamma)^2 \sim E/22]$ where E is the energy seen by an on axis observer given a monochromatic, monodirectional beam that overtakes the scatterer from behind. That is, E is the photon energy in the observer frame prior to scattering.
Accordingly, the luminosity, which from a pencil beam scales as $[1/\Gamma(1-\beta \cos \theta)]^4$ would scale as $(E''/E)^4$, which, in the case of GRB 0170817A, would be $\sim 22^{-4} \sim 4 \cdot 10^{-6}$. Remarkably, this gives a reasonable estimate of the peak luminosity of GRB 170817A, which was of order $2.5 \cdot 10^{47}$ erg s$^{-1}$, relative to the peak luminosity of GRB 090510 ($L_{iso} \simeq1 \cdot 10^{53}$ erg s$^{-1})$, a short GRB that we have previously argued was seen on-axis (Eichler, 2017; Eichler 2014), and sets the standard expectation for an on-axis observer.
Significance of Afterglow Observations
======================================
Most remarkably, however, the radio afterglow observations support the conclusion that there is fact was a more or less normal isotropic equivalent energy output for a short GRB, $E_{iso}\sim 10^{52}$ ergs, beamed away from the observer.[^1] One might suspect, therefore, that the prompt emission is viewed from the same angle as the radio emission, $\gtrsim 0.2$ radians. If the Lorentz factor $\Gamma$ of the scatterer (i.e. the surface of last interaction of the prompt $\gamma$-ray emission) obeys $\Gamma \lesssim 25$, then $\theta$ would indeed be $\gtrsim 4.5/\Gamma$, adequately accounting for the subluminous character of this particular short GRB.
GRB170817A has much in common with GRB150101B (Fong et al, 2016), also believed to have been viewed from a wide viewing angle, in that the afterglow fluence is much higher than that of the prompt emission. Both GRB support the notion that most of the acceleration of the baryonic material that powers the afterglow took place [*after*]{} the prompt emission phase. This fits the picture (Eichler and Manis, 2008; Eichler, Guetta, Manis 2009; Eichler, 2017) that the prompt emission in our direction is aborted by the acceleration of the scattering material to values much greater than $1/\theta$. It connects the unusual brevity of GRB 150101B ($T_{90}$ of only 18 ms) to the unusually high ratio of afterglow fluence to prompt fluence, as both correlate with large $\theta$.[^2]
Now Veres et al (2018) have recently argued that in fact the spectral peak may be somewhat larger at peak luminosity, which would mean, in the context of Eichler (2017), that a smaller value $E/E''$ should be used, perhaps only 10 rather than 22, and that $\theta$ may be a small as $3/\Gamma$, giving a smaller reduction of apparent luminosity due to off-axis kinematic effects.[^3] However, given the uncertainties in the isotropic equivalent luminosity seen by an on-axis observer, this is not a serious concern. The principal uncertainty is that the vast majority of the kinetic energy that powers the afterglow may have entered the scatterer [*after*]{} the peak of the prompt emission, as discussed in Eichler (2017). This is supported by the wide range in the ratio r of afterglow fluence ${\cal F}_{afterglow}$ to prompt fluence ${\cal F}_{prompt}$ over short GRB, ranging from $r\simeq 10^3$ in the cases of highly off-axis viewing angles such as GRB150101B (Fong et al, 2016) down to $r \simeq 10^{-4}$ in the case of GRB090510 which was probably viewed nearly face on. Additional uncertainty can arise from intrinsic scatter in r, apart from viewing angle considerations, especially because nearby GRB are more likely to be at the low end of this scatter, while distant ones are at the high end. Yet more uncertainty arises in the coverage factor of the scattering material ahead of the fireball.
In any case, there now seems to be agreement among theorists that the GRB 170817A was observed way off axis and that the prompt fluence was well below average for a short GRB, but that it was nevertheless accompanied by a long term afterglow of average fluence. As such, it resembles the GRB 150101B, which had a low fluence (though average luminosity) and a huge $\sim 10^3$ afterglow to prompt fluence ratio. Probably GRB 170817A is the even more extreme case as it is viewed even further off axis.
The Transparency Issue
======================
Even within the framework of an off-axis kinematics picture, there may still be open questions. For example there is the question of whether photons that are scattered off-axis (i.e. in the backward hemisphere in the frame of the scatterer) make any further interactions with not-yet scattered ones, or more generally whether pairs are reestablished by the interaction of photons with baryonic material. This depends on the details.
There are several cases that need to be considered.
a\) a baryonic shell impacted from behind by an ultrarelativistic jet. The jet is assumed to be mostly or all photons that peak at several MeV. In the case of GRB 090510, the spectral peak of the photospheric emission was $E_p$ was at $E_p \simeq 4$ MeV. A photon observed at earth of energy $E''$ had an energy $E'$ in the scatterer frame of $E'_o= E''\Gamma(1-\beta \cos \theta)$, whereas the photons in the not yet scattered jet have an energy $E'_j = E/\Gamma(1+\beta)$. The criterion for pair production $E'_oE'_j \ge (2m_ec^2)^2$ is satisfied when $E''E (1-\beta \cos \theta)/(1+\beta) \simeq E''E (1-\beta \cos \theta)/2\ge (2m_e c^2)^2$. For $1-\beta \cos \theta =0.02$, the case of GRB 170817A with $\theta$ taken to be 0.2, the criterion for pair production is that
$$E''E \ge 400(m_e c^2)^2 \sim 200 MeV m_e c^2.$$
So for example if a photon is observed at $E''= 512$ MeV, then it could have pair produced only with a photon that would have been observed at $E \ge 200$ MeV by an on-axis observer. So even if we assume that the photons in the jet had a Comptonization spectrum extending all the way to 200 MeV, if the temperature of those photons is only $2$ MeV, than the fraction of photons that could pair produce would be proportional to the Boltzmann factor $e^{-100}$ which would imply a negligible amount of pairs produced.[^4]
b\) the case where photons are Comptonized with a temperature $T'=E'_p/2$. Here we need to worry about photons at higher energies than the observed ones pair producing with other unobserved photons, because such pairs would block even photons below the pair production threshold. Given the photon isotropic equivalent luminosity $10^{53}L_{iso,53}$ erg s$^{-1}$ of the jet, and the radius of emission, which is taken to be $R=c\delta t''/(1-\cos \theta)=1.5 \cdot 10^{11}$ cm, we compute the optical depth to pairs in Compton equilibrium with the photons of temperature T’, given a comoving energy density of
$$U' = 10^{53}L_{iso,53}{{(1-\beta)}\over{(1+\beta)}}/\left[4\pi c R^2\right]$$
The ratio of pair density to photon energy density is (Svensson, 1984) $$n'/U'=1.2 [8/\pi]^{1/2}[1 + 0.372 y^{-1/2}
+ 0.472 y^{-1} +(3/2 \pi)^{1/2} 1.2y^{-3/2}]^{-1} y^{-3/2} exp[-y]/3kT' \equiv \eta y^{-3/2} e^{-y}/3kT'$$ where $y\equiv m_e c^2/kT'$, and where the numerical factor $\eta$ will turn out to be $\sim 1.7$. So $$\tau =\left[ 10^{53}L_{iso,53}{{(1-\beta)}\over{(1+\beta)}}/12\pi kT' c R\right]\eta y^{-3/2}e^{-y}\sigma_T /\Gamma
\label{transp2}$$
The radius of emission R is constrained by the observation that the delay between the gravitational signal and the GRB of 1.7 s. The quantity $R(1-\cos \theta)/c=0.02R/c $ must therefore be less than 1.7s, so $R\le 2.5 \cdot 10^{12}$ cm. Another constraint comes from the fact that the peak of the emission is only $\sim 0.1$ s, so even if the material emits a pulse of locally infinitesimal duration in our direction the delay between the arrival of the closest and furthest region of the last interaction with baryons, $2R\theta_o (1- \cos \theta)/c$, must be less that 0.1 s, implying that $R\le 1.5 \cdot 10^{11}/\theta_o$ s. Here $\theta_o$ is the opening angle of the jet and may be as small as 0.05. Altogether, $R$ may be at most of order $10^{12}$, and the condition that $\tau$ be less than $\sim 10$ implies that $T'\lesssim 35 $ KeV, and $E_p' = 2 kT' \lesssim 70$ KeV.[**[^5]**]{} If $1-\beta \cos\theta$ can be approximated as $1-\cos \theta \simeq \theta^2/2$, then the observation of a peak frequency $E''$ of about 500 KeV in the observer frame implies
$$E_p' = E_p'' \Gamma(1-\beta \cos \theta) =0.02 \Gamma \cdot 500 \rm KeV \le 70 \rm KeV;$$
whence $\Gamma \lesssim 7$. It is emphasized that this constraint is contingent on the assumption that the pairs and photons are in Compton equilibrium. The observation that the spectrum of GRB 170817A has a soft component below the peak casts doubt on whether this is a good assumption. [Moreover, the peak energy $E_p''$ is extremely uncertain (Veres et al, 2018); if it is only 300 KeV, well within the error bars of Veres et al, then $\Gamma$ is constrained more weakly to be $\lesssim 12$. So the assumption that the photons are in Compton equilibrium with a trace residuum of pairs, though not necessarily the case, would be more constraining; it is just marginally compatible with the rest of the model.[^6] ]{}
c\) the baryons that power the afterglow are already mixed with the on-axis photons at the time of the prompt emission: In this case the afterglow itself establishes the baryon environment through which those photons must escape. The density of baryons, if they power afterglow with about the same luminosity as the prompt emission that would be seen by an on-axis observer, is established by the condition that they must contain enough energy:
$$U = L _{iso} /\left[4\pi c R^2\right] = \Gamma n m_p c^2$$
and $$\tau = n'\sigma_T R/\Gamma = n\sigma_T R/\Gamma^2 = \left[L_{iso} \sigma_T /4\pi Rm_p c^2 \Gamma^3 \right]= 2.2 \cdot 10^8 L_{iso,53}R_{12}^{-1} \Gamma^{-3}$$ so that $\tau\lesssim 1$ implies $\Gamma \gtrsim 600 L_{iso,53}^{1/3} R_{12}^{-1/3}$. In other words, if the observed gamma rays had needed to escape from within the baryons that power the afterglow, then they could do so only if the latter had a Lorentz factor of several hundred, and this would be too ultrarelativistic to give significant amounts of off-axis radiation. This lower limit on $\Gamma$ is well known in the GRB literature.
We thus derive an important result: There cannot be much off-axis emission from within the material that powers the long-term afterglow. It would be too opaque to emit far off axis. Photons generated from within the jet by internal shocks would be dragged away from the observer’s line of sight by the motion and opacity of the baryonic fluid. The off-axis viewing hypothesis for GR 170817A works only if the photons impact the baryons from without and scatter off its surface. We interpret this to mean that the baryons are probably from the merger ejecta and are plowed up by the jet from behind.
Conclusions and Further Discussion
==================================
The surprisingly transparent nature of GRBs, given their high compactness, is perhaps the central mystery of GRBs. Two completely different explanations have been attempted: One is that huge Lorentz factors dilate both comoving time and distance scales of the fireball (Rees and Meszaros, 1992), the effect of which is to lower the density of the outgoing material as well as its true compactness required to account for rapid variability. The other (Levinson and Eichler, 1993) invokes a baryon free corridor - probably connected to an event horizon - that the photons are either produced inside of or scattered into. If they are scattered by swept up material at the leading edge nearly opposite to the direction of the scatterer’s motion (i.e. back towards the central engine that the scatterer is presumed to be receding from) then they are observed at “large” viewing angles” ( i.e. at least several times $1/\Gamma$) relative to the motion of the scatterer.
[A qualitative difference between the two accounts of GRBs is that one - using baryonic kinetic energy as the primary source of energy - puts the energy in baryons which only later manage to generate photons via shocks, whereas the other - a fireball that is nearly baryon free - has the energy flowing into baryons from an essentially non-baryonic fireball and predicts that that the baryons that eventually power the afterglow will be accelerated by the push of a non-baryonic fireball from which the prompt emission originates. The baryon poor fireball scenario thus lends itself to a scenario is which the length of prompt GRB emission may depend on the timescale over which the observer is within the $1/\Gamma$ cone of emission, which in turn depends on the viewing angle of the observer. The dichotomy between short and long GRB, and their different progenitors, in fact, may be that the range of viable viewing angles depends strongly on the progenitor - i.e. merging neutron stars (short GRB) allow viewing from a wider range of angles than collapsars within giant envelopes (long GRB).]{}
It should also be recalled that the combination of acceleration and viewing angle, with few free parameters, has considerable predictive power and , unlike the alternative models, quantitatively explains the Amati relation for long GRB (Eichler and Levinson 2004, 2006), the flat phase of afterglows (Eichler, 2005; Eichler and Jontoff-Hutter, 2005) spectral hard-to-soft evolution (Manis and Eichler 2007, 2008) and the differences between short and hard GRB (Eichler, Guetta and Manis, 2009, Eichler, 2017). In particular, to accommodate the Amati relation extending down to several KeV and $E_{iso} \sim 10^{48}$ erg.s, i.e. almost six orders of magnitude in isotropic equivalent energy, the viewing offset must be as large as $30/\Gamma$ for the softest sources.
Afterglow observations of GRB 170817A provide an unprecedented arena for testing hypothesis that the above features of GRB are due explainable by viewing angle kinematics, because the viewing angle relative to the jet axis is probably larger than for any other GRB to date, and because the large viewing angle is verifiable in several different ways (gravitational wave polarization, radio afterglow, $ etc.$). In an off-axis viewing scenario, photons coming from the jet that end up in our direction must be traveling nearly backward in the frame of the jet material, and here we have carefully considered the question of whether they could get out through the stream of outwardly propagating photons from the central source without pair producing. We find that there is no reason they shouldn’t, unless their energy is much higher than those detected in the NaI detectors of the GBM.
[Using the above considerations the following scenario for prompt emission can be suggested based on figure 6 of Goldstein et al: The GRB begins with $100 \lesssim E_p \lesssim 500 $ KeV, with $7 \lesssim \Gamma \lesssim 12$ and, by the end, where the spectrum peaks at only 10 to 20 KeV, it has been accelerated to $20 \lesssim \Gamma \lesssim 40$. Attributing the softening to acceleration alone would not account quantitatively for the brightness of the soft tail. The soft tail could be due to a light echo off parts of the baryonic shell whose orientation is further from the line of sight. A detailed account of the light curve, though beyond the scope of this paper, is important.]{}
We also find an [*upper*]{} limit on the Lorentz factor if the prompt emission is viewed from a large offset, because, for a given value of $(1-\beta \cos \theta)$, the inverse Doppler factor $\Gamma(1-\beta \cos\theta)$, which relates the comoving photon energy (in the frame of the scatterer) to the observed photon energy, increases with $\Gamma$. When the requirement of transparency bounds the comoving photon energy from above, this provides an upper limit on $\Gamma$.
A viewing offset of 10 or 15 degrees is not consistent with any and all models. We find, for example, that a viewing offset of 0.2 radians is incompatible with the prompt emission being powered by the same pool of kinetic energy that powers the long-term afterglow (e.g. internal shocks tapping about 1/2 of this kinetic energy for prompt emission and leaving the other half for powering the afterglow), for then, in order to power the afterglow, the accompanying electrons would be optically thick to the prompt photons unless their Lorenz factor was above several hundred. But such a high Lorentz factor would give negligible emission at a viewing angle of at least 0.2 radians. The once popular internal shock model for prompt emission is thus challenged unless the opening angle for the observed prompt emission is higher than that of the afterglow core [**(e.g. as in a structured jet or shock breakout model)**]{} in which case the direction of the motion of the shocked material can be nearly along our line of sight.
If the baryonic material that powers the afterglow [*also*]{} powers the prompt emission, then we expect the opening angle of each to be more or less the same. On the other hand, if the material merely [*reflects*]{} the prompt emission, then there is not reason to expect that the latter need scale with the mass of the baryonic material, and the profile of the prompt emission could in fact be wider than the core of the afterglow.
[ Future GW signals from NS mergers may be viewed at even larger angles from the rotational axis. In this case the prompt emission would be even softer and dimmer than for GRB 170817A. In this case wide angle X-ray cameras would be the more suitable means of detecting an accompanying electromagnetic signal (Eichler and Guetta, 2010).]{}
**Acknowledgements**
I thank Dr. N. Globus, A. Levinson, R. Moharana and E. Nakar for helpful conversations This work was supported by funding from the Israel Science Foundation, the Israel-U.S. Binational Science Foundation, and the Joan and Robert Arnow Chair of Theoretical Astrophysics.
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[^1]: This fact had been challenged earlier by advocates (Mooley et al. 2017) of a “choked GRB” interpretation of the low luminosity.
[^2]: GRB 170817A is also unusually short, even among short GRB (Kaneko, et al, 2015) , if the spike, as defined by Kaneko et al (2015) is taken to be the 100 ms peak, and the soft, extended emission is taken to be 2 seconds.
[^3]: They also assume that the ratio of prompt emission on and off- axis varies as $(E/E'')^2$, which is inappropriate for an observer who sees the jet as a pencil beam. They also equate the afterglow energy with the on-axis prompt emission, and this is also without justification. Each of these [unjustified assumptions]{} lead to an overestimate of $E/E''$.
[^4]: We neglect the possibility of a Compton tail of scattered photons extending in energy well beyond the range of the observed photons, because the Compton recoil off the scattering material would soften any such tail. There would be no photons much above $E' \sim mc^2$. Moreover, any pairs that are created are kept out of the body of the jet by radiation pressure.
[^5]: We choose an upper limit of 10 on $\tau$; $1\le \tau \le 10$ is admissible because the low luminosity of GRB 170817A allows an optical depth of up to 10.
[^6]: That the plasma is only marginally transparent at the beginning of the GRB would be attributable to opacity contributing to the delay of the appearance of the GRB following the GW event.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We illustrate recent results concerning the validity of the work fluctuation theorem in open quantum systems \[M. Campisi, P. Talkner, and P. Hänggi, Phys. Rev. Lett. [**102**]{}, 210401 (2009)\], by applying them to a solvable model of an open quantum system. The central role played by the thermodynamic partition function of the open quantum system, – a two level fluctuator with a strong quantum nondemolition coupling to a harmonic oscillator –, is elucidated. The corresponding quantum Hamiltonian of mean force is evaluated explicitly. We study the thermodynamic entropy and the corresponding specific heat of this open system as a function of temperature and coupling strength and show that both may assume negative values at nonzero low temperatures.'
author:
- Michele Campisi
- Peter Talkner
- Peter Hänggi
title: 'Thermodynamics and Fluctuation Theorems for a Strongly Coupled Open Quantum System: An Exactly Solvable Case'
---
Introduction
============
Exact results about nonequilibrium fluctuations in nanosystems, such as the Jarzynski equality [@Jarz97] and the Tasaki-Crooks fluctuation theorem [@Tasaki:2000pi; @Crooks:1999fq] have recently attracted a great deal of attention in the burgeoning field of nonlinear fluctuation relations. These results were first derived for classical systems and later for quantum systems that are either isolated or weakly coupled to their environment [@Tasaki:2000pi; @Mukamel:2003ip; @De-Roeck:2004cq; @Esposito:2006by; @Andrieux08; @CrooksJSM08; @Talkner:2007sf; @Talkner:2008mr; @TH2007]. However, often the interaction with the environment does play an important role which cannot be neglected in real experimental situations. For this reason more attention has been recently devoted to the strong coupling regime, both classically [@Jarzynski:2004pv], and quantum mechanically [@CHT-PRL-09]. In this regime, the driven system of interest (with Hamiltonian $\hat{H}_S(t)$), strongly couples to a bath ($\hat{H}_B$), via a non negligible interaction term $\hat{H}_{SB}$: $$\hat{H}(t)=\hat{H}_S(t)+\hat{H}_{SB}+\hat{H}_{B} \; .
\label{eq:Ham}$$ For the applicability of work and fluctuation theorems, the correct choice of the statistical mechanical description of an open quantum system in terms of the proper thermodynamic partition function, is of decisive importance. The bare system partition sum $$Q_S(t)=\Tr_S{e^{-\beta \hat{H}_S(t)}}
\label{eq:Q(t)intr}$$ clearly fails to account for the effects of the environment on the system. It rather is the open system thermodynamic partition function $$Z_S(t) = \frac{Y(t)}{Z_B}
\label{eq:Zs=Z-Zb}$$ which consistently accounts for these effects [@CHT-PRL-09; @HIT_NJP08; @Ingold:2008mz; @Horhammer; @Fundamentals; @Haenggi:2006br; @Ingold:2002pc; @Theo2002; @HaenggiQTransp98; @Grabert:1988et; @Ford:1985it; @Grabert:1984ux; @Caldeira:1983aj; @FeynStatMech; @Feynman:1963qm]. Here $Y(t)$ denotes the total system partition function; i.e., $$Y(t)={\Tr
e^{-\beta(\hat{H}_S(t)+\hat{H}_{SB}+\hat{H}_{B})}}
\label{eq:Y(t)-intr}$$ and $Z_B$ the bare bath partition function $$Z_B={\Tr_{B} e^{-\beta
\hat{H}_{B}}}
\label{eq:Z_B-intr}$$ The time $ t $ merely specifies the values of the external parameters as they occur in the course of the driving protocol at the time $t$. The symbols $\Tr_S, \Tr_B, \Tr$ denote traces over system, bath, and total system respectively. The symbol $\beta=(k_BT)^{-1}$ indicates the inverse thermal energy, with $k_B$ Boltzmann constant and $T$ the temperature. This temperature is provided via vanishingly small weak contact with a large (super)-bath, which allows for a statistical mechanical treatment.
The adoption of the thermodynamic partition function $Z_S(t)$, and the corresponding free energy $F_S(t)=-\beta^{-1}\ln Z_S(t)$ allows to obtain the Jarzynski equality $$\langle e^{-\beta w} \rangle = e^{-\beta \Delta F_S}
\label{eq:JE}$$ valid irrespectively of the coupling strength [@CHT-PRL-09].
In the following we exemplify this result by applying it to a simple model Hamiltonian of an open quantum system, Sec. \[sec:model\], \[sec:FT\]. We next illustrate the equilibrium thermodynamics of that open system, by computing its Hamiltonian of mean force, its entropy and specific heat, see Sec. \[sec:EqTher\]. Remarks and conclusions are drawn in Sec. \[sec:conclusions\].
\[sec:model\] A Two Level Fluctuator-Oscillator Model
=====================================================
We consider the following Hamiltonian describing a two level system and a harmonic oscillator interacting with each other: $$\hat{H}(t)= \frac{\varepsilon(t)}{2} \hat{\sigma}_z \otimes \hat{1}_B + \hat{1}_S \otimes \Omega \left(\hat{a}^\dag \hat{a}+\frac{1}{2}\right) + \chi \hat{\sigma}_z \otimes \left(\hat{a}^\dag \hat{a}+\frac{1}{2}\right).
\label{eq:H}$$ Here $\hat{\sigma}_z$ is a Pauli matrix of the two level system, $\hat{a}^\dag$ and $\hat{a}$ are raising and lowering operators of the harmonic oscillator, $\varepsilon(t),\Omega,\chi$ are the two level system energy spacing, the oscillator energy quantum and the coupling energy, respectively. The parameter $\chi$ can assume positive and negative values whereas $\varepsilon(t)$ and $\Omega$ are strictly positive. The oscillator energy quantum $\Omega$ is related to the oscillator frequency $\omega$, via Planck’s constant $\Omega = \hbar \omega$. We consider the two level system (also referred to as the qubit throughout the text) as our system of interest ($\hat{H}_S(t)={\varepsilon(t)} \hat{\sigma}_z/2\otimes \hat{1}_B$), and the oscillator as our stylized, “minimal” bath; i.e. $\hat{H}_B=\Omega (\hat{a}^\dag \hat{a}+1/2)\otimes \hat{1}_S$. The operators $\hat{1}_{S}$ and $\hat{1}_B$ denote the identity operators acting on the system and bath Hilbert spaces, respectively. We require $|\chi|<\Omega$, which ensures that the total Hamiltonian is bounded from belowguaranteeing stability of the total system (from Eq. (\[Ens\]), $\Omega + \chi s$ must be positive in order that the smallest eigenvalue be finite). The two level system energy spacing $\varepsilon$ is assumed to depend on time according to some pre-specified protocol. This model Hamiltonian has the peculiarity that the interaction Hamiltonian commutes with both system and bath Hamiltonians, implying a so called quantum nondemolition coupling: $$[\hat{H}_S(t) \otimes \hat{1}_B,\hat{H}_{SB}]=[\hat{1}_S\otimes \hat{H}_B,\hat{H}_{SB}]=0 \ .$$
The time-instantaneous energy eigenvalues assume the form $$E_{n,s}(t) = \frac{\varepsilon(t)}{2} s + \Omega \left(n+\frac{1}{2}\right) + \chi s \left(n+\frac{1}{2}\right)
\label{Ens}$$ $s = \pm 1, \; n = 0,
1, 2, \ldots$ . We remark that the corresponding instantaneous eigenstates $|n,s\rangle$ do not depend on time.
The partition function $Y(t)$ of the total system becomes, with Eq. (\[eq:Y(t)-intr\]) $$Y(t)= \sum_{s,n} e^{-\beta E_{n,s}(t)} = q_+(t)+q_-(t)
\label{eq:Y(t)}$$ where $$q_{\pm}(t) = \frac{e^{-\beta \Omega/2} e^{\mp \beta (\varepsilon(t)+\chi)/2}}{1-e^{-\beta(\Omega \pm \chi)}} \ .$$ The bare bath partition function $Z_B$ is, with Eq. (\[eq:Z\_B-intr\]): $$Z_{B} = \sum_{n} e^{-\beta \Omega (n+1/2)}= \frac{1}{2\sinh( \beta \Omega/2)} \ .
\label{eq:Z_B}$$ Then the thermodynamic partition function $Z_S(t)$ of the open system becomes, according to Eq. (\[eq:Zs=Z-Zb\]): $$Z_{S}(t) = 2(q_+(t)+q_-(t))\sinh(\beta \Omega/2) \ .
\label{eq:Z_S(t)}$$ Note that the open system thermodynamic partition function differs substantially from the bare system partition sum $Q_S(t)$, which reads, with Eq. (\[eq:Q(t)intr\]): $$Q_S(t) = \sum_{s} e^{-\beta \varepsilon(t) s/2}
= 2 \cosh ( \beta {\varepsilon(t)}/{2}) \ .
\label{eq:Q_S(t)}$$ In particular the thermodynamic partition function consistently accounts for the presence of the oscillator and the interaction, as it depends on $\Omega$ and $\chi$, whereas the partition sum $Q_S(t)$ does not.
\[sec:FT\]Work and Fluctuation Theorems
=======================================
For a prescribed protocol of the two level spacing $\varepsilon(t)$ $t_0\leq t \leq t_f$, the work $w$ performed on the two level system is distributed according to the probability density function $p_{t_f, t_0}(w)$, given by [@Talkner:2007sf]: $$\begin{split}
p_{t_f, t_0}(w) =\sum_{m,n=0}^{\infty} \sum_{r,s=\pm1} & \delta(w-(E_{m,r}(t_f)-E_{n,s}(t_0))) \\ & \times P(m,r|n,s)\frac{e^{-\beta E_{n,s}(t_0) }}{Y(t_0)}
\end{split}$$ where $\delta(x)$ denotes the Dirac delta function and $P(m,r|n,s)$ is the transition probability to jump from the eigenstate $|n,s\rangle$ of the total Hamiltonian at time $t_0$ to the eigenstate $|m,r\rangle$ at time $t_f$: $$P(m,r|n,s)= |\langle m,r| \hat{U}_{t_f,t_0} |n,s\rangle |^{2}
\label{eq:p(mr|ns)}$$ with $\hat{U}_{t_f,t_0}=\mathcal{T}\exp (-i\int_{t_0}^{t_f} dt\hat{H}(t)/\hbar)$ denoting the time evolution operator. The model Hamiltonian in Eq. (\[eq:H\]), commutes with itself at different times, so the time ordered exponential reduces to an ordinary exponential $$\begin{split}
\hat{U}_{t_f,t_0} = \exp \left[-\frac{i}{\hbar} \left( \int_{t_0}^{t_f} dt \frac{\varepsilon(t)}{2} \hat{\sigma}_z \otimes \hat{1}_B \right. \right. \\
+ \hat{1}_S \otimes \Omega \left(\hat{a}^\dag \hat{a}+\frac{1}{2}\right) (t_f-t_0)\\
+ \left. \left. \chi \hat{\sigma}_z \otimes \left(\hat{a}^\dag \hat{a}+\frac{1}{2} \right)(t_f-t_0) \right)\right]
\end{split}$$ By inserting this expression into Eq. (\[eq:p(mr|ns)\]), one sees that no transition takes place $$P(m,r|n,s)= \delta_{m,n}\delta_{r,s}$$ with $\delta_{m,n}$ denoting the Kronecker symbol. This is of course to be expected since an interaction that commutes with the free evolution does not cause any transition. Thus, for the work probability density one obtains: $$\begin{split}
p_{t_f, t_0}&(w) = \\
& \frac{q_+(t_0)\delta (w-\Delta\varepsilon/2)+q_-(t_0)\delta(w+\Delta\varepsilon/2)}{q_+(t_0)+q_-(t_0)}
\end{split}
\label{eq:p(w)}$$ where $\Delta\varepsilon=
\varepsilon(t_f)-\varepsilon(t_0)$. By exchanging $t_0$ with $t_f$ one obtains the backward pdf of work $p_{t_0, t_f}(w)$, corresponding to the backward protocol $\bar{\varepsilon}(t)=\varepsilon(t_f+t_0-t)$. After some calculations one obtains the following expression for their ratios: $$\frac{p_{t_f, t_0}(w)}{p_{t_0, t_f}(-w)} = e^{\beta w} \frac{\cosh(\beta \frac{\varepsilon(t_f)+\chi}{2})-e^{-\beta \Omega}\cosh(\beta \frac{\varepsilon(t_f)-\chi}{2})}{\cosh(\beta \frac{\varepsilon(t_0)+\chi}{2})-e^{-\beta \Omega}\cosh(\beta \frac{\varepsilon(t_0)-\chi}{2})}$$ where we recognize that the ratio on the right hand side is equal to $Z_S(t_f)/Z_S(t_0)$, as predicted by the work fluctuation theorem for arbitrary open quantum systems [@CHT-PRL-09]. Using (\[eq:p(w)\]) we also obtain the following expression for the Jarzynski exponentiated work: $$\langle e^{-\beta w}\rangle=\frac{\cosh(\beta \frac{\varepsilon(t_f)+\chi}{2})-e^{-\beta \Omega}\cosh(\beta \frac{\varepsilon(t_f)-\chi}{2})}{\cosh(\beta \frac{\varepsilon(t_0)+\chi}{2})-e^{-\beta \Omega}\cosh(\beta \frac{\varepsilon(t_0)-\chi}{2})}
\label{eq:JEb}$$ that is, $$\langle e^{-\beta w}\rangle=\frac{Z_S(t_f)}{Z_S(t_0)}
\label{eq:JE2}$$ as predicted by the Jarzynski equality for arbitrary open quantum systems in Eq. (\[eq:JE\]) [@CHT-PRL-09].
By comparison of Eqs. (\[eq:Q\_S(t)\]) and (\[eq:JEb\]) one observes that: $$\langle e^{-\beta w} \rangle \neq \frac{Q_S(t_f)}{Q_S(t_0)}.$$ This result is in contrast to recent claims reported by Teifel and Mahler [@Teifel:2007qr], according to which the averaged exponentiated work should be identical to the ratio of partition sums $Q_S$ independently of coupling strength, provided the interaction commutes with both system and bath Hamiltonians as it is the case with the present study.
![Dimensionless difference between renormalized qubit’s energy spacing and original qubit’s energy spacing, $\Delta/\varepsilon$ (solid line, Eqs. (\[eq:varepsilon\*\],\[eq:delta\])), and global dimensionless shift of the energy spectrum, $\gamma/\varepsilon$ (dashed line, Eq. (\[eq:gamma\])), as functions of (a) the dimensionless coupling strength $\chi/\varepsilon$ (top panel), (b) dimensionless oscillator’s energy quantum $\Omega/\varepsilon$ (bottom panel). The graphs correspond to a temperature of $T=50$mK, and the experimental values employed in Ref. [@Schuster:2007ph] $\varepsilon/2\pi\hbar=6.9$ GHz, (a) $\Omega /2 \pi \hbar =5.7$GHz (top panel), (b) $\chi /\pi \hbar =-17$MHz (bottom panel).[]{data-label="fig:1"}](Figure1.eps){width="8cm"}
\[sec:EqTher\]Equilibrium Thermodynamics
========================================
We turn now to the study of the equilibrium thermodynamics of the open two level system. This means that we now keep $\varepsilon$ fixed and study the time independent Hamiltonian $$\hat{H}= \frac{\varepsilon}{2} \hat{\sigma}_z \otimes \hat{1}_B +
\hat{1}_S \otimes \Omega \left(\hat{a}^\dag \hat{a}+\frac{1}{2}\right) +
\chi \hat{\sigma}_z \otimes \left(\hat{a}^\dag \hat{a}+\frac{1}{2}\right).
\label{eq:H-time-ind}$$
The Hamiltonian of Mean Force
-----------------------------
A fundamental quantity that is closely related to the open system partition function $Z_S$ is the quantum Hamiltonian of mean force [@CHT-PRL-09] $$\hat{H}^*:= -\frac{1}{\beta}\ln \frac{\Tr_B
e^{-\beta(\hat{H}_S+\hat{H}_{SB}+\hat{H}_{B})}}{Z_B}.
\label{eq:H*}$$ It generalizes the potential of mean force commonly employed in reaction rate theory [@HTB1990] and implicit solvent models [@Roux19991]. The Hamiltonian of mean force is the effective Hamiltonian that describes the open system at equilibrium with the environment according to the equation: $$Z^{-1}_{S} e^{-\beta \hat{H}^*}= Y^{-1} \Tr_B e^{-\beta
\hat{H}}\;.
\label{rhoS}$$ It hence determines the reduced density matrix of the open system, $\rho_S$, in thermal equilibrium according to $\rho_S=Z^{-1}_{S} e^{-\beta \hat{H}^*}$. The calculation of $\hat{H}^*$ in general is a difficult task. However for the model Hamiltonian in Eq. (\[eq:H-time-ind\]), the calculation is straightforward and leads to $$\hat{H}^* = \frac{\varepsilon^*}{2} \hat{\sigma}_z + \gamma \hat{1}_S
\label{eq:H*special}$$ where $$\varepsilon^* = \varepsilon + \chi + \frac{2}{\beta} \operatorname{artanh}\left( \frac{e^{-\beta \Omega} \sinh(\beta \chi)}{1-e^{-\beta \Omega} \cosh(\beta \chi)} \right)
\label{eq:varepsilon*}$$ is the renormalized level spacing, and $$\gamma = \frac{1}{2 \beta}\ln \left( \frac{ {1-2e^{-\beta \Omega} \cosh(\beta \chi) + e^{-2 \beta \Omega}} }{(1-e^{-\beta \Omega})^2} \right)
\label{eq:gamma}$$ specifies a global shift of the spectrum. In obtaining Eq. (\[eq:H\*special\]) we used the identity $e^{a \hat{\sigma}_z}=\cosh(a)\hat{1}_S+\sinh(a)\hat{\sigma}_z$. When $\chi\rightarrow 0$, the renormalized spacing tends to the original spacing $\varepsilon$, and the offset $\gamma$ vanishes, so that $H^*$ tends to the bare system Hamiltonian $H_S$, as expected. Fig. \[fig:1\](a) displays $\gamma$ as well as the amount of renormalization $$\Delta := \varepsilon^* -\varepsilon
\label{eq:delta}$$ which is independent of the bare spacing $\varepsilon$, as functions of the coupling strength $\chi$, for $|\chi| < \Omega$. These quantities are displayed in non-dimensional units where energies are rescaled by $\varepsilon$. As $|\chi|/\varepsilon$ approaches the stability limit $\pm \Omega/\varepsilon$, $\Delta/\varepsilon$ and $\gamma/\varepsilon$ diverge, while they vanish as the coupling $\chi/\varepsilon$ approaches zero. From Eq. (\[eq:varepsilon\*\]), we note that, given certain values of the spacing $\varepsilon$ and of $\Omega$, there exists a value of $\chi$ for which the renormalized energy spacing $\varepsilon^*$ vanishes, meaning that an effective degeneracy of the qubit is induced by the presence of the oscillator. In Fig. \[fig:1\](b), $\Delta/\varepsilon$ and $\gamma/\varepsilon$ are plotted as functions of $\Omega/\varepsilon$, for fixed $\beta$ and $\chi$, and for $\Omega/\varepsilon > |\chi|/\varepsilon$.
The graphs in Fig. \[fig:1\] correspond to values of $\Omega$ and $\chi$ that match the regime of values used in an experimental implementation of the model Hamiltonian in Eq. (\[eq:H-time-ind\]) with superconducting circuits, as it has been recently reported [@Schuster:2007ph]. In that experiment $|\chi|$ is about two orders of magnitudes smaller than $\Omega$, thus the leading corrections to the energy spacing are of first order and those of the shift $\gamma$ are at most of second order in $\chi/\varepsilon$.
![Contour plot of dimensionless entropy $S_S/k_B$, Eq. (\[eq:Ss\]), as a function of dimensionless temperature $k_B T/\varepsilon= 1/(\beta \varepsilon)$, and dimensionless interaction strength $\chi/\varepsilon$, for (a) $\Omega/\varepsilon =3 $ (top panel), (b) $\Omega/\varepsilon =1/3 $ (bottom panel). The entropy is nowehere negative in panel (a). In panel (b), it assumes negative values in the region, labelled as $S_S<0$, enclosed by the level line $S_S=0$ (thick light blue line).[]{data-label="fig:2"}](Figure2.eps){width="8.5cm"}
At low temperatures we find the following limiting results: $$\begin{aligned}
\lim_{\beta \rightarrow \infty}\gamma &=&0 \label{eq:LimGamma} \\
\lim_{\beta \rightarrow \infty}\varepsilon^* &=& \varepsilon + \chi \label{eq:LimVarepsilon*}\\
\lim_{\beta \rightarrow \infty}\frac{\partial^k}{\partial \beta^k}\varepsilon^* &=& 0 \qquad k=1,2,3 ... \label{eq:LimDersVarepsilon*}\end{aligned}$$ From the previous two equations we deduce that the degeneracy of the spectrum occurs at $T=0$ for the special value $\chi= - \varepsilon$. In the following we will come back to the effect of this degeneracy on the system’s entropy and specific heat.
Thermodynamic Entropy
---------------------
From the partition function, $Z_S$, one obtains the free energy: $$F_S = -k_B T \ln Z_S = -(1/\beta) \ln Z_S,$$ and the entropy: $$S_S = -\frac{\partial F_S}{\partial T} = k_B \beta^2 \frac{\partial F_S}{\partial \beta} \ .
\label{eq:Ss}$$ In Fig. \[fig:2\](a) the entropy following from Eq. (\[eq:Z\_S(t)\]) with $\varepsilon(t)=\varepsilon$, is displayed as a function of dimensionless temperature $k_B T/\varepsilon= 1/(\beta \varepsilon)$ and dimensionless coupling strength, $\chi / \varepsilon$ for a fixed value of rescaled oscillator energy quantum $\Omega/\varepsilon$, larger than $1$. As $\chi/\varepsilon$ approaches the instability values $\pm\Omega/\varepsilon$ the entropy diverges. For all values of $\chi/\varepsilon$, the entropy vanishes at zero temperature in agreement with the third law. An exception is at the special case $\chi/\varepsilon=-1$, where the ground state assumes a finite degeneracy. Put differently, for $\chi/\varepsilon=-1$, the zero temperature entropy is no longer zero but assumes the finite positive value $k_B \ln 2$ [@Planck1910; @Giauque1928; @Giauque1933; @Pauling1935]. This $k_B \ln 2$ term is a consequence of the fact that in the limit of zero temperature and for $\chi=-\varepsilon$, the effective spacing $\varepsilon^*$ and all its higher order derivatives with respect to temperature vanish (see Eq. (\[eq:LimVarepsilon\*\],\[eq:LimDersVarepsilon\*\])). At finite temperatures there are values of $\chi$ for which the spacing $\varepsilon^*$ vanishes, however these do not coincide with the values of $\chi$ for which the entropy is $k_B \ln 2$, since then the first derivative of $\varepsilon^*$ with respect to $\beta$ does not vanish and consequently yields a contribution to the entropy.
Fig. \[fig:2\](b) depicts the entropy as a function of dimensionless temperature $k_B T/\varepsilon= 1/(\beta \varepsilon)$ and dimensionless coupling strength, $\chi / \varepsilon$, for a fixed value of rescaled oscillator energy quantum $\Omega/\varepsilon$, smaller than $1$. The most prominent feature in this case $\Omega/\varepsilon<1$ is the appearance of a region of *negative entropy* for small values of $k_B T$ and negative coupling (the region labeled as $S_S<0$ in Fig. \[fig:2\](b)). Interestingly, the experiment reported in Ref. [@Schuster:2007ph] lies in this region $\Omega<\varepsilon$, $\chi <0$ where the entropy may become negative. For the parameter values reported therein, a negative entropy is expected below $\sim 22$ mK. From Fig. \[fig:2\](b) we notice that, for positive $\chi$, the entropy vanishes at absolute zero temperature, in accordance to the third law of thermodynamics [@Planck1910], and reaches a plateau at high temperatures, without becoming negative. For negative $\chi$, it vanishes as well at zero temperature, however with increasing low temperatures, entropy first decreases until it reaches a negative minimum, and then increases until it approaches a positive plateau-value at high temperatures. This behaviour is further illustrated in Fig. \[fig:3\].
Independently of the sign of $\chi$, at high temperature the entropy reaches the asymptotic value: $$\lim_{\beta\rightarrow 0} S_S = k_B \ln
\left(\frac{2\Omega^2}{\Omega^2-\chi^2} \right)
\label{eq:HighTempEntr}$$ which notably does not depend on the spacing $\varepsilon$. For $\chi=0$, the high temperature entropy in Eq. (\[eq:HighTempEntr\]) becomes equal to $k_B \ln 2$, reflecting the fact that spin up and spin down states become equally populated at infinite temperature. For $\chi \neq 0$, it assumes values larger than $k_B \ln 2$, and diverges for $|\chi|$ approaching $\Omega$.
![Dimensionless entropy $S_S/k_B$ as a function of dimensionless temperature $k_B T/\varepsilon= 1/(\beta \varepsilon)$, for $\Omega/\varepsilon=1/3$, and $\chi/\varepsilon=-1/6 $ (solid line), $\chi/\varepsilon=1/6 $ (dashed line). The dotted line is the asymptotic value calculated from Eq. (\[eq:HighTempEntr\]).[]{data-label="fig:3"}](Figure3.eps){width="8.5cm"}
![Contour plot of dimensionless specific heat $C_S/k_B$, Eq. (\[eq:Cs\]), as a function of dimensionless temperature $k_B T/\varepsilon= 1/(\beta \varepsilon)$, and dimensionless interaction strength $\chi/\varepsilon$, for (a) $\Omega/\varepsilon =3 $ (top panel), (b) $\Omega/\varepsilon =1/3 $ (bottom panel). The specific heat is nowehere negative in panel (a). In panel (b), it assumes negative values in the region, labelled as $C_S<0$, enclosed by the level line $C_S=0$ (thick light blue line).[]{data-label="fig:4"}](Figure4.eps){width="8.5cm"}
Specific heat
-------------
From the entropy (\[eq:Ss\]) one obtains the the specific heat of the open two level system: $$C_S = T \frac{\partial S_S}{\partial T}=-\beta \frac{\partial S_S}{\partial \beta}
\label{eq:Cs}$$ Figs. \[fig:4\](a) and\[fig:4\](b) represent the specific heat as a function of dimensionless temperature $k_B T/\varepsilon= 1/(\beta \varepsilon)$ and dimensionless coupling strength, $\chi / \varepsilon$ for fixed values of rescaled oscillator energy quantum $\Omega/\varepsilon$.
In Fig. \[fig:4\](a) $\Omega$ is larger than $\varepsilon$. For values $| \chi/ \varepsilon|<\Omega/\varepsilon$, the specific heat vanishes at zero temperature. With growing temperatures, it first increases, then reaches a maximum and finally goes again to zero. The maximum occurs at decreasing temperatures as $\chi/ \varepsilon \rightarrow-1$. As $\chi/ \varepsilon$ approaches $-1$ we also see the appearance of a second maximum for larger values of $k_B T$. These features are further illustrated in Fig. \[fig:5\](a).
![Dimensionless specific heat $C_S/k_B$ as a function of rescaled temperature $k_B T/\varepsilon=1/(\beta \varepsilon)$, for various values of $\chi/ \varepsilon$ and (a) $\Omega/\varepsilon=3$ (top panel), (b) $\Omega/\varepsilon=1/3$ (bottom panel).[]{data-label="fig:5"}](Figure5.eps){width="8.5cm"}
The specific heat landscape changes drastically for $\Omega<\varepsilon$, Fig. \[fig:4\](b). The most relevant feature in this parameter range is the appearance of a region of *negative specific heat* at low temperatures and negative $\chi/\varepsilon$ (the region labelled as $C_S<0$ in Fig. \[fig:4\](b)). From Fig. \[fig:4\](b) we observe that, for $0< \chi/ \varepsilon < \Omega/\varepsilon$ the specific heat starts from zero at zero temperature, reaches a maximum and goes to zero again at high temperatures. For $-\Omega/\varepsilon< \chi / \varepsilon < 0$ the specific heat starts at $T=0$ from zero, reaches a negative minimum with increasing temperature, then a positive maximum and finally goes to zero at high temperature. These features are further illustrated in Fig. \[fig:5\](b). From Fig. \[fig:5\](b) we also notice that the curves corresponding to positive $\chi$ all cross each other within a very small temperature range around $k_B
T/\varepsilon \sim 0.21$ for the given value $\Omega/\varepsilon =
1/3$. Indeed, for $k_B T/\varepsilon = 0.21$, $\Omega/\varepsilon
= 1/3$, and $\chi/ \varepsilon$ ranging from $0$ to $\Omega/\varepsilon$, the specific heat is almost constant (with variations within 5 % of its value). An analogous situation happens also for other values of $\Omega/\varepsilon <1$, showing that in this regime one should expect the presence of a narrow temperature range for which the specific heat is not very sensitive to changes in coupling strength $\chi$, as long as this remains positive.
![Dimensionless specific heat $C_S/k_B$ as a function of rescaled temperature $k_B T/\varepsilon=1/(\beta \varepsilon)$ for $\Omega/\varepsilon =1/3$ and various small values of $|\chi/\varepsilon| - \Omega/\varepsilon$.[]{data-label="fig:6"}](Figure6.eps){width="8.5cm"}
Regardless of whether $\Omega$ is larger or smaller than $\varepsilon$, the specific heat approaches a unique functional form in the limits as $\chi$ approaches $\pm \Omega$. This limiting function can be calculated analytically: $$\lim_{\chi \rightarrow \pm \Omega} C_S = k_B \left(1- \left[
\frac{2k_B T}{\Omega}\sinh \left( \frac{\Omega}{2k_B T}
\right)\right]^{-2}\right) \label{eq:limCsChi2Omega}$$ The fact that it does not tend to zero at zero temperature is not in contrast with the third law of thermodynamics, since for $\chi =\pm \Omega$ the system is no longer stable.
Fig. \[fig:6\] depicts the behavior of the specific heat as $\chi/\varepsilon \rightarrow \pm \Omega/\varepsilon$ for $\Omega/\varepsilon <1$. The convergence to the limiting function in Eq. (\[eq:limCsChi2Omega\]) as $\chi/\varepsilon \rightarrow +\Omega/\varepsilon$ is quite fast compared to the much slower convergence in the other limit $\chi/\varepsilon \rightarrow -\Omega/\varepsilon$. The approach to the limit is qualitatively very different in the two cases. In both cases the specific heat vanishes at low temperature and it approaches the limiting curve in Eq. (\[eq:limCsChi2Omega\]) for large temperatures. However in the case $\chi/\varepsilon \sim -\Omega/\varepsilon$, the specific heat displays a drastic peak at intermediate temperatures. As $\chi/\varepsilon$ approaches $-\Omega/\varepsilon$, the peak becomes increasingly pronounced while getting closer to the origin of the temperature axis. In the limit $\chi/\varepsilon \rightarrow -\Omega/\varepsilon$, eventually the peak becomes a delta singularity at zero temperature. This singularity contributes with a finite term $\lambda$ to the total heat $Q=\int C_S dT$, which, in analogy with first order phase transitions, can be interpreted as a latent heat.
\[sec:conclusions\]Conclusions
==============================
We illustrated the validity of the Jarzynski equality and the work fluctuation theorem in the strong coupling regime, for the model Hamiltonian (\[eq:H\]). The central role is played by the thermodynamic partition function of the open system, that incorporates the interaction of the system of interest with its environment. The influence of the interaction is of major importance even in the seemingly trivial case in which the system bath interaction Hamiltonian commutes with both the bath and the system Hamiltonians, notwithstanding claims to the contrary [@Teifel:2007qr]. We computed the Hamiltonian of mean force for this model explicitly and studied its equilibrium thermodynamics. In particular we discussed its entropy and its specific heat as functions of temperature and other system parameters. Like for other strongly coupled system [@HIT_NJP08; @Ingold:2008mz] these two quantities can become negative at low temperature. Despite entropy and specific heat may become negative, they vanish at zero temperature, in accordance with the third law of thermodynamics. The only exception to this, is for the special value of coupling strength $\chi$ exactly equal to $-\varepsilon$, for which the zero temperature entropy is equal to $k_B \ln 2$. This result is, however, not in contradiction with the third law but rather corroborates this law; this is so because the two level fluctuator becomes degenerate in this case, as is clearly indicated by the Hamiltonian of mean force.
Interestingly, recent experiments in circuit cavity quantum electrodynamics [@Schuster:2007ph] use a parameter regime where negative entropy and specific heat may appear. For the architecture presented in [@Schuster:2007ph], these are expected below $\sim 22$ mK and $\sim 20$ mK respectively. In cavity quantum electrodynamics the Hamiltonian in Eq. (\[eq:H-time-ind\]) is obtained from the time-independent Jaynes-Cummings model Hamiltonian in the rotating wave approximation *and* dispersive regime [@Schleich]. These conditions imply weak coupling $|\chi| \ll \Omega,\varepsilon$, which in fact is the case for Ref. [@Schuster:2007ph]. Whether a Hamiltonian of the type in Eq. (\[eq:H\]), with time- dependent $\varepsilon(t)$, and/or possibly strong coupling can be implemented with superconducting circuits remains an open problem.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank David Zueco for fruitful discussions. Financial support by the German Excellence Initiative via the [*Nanosystems Initiative Munich*]{} (NIM) and the Volkswagen Foundation (project I/80424) is gratefully acknowledged.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We found that multiple circular walls (MCW) can be generated on a thin film of a nematic liquid crystal through a spiral scanning of a focused IR laser. The ratios between the radii of adjacent rings of MCW were almost constant. These constant ratios can be explained theoretically by minimization of the Frank elastic free energy of nematic medium. The director field on a MCW exhibits chiral symmetry-breaking although the elastic free energies of both chiral MCWs are degenerated, i.e., the director on a MCW can rotate clockwise or counterclockwise along the radial direction.'
address: 'Department of Physics, Graduate School of Science, Kyoto University & Spatio-temporal Project, ICORP, JST, Kyoto 606-8502, Japan\'
author:
- 'M. Kojima'
- 'J. Yamamoto'
- 'K. Sadakane'
- 'K. Yoshikawa'
bibliography:
- 'bibliography.bib'
title: Generation of multiple circular walls on a thin film of nematic liquid crystal by laser scanning
---
,
,
,
Received 11 February 2008; in final form 2 Marc 2008h
Introduction
============
Liquid crystal (LC) phases bring about a rich variety of textures as visualized by polarizing microscopy [@Dierking]. This polymorphism in the texture results from the inhomogeneity in molecular alignment due to defects in the LC [@deGennes]. Over the past few decades, much attention has been paid to colloidal dispersions in nematic LC because of their peculiar properties, such as a topological defect around a particle and long-range interaction among the particles [@Poulin_Science; @Poulin_PRE; @Stark]. This long-range interaction also results from inhomogeneity in the molecular alignment. The averaged molecular alignment is expressed as the director [@deGennes]. It has been found that a strong laser beam induces a distortion in the director field of nematic LC [@Durbin]. Especially, a linearly polarized laser can orient the director in the illuminated region in the direction of laser polarization [@Gibbons; @Simons]. It has been reported that defects in LC can be modified by use of laser manipulation [@Janossy; @Hotta]. Recently, M[ǔ]{}sevic et al. combined the property of a defect around a colloidal particle suspended in nematic LC with distortion in the nematic LC induced by a laser beam. They found that a particle with a lower refractive index than that of the surrounding nematic medium can be picked up with optical tweezers [@Musevic], although such a particle cannot be trapped in isotropic media. [Š]{}karabot et al. showed theoretically that such extraordinary trapping is achieved through the interaction between the laser-induced distortion in the director field and a topological defect near the particle [@Skarabot]. Thus, the interaction between a defect in LC and a laser beam produces various unique phenomena. Here, we tried to generate a new pattern in a nematic LC by using the interaction between a defect and a laser beam.
Materials and methods
=====================
The nematic material 5CB (Tokyo Chemical Industry co., Japan) was put into a microtube. Pure water was dispersed in the microtube. The nematic containing water droplets was vortexed. The microtube was then centrifuged for a short period to eliminate large water droplets. The nematic containing micron-sized droplets was placed between glass slides with a thickness a few $\rm\mu m$. The thickness was estimated by dividing the volume of nematic by the contact area on the glass surface. The nematic was sheared to easily give a Schlieren texture, and the glass slides were baked at 500 $^\circ$C for an hour before use.
Observations were performed through a polarizing microscope (converted IX70, Olympus, Japan) equipped with a $\times$ 100 oil immersion objective lens (UPlan Apo IR, N.A. 1.35, W.D. 0.1 $\rm mm$, Olympus, Japan). A linearly polarized Nd:YAG laser with a focus of $\sim$ 1 micrometer was introduced to the nematic by a dichroic mirror and the objective lens. The linearly polarized laser was constructed with a randomly polarized laser source (JOL-D8PK-Y, JENOPTIK, Germany) and a polarization beam splitter. The direction of laser polarization was controlled by a half-wave plate. The laser emission power was 2.0 W, as calibrated just before the objective lens. The observation and laser irradiation area in the nematic were controlled with a motorized stage (BIOS-302T, Sigma-koki, Japan).
Results
=======
Figure \[steady\] shows the responses of a brush distributed from a wedge disclination for horizontally and vertically polarized laser beams. The disclination was pinned to the glass substrate by chance. The strength of the disclination was -1/2, as judged from the response in the texture for a simultaneous rotation of the polarizer and analyzer of the polarizing microscopy while maintaining crossed nicols. When the brush was illuminated by a horizontally polarized laser beam (Fig. \[steady\] (a)), the texture showed a minute change (Fig. \[steady\] (b)). On the other hand, with a vertically polarized laser beam, the illuminated brush was repelled from the beam spot and the texture was completely changed from Fig. \[steady\] (c) to Fig. \[steady\] (d). The responses of the neighboring brush connected to an identical disclination core for polarized beams were inverted: the neighboring brush was only repelled from the horizontally polarized laser beam. When the laser was shut off, the conformation of the brush returned back to the texture seen before the laser irradiation. The distances between the repelled brush and laser spot were distributed broadly. We confirmed experimentally that brushes which grew from disclination cores ($\pm$ 1/2 and $\pm$ 1) are repelled from either a horizontally or vertically polarized beam spot. The responses of a disclination core to a linearly polarized laser beam have been reported by Hotta et al. [@Hotta].
Figure \[dynamical\] shows the emergence of a single-ring pattern. The real time movie of the process is available from the internet [@movie1]. The single-ring pattern centered on the beam spot was generated by a laser scanning along the trajectory depicted schematically in Fig. \[dynamical\] (A). We have chosen the velocity of the laser scan so as to grasp the brush in a steady manner. Actually, the spot was moved step-by-step, where typical one step is $\sim$ 20 $\rm\mu m$, and during the step-motion the speed was chosen $\sim$ 50 $\rm\mu m$/s. The present disclination core is identical to that in Fig. \[steady\]. Figure \[dynamical\] (B) (a)-(f) show snapshots at each point on the scanning trajectory in Fig. \[dynamical\] (A). The beam spot passed through without changing the conformation of the brush (Fig. \[dynamical\] (a)), whereas the neighboring brush was repelled from the beam spot(Fig. \[dynamical\] (b)). When the spot was further broken into the brush, the plucked part of the brush spontaneously closed and a dark ring centered on the beam spot was formed(Fig. \[dynamical\] (c)-(d)). The dark ring kept up with the motion of the laser spot with deformation from a complete circle. When the beam spot came to rest, the dark ring relaxed into a symmetrical circular form (Fig. \[dynamical\] (e)). The laser spot coated with the dark ring repelled the brush (Fig. \[dynamical\] (f)), which was not repelled in the case of a bare laser spot (Fig. \[dynamical\] (a)). We have succeeded in generating this pattern only from $\pm$ 1/2 disclinations. When the laser irradiation is shut off, the generated rings disappear within the order of 0.1 s. The ring pattern could not be generated when the velocity of the laser spot was above 100 $\rm\mu m$/s. In addition, the generated ring patterns broke down when the velocity of the laser spot was above 100 $\rm\mu m$/s. The sizes of the dark rings have a broad distribution.
Figure \[multi\_ring\] shows the appearance of a triple-ring pattern when the laser spot follows a trajectory $\sim$ a $+$ 1/2 disclination core. The real time movie of the process is available from the internet [@movie2]. The spot was moved step-by-step, where the typical one step is $\sim$ 20 $\rm\mu m$, and during the step-motion the speed was chosen below 30 $\rm\mu m$/s. The scanning process corresponds to three iterations of the pattern used to generate a single-ring pattern. The number of rings increased when the beam spot passed across the brushes (Fig. \[multi\_ring\] (B) (a)-(d)). The patterns typically measure dozens of micrometers, which is more than 10-fold larger than the size of the beam spot. When the laser irradiation is shut off, the patterns shrink toward the center. The extinction time of the pattern is on the order of a second, which is much longer than that of a single-ring pattern. In Fig. \[quad\] is shown the example of a quadruple-ring, which was generated with a similar procedure. In a generating process of the multiple-ring pattern, a new ring is formed outside of the existing multiple-ring pattern. When the new ring is created, the inner rings shrink in size. The radii of dark rings have a broad size distribution. The extinction time of a multiple-ring pattern due to shutting-off of the laser is extended with an increase in the number of rings.
![ (color online). Generation of a single-ring wall by laser scanning. (A) Schematic representation of the trajectory of laser scanning around a disclination. (B) Snapshots at the points labeled in (A). (a) The laser beam illuminates the region near the brush, where the director of the LC is parallel to the laser polarization. (b) The laser beam repels the brush, where the director is perpendicular to the laser polarization. (c) The beam spot bulldozes out part of the brush. (d) The plucked brush is closed spontaneously and a ring pattern emerges. (e) The ring is a complete circle while the beam spot is at rest. (f) The laser spot covered by a circular wall repels the brush, although the director on the brush is parallel to the laser polarization. When the laser beam is shut off, the ring pattern disappears immediately. The white arrow in (B)(a) represents the position of the beam spot in the observation area.The position is common in (a)-(f). Other symbols are identical to those in Fig. \[steady\][]{data-label="dynamical"}](fig2.eps)
![ (color online). Quadruple walls generated from a plural number of disclinations through a laser scanning. []{data-label="quad"}](fig4.eps)
Discussion
==========
Let us discuss the mechanism of the change in the conformation of the brush in Figs. \[steady\] and \[dynamical\]. Since the nematic had a planar configuration, we consider the system to be two-dimensional. The interaction between the local director and the laser beam can be interpreted in terms of the change in the dielectric free energy $\Delta F_E$, which can be written as [@Simons] $$\Delta F_E =\int {\rm d}\bm{r} (- \frac{\Delta \epsilon}{4 \pi} |\bm {n}(\bm{r}) \cdot \bm{E}(\bm{r})|^2)$$ where $\Delta \epsilon = \epsilon_\parallel - \epsilon_\perp$ is the anisotropy in the dielectric constants between parallel $\epsilon_\parallel$ and perpendicular $\epsilon_\perp$ to the director. The symbols $\bm{n}(\bm{r})$ and $\bm{E}(\bm{r})$ represent the director and oscillating electric field, respectively, of the incident laser beam at position $\bm{r}$. Since the nematic phase of 5CB has a positive value in $\Delta \epsilon$ ($\fallingdotseq$ 12 $\epsilon_0$, where $\epsilon_0$ is the dielectric constant of vacuum [@dielectric] ), the director in the illuminated region prefers to be oriented along the direction of laser polarization.
In our experiments, the optical torque for the director is strong enough to completely orient the illuminated director along the direction of laser polarization, because the beam spot appeared as black circles for both vertically and horizontally polarized laser beams. As a consequence, the illuminated director behaves as a boundary condition for the director field. When the illuminated director and polarization of the incident laser beam are parallel, the texture remains constant, because the illuminated director is already suited for the laser-induced boundary condition. When the director and laser polarization are perpendicular, the illuminated director is forced to rotate along the direction of polarization of the incident laser. As a result, the director field and texture change so as to satisfy the laser-induced boundary condition. We noted that the distances between the repelled brush and the laser spot are not constant in the experiments. This suggests that the interaction between the repelled brush and the laser spot is affected by the laser polarization and director distortion due to other disclination cores, distributed in the outside area of the photographs.
Figure \[multi\_ring\] (d’) shows the intensity profile of Fig. \[multi\_ring\] (d). The radii $r_i$, which have maximal values in the intensity profile, are approximately $r_1$ = 0.5 $\rm\mu m$, $r_2$ = 1.8 $\rm\mu m$ and $r_3$ = 6.0 $\rm\mu m$, where the subscript indicates $i$ = 1, 2, 3. In the same way, the radii $r'_i$, which take minimal values, are approximately $r'_1$ = 1.0 $\rm\mu m$, $r'_2$ = 3.4 $\rm\mu m$ and $r'_3$ = 11.0$\rm\mu m$. Thus, the ratios $r_i/r'_i$ are found to be $\sim$ 1/2. We have confirmed through the experiments that this ratio is constant for each triple-ring pattern although the radii of the rings exhibit dispersion. We calculate the elastic free energy of the director distortion to explain this trend in the triple-ring patterns. For simplicity, we assume that the director field on a multiple-ring pattern has cylindrical symmetry on a two-dimensional plane. Thus, the director $\bm{n}$ on a multiple-ring pattern depends only on the distance $r$ from the center of the beam spot. Therefore, the director field on a multiple-ring pattern is expressed as ${\bm n}=(n_x, n_y, n_z)= (\cos \psi(r), \sin\psi(r), 0)$, where $\psi$ is the azimuthal angle of the director. With one constant approximation of the elastic constants [@deGennes], the elastic free energy $F_{\rm ela}$ can be written as $$F_{\rm ela}=\int \frac{1}{2} K(\partial_\alpha n_\beta\partial_\alpha n_\beta){\rm d}\bm{r}.$$ where $K$ is the elastic constant of nematic medium. We consider the boundary conditions of a multiple-ring pattern as follows. Since the director on the beam spot is adjusted to the direction of laser polarization in the experiments, we adopt $\psi(r_{\rm c})=\psi_{\rm c}$ as an inner boundary condition, where $r_{\rm c}$ corresponds to the radius of the beam spot ($2r_{\rm c}\simeq$ wavelength $\lambda$, $\sim$ 1 $\rm\mu m$) and $\psi_{\rm c}$ is the azimuth angle of the direction of laser polarization. The parameter $r_{\rm b}$ is introduced as a cut-off length at which the director recovers the orientation angle $\psi_{\rm b}$ in the bulk, where $r_{\rm b}$ is several tens of $\rm\mu m$. If we minimize the elastic free energy, $\psi(r)$ is given as $$\label{director_sol}
\psi(r) =( \psi_{\rm b} - \psi_{\rm c} )\frac{\log(r/r_{\rm c})}{\log (r_{\rm b}/r_{\rm c})} + \psi_{\rm c}$$ Eq. (\[director\_sol\]) is independent of the elastic constant $K$. Eq. (\[director\_sol\]) indicates constant ratios between $r_i$ and $r'_i$ in the intensity profile. The difference $\psi(r_i) - \psi(r'_i)$ may be $\pi/4$ or $-\pi/4$. Thus, we have $$\log (r_i/r'_i)=\log(r_{\rm b}/r_{\rm c})(\psi(r_i) - \psi(r'_i))/(\psi_{\rm b}-\psi_{\rm c})$$ where the right-hand term is a constant. In the results shown in Figs.\[dynamical\] and \[multi\_ring\], the properties of a ring pattern, such as the size and extinction time, may depend on $r_{\rm b}$ and $\psi_{\rm b}$ originated in other disclinations. To control the size of the pattern, it is essential to control the distortion in the director field far from the laser spot.
Eq. (\[director\_sol\]) gives the director field on the pattern. The director on the pattern rotates along the radial direction. In the texture of a multiple-ring pattern, the director on a dark ring is perpendicular to that of the adjacent rings. Thus, a multiple-ring pattern is found as multiple circular walls (MCW) [@deGennes].
Figure \[simulated\_ring\] shows the existence of a chiral pair in MCWs calculated from Eq. (\[director\_sol\]). There are clockwise (Fig. \[simulated\_ring\] (a)) and counterclockwise (Fig. \[simulated\_ring\] (b)) MCWs, where the intensity profiles of Fig. \[simulated\_ring\] (a) and (b) are identical. In addition, the elastic energies of MCW in Fig. \[simulated\_ring\] (a) and (b) are also degenerated. In the experiments, we observed the chirality of MCWs by using the analyzer rotation technique [@Tabe03]. This chirality in MCWs is controlled by choosing the proper trajectory of laser scanning.
There have been several reports of target patterns with many rings in a thin film of SmC LC [@Tabe03; @Cladis85; @Cladis95; @Stannarius; @uto]. If we compare our results to those reports, the C-director of the SmC phase plays the role of the director of the nematic. Especially, an expression similar to Eq. (\[director\_sol\]) was previously obtained for the azimuth angle of C-director [@Cladis85].
![ (color online). Existence of a chiral pair in MCW. The intensity of both MCWs are based on Eq. (\[director\_sol\]). The director fields (blue lines) are calculated from Eq. (\[director\_sol\]). (a) Clockwise MCW ($\psi_{\rm b}$=-1.75$\pi$). (b) Counterclockwise MCW ($\psi_{\rm b}$=1.75$\pi$). The parameters $r_{\rm b}/r_{\rm c}$=32.0, $\psi_{\rm c}$=0 are the same in the two cases. In the experiments, chirality is determined by the trajectory of scanning.[]{data-label="simulated_ring"}](fig5.eps)
Conclusion
==========
We have reported a novel method for generating MCWs, which are a stable director field in nematic LC, through the use of proper laser scanning. We have shown that the chirality of MCWs can be controlled by choosing a suitable trajectory of laser scanning.
This work was supported by Technology of Japan and by a Sasakawa Scientific Research Grant (No. 19-643) from The Japan Science Society, Grant-in-aid for young researchers from Kyoto University Venture Business Laboratory (KU-VBL) and a Grant-in-Aid for Scientific Research on Priority Areas (No. 17076007) from the Ministry of Education, Culture, Sports and Science.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Transiting planets are generally close enough to their host stars that tides may govern their orbital and thermal evolution of these planets. We present calculations of the tidal evolution of recently discovered transiting planets and discuss their implications. The tidal heating that accompanies this orbital evolution can be so great that it controls the planet’s physical properties and may explain the large radii observed in several cases, including, for example, TrES-4. Also because a planet’s transit probability depends on its orbit, it evolves due to tides. Current values depend sensitively on the physical properties of the star and planet, as well as on the system’s age. As a result, tidal effects may introduce observational biases in transit surveys, which may already be evident in current observations. Transiting planets tend to be younger than non-transiting planets, an indication that tidal evolution may have destroyed many close-in planets. Also the distribution of the masses of transiting planets may constrain the orbital inclinations of non-transiting planets.
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Submitted 2008 Jul 3
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Introduction
============
Most close-in planets, and thus most transiting planets, have likely undergone significant evolution of their orbits since the planets formed and the gas disks dissipated. Jackson et al. (2008a) showed that the initial eccentricities of close-in planets were likely distributed in value similarly to the eccentricities of planets far from their host stars. Current eccentricities, as well as reduced semi-major axes, result from subsequent tidal evolution. Here we apply similar calculations of tidal evolution to transiting planets discovered more recently. In all cases, orbital evolution has been significant.\
Tides heat planets as they change their orbits. Jackson et al. (2008b) computed the heating that accompanies tidal circularization, showing that many close-in planets experience large and time-varying internal heating. They noted that tidal heating has been large enough recently enough that it may explain the anomalously large radii of many close-in planets. Since then, as more transiting planets have been discovered, the number of planets with anomalously large radii has increased. Here we compute the tidal-heating histories for recently discovered transiting planets.\
Tidal evolution can also affect the probability for transits to be observable from Earth because it changes the orbits. As a result, tidal evolution may introduce biases into transit observations as we discuss in Section 4 below.
Orbital Evolution
=================
The distribution of eccentricities $e$ of extra-solar planets is very uniform over the range of semimajor axis values $a > 0.2$ AU (Jackson et al. 2008a). Values for $e$ are relatively large, averaging 0.3 and broadly distributed up to near 1. For $a < 0.2$ AU, eccentricities are much smaller (most $e < 0.2$), a characteristic widely attributed to damping by tides after the planets formed and the protoplanetary gas disk dissipated. However, estimates of the tidal damping often consider the tides raised on the planets, while ignoring the tides raised on the stars. Results depend on assumed specific values for the planets’ poorly constrained tidal dissipation parameter $Q_p$. Perhaps most important, the strong coupling of the evolution of $e$ and $a$ is often ignored.\
[@Jacksonetal08a] integrated the coupled tidal evolution equations for $e$ and $a$ over the estimated age of many close-in planets. There we found that the distribution of initial (i.e. immediately after completion of formation and gas-disk migration) $e$ values of close-in planets do match that of the general population if stellar and planetary $Q$ values are $10^{5.5}$ and $10^{6.5}$, respectively, however the results are nearly as good for a wide range of values of the stellar $Q$. The accompanying evolution of $a$ values shows most close-in planets had significantly larger $a$ at the start of tidal evolution. The earlier gas-disk migration did not bring most planets to their current orbits. Rather, the current small values of $a$ were only reached gradually due to tides over the lifetime of the systems.\
Here we update those calculations by including more recently discovered transiting planets. The results highlight the importance of accounting for both the effects of the tide raised on the star and for the strong coupling between the evolution of semi-major axis and eccentricity. The results also show that neglecting either of these effects can lead to incorrect inferences about transiting (or other) close-in planets, such as beliefs that orbits must have been circularized.\
Fig. \[Fig1\] shows the past tidal evolution of recently discovered transiting planets. We include only planets for which a best-fit non-zero eccentricity has been published. We show the tidal evolution of $e$ and $a$, calculated as described by [@Jacksonetal08a], going back in time from their current nominal orbital elements.\
![Orbital evolution of all transiting planets with best-fit $e \neq 0$, discovered since [@Jacksonetal08a]. Current orbital elements are at the lower left end of each path. Tick marks are spaced at intervals of 500 Myr. Black dots show the age of each system for which an estimate is available. The age of OGLE-TR-211 b (called TR-211 in the figure) is not known; the black spot near the corresponding trajectory is for HAT-P-4 b. Orbital and physical parameters are taken from [@Alonsoetal08], [@Bakosetal07a], [@Kovacsetal07], [@Noyesetal08], [@Udalskietal08], [@Pollaccoetal08], [@Andersonetal08], [@Christianetal08], [@Joshietal08], and [@Winnetal08]. With $a = 0.1589$ AU, HD 17156 b ([@Gillonetal08]) is off the right-hand side of the plot and experiences negligible tidal evolution.[]{data-label="Fig1"}](Fig1.jpg){width="3.4in"}
As noted by [@Jacksonetal08a], the concavity of the $e$-$a$ trajectories reflects the dominant tide, either the tide raised on the planet, or the tide raised on the star. Where a trajectory is concave down (usually when $e$ is large), the effects of the tide raised on the planet dominates. Where a trajectory is concave up (usually when $e$ is small), the tide raised on the star dominates. In the latter case orbital angular momentum is transferred to the host star’s rotation, resulting in a spin-up of the star (such as may be the case for $\tau$ Boo b ([@Henryetal00]). In most of the trajectories in Fig. \[Fig1\], there is a transition from dominance by the tide on the planet to dominance by the tide on the star, as $e$ passes from large ($> 0.4$) to small values. In these cases, neglecting the effects of the tide raised on the star would underestimate the rate of tidal evolution of $e$ and $a$, especially later in the evolution, when eccentricities are small.\
In fact, ignoring the coupling of $a$ and $e$ evolution has been an implicit feature of a very common approach to estimating the damping of $e$ values. There are numerous examples in the literature, as reviewed by [@Jacksonetal08a], of computing and applying a “circularization timescale”, $e/(de/dt)$, which is incorrectly assumed to apply over a significant part of the tidal evolution. The actual change of $e$ over time is often quite different from these “circularization timescale” considerations, due to the coupled changes in $a$, as discussed by [@Jacksonetal08a].\
The problematic “circularization timescale” approach continues to be inappropriately applied for constraining the orbits of recently discovered transiting planets (e.g., [@Collier-Cameronetal07], [@ODonovanetal07], [@Gillonetal07], [@Pontetal07], [@Bargeetal08], [@Burkeetal08], [@Christianetal08], [@Johnsonetal08], [@JohnsKrulletal07], [@McCulloughetal08], [@Nutzmanetal08], [@Weldrakeetal08], and [@Joshietal08]). Consider the example of HAT-P-1 b. [@Johnsonetal08] obtained an orbital fit to the observations that admits an eccentricity as large as 0.067. However, on the basis of an estimate of the “circularization timescale”, they then assumed that $e = 0$. Their estimate for the circularization timescale, based on an assumed $Q_p = 10^6$, is 0.23 Gyr, which is less than 10% of the estimated age (2.7 Gyr) of the system. Based on those numbers, it is reasonable to expect the current $e$ to be several orders of magnitude smaller than its initial value. Even if we choose the larger value of $Q_p = 10^{6.5}$ suggested by the results of [@Jacksonetal08a], the circularization timescale only increases to 0.6 Gyr, still short enough that $e$ should have damped to less than 0.01 during the lifetime of this system. (If we also include the effect of tides raised on the star, the circularization timescale is even shorter, although this factor is negligible in this particular case.) But recall that the circularization-timescale approach does not take into account the coupled tidal evolution of $a$ along with $e$. In effect, the decrease in a over time means that $e/(de/dt)$ cannot be treated as a constant. According to Fig. \[Fig1\], when the coupling of $a$ and $e$ is taken into account, $e$ could have started at $<$ 0.6 (an unremarkable value for a typical extra-solar planet), and it would still have a value of 0.09 (the best-fit value previously reported by [@Bakosetal07a]) at the present time. Clearly the circularization-timescale method, as it is commonly used, can drastically over-estimate the damping of orbital eccentricities. For HAT-P-1 b and many other cases, it has incorrectly constrained eccentricity values.\
At best, the circularization timescale $e/(de/dt)$ can only describe the current rate of evolution, so it is not relevant to the full history of the tidal evolution. Even if one is only interested in the current rate, there are pitfalls. One, of course, is uncertainty in the appropriate values of $Q$. For example, in a discussion of WASP-10b, [@Christianetal08] found the persistence of substantial eccentricity to be surprising. Christian et al. calculated a circularization timescale substantially less than 1 Gyr for $Q_p = 10^5$ to $10^6$. In fact, with $Q_p = 10^6$, the damping timescale is about 1 Gyr, and with $Q_p = 10^{6.5}$, it is 3 Gyr, so it is not clear why the observed value (about 0.057) is surprising. The age of this system also has been estimated at 1 Gyr, so the current eccentricity would only be potentially problematic if $Q_p$ had the unlikely value of $10^5$.\
Another pitfall involves neglecting the effect of tides raised on the star by the planet, as was done by Johnson et al. and by Christian et al., and indeed in most of the papers cited above. For HAT-P-1 b they are not important. But for WASP-10 b, using the favored values $Q_p = 10^{6.5}$ and $Q_* = 10^{5.5}$, we find that including the effect of tides on the star, the circularization timescale drops from 3 Gyr to 0.5 Gyr. Evidently, the damping may be dominated by tides raised on the star. This dominance is also evident in Fig. \[Fig1\], where the evolution trajectory has been concave upward since the formation of the system. Also, in this case the timescale for damping $e$ is similar whether we simply use the circularization-timescale estimate or account for the coupled evolution of $e$ and $a$ (Fig. \[Fig1\]), because the system happens to be young. More generally, however, it is a mistake to rely on circularization-timescale estimates. Rather it is essential to include the full coupled equations for tidal evolution.\
Based on estimates of short “circularization timescales”, observers commonly assume $e = 0$ in fitting the orbits of close-in planets, rather than reporting observational upper limits on $e$. In fact, $e$-damping is probably much slower, as shown in Fig. \[Fig1\]. Thus those observers fitting data to possible orbits should not discount substantial $e$-values on the basis of uncertain tidal models, but should solve for the best fit (and range of uncertainty) to their observations. Those results will help constrain the histories of these systems. Moreover, even small $e$-values may have important implications for the physical properties of the planets as discussed in the next section.\
In the above discussion we focus on two papers that provide examples of how misleading it may be to neglect tides raised on the star or the coupling of the evolution of $a$ with $e$. We cite these particular cases because the circumstances happen to provide good illustrative examples. However, it should be understood that those authors were following what has come to be a widespread standard procedure as observers try to reduce the range of uncertainty of orbital elements implied by their data. All observers should adopt a more complete tidal model before over-interpreting possible constraints.
Tidal Heating
=============
Extra-solar planets close to their host stars have likely undergone significant tidal evolution since the time of their formation. Tides probably dominated their orbital evolution once the dust and gas cleared away, and as the orbits evolved, there was substantial tidal heating within the planets. The tidal heating history of each planet may have contributed significantly to the thermal budget governing the planet’s physical properties, including its radius, which in many cases may be measured by observing transits. Typically, tidal heating first increases as a planet moves inward toward its star and then decreases as its orbit circularizes.\
[@Jacksonetal08b] computed the plausible heating histories for several planets with measured radii, using the same tidal parameters for the star and planet that have been shown to reconcile the eccentricity distribution of close-in planets with other extra-solar planets. For several planets whose radii are anomalously large, we showed how tidal heating may be responsible. For one case, GJ 876 d ([@Riveraetal05]), tidal heating may have been so great as to preclude its being a solid, rocky planet. We concluded that theoretical models of the physical properties of any close-in planet should consider the possible role of tidal heating, which is time-varying, and can be quite large.\
Here we present heating rates and histories for transiting planets discovered more recently, and then discuss their implications. First, we calculate the possible tidal heating of planets whose eccentricities are reported to be zero (often because of the problematic circularization timescale discussed in Section 2 above). Because observations allow the possibility of non-circular orbits, we calculate tidal heating histories assuming various small but non-zero current eccentricities. Even very small current eccentricities ($<$ 0.01) may have dramatic implications in many cases. Second, we calculate heating histories of planets for which there is some non-zero eccentricity reported. We show that tidal heating is likely large in many cases and should be incorporated into physical models of planetary radii. We show that tidal heating may help reconcile discrepancies between predicted and observed planetary radii in many cases.
Planets with reported zero eccentricities
-----------------------------------------
Fig. \[Fig2\] shows the tidal heating history of TrES-4, the transiting planet with the largest radius identified to date [@Mandushevetal07]. We have computed these rates as described by [@Jacksonetal08b]. Although Mandushev et al. assume $e = 0$ for this planet, we show here (as for all the planets discussed in this section) heating curves for which we assume current $e$-values of 0.001, 0.01, and 0.03. The current heating rate (left edge of the graph) is greatest for the largest $e$ and smallest for smallest $e$, so the curves are readily identified. If it is available, an estimate for the age of the planet is shown as a vertical line. The “(I)" next to the planet’s name indicates that the observed radius is reported to be inflated relative to a theoretical prediction that neglected tidal heating. For planets that are reported not to be inflated, in subsequent plots, we put an “(N)" next to the name.
![Tidal heating for TrES-4. Orbital and physical parameters are taken from [@Mandushevetal07].[]{data-label="Fig2"}](Fig2.jpg){width="3.4in"}
![Tidal heating for CoRoT-Exo-1 b, WASP-4 b, WASP-1 b, and TrES-3, all of which are reported to be inflated. Orbital and physical parameters are taken from [@Bargeetal08], [@Wilsonetal08], [@Collier-Cameronetal07], and [@ODonovanetal07], respectively.[]{data-label="Fig3"}](Fig3.jpg){width="3.4in"}
As shown in Fig. \[Fig2\], for TrES-4 even if the current $e < 0.01$, tidal heating could have been $> 10^{19}$ W for much of the past billion years. That much recent heating may be sufficient to pump up the radii of close-in extra-solar planets, as discussed by [@Jacksonetal08b]. Thus, it may help explain the anomalously large radius of TrES-4. [@Liuetal08], motivated by [@Jacksonetal08b], obtained similar results, although they assumed a current $e$ of 0.04 and $Q_p = 10^5$.\
The shape of TrES-4’s heating curve is characteristic of many planetary heating curves. As tidal dissipation of orbital energy reduces $a$, the heating rate increases (time goes forward towards the left in these graphs). However, as $a$ drops, the rate of orbital circularization also increases. Eventually $e$ drops enough that tidal heating slows. (See Equation 1 from [@Jacksonetal08b].) This increase and subsequent decrease in the heating rate results in a peak in the heating for many planets, including TrES-4. [@Jacksonetal08c] showed that the peak occurs at $e = 0.34$ for cases where tides on the planet dominate the tidal evolution and orbital angular momentum is conserved.\
Figs. \[Fig3\] through \[Fig5\] show (similarly to Fig. \[Fig2\]) the heating for several other transiting planets whose reported $e = 0$. In the cases of CoRoT-Exo-1 b, WASP-4 b, WASP-1 b, TrES-3, HAT-P-5 b, and HAT-P-7 b (Figs. \[Fig3\] and \[Fig4\]), the radii have been reported to be inflated. Our results show recent (within the last Gyr) tidal heating exceeds $10^{19}$ W even for the small current eccentricities we consider. In these cases, it seems that tidal heating may contribute significantly to the inflated radii.
![Tidal heating of HAT-P-3 b, -5 b and -7 b. Orbital and physical parameters are taken from [@Torresetal07], [@Bakosetal07c], and [@Paletal08], respectively.[]{data-label="Fig4"}](Fig4.jpg){width="3.4in"}
![Tidal heating of OGLE-TR-182 b, XO-2 b and XO-4 b. Orbital and physical parameters are taken from [@Pontetal07], [@Burkeetal08], and [@McCulloughetal08], respectively.[]{data-label="Fig5"}](Fig5.jpg){width="3.4in"}
For HAT-P-3 b and OGLE-TR-182 b, the radii are reportedly not inflated. The tidal heating rates in Figs. \[Fig4\] and \[Fig5\] do not allow the tidal heating to exceed $10^{19}$ W at any time during the past billion years, which may be consistent with the uninflated radius.\
HAT-P-3 b’s radius is so small, in fact, that its discoverers ([@Torresetal07]) suggest that it may have a rocky core with a mass about 75 Earth masses. In the absence of tidal heating, such a large, rocky core would probably account for the small radius. However, such a large core could reduce the planet’s effective $Q_p$ by orders of magnitude. $Q_p$ for a rocky planet is probably $ \sim 10^2$. Such a value would yield much larger tidal heating than illustrated here. Similarly, any transiting planet whose observed radius seems to require the presence of a large, rocky core may be a candidate for severe tidal heating [@Jacksonetal08b]. This heating might affect the planet’s radius and physical state in ways that have not yet been modeled.\
XO-2 b does not have an inflated radius, which suggests that $e < 0.01$. Otherwise, according to Figs. \[Fig5\], there might have been enough recent tidal heating to have pumped up the radius. For XO-4 b, the range of eccentricities we consider does not allow heating to exceed $10^{18}$ W, but the radius has been reported to be inflated [@McCulloughetal08]. To explain the large radius by tidal heating would require the eccentricity of XO-4 b to exceed 0.03 or its $Q_p$ value to be smaller than we have assumed. Further study of this planet might include consideration of whether the transit data admit $e > 0.03$.
Planets with reported non-zero eccentricities
---------------------------------------------
In Figs. \[Fig6\] and \[Fig7\], we illustrate the tidal heating histories for planets for which some non-zero eccentricity values have been reported. We consider the range of tidal heating allowed by the observational uncertainties of $a$ and $e$. For each of these planets, the heating curves are shown for nominal current $a$ and $e$ values and for sets of $a$ and $e$ (within the range of observational uncertainty) that give the maximum and minimum current tidal heating rates. For CoRoT-Exo-2 b and OGLE-TR-211 b, the uncertainty extends to zero eccentricity, so the corresponding minimum heating rates are zero. Note that, in many other cases as well, the sources of orbital elements suggest that their observations may be consistent with a circular orbit, in which case past and present tidal heating would be small. Heating histories will become more reliable as future observational work yields better determination of orbital eccentricities.
![Tidal heating of WASP-3 b, CoRoT-Exo-2 b, HAT-P-1 b, HAT-P-6 b, and XO-3 b. Orbital and physical parameters are taken from [@Pollaccoetal08], [@Alonsoetal08], [@Bakosetal07a], [@Noyesetal08], and [@Winnetal08], respectively. []{data-label="Fig6"}](Fig6.jpg){width="3.4in"}
![Tidal heating of OGLE-TR-211 b, HAT-P-4 b and WASP-10 b. Orbital and physical parameters taken from [@Udalskietal08], [@Kovacsetal07], and [@Christianetal08], respectively. []{data-label="Fig7"}](Fig7.jpg){width="3.4in"}
All of the planets in Figs. \[Fig6\] and \[Fig7\] are reported to have inflated radii. In each case, the tidal heating exceeds $10^{19}$ W for some allowed value of the eccentricity. These results are consistent with our suggestion that tidal heating, either recent or current, may be responsible for the inflated radii seen in many planetary transits.
Transit Probabilities
=====================
The geometric probability for a planet to transit its host star increases the closer a planet is to the star. Consequently, the transit probability, $P$, is related to the planet’s orbital semi-major axis and eccentricity, as given by [@Barnes07]: $$P = \frac{R_* + R_p}{a (1-e^2)}$$ where $R_*$ is the stellar radius. This expression shows that we are more likely to observe planets transit when they have small $a$ and/or large $e$. Because tidal evolution changes $a$ and $e$, it also changes the transit probabilities of tidally evolving planets. The change in transit probability depends on a planet’s specific trajectory through $e$-$a$ space and thus depends sensitively on the physical and orbital parameters of the system.\
As a result, tidal evolution may introduce certain biases in transit statistics that will become more apparent as more transiting planets are discovered, as we describe below.
Evolution of Transit Probabilities
----------------------------------
Applying our tidal evolution calculations (e.g. Fig. \[Fig1\]) to Eqn. (1) above yields the evolution of transit probabilities. For example, Fig. \[Fig8\] illustrates the history of transit probabilities for three transiting planets. For LUPUS-TR-3 b, the transit probability is nearly constant over time, while the others show dramatic increases as the present time is approached. The difference arises from the relative contribution of tides raised on the stars. For close-enough-in planets, because a tide raised on a star exerts a negative torque on the planet’s orbit, the orbital angular momentum $L$ drops with time. The denominator in Eqn. (1) is proportional to $L^2$, so the transit probability increases with time. The magnitude of the effect is greater for planets with larger masses (because they raise larger tides on the star) and for stars with larger radii. LUPUS-TR-3 b has a much smaller mass relative to its star than CoRoT-Exo-3 b, and its star has a much smaller radius than HAT-P-3. As a result, the tide raised on LUPUS-TR-3 exerts a smaller torque, and $L$ is nearly constant, which explains the constant probability in Fig. \[Fig8\].
![Time evolution of transit probabilities for LUPUS-TR-3 b, HAT-P-7 b, and CoRoT-Exo-3 b. Orbital and physical parameters taken from [@Weldrakeetal08], [@Paletal08], and the exoplanets catalog located at exoplanets.eu, respectively.[]{data-label="Fig8"}](Fig8.jpg){width="3.4in"}
Masses and Ages of Transiting Planets
-------------------------------------
This result has important implications for the physical properties expected for transiting planets. Because the transit probabilities of planets that raise large tides on their stars increase with time, we might expect that transiting planets will tend to be more massive than non-transiting planets, and we might see them more often orbiting stars with larger radii.\
Fig. \[Fig8\] seems to suggest that transiting planets will be found preferentially when they are older, after tidal evolution has enhanced the transit probabilities. To test whether this trend is evident in the observed population, we compare all transiting planets to all non-transiting planets for which there is some estimate of the stellar age. Fig. \[Fig9\] shows the ratio of the planet’s mass ($M_p$) to its host star’s mass ($M_*$) vs. the best estimate of the stellar age.
![Distribution of planetary mass ratios and stellar ages for planets with a $<$ 0.2 AU. Black squares represent transiting planets, open squares non-transiting planets. The dashed line is drawn by hand but appears to define a region that contains no extra-solar planets. Data are taken from the sources listed in previous captions and from [@Butleretal06], [@Bonfilsetal07], [@GnS98], [@Bakosetal07b], [@FnV05], [@Geetal06], [@DaSilvaetal06], [@Nutzmanetal08], [@Johnsonetal06], [@Knutsonetal07], [@Pepeetal04], [@Gillonetal07], [@Holmanetal06], and [@Burkeetal08].[]{data-label="Fig9"}](Fig9.jpg){width="3.4in"}
Only one transiting planet is older than 6 Gyr (XO-5 b), whereas there are substantial fraction of non-transiting planets from 6 Gyr old to $>$ 10 Gyr. This result runs contrary to the idea that we would preferentially observe older transiting planets.\
This result can be explained by the fact that, while tidally-evolved, close-in planets have high probabilities to be transiting, this condition may be short-lived as the tides raised on the star cause a planet’s semi-major axis to drop faster the closer a planet is to the star. Planets that are close enough to have large transit probabilities have limited lifetimes ([@Jacksonetal08d]). Very quickly, tides drag a potentially transiting planet into the Roche zone of the star and tear the planet apart.\
The dearth of planets above the dashed line in Fig. \[Fig9\] also corroborates this scenario. The effects of the tide raised on the star are greater for planets with larger mass ratios, so these planets have shorter lifetimes. For example, a typical planet with a mass ratio $\geq$ $10^{-3}$ may not last more than 10 Gyr because Fig. \[Fig9\] shows a distinct lack of such planets.\
Fig. \[Fig9\] also shows several young transiting planets with mass ratios clustered around $10^{-2}$. For planets with such a large mass ratio, the effects of the tide on the star will preferentially enhance the transit probabilities, as discussed in the previous section. The larger fraction of transiting planets relative to non-transiting planets in this region of the figure is also evidence of the potential bias that tidal evolution introduces into transit observations.\
Fig. \[Fig9\] shows a lower cut-off in the mass-ratios of transiting planets at about $3 \times 10^{-4}$, while the mass ratios of non-transiting planets extend down to much lower values. However, for the non-transiting planets, their mass values come from radial-velocity observations, and are thus minimum masses. The transiting population probably gives a more accurate representation of the actual distribution of planetary masses. The difference between the two populations may give a sense of the inclination of the orbital planets of the non-transiting planets. The orbits of planets with $M_p/M_* < 10^{-4}$ are likely significantly inclined relative to our line of sight. Moreover, these results suggest the processes of planet formation tend to form hot Jupiters with a planet-to-star mass ratio $10^{-3}$. We encourage theorists to develop models that explain this mass ratio.
Conclusions
===========
Tidal evolution of the orbits of close-in planets involves coupling between semi-major axis and eccentricity. The effects of tides on both the host star and planet can be important and should be considered. Many of the recently discovered transiting planets have likely undergone significant tidal evolution, and in most cases, both changes in semi-major axis and the effects of the tide on the star have made important contributions to the evolution of orbital eccentricities. Observers should not discount the full range of possible current eccentricities based on overly simplified theories.\
Significant tidal heating accompanied the orbital evolution for most recently discovered transiting planets. In many cases, recent tidal heating rates have been large. These examples support our earlier suggestion that tidal heating may explain the otherwise surprising large radii of close-in planets. In cases for which the planet’s radius seems unaffected by heating, we may be able to place upper limits on the eccentricity. In cases for which heating may be required to account for a planet’s anomalously large radius, we may be able to place lower limits on the eccentricity.\
Tides raised on the star by a planet also increase the transit probability over time. The increase in probability depends on the physical and orbital parameters of the system, so tidal evolution likely introduces biases into transit observations. We may be more likely to detect the transits of planets with large masses relative to their stars and planets whose host stars have large radii. Current transit statistics seem to bear out these predictions. Transiting planets tend to have greater masses (relative to their stars) than the minimum mass values for radial-velocity planets, a result indicative of the distribution of orbital inclinations relative to our line of sight. As planets age and move closer to their star, their transit probability spikes, but only for a short time before the planet is destroyed, because close-in planets experience the strongest tidal effects. This process may explain why transiting planets tend to be younger than non-transiting planets. Tidal evolution is a key process in developing the orbital, physical, and statistical properties of planetary systems.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present an Angle-Resolved PhotoElectron Spectroscopy study of the changes in the electronic structure of electron doped Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ across the superconducting phase transition. By changing the polarization of the incoming light, we were able to observe the opening of the gap for the inner hole pocket $\alpha$, and to compare its behavior with the outer hole-like band $\beta$. Measurements along high symmetry directions show that the behavior of $\beta$ is consistent with an isotropic gap opening, while slight anisotropies are detected for the inner band $\alpha$. The implications of these results for the $s\pm$ symmetry of the superconducting order parameter are discussed, in relation to the nature of the different iron orbitals contributing to the electronic structure of this multiband system.'
author:
- 'B. Mansart'
- 'E. Papalazarou'
- 'M. Fuglsang Jensen'
- 'V. Brouet'
- 'L. Petaccia'
- 'L. de’ Medici'
- 'G. Sangiovanni'
- 'F. Rullier-Albenque'
- 'A. Forget'
- 'D. Colson'
- 'M. Marsi'
title: 'Opening of the superconducting gap in the hole pockets of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ as seen via Angle-Resolved PhotoElectron Spectroscopy.'
---
After the recent discovery of iron-based superconducting compounds [@Kamihara2008; @Rotter2008], several experimental and theoretical studies tried to explain the mechanisms underlying their high critical temperatures (T$_c$). To this end, the symmetry of the order parameter is one critical piece of information [@Hirschfeld2011]. For instance, considerable interest was raised by early theoretical predictions [@Mazin2008; @Kuroki2008], describing the possibility of a $s\pm$ symmetry with a gap following the law $\Delta=\Delta_0(cos(k_x)+ cos(k_y))$, implying a dephasing of $\pi$ between hole and electron pockets of the Fermi surface.
Experimentally, Angle-Resolved PhotoElectron Spectroscopy (ARPES) has been widely used in this search, thanks to its capability of resolving in k-space the electronic structure of materials [@Damascelli2003]. This technique has been employed to measure the amplitude of the superconducting gap (SC gap) in all the existing iron-based compound families: the 1111 (RE(O$_{1-x}$F$_x$)FePn, RE being a rare-earth and Pn a pnictogen) [@Sato2008; @Aiura2008], 11 (FeSe$_{1-x}$, FeSe$_{1-x}$Te$_x$ and FeTe$_{1-x}$S$_x$) [@Nakayama2010], 111 (LiFeAs and CaFeAs) [@Borisenko2010; @Umezawa2012], in the iron selenides A$_x$Fe$_2$Se$_2$ (A=Cs, K) [@Zhang2011] and in the 122 compounds (doped AEFe$_2$As$_2$, AE being an alcaline earth metal).
In particular, this latter family has been extensively studied, due also to the availability of high quality samples. The 122 compounds present a Fermi surface composed of three hole pockets (two almost degenerate inner $\alpha$ bands and an outer $\beta$ band) located around the $\Gamma Z$ direction, and two electron pockets ($\gamma$ and $\delta$) located around $XA$. The Fermi surface and the folded first Brillouin zone in two dimensions are shown in Fig. \[images\] (d). In a recent study, we showed that selective measurements of these bands are possible with polarization-dependent ARPES, which also gives insight into the orbital character for each one of them [@Mansart2011]. The highest T$_c$ in this family is obtained in optimally hole-doped Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$, for which several ARPES experiments showed that the SC gap amplitude was band-dependent [@Ding2008; @Wray2008; @Nakayama2009; @Nakayama2011], and presented a three-dimensional character for one of the hole pockets [@Zhang2010; @Xu2011]. However, another recent study showed an orbital independent SC gap amplitude in both hole-doped Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ and isovalent BaFe$_{2}$(As$_{1-x}$P$_x$)$_2$ [@Shimojima2011], raising the question of a possible orbital-fluctuations mediated superconductivity mechanism.
Only one ARPES measurement of the SC gap has been reported so far in the electron-doped system Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ [@Terashima2009]. The SC gap was measured to be larger in $\beta$ than in $\gamma$; no information was obtained on the $\alpha$ gap, since this band could not be seen to cross the Fermi level. Interestingly, the SC gap appeared isotropic for both $\gamma$ and $\beta$.
We report here an observation of momentum and band-dependent SC gaps using polarization-dependent ARPES in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$, which allowed us to detect the gap opening across T$_c$ for both $\alpha$ and $\beta$. These results confirm that the order parameter is consistent with a $s\pm$ symmetry. A slight anisotropy in the behavior of $\alpha$ along $\Gamma X$ and $\Gamma M$ could be detected, and we discuss how this may require the introduction of multiple harmonic terms for the description of the order parameter.
We studied single crystal samples of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$, x=0.07 with T$_c$=24.5 K, grown by the self-flux method [@Rullier-Albenque]. They were fully characterized and oriented prior to our measurements. ARPES experiments were performed on the BaDElPh beamline at the Elettra synchrotron light source [@Petaccia2009], which is optimized for photon energies $h\nu$ below 10 eV, allowing bulk-sensitive measurements; all the results presented in this article have been obtained at a photon energy of 9 eV. The photoelectron analyser was a SPECS Phoibos 150, with an energy resolution of 5 meV, and an angular resolution of $0.1^{\circ}$. Measurements were performed on high quality surfaces cleaved in-situ under ultra-high vacuum with pressure better than $5\times10^{-11}$ mbar. The reproducibility of all the experimental results presented here was verified via multiple thermal cycles across T$_c$, yielding consistent results.
![ARPES images acquired at T=14 K along the $\Gamma M$ sample orientation, (a) in p-polarization and (b) in s-polarization; (c) Momentum Dispersion Curves extracted from the images at E-E$_F$ = -10 meV. (d) experimental geometries for measurements along $\Gamma X$ and $\Gamma M$.[]{data-label="images"}](fig1.png){width="0.9\linewidth"}
ARPES images in the $\Gamma M$ direction are presented in Fig. \[images\] (a) and (b). Three hole-like pockets are visible, two almost degenerate inner $\alpha$ bands (that in our previous work [@Mansart2011] were named $\alpha_{\sigma}$ and $\alpha_{\pi}$) and an outer $\beta$ band, as shown in the Momentum Dispersion Curves (MDCs) in Fig. \[images\] (c). As shown in our previous study [@Mansart2011], photon polarization and $k_z$ dispersion effects make it possible to obtain information on their orbital character. Namely, even orbitals with respect to the photoemission plane are measureable with p-polarized photons, while odd ones are visible only with s-polarized photons. Our experimental geometry is depicted in Fig. \[images\] (d), and the notations for sample orientation and photon polarization are the same as in Ref. : in particular, $\Gamma X$ corresponds to the nearest neighbours Fe-Fe bond direction. In the $k_z\cong0.8~\left[4\pi/c\right]$ plane, probed with h$\nu$ = 9 eV, and along $\Gamma M$, $\beta$ has a marked $d_{z^2}$ character, which in our experimental configuration is even and thus measurable only in p-polarization. On the other hand, the $\alpha$ bands are formed by combinations of $d_{xz}$ and $d_{yz}$ orbitals, and are detectable for every sample orientation and polarization. Consequently, in p-polarization we simultaneously detect both the even $\alpha$ band and $\beta$, with $\beta$ having an overwhelming spectral weight. Conversely, in s-polarization $\beta$’s photoemission yield is suppressed by matrix element effects, and the odd $\alpha$ band can be measured in a very accurate way, which was not possible in the previous gap study on this system [@Terashima2009].
![(a)-(c) Energy Dispersion Curves for the $\alpha$ and $\beta$-bands, along $\Gamma X$ (a-b) and $\Gamma M$ (e-f); the insets show in more detail the Fermi level region. (c-d) and (g-h) Corresponding differences between superconducting phase and normal state. Red dots in (g-h) correspond to simulations using a BCS-like function, giving for $\beta$ a coherent quasiparticle with width 3.5 meV and centered at 2 meV, and for $\alpha$ the same width but at 3.5 meV away from the Fermi level.[]{data-label="gap_GM"}](fig2.png){width="0.9\linewidth"}
In Fig. \[gap\_GM\], we present the Energy Dispersion Curves (EDCs), extracted from ARPES images along $\Gamma M$ and $\Gamma X$, at the Fermi wavevector corresponding to each band and for two temperatures across the superconducting (SC) phase transition. Upon cooling down, a shift of the Fermi Leading Edge (FLE) is observed, corresponding to the opening of the SC gap, in every band. This shift can be also seen in the differences between the SC and normal state EDCs, shown in Fig. \[gap\_GM\] (c), (d), (g) and (h). The simple thermal contribution to the Fermi level broadening - different at T=14 K and T=30K - would give a symmetric peak-valley feature in the differences. For our data, the zero-level crossing happens at energies lower than the Fermi energy (indicated by an arrow in Fig. \[gap\_GM\] (g)), which is a very sensitive indication of the shift of the leading edge. Since no prominent quasiparticle peak could be observed in the SC state, we did not try to evaluate the gap value by modelling the spectral lineshape [@Norman1998; @Campuzano1996].
We preferred to carefully fit the EDCs with Fermi functions in order to precisely quantify the FLE shift - which is proportional to the gap - and measure this value for high symmetry directions in k-space. The fitting functions together with the experimental spectra for the $\alpha$ band along $\Gamma M$ are shown in Fig. \[fitting\] (a)-(b), and the difference between the two fitting functions across T$_c$ in Fig. \[fitting\] (c). The FLE shifts obtained by this procedure are the following: along $\Gamma M$, we obtain for $\beta$ 0.5$\pm$0.4 meV and for $\alpha$ 1.3$\pm$0.4 meV. On the other hand, along $\Gamma X$ the FLE shifts are 0.55$\pm$0.4 for $\beta$ and 0.6$\pm$0.4 for $\alpha$, as summarized in Fig. \[theo\] (c).
![Energy Dispersion Curves fitting by a Fermi function, for $\alpha$ band along $\Gamma M$ at (a) 30 K and (b) 14 K. (c) Fitting functions with their difference; the Fermi level crossing is shifted towards negative energies, indicating the opening of the superconducting gap.[]{data-label="fitting"}](fig3.png){width="1\linewidth"}
This small but unambiguous difference between the FLE shifts of $\alpha$ and $\beta$ deserves further attention, but first the question should be addressed if other factors than the opening of the SC gap can affect such shifts. One important factor that should be kept in mind is that the FLE is directly affected by the scattering rate: the stronger the scattering, the broader the superconducting coherent peak, and thus the smaller the Fermi leading edge. The scattering can be in general band- and momentum-dependent, and may produce an effective gap anisotropy along $\Gamma M$ and $\Gamma X$.
We tried to evaluate these possible effects on our data by using a BCS-like function to fit the differences in the EDC’s presented in Fig. \[gap\_GM\] (g) and (h), in order to model the transfer of spectral weight into the superconducting coherent peak for the two bands along $\Gamma M$. As already mentioned, the absence of a marked SC gap makes it hard to unambiguously fit the superconducting coherent peak. Once the hypothesis on the SC peak line shape is made (BCS in our case), fitting the SC-normal differences is instead a very sensitive way of obtaining information on the coherent peak position - the price to pay is of course the absence of any information on the overall density of states in the two phases. Our fits show that the differences for both $\alpha$ and $\beta$ can be correctly reproduced only by using the same values for the full width of the coherent peak (3.5 meV) in the two cases, but with different values for its position (3.5 meV and 2 meV for $\alpha$ and $\beta$, respectively). This suggests that the difference in scattering rate should not be regarded as the main factor in determining the inequivalent behavior for $\alpha$ and $\beta$ along $\Gamma M$; furthermore, it corroborates the hypothesis that the FLE shift is proportional to the SC gap, and indicates a slight anisotropy in the gap opening for the $\alpha$ band.
Before discussing the possible implications of this small anisotropy on the superconducting order parameter, we present in Fig. \[theo\] (a) a schematic view of the hole-like pockets, together with the contour lines expected for an $s\pm$ order parameter with a $cos(k_x)+cos(k_y)$ behavior, and we show the corresponding k-dependence of the gap amplitude in Fig. \[theo\] (c), in dotted lines. In Fig. \[theo\] (b) and (d) we present a similar situation with larger Fermi surfaces, which would be representative of a hole-doped compound.
One should observe that the contour lines of the order parameter magnitude evolve from a circular shape around the $\Gamma$ point to a diamond shape (where its value is zero) for larger momenta. Therefore its magnitude on a small circular Fermi surface around $\Gamma$ (green curve in Fig. \[theo\] (a) and corresponding gap in Fig. \[theo\] (c)) will be nearly constant, while a flower-shaped modulation is expected for larger circular Fermi surfaces (blue curve in Fig. \[theo\] (b) and gap in Fig. \[theo\] (d)). Also the overall magnitude is expected to be smaller in the latter case, because the Fermi surface approaches the node line (red dashed contour lines). This scenario is indeed found for the hole-doped pnictides of the 122 family [@Wray2008] (see Fig. \[theo\], (b) and (d)).
Conversely, our results do not fit into this simple picture: the experimental FLE shift values (blue circles in Fig.\[theo\] (c)) for the larger Fermi surface ($\beta$, dark blue) are quite isotropic , while the experimental FLE shift values (green squares in Fig.\[theo\] (c)) for the smaller Fermi surface ($\alpha$, in green in Fig.\[theo\] (a)) are compatible with a flower-like modulation.
Our results are consistent with the symmetry expected within the $s\pm$ scenario [@Mazin2008; @Kuroki2008]. However, they also indicate that the simplest version of this model, in which the superconducting order parameter follows a pure cos($k_x$)$+$cos($k_y$) law, has to be extended [@Kuroki2008]: in order to account for our findings, in particular for the modulation of the $\alpha$ band, a band dependent order parameter is needed.
One possibility for this is to have higher harmonics contributing to the order parameter, with orbital dependent weighting prefactors. Higher harmonics are needed for the gap on the inner Fermi surface to modulate, because they have more node lines, and they are closer to $\Gamma$; these harmonics have however to contribute much less to the gap of the outer Fermi surface. Such strong orbital dependence is in line with the analysis by Kuroki [*et. al*]{} [@Kuroki2008].
![(a) Inner ($\alpha$, green) and outer ($\beta$, blue) hole-like pockets in the folded Brillouin zone of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$; the black lines represent the order parameter contour lines corresponding to the $s\pm$ symmetry: the nodal lines (zero value) are the red dashed lines. (b) Example of a similar situation with larger Fermi surfaces (hole-doped compounds). (c) expected gap amplitude dispersion (in dotted lines) for the situation described in (a), and experimental points; green squares correspond to the $\alpha$ band and blue circles to the $\beta$ one. The green dashed line is a guide to the eye illustrating a behavior for $\alpha$ compatible with the experimental data (d) expected gap amplitude dispersions for the situation described in (b).[]{data-label="theo"}](fig4.png){width="0.95\linewidth"}
A few considerations should be made on the differences observed for the electron and hole doped compounds of the 122 family. First, as just discussed, the order parameter can be described with a single harmonic term for the hole doped systems, while multiple terms might be necessary for the electron doped ones. Then another important difference concerns the relation between gap amplitude and Fermi surface nesting: in hole doped Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$, the largest SC gap opens in the hole pocket presenting the most efficient nesting with electron pockets [@Nakayama2011], which can be easily explained by inter-band scattering favoring a larger SC order parameter amplitude.
In the case of electron-doped Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$, nesting with the electron pockets was found for the $\beta$ band [@Brouet2009], and this doesn’t seem to be correlated to a larger gap; it may be related to a stronger scattering rate for $\beta$ than for $\alpha$, which would affect the effective FLE shift, but as discussed above this is unlikely to be the main factor emerging from our experimental data. Furthermore, it should be kept in mind that the gap values presented here were measured for a $k_z$ different from the values explored in ref. [@Brouet2009]. Since Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ presents a marked three-dimensional character, an extensive $k_z$ dependent measurement of the gap (as performed in K-doped 122 compounds [@Zhang2010; @Xu2011]), would be needed to get clear indications on its relation with nesting.
The differences between hole and electron doped compounds may come from the multi-orbital character of their electronic structure. While the inner hole pockets of K-doped 122’s are formed by $d_{xz}$ and $d_{yz}$, the outer band is constituted of $d_{xy}$ [@Zhang2010]. On the other hand, the two inner bands of Co-doped 122’s are constituted of $d_{xz}$ and $d_{yz}$, but the outermost one is hybridized in a more complex way, especially with $d_{z^2}$ away from $k_z\cong1~\left[4\pi/c\right]$ [@Mansart2011]. It should be noted that a recent ARPES study on both hole-doped and isovalent-doped 122 compounds by Shimojima *et al.* shows evidence of an orbital independent order parameter amplitude for the different hole-like pockets [@Shimojima2011], while for the electron doped system presented here we find instead indications of orbital dependence. The importance of these considerations should trigger more studies on the symmetry of the order parameter comparing electron- and hole-doped compounds of the 122 family. For ARPES experiments, it would be particularly important to compare more extensive experimental results with detailed calculations of the electronic structure of these compounds: this would allow to unambiguously extract quantitative values for the SC gap, going beyond the level of uncertainty coming from the use of FLE shifts and taking into account the density of states of these material when modelling the transfer of spectral weight from the normal to the superconducting state.
In conclusion, we performed an angle resolved photoemission study of the hole-like Fermi surface above and below the superconducting phase transition in optimally doped Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ along high symmetry directions in k-space. The analysis of the Fermi leading edge shifts confirms that the SC gap is isotropic for the outer hole pocket $\beta$, while it shows slight anisotropies for the inner one, $\alpha$. These results are consistent with a $s\pm$ order parameter following a cos($k_x$)$+$cos($k_y$) behavior, and suggest that multiple harmonic terms may be included in its description. More comparative studies between hole and electron doped materials of the 122 family appear necessary to completely understand the implications of these results, and the dependence of the superconducting order parameter on the nature of the various orbitals contributing to the physics of these multiband materials.
The authors gratefully acknowledge D. Lonza for his technical assistance during the experiments, and E. Cappelluti and M. Capone for interesting discussions. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement nº 226716. L. de’ Medici is financially supported by Agence Nationale de la Recherche under Program No. ANR-09-RPDOC-019-01 and by RTRA Triangle de la Physique.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show the way in which the self-consistent Ornstein-Zernike approach (SCOZA) to obtaining structure factors and thermodynamics for Hamiltonian models can best be applied to two-dimensional systems such as monolayer films. We use the nearest-neighbor lattice gas on a square lattice as an illustrative example.'
address: |
$^1$Department of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794-3800\
$^2$Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400
author:
- 'Chi-Lun Lee$^1$ and George Stell$^2$'
title: SCOZA for Monolayer Films
---
-0.1in
Introduction
============
The self-consistent Ornstein-Zernike approach (SCOZA) was introduced by Høye and Stell[@scoza1] as an approximation method specifically tailored to obtain structure factors and thermodynamics for Hamiltonian models in three or more spatial dimensions. It was subsequently found by Høye and Borge[@ising2d] that the SCOZA yields extremely accurate results for the two-dimensional lattice gas as well, when appropriately used, thus opening the way toward the use of SCOZA in treating thin-film problems. In this article we summarize the two-dimensional SCOZA results. We point out that those results for systems considered in the thermodynamic limit are strikingly similar to the results that would be found in an exact analysis of two-dimensional systems that are finite[@mccoy] or semi-infinite[@onsager]. We note why this is to be expected, and using the behavior of the specific heat as a criterion, we find the size of the finite and semi-infinite systems that yield the best match to the SCOZA results for the infinite square lattice.
Background
==========
The SCOZA is based on an ansatz used by Ornstein and Zernike[@oz], which is that the direct correlation function $c({\bf r})$ introduced by those authors has the range of the pair potential. In SCOZA, this ansatz is used along with a core condition that guarantees that the two-body distribution function $g({\bf r})$ must be zero for values of ${\bf r}$ for which the pair potential is infinite. In a lattice gas, the core-condition implies the exclusion of multiple occupancy of a single lattice site or cell; for the equivalent Ising model in which the spin variable at site $i$ is +1 or -1, the corresponding condition is that $\langle s_i s_j\rangle_{i=j} = 1$, which simply reflects the fact that the spin must be pointing either up or down with probability one.
An analysis of the Ornstein-Zernike formalism made by one of the authors some time ago[@stell] showed that the core condition plus the assumption that $c({\bf r})$ is proportional to the pair potential implies that for short-ranged potentials, a two-dimensional system can not have a critical point at nonzero temperature. (In three or more dimensions there is no such restriction on criticality.) One knows however, that in two dimensions, systems such as the nearest-neighbor lattice gas do in fact have a critical point at nonzero $T_c$. Hence SCOZA did not initially appear to be a promising method for treating arbitrarily large two-dimensional1 systems. Its apparent unsuitability is also consistent with the observation[@stell; @scoza2] that the assumption that $c({\bf r})$ has the range of the potential implies that for short-ranged potentials, the critical exponent $\eta = 0$, so that at a critical point at $T_c \neq 0$, where on expects $g({\bf r}) -1 \approx const./r^{d-2+\eta}$, SCOZA would yield $g({\bf r}) -1 \approx const./r^{d-2}$. In three dimensions, in which $\eta \approx 0.03$, and $d-2+\eta \approx 1.03$, this leads to negligible error. But for $d = 2$, in which $\eta = 1/4$, it represents the difference between $g({\bf r}) -1 \approx const.
/r^{1/4}$ and a $g({\bf r}) -1$ that does not appropriately decay with increasing $r$.
However, as we shall see below, in two dimensions, the SCOZA results for a square lattice of infinite extent are strikingly similar in some respects to exact results for either an $N\times
N$ lattice or an $N \times \infty$ lattice, with $N \approx 22$.
It is not hard to understand why the assumption of a $c({\bf
r})$ for an infinite square lattice yields results that mimic exact results for a finite system. For a finite system, $c({\bf
r})$ is limited in range by the finite boundaries of the system. One also knows that in an exact analysis of an infinite one-dimensional system with short-ranged potential one finds no critical behavior for non-zero temperature. From these qualitative statements however, it is not clear what values of $N$ in an $N\times N$ or $N\times \infty$ lattice will give rise to exact results that most closely match SCOZA results for an infinite square lattice. As we shall see in Section IV, when one uses the behavior of the specific heat as a criterion, $N$ turns out to be around $22$.
Theory
======
In the following we consider the two-dimensional square lattice gas, which is isomorphic to the two-dimensional Ising model. The potential between particles is $$v({\bf r_i}-{\bf r_j}) = \left\{
\begin{array}{ll}
+\infty, & \mbox{${\bf r_i} = {\bf r_j}$}\\
-w, & \mbox{i, j nearest neighbors}\\
0, & \mbox{otherwise.}
\end{array}
\right.$$ For convenience $w$ is scaled to be $1$ in our calculations. In this convention the internal energy per spin for the Ising model $U$ is related to the internal energy per particle for the lattice gas $u$ via the following relation: U = u +12 q - q , where $q$ is the number of nearest neighbors ($q = 4$ for the square lattice), and $\rho$ is the density for the lattice gas.
SCOZA is based on the enforcement of thermodynamic consistency between different routes to thermodynamics. This imposes the following relation: = , \[consistency\] where $\beta = 1/T$, the inverse temperature, $\rho^{-2} \chi$ is the isothermal compressibility, and $u$ is the internal energy per particle for the lattice gas. These quantities can be given in terms of correlation functions through fluctuation theory, which yields \^[-1]{}= , \[chi1\] and through the ensemble average of the Hamiltonian, which yields u = -qg\_1 = -q(1+h\_1) , \[energy\] where $h({\bf r}) \equiv g({\bf r}) - 1$, and $g({\bf r})$ is the two-body distribution function. Here $g_1$ and $h_1$ represent the functional values of $g({\bf r})$ and $h({\bf r})$ at nearest-neighbor positions, and $\tilde{h}({\bf k})$ is the Fourier transform of $h({\bf r})$, which is related to the direct correlation function $c({\bf r})$ by the Ornstein-Zernike equation: h([**r\_i**]{}) = c([**r\_i**]{}) + \_[j]{} c([**r\_j**]{}) h([**r\_i**]{}-[**r\_j**]{}) , \[OZ\] or in the Fourier-transformed space: 1+([**k**]{}) = . \[fourier\]
The above relations are exact. In order to proceed we shall approximate the form of the direct correlation function $c({\bf r})$ by using the ansatz introduced by Ornstein and Zernike (OZ) that $c({\bf r})$ has the range of the pair potential. In SCOZA this can be done by generalizing somewhat the mean spherical approximation (MSA), in which ([**k**]{}) = c\_0 + q c\_1 ([**k**]{}) , \[SCOZA\] where $\Phi ({\bf k})$ is the nearest neighbor sum, ([**k**]{}) = , , and $c_0$ and $c_1$ are functions of $(\rho,\beta)$. Eq. (\[SCOZA\]) is the OZ ansatz applied to the lattice gas. The MSA is the special case obtained by setting $c_1 = \beta$ and adjusting $c_0$ to be compatible with the core condition that assigns zero probability to multiple occupancy of a single site. In SCOZA one instead adjusts $c_1$ to insure self consistency between Eq. (\[chi1\]) and (\[energy\]).
The core condition $h(0)= -1$ implies a relation between $c_0$ and $c_1$ through the OZ equation: 1-&=&\
&=&\
&& , \[core\] where $z \equiv q\rho c_1/(1-\rho c_0)$. $P(z)$ is the value for the lattice Green function $P(z,{\bf r})$ at ${\bf r} = 0$. For a two-dimensional square lattice we have P(z) = \_[0]{}\^ = K(z) , where $K(z)$ is the complete elliptic integral of the first kind. From Eq. (\[core\]) we have c\_0 = , and c\_1 = . By taking ${\bf r_i} = 0$ in Eq. (\[OZ\]), we get -1=h(0)=c\_0-c\_0 + qc\_1 h\_1 , so h\_1 = -\[1+(1-)c\_0\] = - .
After substitutions Eq. (\[chi1\]) and (\[energy\]) become \^[-1]{} = \[chi2\] and u = -12 q (- (1-) ) . From Eq. (\[chi2\]) we conclude that in SCOZA the criticality, if any, occurs at $z = 1$, since at the critical point one has $\chi^{-1}= 0$, and we have $P(z) > 0$ for all $z$.
Finally, by applying the thermodynamic consistency via Eq. (\[consistency\]), we get the SCOZA partial differential equation: \[(1-z)P(z)\] = - - q . The boundary conditions are $z=0$, i.e., $P(z)=1$, at $\beta =0$ and $\rho = 0, 1$.
Specific Results and Discussion
===============================
Since for the two-dimensional square lattice $P(z)$ diverges when $z \rightarrow 1$, the renormalized inverse temperature parameter $c_1$ also diverges. As a result, SCOZA fails to predict a true critical point above zero temperature in this case. Nevertheless, the SCOZA results for $u$ match the exact Onsager expression[@onsager] for $u$ remarkably well over the whole temperature range. Instead of having an infinite slope at the exact critical temperature, the SCOZA slope achieves its maximum at a temperature within a fraction of a percent of the exact value. The nonsingular but near-singular behavior near the ideal transition temperature makes our results strikingly similar to the exact results for a finite-size Ising model on a square lattice, $N\times N$[@mccoy], or a finite-width strip, $N\times
\infty$[@onsager], for an $N$ a bit greater than 20.
In Fig. 1 we plot the negative internal energy $-U$ versus the inverse temperature $\beta$ along the critical isochore $\rho =
\rho_c =1/2$, i.e., magnetization being equal to 0, for the comparison between SCOZA and both infinite- and finite-size Ising exact results. The comparison is made even clearer by plotting their residuals in Fig. 2. We find that the deviations between SCOZA and the other results are very small over the whole temperature range. The deviations get larger near the critical point $\beta = \beta_c$, although the largest deviation is still within 3 percent. Comparing with finite-size and finite-width exact results, we find this deviation gets minimized when we choose $N=22$ for an $N\times N$ Ising model or $N=21$ for an $N\times \infty$ Ising model.
In Fig. 3 we plot the specific heat versus $\beta$ along the critical isochore. The SCOZA result has a specific heat that stays finite at its maximum, as all the finite-size and finite-width specific heats do in an exact theory. In this comparison we again find that the SCOZA infinite-lattice result has great resemblance to the $22\times 22$ or $21\times \infty$ Ising model. The SCOZA curve matches well with the $21\times \infty$ Ising model for $\beta < \beta_c$. But for $\beta > \beta_c$ the SCOZA result has somewhat less difference to the $22\times 22$ solution. The maximum of SCOZA curve occurs at $\beta_{SCOZA} = 1.758$, whereas the exact critical point for the infinite Ising model is $\beta_c
= 1.763$. The maximum for the $22\times 22$ and $21\times \infty$ Ising models occur at $\beta_{22\times 22} = 1.735$ and $\beta_{21\times \infty} = 1.764$, respectively. The deviations between these maximum temperatures and the exact critical temperature is of the same order for the SCOZA and $21\times
\infty$ results, whereas it is a bit larger for the $22\times 22$ case. In this regard the SCOZA is a bit more similar to the $21\times \infty$ model.
For each fixed value of $\rho$, we define the temperature where the specific heat is at its maximum as the pseudo singular temperature. Here we try to determine a ’pseudo’ spinodal curve by collecting the set of $(\rho, T)$ points that correspond to those pseudo singularities. This ’pseudo’ spinodal curve is shown in Fig. 4. Note that the points on the curve don’t mark real singularities, and both the specific heat and the compressibility remain finite, but very large, at these points. Furthermore, near the pseudo critical point $\rho = \rho_c = 1/2$, by defining $\Delta \rho = \rho-\rho_c$, $\Delta T=T_c - T$, we find \~(T)\^[\_[spinodal]{}]{} , with a classical exponent $\beta_{spinodal} = 1/2$ when $T < T_c$.
From Eqs. (\[fourier\]) and (\[SCOZA\]) we find that in SCOZA the function $1+\rho \tilde{h}({\bf k})$ has a form proportional to the Fourier transform of the lattice Green function $P(z, {\bf k})$. Hence we have $\delta ({\bf r}) + \rho h({\bf r}) \propto P(z,{\bf r})$. From the fact that h([**r**]{}) \~P(z,[**r**]{}) \~(-2r) when ${\bf r} \rightarrow \infty$[@montroll], we find in SCOZA the correlation length = 12 . In Fig. 5 we plot the correlation length $\xi$ versus temperature at the critical isochore. Since for the two-dimensional SCOZA scheme $z \rightarrow 1$ only when $T
\rightarrow 0$, the correlation length keeps finite at the exact critical temperature $T_c$ and only diverges when $T\rightarrow
0$. However, when $T=T_c$ we have $\xi \simeq 44$ according to the SCOZA scheme, which already indicates strong correlations. Furthermore, the correlation length increases sharply right below $T=T_c$.
The authors gratefully acknowledge the support of the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U. S. Department of Energy.
[99]{} Høye J S and Stell G 1977 [*J. Chem Phys.*]{} [**67**]{} 439; ibid 1977 [*Mol. Phys.*]{} [**52**]{} 1071. Høye J S and Borge A 1998 [*J. Chem. Phys.*]{} [ **108**]{} 8830. McCoy B M and Wu T T 1973 [*The Two-Dimensional Ising Model*]{} (Harvard Univ. Press). Onsager L 1944 [*Phys. Rev.*]{} [**65**]{} 117. Ornstein L S and Zernike F 1914 [*Proc. Akad. Sci. (Amsterdam)*]{} [**17**]{} 793. Stell G 1969 [*Phys. Rev.*]{} [**184**]{} 135. Dickman R and Stell G 1996 [*Phys. Rev. Lett.*]{} [**77**]{} 996. Montroll E W, Potts R B and Ward J C 1963 [*J. Math. Phys.*]{} [**4**]{} 308.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present deep spatially-resolved optical spectroscopy of the NW companion galaxy of the quasar BR 1202-0725 at $z=4.7$. Its rest-frame UV spectrum shows star-forming activity in the nuclear region. The Ly$\alpha$ emission profile is symmetric with wavelength while its line width is unusually wide (FWHM $\simeq 1100$ km s$^{-1}$) for such a high-$z$ star-forming galaxy. Spectrum taken along the Ly$\alpha$ nebula elongation, which is almost along the minor axis of the companion host galaxy, reveals that off-nuclear Ly$\alpha$ nebulae have either flat-topped or multi-peaked profiles along the extension. All these properties can be understood in terms of superwind activity in the companion galaxy. We also find a diffuse continuum component around the companion, which shows similar morphology to that of Ly$\alpha$ nebula, and is most likely due to scattering of the quasar light at dusty halo around the companion. We argue that the superwind could expel dusty material out to the halo region, making a dusty halo for scattering.'
author:
- 'Youichi Ohyama, Yoshiaki Taniguchi, & Yasuhiro Shioya'
title: 'Subaru Deep Spectroscopy of a Star-forming Companion Galaxy of BR 1202-0725 at $z=4.7$ [^1] '
---
INTRODUCTION
============
BR 1202$-$0725 at $z=4.7$ is one of well-studied quasars at high redshift (e.g., Kennefick, Djorgovski, & Meylan 1996; Ohta et al. 1996; Omont et al. 1996; Hu, McMahon, & Egami 1996; Petitjean et al. 1996; Storrie-Lombardi et al. 1996). A faint companion galaxy is also found to be associated with BR 1202-0725 (Djorgovski 1995). This is located at 2.3 NW of the quasar (hereafter NW companion) and shows an extended Ly$\alpha$ nebula almost at the same redshift as that of the quasar (Hu et al. 1996; Petitjean et al. 1996; Storrie-Lombardi et al. 1996). Previous optical spectroscopies show that it is a star-forming galaxy with only a narrow Ly$\alpha$ emission (no strong metal emission lines, such as N [v]{}, C [iv]{}) (e.g., Petitjean et al. 1996; Hu, McMahon, & Egami 1997; Fontana et al. 1998). The redshifted \[O [ii]{}\] emission is also detected on the galaxy at near-infrared wavelength (Ohta et al. 2000; see also Pahre & Djorgovski 1995 for earlier measurement with only an upper-limit value). Interestingly, redshifted CO emission is detected at another location near the quasar (4 NW of the quasar), being $\simeq 2$ away from the NW companion (hereafter, 2nd CO emitter), as well as on the quasar itself (Omont et al. 1996; Carilli et al. 2002; see also Ohta et al. 1996). There are lines of evidence for vigorous star-forming activities at quasar and 2nd CO emitter (e.g., massive content of molecular gas, high excitation temperature of CO lines, optical-FIR SED which is typical of vigorous star-forming galaxies) (e.g., Benford et al. 1996; Omont et al. 1996; Ohta et al. 1996; Ohta et al. 1998; Ohta et al. 2000; Yun et al. 2000; Carilli et al. 2002). Therefore, the “BR 1202-0725 group” seems to contain various kinds of young star-forming objects, and is suitable for investigating star-forming activities in early universe.
Among the three objects in the BR 1202-0725 group, the NW companion would give us an unique opportunity to investigate nature of star formation in a young galaxy because this galaxy appears to be free both from extremely dusty environment and from intense quasar light. Therefore we have made very deep optical spectroscopy of the NW companion with the 8.2m Subaru Telescope, and present our new results in this paper. We assume $\Omega_{\rm M}=0.3, \Omega_\Lambda=0.7, H_0=70$ km s$^{-1}$ Mpc$^{-1}$ in this paper. 1 corresponds to 6.7 kpc, in the adopted cosmology.
OBSERVATION AND DATA REDUCTION
==============================
We used the FOCAS (Kashikawa et al. 2002) attached at Cassegrain focus of the Subaru Telescope (Iye et al. 2004) on Feb. 14 and 15, 2004. A VPH grism (with 600 grooves mm$^{-1}$ and 6500Å central wavelength) and an order-sorting filter Y47 were used to cover a wavelength range from 5950Å to 8400Å. With a combination of a 0.8 width longslit, this setting provides a spectral resolution of $R=1700$ (measured with sky lines) near the redshifted Ly$\alpha$ ($\simeq 6930$Å). The CCDs were binned onchip to $3\times 2$ (0.3 $\times$ 0.2 in space and wavelength directions, respectively). The slit was placed on the companion at two position angles (PAs), at PA$=-38.08^{\circ}$ (along the quasar and the NW companion: Hu et al. 1996; hereafter, NW-SE slit) on the first night, and at PA$=-128.08^{\circ}$ (perpendicular to the NW-SE slit; hereafter, NE-SW slit) on the second night (Figure 1). We took eight 30 minute exposures per slit PA (four hours in total). A small nodding of the target along the slit was applied during each exposure. Seeing size was $\simeq 0.5$$-0.8$ FWHM during the observations, and was typically 0.6 FWHM.
Data reduction was made in a standard manner, i.e., bias subtraction, flat fielding, wavelength calibration with Th-Ar arc lines, sky subtraction, and spectral sensitivity calibration with a standard star GD153, were applied. Atmospheric absorption features were corrected with another spectrum of GD153, taken with exactly the same spectroscopy setting as that for the companion. Then, a one-dimensional nuclear spectrum was extracted from the two-dimensional spectra in the following way. First, two nuclear spectra taken at each slit PA were extracted over 0.6 aperture along the slit around the peak, and then two spectra were coadded. Then, a contamination of the bright quasar light is corrected, by subtracting the scaled quasar spectrum from the observed spectrum. Here the scale of the quasar spectrum was estimated by measuring the quasar flux at the same distance to the companion but at another side of the quasar (2.3 SE of the quasar) on the spectrum taken along the NW-SE slit. Figure 2 shows the final companion nuclear spectrum as well as the quasar one. Further, we deduced one-dimensional spatial flux distribution of both continuum and Ly$\alpha$. For each slit PA, two-dimensional spectrum is averaged over the wavelength range of $\lambda > \lambda_{\rm Ly\alpha}$ and just around $\lambda_{\rm Ly\alpha}$ to obtain spatial flux distributions of continuum and Ly$\alpha$, respectively (Figure 3).
RESULTS
=======
The nuclear spectrum (Figure 2) is composed of a narrow Ly$\alpha$ emission, almost flat continuum emission at $\lambda > \lambda_{\rm Ly\alpha}$, and partially absorbed continuum at $\lambda < \lambda_{\rm Ly\alpha}$. The Ly$\alpha$ emission is detected at $\lambda_{\rm peak}=6932$Å (or $z=4.7026$), and its width is $\simeq 1100$ km s$^{-1}$ FWHM, being consistent with previous results (Petitjean et al. 1996; Fontana et al. 1998). The line profile is almost symmetric along wavelength, except for a narrow absorption line at blue side of the profile ($z=4.687$, or $\Delta V\equiv V-V_{\rm Ly\alpha~peak}\simeq -800$ km s$^{-1}$), and can indeed be reproduced with a Gaussian emission affected by a single absorption line (Figure 2). We point out that, although Petitjean et al. (1996) showed the blue-deficient asymmetric profile of Ly$\alpha$, such profile seems to be a result of unresolved blue absorption in their lower-quality spectrum.
The nuclear Ly$\alpha$ flux is $f$(Ly$\alpha$ nuc.)$=6.5 \times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$, which is significantly smaller than that of the total one ($f$(Ly$\alpha$ total)$=2.7 \times 10^{-16}$ erg s$^{-1}$ cm$^{-2}$). Note that the total flux, rather than the nuclear one, is consistent with the previously reported values (Hu et al. 1996; Petitjean et al. 1996; Fontana et al. 1998). The continuum spectrum at red side of Ly$\alpha$ ($\lambda > \lambda_{\rm Ly\alpha}$) shows neither absorption nor emission lines within the observed wavelength range, although rather poor S/N of the continuum spectrum hampered detection of rather week absorption lines. The continuum flux at 1400Å, without extinction correction, is $f(\rm continuum)=7.3 \times 10^{-32}$ erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$. At blue side of Ly$\alpha$ ($\lambda < \lambda_{\rm Ly\alpha}$), the spectrum shows absorbed continuum, which looks very similar to the quasar one showing rich Ly$\alpha$ absorptions and a DLA system (e.g., Storrie-Lombardi et al. 1996).
We found that both the Ly$\alpha$ and the continuum emissions come from extended region out to $3$$ - 4$ from their peaks along the NE-SW slit, and their flux peaks coincide with each other (Figure 3). This result is slightly different from that of Hu et al. (1996), where they showed that the peak of the Ly$\alpha$ nebula is located about 0.6 E from the continuum peak. However, the nebula elongation (NE) appears to be closer to the direction of the peak displacement (E) mentioned by Hu et al. (1996). The Ly$\alpha$ nebula seems to show more centrally concentrated flux distribution comparing with that of the continuum along the slit, especially at the SW side of the companion. The NE nebula is brighter and more extended from the nucleus (distance from the nucleus: $r\simeq 4$ or $\simeq 27$ kpc). At $r= 0.5$$-1.5$ NE, the emission shows flat-topped profile, which is remarkably different from one at nucleus (Figure 4). We found that the profile can be reproduced by a combination of two Gaussian components (blue and red) at $\Delta V \simeq \pm 400$ km s$^{-1}$, and the line width of each component is $600-1200$ km s$^{-1}$ FWHM. At even outer NE regions ($r=2$$-3.5$ NE), the emission shows more complicated shapes, and the profile is probably composed of three velocity components (a near-systemic one at $\Delta V \simeq +200$ km s$^{-1}$, a blue one at $\Delta V \simeq -700$ km s$^{-1}$, and a red one at $\Delta V \simeq +500$ km s$^{-1}$) (see spectrograms in Figure 1, as well as Ly$\alpha$ line profiles in Figure 4). The overall line width, including these three components, is as wide as $\sim 1500$ km s$^{-1}$, and the mean velocity is close to the systemic velocity ($|\Delta V| \lesssim 200$ km s$^{-1}$). At another side of the companion, the SW nebula, is fainter and more compact in space ($\simeq 3$ or $\simeq 20$ kpc). At $r=0.5$$-2.5$ SW, the profile looks similar to that at outer NE nebula, although the profile looks composed of two brighter components at blue ($\Delta V\simeq -400$ km s$^{-1}$) and near-systemic velocity ($\Delta V \simeq 0$ km s$^{-1}$) as well as a possible fainter component at red ($\Delta V \sim +500$ km s$^{-1}$) which is barely visible on a smoothed spectrogram.
Compared with the NE-SW extension, the extension along the NW-SE slit is much less prominent, although Ly$\alpha$ nebula is slightly more extended than that for the continuum. There is an extension toward NW (toward 2nd CO emitter) out to $r \simeq 2$ ($\sim 14$ kpc). The profile there looks more complicated than that the NE nebula, and is probably composed of two narrow components with overall velocity extent of as wide as 1000 km s$^{-1}$ at its mean velocity of $\Delta V \sim +500$ km s$^{-1}$. At another side of the quasar toward SE, fainter and narrow ($\sim 500$ km s$^{-1}$ FWHM) extension is detected ($r \simeq 1$, or $\simeq 7$ kpc). There is no major velocity structure along the extension ($|\Delta V| \lesssim 200$ km s$^{-1}$), although details of its kinematical properties are not well known due to contamination of brighter quasar light. These properties (extent and velocity structure) of the nebula along the NW-SE slit are consistent with the report of Fontana et al. (1998). We also note that there is a nebula extension even at SE of the quasar ($r \sim 4$) at $\Delta V \sim -500$ km s$^{-1}$ (see a spectrogram in Figure 1). Since we do not have any detailed information for it, no discussions will be made on it.
DISCUSSION
==========
Star-Formation Activity of the NW Companion
-------------------------------------------
The NW companion shows the following distinct characteristics: (1) It has a narrow \[FWHM(NW companion)$<<$FWHM(quasar)\] Ly$\alpha$ emission, whose redshift is very close to that of quasar. (2) Its continuum spectrum shows a step in flux between blue and red sides of Ly$\alpha$. (3) It has no strong metal emission lines, such as N [v]{} and C [iv]{}, being suggestive of AGN activity. All these indicate that the companion is a star-forming galaxy which is physically associated with BR 1202-0725 quasar. We estimate the star-forming rate (SFR) to be $\simeq 13$ $M_{\rm \sun}$ yr$^{-1}$ based on the relation of Kennicutt (1998) and the estimated continuum luminosity at 1500Å from our measurement at 1400Å. Note that the Ly$\alpha$ luminosity seems not useful for estimating SFR, because Ly$\alpha$ emission from the extended nebula, which is likely to be excited by shock, may contribute to the nuclear Ly$\alpha$ flux (see later sections for details).
Properties of the companion host galaxy is examined by using a stellar continuum color of $I$ and $K$. We do not use $R$ data, because it includes a strong Ly$\alpha$ emission and bluer-than-Ly$\alpha$ continuum, both of which are difficult to be modeled. We adopt $I=24.1\pm 0.2$ and $K=23.4\pm 0.4$ from Fontana et al. (1996) and Hu et al. (1996), and a contribution of redshifted \[O [ii]{}\]$\lambda$3727 in $K$ is corrected based on \[O [ii]{}\] equivalentwidth measured by Ohta et al. (2000). Continuum emission from the ionized gas is neglected here, because such continuum emission is generally much fainter than that of stellar continuum in starburst galaxies. We adopt a synthetic stellar spectrum model “starburst 99” (Leitherer et al. 1999) to match the observed color. We adopt instantaneous burst models (IMF slope $\alpha=2.35$, $M_{\rm up}=100$ M$_{\rm \odot}$, $M_{\rm low}=1$ M$_{\rm \odot}$, $Z=1/5Z_{\rm \sun}$ or $Z=1/20Z_{\rm \sun}$) with various ages (5-50 Myr). Here we fix IMF parameters, and ignore the reddening effect for simplicity. By comparing models and the observed color, we found that models of lower metallicity ($\simeq 1/20 Z_{\rm \sun}$) and younger age (10 Myr) better represent the observations (Figure 5). It seems important to note that the observed blue UV color can only be reproduced if almost no reddening is applied on the young stellar population with bluest color.
Nature of the Extended Ly$\alpha$ Nebula
----------------------------------------
### Kinematic Properties of the Nebula
Although the NW companion shows characteristic properties for star-forming galaxies, it also shows unusual properties for such a high-$z$ star-forming galaxy: (1) At the nucleus, Ly$\alpha$ emission profile is wide ($\simeq 1100$ km s$^{-1}$ FWHM) and almost symmetric in wavelength, being different from narrower ($200-400$ km s$^{-1}$ FWHM) and blue-deficient asymmetric profiles which are typical for such a high-$z$ star-forming galaxy (e.g., Dawson et al. 2002; Ajiki et al. 2002; Kodaira et al. 2003; Rhoads et al. 2003; see for a review Taniguchi et al. 2003 and references therein). (2) At the off-nucleus, the companion show an extended Ly$\alpha$ nebula with a complicated velocity structure, whose overall velocity width is as wide as $\sim 1500$ km s$^{-1}$ . If we assume that an extended nebula with such a violent kinematical status surrounds the companion, we could expect that the high velocity component ($|\Delta V| > 300$ km s$^{-1}$) on the nucleus arises from the nebula along our line-of-sight toward the companion nucleus. If this is the case, the NW companion is likely composed of a usual star-forming galaxy and the surrounding extended nebula with violent kinematical status.
Here a question arises as what is the origin of the extended nebula. There seems several possibilities: (1) scattering of the quasar light by circumgalactic gas, (2) photoionization of circumgalactic gas by the quasar UV light, (3) spatially extended star-formation of the companion, (4) cooling radiation from protogalaxies within dark matter halos (e.g., Haiman, Spaans, & Quataert 2000), and (5) galaxy-scale shock heating. First possibility can be rejected due to different spectrum shape between the nebula and the quasar. Second possibility seems less likely since no strong emission lines suggestive of AGN excitation, such as N [v]{}, are detected. Third and fourth possibilities seem also less likely since they could not explain straightforwardly the observed kinematical structure of the nebula, such as wide line width and flat-topped/multi-peaked profiles. Also, the third possibility is less likely because the host galaxy elongates along NW-SE directions (Hu et al. 1996), being perpendicular to the direction of the nebula elongation. Therefore, the last possibility seems most feasible, since shock could be excited within the nebula showing the observed violent kinematical structure, and it can emit intense Ly$\alpha$ emission (e.g., Shull & McKee 1979), if we assume that some kind of mechanism works to produce a violent internal motion of the nebula.
We showed that the Ly$\alpha$ profile in the NE nebula can be composed of two (blue and red) or three (blue, near-systemic, and red) components at inner and outer parts of the nebula, respectively (see section 3). We note that the inner nebula, which appears to show only two components, could be composed of three components, and the fainter and narrower near-systemic one was missed due to nearby brighter and wider components. The SW nebula also seems to have two (blue and near-systemic) components, and a possibly fainter red component. One may think that these nebulae show similar kinematical properties, and might form a single, large, elongated nebula at both sides of the companion along its minor axis. An idea to explain a pair of blue and red components at a same position within the nebula is to introduce an expanding or contracting shell of ionized gas, in which each component comes from either front or back side of the shell. A near-systemic component may be attributed to other component, e.g., a dusty halo without violent motion or scattering the companion nuclear light, although we can not discriminate these ideas.
### The Superwind Model
Because of the vigorous star-formation activity of the NW companion, it seems natural to expect a superwind activity associated with the companion. The superwind blows at later phase of the starburst evolution when the large number of OB stars die as supernovae (SNe) and release huge kinetic energy into circumnuclear region, and a circumnuclear bubble of shock-heated hot ionized gas expands eventually out to halo area (see, e.g., Heckman, Armus, & Miley 1990 for details). During the course of expansion, the bubble interacts with dense gas within the galactic disk, and it preferentially elongates along the disk polar direction where the density is smallest. The nebula emits UV-optical emission lines (including Ly$\alpha$) by shock heating occurring around the hot gas bubble where it interacts with ambient cold matter. Therefore, the superwind nebula often forms an expanding shell or bubble, being capable of producing a pair of blue- and red-shifted nebula emissions. Since the companion shows a “linear” or highly elongated disk-like structure in continuum along NW-SE direction (Hu et al. 1996), the superwind nebula, if any, would show elongation along NE-SW direction, being consistent with the observation. Superwind nebula can extend as large as a few kpc $-$ several tens kpc, and the wind velocity can be as fast as a few $-$ several hundreds km s$^{-1}$ (e.g., Heckman et al. 1990). Therefore, the superwind model can explain velocity structure and morphology of the nebula in a qualitative way. An axis of the superwind outflow is likely to be close to the sky plane, as expected from the highly elongated appearance of the host galaxy, and this could help to explain weak velocity shear along the nebula extension.
According to the galactic wind theory (e.g., Arimoto & Yoshii 1987), the star formation ceases when the galactic wind blows out. Therefore, the youngest stellar population, which has born when the wind just started to blow out, will contribute most to the observed UV light due to largest UV luminosity/mass ratio, if SFR is almost constant while star-bursting. Since the blue UV color indicates an age of the stellar population as $\sim 10$ Myr, the wind age (or timescale of the wind propagation) is likely to be $\sim 10$ Myr. If this is the case, the superwind nebula could extend out to $\simeq 4$ kpc/$f$ (or $\simeq 0.6$/$f$), assuming a constant wind speed of $V_{\rm exp} =$ ($1/2$ of velocity separation of blue- and red-Gaussian components at off-nuclear region)/$f \simeq 400$ km s$^{-1}$/$f$, where $f$ is a geometrical conversion factor for calculating transverse velocity from the line-of-sight velocity. This estimate is consistent with the observed typical nebula size ($\simeq 3$) if $f$ is $\sim 1/5$. Therefore, all information (stellar color, line width, and nebula size) can be understood in a context of the superwind model, although all the estimates are based on a simple and/or order-of-magnitude calculation.
Origin of Diffuse Continuum Component around the Companion
----------------------------------------------------------
We showed that the continuum emission extends spatially along the NE-SW slit over $\pm 3$ from the companion nucleus. Since the Ly$\alpha$ nebula shows more concentrated flux distribution around the nucleus, this component should show a continuum-dominated spectrum at $\lambda > \lambda_{\rm Ly\alpha}$, especially at SW side of the companion (Figure 3). Therefore, neither of scattering of the companion light, photoionization either by the quasar nor the companion are likely as an origin of the component, since all these models would create a Ly$\alpha$-dominated spectrum. Therefore, the most likely origin of the component is a scattering of the quasar light at circumgalactic matter.
The HST image also reveals a diffuse continuum structure around the companion (Figure 1). This component seems to show similar morphology to that of the Ly$\alpha$ nebula, i.e., the elongation is found along NE-SW direction but not toward NW (information toward SE is lost due to bright quasar light). Therefore, the medium for the quasar light scattering is likely to be associated with the NW companion. Here, a question arises as how such a component is created around the companion. There are several possibilities: (1) tidal structures around the companion made during merging process of the companion, quasar, and the 2nd CO emitter, (2) pre-galactic clumps remaining around the companion, (3) dusty ejecta from the companion, and (4) genuine dusty halo of the companion. Although we could not discriminate these possibilities, the third possibility seems more likely, because neutral matter could be transferred from dusty nuclear region out to halo as well as the ionized gas within a context of superwind model (Heckman et al. 2000 and references therein). If this is the case, we can explain similar morphological properties between Ly$\alpha$ nebula and the continuum structure. Because there are plenty of molecular gas (and hence dusts) within the BR 1202-0725 group (at quasar and 2nd CO emitter), it seems natural to consider that the star formation occurred in a dusty environment also in the companion. If we assume that the dusty circumnuclear material has been expelled by the superwind outflow, we might explain a current weak-reddening nuclear environment of the companion.
[^1]: Based on data collected at the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We theoretically investigate magnon-phonon hybrid excitations in two-dimensional ferromagnets. The bulk bands of hybrid excitations, which are referred to as magnon-polarons, are analytically shown to be topologically nontrivial, possessing finite Chern numbers. We also show that the Chern numbers of magnon-polaron bands and the number of band-crossing lines can be manipulated by an external magnetic field. For experiments, we propose to use the thermal Hall conductivity as a probe of the finite Berry curvatures of magnon-polarons. Our results show that a simple ferromagnet on a square lattice supports topologically nontrivial magnon-polarons, generalizing topological excitations in conventional magnetic systems.'
author:
- Gyungchoon Go
- Se Kwon Kim
- 'Kyung-Jin Lee'
title: 'Topological Magnon-Phonon Hybrid Excitations in Two-Dimensional Ferromagnets with Tunable Chern Numbers'
---
#### Introduction—
Since Haldane’s prediction of the quantized Hall effect without Landau levels [@Haldane1988], intrinsic topological properties of electronic bands have emerged as a central theme in condensed matter physics. The band topology can be characterized by emergent vector potential and associated magnetic field defined in momentum space for electron wavefunctions, called Berry phase and Berry curvature, respectively [@Xiao2010]. The Berry curvature is responsible for various phenomena on electron transport such as anomalous Hall effect [@Onoda2006; @Sinitsyn2007] and spin Hall effect [@Murakami2003; @Kane2005; @Bernevig2006]. In addition, nontrivial topology of bulk bands gives rise to chiral or helical edge states according to the bulk-boundary correspondence [@Hatsugai1988].
Recently, research on the effects of Berry curvature on transport properties, which was initiated for electron systems originally, has expanded to transport of collective excitations in various systems. In particular, magnetic insulators, which gather great attention in spintronics due to their utility for Joule-heat-free devices [@Kajiwara2010], have been investigated for nontrivial Berry phase effects on their collective excitations [@Katsura2010; @Onose2010; @Matsumoto2011; @Matsumoto2011b; @Shindou2013; @Zhang2010]: spin waves (magnons) and lattice vibrations (phonons). Previous studies exclusively considering either only magnons or only phonons showed that they can have the topological bands of their own, thereby exhibiting either the magnon Hall effect in chiral magnetic systems [@Katsura2010; @Onose2010; @Matsumoto2011; @Matsumoto2011b; @Shindou2013] or the phonon Hall effect [@Zhang2010] when the Raman spin-phonon coupling is present.
Interestingly, the hybridized excitation of magnons and phonons, called a magnetoelastic wave [@Ogawa2015] or magnon-polaron [@KikkawaPRL2016], is able to exhibit the Berry curvature and thus nontrivial topology due to magnon-phonon interaction [@Takahashi2016; @Zhang2019; @Park2019], even though each of magnon system and phonon system has a trivial topology. In noncollinear antiferromagnets, the strain-induced change (called striction) of the exchange interaction is able to generate the nontrivial topology in the magnon-phonon hybrid system [@Park2019]. In ferromagnets, which are of main focus in this work, nontrivial topology of magnon-polarons is obtained by accounting for long-range dipolar interaction [@Takahashi2016]. In addition, in ferromagnets with broken mirror symmetry, the striction of Dzyaloshinskii-Moriya (DM) interaction leads to topological magnon-polaron bands [@Zhang2019].
![(a) The schmematic illustration of the magnon and phonon system. The ground state of the magnetization is given by the uniform spin state along the $z$ axis (red arrow). (b) The Chern number of our magnon-phonon hybrid system. $H_{\rm eff}$ represents the effective magnetic field including the anisotropy field and the external magnetic field, $H_{\rm eff} = K_z S + \cal B$. Here we use the parameters $S = 3/2$, $\hbar \omega_0 = 10$ meV, and $M c^2 = 5\times 10^{10}$ eV. []{data-label="fig:1"}](fig1.pdf){width="86mm"}
In this Letter, we theoretically investigate the topological aspects of the magnon-phonon hybrid excitation in a simple two-dimensional (2D) square-lattice ferromagnet with perpendicular magnetic anisotropy \[see Fig. \[fig:1\](a) for the illustration of the system\]. Several distinguishing features of our model are as follows. Our model is optimized for atomically thin magnetic crystals, *i.e.*, 2D magnets. The recent discovery of magnetism in 2D van der Waals materials opens huge opportunities for investigating unexplored rich physics and future spintronic devices in reduced dimensions [@McGuire2015; @Zhang2015; @Lee2016; @Gong2017; @Huang2017; @Bonilla2018; @OHara2018; @Fei2018; @Deng2018; @Burch2018; @Gibertini2019]. Because we consider 2D model, we ignore the non-local dipolar interaction, which is not a precondition for a finite Berry curvature in 2D magnets. Moreover, the Berry curvature we find does not require a special spin asymmetry such as the DM interaction nor a special lattice symmetry: Our 2D model description is applicable for general thin film ferromagnets. Therefore, we show in this work that even without such long-range dipolar interaction, DM interaction, or special lattice symmetry, the nontrivial topology of magnon-phonon hybrid can emerge by taking account of the well-known magnetoelastic interaction driven by Kittel [@Kittel1958]. As the Kittel’s magnetoelastic interaction originates from the magnetic anisotropy, which is ubiquitous in ferromagnetic thin film structures [@Dieny], our result does not rely on specific preconditions but quite generic. Furthermore, we show that the topological structures of the magnon-polaron bands can be manipulated by effective magnetic fields via topological phase transition. We uncover the origin of the nontrivial topological bands by mapping our model to the well-known two-band model for topological insulators [@Bernevig2006], where the Chern numbers are read by counting the number of topological textures called skyrmions of a certain vector in momentum space. At the end of this Letter, we propose the thermal Hall conductivity as an experimental probe for our theory.
#### Model—
Our model system is a 2D ferromagnet on a square lattice described by the Hamiltonian $$\label{eq:htot}
H = H_{\rm mag} + H_{\rm ph} + H_{\rm mp} \, ,$$ where the magnetic Hamiltonian is given by $$\begin{aligned}
H_{\rm mag} = -J \sum_{\langle i,j\rangle} {{\bf S}}_i\cdot {{\bf S}}_j - \frac{K_z}{2} \sum_i S_{i,z}^2 - {\cal B} \sum_i S_{i,z},\end{aligned}$$ where $J > 0$ is the ferromagnetic Heisenberg exchange interaction, $K_z >0 $ is the perpendicular easy-axis anisotropy, and $\cal B$ is the external magnetic field applied along the easy axis. Throughout the paper, we focus on the cases where a ground state is the uniform spin state along the $z$ axis: $\mathbf{S}_i = \hat{\mathbf{z}}$. The phonon system accounting for the elastic degree of freedom of the lattice is described by the following Hamiltonian: $$\begin{aligned}
\label{Hph}
H_{\rm ph} = \sum_i \frac{{{\bf p}}_i^2}{2M} + \frac{1}{2} \sum_{i,j,\alpha,\beta} u_i^\alpha \Phi_{i,j}^{\alpha,\beta} u_j^\beta,\end{aligned}$$ where ${{{\bf u}}}_i$ is the displacement vector of the $i$th ion from its equilibrium position, ${{\bf p}}_i$ is the conjugate momentum vector, $M$ is the ion mass, and $\Phi_{i,j}^{\alpha,\beta}$ is a force constant matrix. The magnetoelastic coupling is modeled by the following Hamiltonian term [@Kittel1958; @Thingstad2019]: $$\begin{aligned}
\label{mpint}
H_{\rm mp} = \kappa \sum_i \sum_{{{\bf e}}_i} \left({{\bf S}}_i \cdot {{\bf e}}_i \right) \left(u^z_i - u^z_{i+{{\bf e}}_i} \right),\end{aligned}$$ where $\kappa$ is the strength of the magnon-phonon interaction and ${{\bf e}}_i$’s are the nearest neighbor vectors. Equation describes the magnetoelastic coupling as a leading order in the magnon amplitude, where the in-plane components of the displacement vector do not appear.
We note here that our model Hamiltonian does not include the dipolar interaction and the DM interaction, distinct from the model considered in Refs. [@Takahashi2016] and [@Zhang2019]. Because the above-mentioned interactions are absent in our model, neither ferromagnetic system nor elastic system exhibits the thermal Hall effect when they are not coupled. In other words, they are invariant under the combined action of time-reversal ($\cal T$) and spin rotation by $180^\circ$ around an in-plane axis ($\cal C$) [@Zhang2019]. It is the magnetoelastic coupling term $H_{\rm mp}$ that breaks the combined symmetry $\cal T \cal C$ and thus can give rise to the thermal Hall effect as will be shown below.
#### Magnon-phonon hybrid excitations—
We first diagonalize the magnetic Hamiltonian $H_{\rm mag}$ and the phonon Hamiltonian $H_{\rm ph}$ separately, and then obtain the magnon-phonon hybrid excitations, which are called magnon-polarons, by taking account of the coupling term $H_{\rm mp}$.
The magnetic Hamiltonian is solved by performing the Holstein-Primakoff transformation $S^x_i \approx (\sqrt{2S} / 2) (a_i + a^\dag_i)$, $S^y_i \approx (\sqrt{2S} / 2i) (a_i - a^\dag_i)$, $S^{z}_i = S - a^\dag_i a_i$, where $a_i$ and $a_i^\dagger$ are the annihilation and the creation operators of a magnon at site $i$. By taking the Fourier transformation, $a_i = \sum_{{{\bf k}}} e^{i {{\bf k}}\cdot {{\bf R}}_i} a_{{{\bf k}}} / \sqrt{N}$, where $N$ is the number of sites in the system, we diagonalize the magnetic Hamiltonian in the momentum space: $$\begin{aligned}
H_{\rm mag} = \sum_{{{\bf k}}} \hbar \omega_m({{\bf k}}) a^\dag_{{{\bf k}}} a_{{{\bf k}}},\end{aligned}$$ where the magnon dispersion is given by $\omega_m ({{\bf k}}) = [2JS\left(2-\cos{k_x} - \cos{k_y}\right) + K_zS + {\cal B}]/\hbar$.
For the elastic Hamiltonian $H_{\rm ph}$, it is also convenient to describe in the momentum space: $$\begin{aligned}
H_{\rm ph} = \sum_{{{\bf k}}} \left[\frac{{p}^z_{-{{\bf k}}} {p}^z_{{{\bf k}}}}{2M} + \frac{1}{2} {u}^z_{-{{\bf k}}} \Phi({{\bf k}}) {u}^z_{{{\bf k}}}\right],\end{aligned}$$ where only nearest-neighbor elastic interactions are maintained as dominant terms and the momentum-dependent spring constant is $\Phi({{\bf k}}) = M\omega_0^2 \left(4 - 2\cos{k_x} - 2\cos{k_y}\right)$, where the characteristic vibration frequency $\omega_0$ corresponds to the elastic interaction between two nearest-neighbor ions. To obtain the quantized excitations of the phonon system, we introduce the phonon annihilation operator $b_{{{\bf k}}}$ and the creation operator $b_{{{\bf k}}}^\dagger$ in such a way that $$\begin{aligned}
\label{eq:d-hmag}
&u^z_{{{\bf k}}} = \sqrt{\frac{\hbar}{M \omega_{p}({{\bf k}})}} \left(\frac{b_{{{\bf k}}} + b^\dag_{-{{\bf k}}}}{\sqrt2}\right),\\
&p^z_{{{\bf k}}} = \sqrt{\hbar M \omega_{p}({{\bf k}})} \left(\frac{b_{-{{\bf k}}} - b^\dag_{{{\bf k}}}}{{\sqrt2} i}\right),\end{aligned}$$ where the phonon dispersion is given by $\omega_{p}({{\bf k}}) = \omega_0 \sqrt{4 - 2\cos{k_x} - 2\cos{k_y}}$. This leads to the following diagonalized phonon Hamiltonian: $$\begin{aligned}
\label{eq:d-hph}
H_{\rm ph} = \sum_{{{\bf k}}} \hbar \omega_{p}({{\bf k}}) \left(b^\dag_{{{\bf k}}} b_{{{\bf k}}} + \frac12 \right) \, .\end{aligned}$$ In terms of the magnon and phonon operators introduced above, the magnetoelastic coupling term is recast into the following form in the momentum space: $H_{\rm mp} = H_{\rm mp1} + H_{\rm mp2}$, where $$\begin{aligned}
\label{eq:d-hmp}
&H_{\rm mp1} =\tilde\kappa \sum_{{{\bf k}}}\left[ a^\dag_{{{\bf k}}} b_{{{\bf k}}}\left(-i \sin k_x + \sin k_y\right)\right] + \text{h.c.} \, ,\\
&H_{\rm mp2} =\tilde\kappa \sum_{{{\bf k}}}\left[ a^\dag_{-{{\bf k}}} b^\dag_{{{\bf k}}}\left(i \sin k_x - \sin k_y\right) \right]+ \text{h.c.} \, ,\end{aligned}$$ with $\tilde \kappa = \kappa \sqrt{\hbar S / (M \omega_{{{\bf k}}})}$. Note that $H_{\rm mp1}$ conserves the total particle number, whereas $H_{\rm mp2}$ does not. Because of $H_{\rm mp2}$, the total Hamiltonian takes the Bogoliubov-de-Gennes (BdG) form.
![The band structure and its topology for $|C| = 1$ case. The band structure for $\kappa = 0$ (a) and $\kappa = 10$ meV/[Å]{} (b). The red dashed line represents the band-crossing points. (c) Berry curvatures of the upper band in log-scale $\Gamma(\Omega^z) = {\rm sign}(\Omega^z) {\rm log} (1+|\Omega^z|)$ for $\kappa = 10$ meV/[Å]{} (d) Schematic illustration of ${{\bf \hat d}} ({{\bf k}})$ for $\kappa = 10$ meV/[Å]{}. The in-plane components (${\hat d}_x, {\hat d}_y$) are shown in red arrows.[]{data-label="fig:C1"}](fig2.pdf){width="86mm"}
The band structure of magnon-phonon hybrid system is obtained by solving the Heisenberg equations with the above results \[Eqs. -\] (see the supplementary information for the detailed calculation and the schematic illustration of the band structure [@supple]). Without magnon-phonon interaction, there are two positive branches consisting of a magnon band and a phonon band. The two bands cross at ${{\bf k}}$ points satisfying $\omega_m ({{\bf k}})= \omega_p ({{\bf k}})$. Different from the conventional Dirac system, there are innumerable band-crossing points which form a closed line. These band-crossing lines are removed by the magnon-phonon interaction $\propto \kappa$, which induces the nontrivial topological property of the bands, characterized by the Berry curvatures. In the BdG Hamiltonian, the Berry curvature is given by [@Zhang2019; @Park2019; @Cheng2016] $$\boldsymbol \Omega_n ({{\bf k}}) = \nabla\times {{{\bf A}}}_n({{\bf k}}) \, ,$$ where ${{\bf A}}_n = i \langle \psi_{n,{{\bf k}}}|{\cal J}\nabla_{{{\bf k}}}|\psi_{n,{{\bf k}}} \rangle$ and $\psi_{n,{{\bf k}}}$ are the $n$-th eigenstates (see supplementary information for details). The topological property of the whole system is determined by the Chern number of bands, which is the integral of the Berry curvature over the Brillouin zone [@Qi2008]. In Fig. \[fig:1\](b), we show the Chern number of our bosonic system with nonzero magnon-phonon interaction $\kappa$. In our system, the Chern number can be one of three integers (0, 1, and 2) depending on the effective magnetic field $H_{\rm eff} = K_z S + \cal B$ and exchange interaction $J$. This is one of our central results: The magnon-polaron bands in a 2D simple square-lattice ferromagnet are topologically nontrivial even in the absence of dipolar or DM interaction and their topological property can be controlled by the effective magnetic field.
![The band structure and its topology for $|C| = 2$ case. The band structure for $\kappa = 0$ (a) and $\kappa = 10$ meV/[Å]{} (b). The red dashed line represents the band-crossing points. (c) Berry curvatures of the upper band in log-scale $\Gamma(\Omega^z) = {\rm sign}(\Omega^z) {\rm log} (1+|\Omega^z|)$ for $\kappa = 10$ meV/[Å]{}. (d) Schematic illustration of ${{\bf \hat d}}({{\bf k}})$ for $\kappa = 10$ meV/[Å]{}. The in-plane components (${\hat d}_x, {\hat d}_y$) are shown in red arrows.[]{data-label="fig:C2"}](fig3.pdf){width="86mm"}
#### Origin of the topological property—
The origin of the nontrivial magnon-polaron bands obtained above can be understood through the mapping our system to the well-known model for two-dimensional topological insulators such as HgTe [@Bernevig2006; @Qi2008]. Considering $H_{\rm mp}$ as a weak perturbation with unperturbed Hamiltonian with well-defined energies of magnons and phonons, the effect of particle-number-nonconserving component $H_{\rm mp2}$ on the band structure is much smaller than that of particle-number-conserving part $H_{\rm mp1}$. Neglecting $H_{\rm mp2}$, the total Hamiltonian is simplified into a single-particle two-band Hamiltonian $$H \approx \sum_{{{\bf k}}} \left(
\begin{array}{cc}
a^\dag_{{\bf k}} & b^\dag_{{\bf k}} \\
\end{array}
\right)
{\cal H_{{{\bf k}}}} \left(
\begin{array}{c}
a_{{\bf k}} \\
b_{{\bf k}} \\
\end{array}
\right)
,$$ where $$\begin{aligned}
\label{Hsimp}
{\cal H_{{{\bf k}}}} = \left(
\begin{array}{cc}
\hbar \omega_{m}({{\bf k}}) & \tilde\kappa (\sin k_y - i \sin k_x) \\
\tilde\kappa (\sin k_y + i \sin k_x) & \hbar \omega_{p}({{\bf k}}) \\
\end{array}
\right)
.\end{aligned}$$ In terms of the Pauli matrices $\boldsymbol \sigma = (\sigma_x, \sigma_y, \sigma_z)$, we write Eq. in a more compact form $$\begin{aligned}
{\cal H_{{{\bf k}}}} = \frac\hbar2 \left[\omega_m ({{\bf k}}) + \omega_p ({{{\bf k}}})\right] I_{2\times 2} + {{\bf d}} ({{\bf k}})\cdot \boldsymbol \sigma,\end{aligned}$$ where $$\begin{aligned}
{{\bf d}} ({{\bf k}}) = \left(\tilde \kappa \sin k_y, \tilde \kappa \sin k_x, \frac\hbar2 \left(\omega_m ({{\bf k}}) - \omega_p ({{{\bf k}}})\right)\right).\end{aligned}$$ The band structure for the above Hamiltonian is given by $$\begin{aligned}
E_\pm ({{\bf k}}) = \frac\hbar2 \left[\omega_m ({{\bf k}}) + \omega_p ({{{\bf k}}})\right] \pm |{{\bf d}} ({{\bf k}})|.\end{aligned}$$ In terms of ${{\bf d}}$ vectors, the Berry curvature is written explicitly as $$\Omega^z_\pm ({{\bf k}}) = \mp \frac12 {{\bf \hat d}} ({{\bf k}})\cdot\left(\frac{\partial {{\bf \hat d}} ({{\bf k}})}{\partial {k_x}}\times\frac{\partial {{\bf \hat d}}({{\bf k}})}{\partial {k_y}}\right) \, .$$ The corresponding expression for the Chern number is given by [@Qi2008; @Volovik1988; @Qi2006] $$\begin{aligned}
\label{CN}
C_\pm = \frac{1}{2\pi} \int dk_x dk_y \Omega^z_\pm ({{\bf k}})\, ,\end{aligned}$$ which is the skyrmion number of the ${{\bf d}}$ vector [@Qi2008], counting how many times ${{\bf \hat d}}$ wraps the unit sphere in the Brillouin zone. From Eq. , we read that the magnon band and phonon band cross at ${{\bf k}}$ points satisfying $\omega_m ({{\bf k}})= \omega_p ({{\bf k}})$ without the magnon-phonon interaction. These band crossing points are opened by the magnon-phonon interaction $\propto \kappa$ and the finite Berry curvatures are induced near the gap opening region. After integrating the Berry curvatures over the Brillouin zone, we obtain $C_\pm = 0$, $\pm 1$ or $\pm 2$. The two-band model has almost identical band structures and Berry curvatures to those of full Hamiltonian, where $H_{{\rm mp}2}$ is additionally considered (see the supplemental information).
In Fig. \[fig:C1\] and Fig. \[fig:C2\], we show that the bulk band structures and their topological properties for $|C| = 1$ and $|C| = 2$, respectively. For calculation, we use the parameters of the monolayer ferromagnet ${\rm CrI}_3$ in Ref. [@Huang2017; @Zhang2015; @Lado2017; @Zhang2019] ($J = 2.2$ meV, $K_z = 1.36$ meV, $S = 3/2$, and $M c^2 = 5\times 10^{10}$ eV). The force constant between the nearest-neighbor phonon is assumed as $\hbar \omega_0 = 10$ meV. The external magnetic field ${\cal B} = -0.1$ meV is chosen for Fig. \[fig:C1\] and ${\cal B} = 0.1$ meV is chosen for Fig. \[fig:C2\]. In Fig. \[fig:C1\](a), we find a band-crossing line (red dashed line) which is removed by the magnon-phonon interaction \[Fig. \[fig:C1\](b)\]. In this case, a dominant contribution of the Berry curvature comes from vicinity of the $\Gamma$-point \[Fig. \[fig:C1\](c)\]. An intuitive way to verify the topological nature of the system is the number of skyrmions of the unit vector ${{\bf \hat d}} ({{\bf k}})$. In Fig. \[fig:C1\](d), we find that there is a skyrmion at the $\Gamma$-point corresponding to $|C| = 1$. By changing the sign of external magnetic field $\cal B$, we can modify the band structure with two band-crossing lines \[Fig. \[fig:C2\](a)\]. In this case, the dominant contribution of the Berry curvature comes from vicinity of the $\Gamma$- and $\rm M$-points \[Fig. \[fig:C2\](c)\]. In terms of ${{\bf \hat d}} ({{\bf k}})$, we find that one skyrmion is located at $\Gamma$-point and the other skyrmion is at $\rm M$-point corresponding to $|C| = 2$ \[Fig. \[fig:C2\](d)\].
#### Thermal Hall effect—
The finite Berry curvatures of magnon-phonon hybrid excitations give rise to the intrinsic thermal Hall effect as shown below. The semiclassical equations of motion for the wave packet of magnon-phonon hybrid are given by [@Sundaram1999; @Xiao2010] $$\begin{aligned}
\dot {{{\bf r}}}_n = \frac{1}{\hbar} \frac{\partial E_n({{\bf k}})}{\partial {{\bf k}}} - \dot {{{\bf k}}} \times {\boldsymbol \Omega}_n({{\bf k}}), \quad \hbar \dot{{{\bf k}}} = - \nabla U({{\bf r}}),\end{aligned}$$ where $U({{\bf r}})$ is the potential acting on the wave packet which can be regarded as a confining potential of the bosonic excitation. Near the edge of sample, the gradient of the confining potential produces the anomalous velocity, $\nabla U({{\bf r}}) \times {\boldsymbol \Omega}_n({{\bf k}})$. In equilibrium, the edge current circulates along the whole edge and net magnon current is zero along any in-plane direction. However, if the temperature varies spatially, the circulating current does not cancel, which causes the thermal Hall effect [@Matsumoto2011].
The Berry-curvature-induced thermal Hall conductivity is given by [@Matsumoto2011; @Matsumoto2011b] $$\begin{aligned}
\kappa^{xy} = -\frac{k_B^2 T}{\hbar V} \sum_{n, {{\bf k}}} c_2(\rho_{n,{{\bf k}}}) \Omega_n^z({{\bf k}}),\end{aligned}$$ where $c_2(\rho) = (1+\rho) \ln^2 [(1+\rho) / \rho] - \ln^2 \rho - 2 {\rm Li}_2(-\rho)$, $\rho_{n,{{\bf k}}} = [e^{(E_n({{\bf k}}))/{k_B T} - 1}]^{-1}$ is the Bose-Einstein distribution function with a zero chemical potential, $k_B$ is the Boltzmann constant, $T$ is the temperature, and ${\rm Li}_2 (z)$ is the polylogarithm function. In Fig. \[fig:THC1\](a), we show the dependence of thermal Hall conductivity on the effective magnetic field $H_{\rm eff}$ at different temperatures $T$.
![(a) Dependence of thermal Hall conductivity on the effective magnetic field $H_{\rm eff}$ and with different temperatures $T$ using the parameters in the main text. (b) Dependence of Chern number on the effective magnetic field $H_{\rm eff}$ using the parameters in the main text.[]{data-label="fig:THC1"}](fig4.pdf){width="86mm"}
For small $H_{\rm eff}$, the thermal Hall conductivity increases with increasing $H_{\rm eff}$. However, for large $H_{\rm eff}$, it decreases with increasing $H_{\rm eff}$. This behavior of the thermal Hall conductivity can be understood through the Chern number of magnon-polaron bands depicted in Fig. \[fig:THC1\](b). The absolute value of the Chern number is 1 for small $H_{\rm eff}$, then it jumps up to 2 for a certain value of $H_{\rm eff}$ and vanishes for large $H_{\rm eff}$.
#### Discussions—
In this paper, we investigate the topology of the magnon-polaron bands in a simple 2D ferromagnet without long-range dipolar interaction and DM interaction. In our model, the topological structure can be controlled by the effective magnetic field which changes the number of band-crossing lines. Using a perturbation approach, we develop a two-band model Hamilitonian which provides an intuitive understanding of the topological structure of the model. In the two-band model, the nontrivial topology of the magnon-polaron bands are reflected in the skyrmion number of ${{\bf d}}({{\bf k}})$ in momentum space. As an experimental demonstration, we propose that the thermal Hall conductivity arises from the non-trivial topology of the magnon-polaron bands. The thermal Hall conductivity depends on the effective magnetic field which can be manipulated by the external magnetic field or voltage-induced magnetic anisotropy change [@Maruyama2009]. Our results show that the magnetoelastic interaction generates nontrivial topology in simple 2D ferromagnets with topological tunability, suggesting the ubiquity of topological transports in conventional magnetic systems with reduced dimensions.
K.-J. L. acknowledges a support by the National Research Foundation (NRF) of Korea (NRF-2017R1A2B2006119). G.G. acknowledges a support by the NRF of Korea (NRF-2019R1I1A1A01063594). S.K.K. was supported by the startup fund at the University of Missouri.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Using perturbative techniques, we investigate the existence and properties of a new static solution for the Einstein equation with a negative cosmological constant, which we call the deformed black hole. We derive a solution for a static and axisymmetric perturbation of the Schwarzschild$-$anti-de Sitter black hole that is regular in the range from the horizon to spacelike infinity. The key result is that this perturbation simultaneously deforms the two boundary surfaces—i.e., both the horizon and spacelike two-surface at infinity. Then we discuss the Abbott-Deser mass and the Ashtekar-Magnon one for the deformed black hole, and according to the Ashtekar-Magnon definition, we construct the thermodynamic first law of the deformed black hole. The first law has a correction term which can be interpreted as the work term that is necessary for the deformation of the boundary surfaces. Because the work term is negative, the horizon area of the deformed black hole becomes larger than that of the Schwarzschild$-$anti-de Sitter black hole, if compared under the same mass, indicating that the quasistatic deformation of the Schwarzschild$-$anti-de Sitter black hole may be compatible with the thermodynamic second law (i.e., the area theorem).'
author:
- Hirotaka Yoshino
- Tohru Ohba
- Akira Tomimatsu
title: 'Static black holes with a negative cosmological constant: Deformed horizon and anti$-$de Sitter boundaries'
---
Introduction
============
Recently, spacetimes with a negative cosmological constant $\Lambda$ have attracted a lot of attention in various contexts, such as the AdS/conformal field theory (CFT) correspondence (see [@AGMOO00] for a recent review) or the Randall-Sundrum brane world scenario [@RS99]. One of the necessary investigations in these contexts would be to analyze the classical feature of the spacetime with negative $\Lambda$, such as the black hole physics in these spacetimes.
Black hole physics in spacetimes with $\Lambda<0$ has a remarkable feature: the spatial topology of the black hole horizon is not necessarily spherical even in stationary four-dimensional spacetimes. In the 1990s, new solutions which represent the black holes with nonspherical topology horizons were discovered [@Lem95; @Lem95-2; @HL95; @LZ96; @CZ96; @ABHP96; @Bri96; @Man97; @KMV98] in addition to the well-known solutions: i.e., the Schwarzschild$-$anti-de Sitter black hole and the Kerr$-$anti-de Sitter black hole. For example, Lemos constructed a static solution of an infinitely large planar black hole for the four-dimensional spacetime with $\Lambda<0$ [@Lem95]. He also pointed out that the planar solution also represents the cylindrical black hole or the toruslike black hole by appropriate compactifications of this spacetime. By these procedures, the topology of the black hole horizon becomes the cylinder ($R\times S^1$) or the torus ($S^1\times S^1$), and conformal spacelike infinity has the same topology as the horizon. Lemos generalized his static cylindrical black hole solution to the rotating one [@Lem95-2]. The generalization including charge and dilaton can be found in [@HL95; @LZ96; @CZ96]. There exists another solution of the black hole with unusual topology. Åminneborg [*et al.*]{} constructed a solution of a black hole whose horizon is a Riemann two-surface and can take an arbitrary genus value $g>1$ by appropriate compactifications [@ABHP96]. The topology of the conformal spacelike infinity is the same as that of the horizon due to this compactification. The generalization for the charged case can be found in [@Bri96; @Man97], and rotating topological black holes were introduced by Klemm [*et al.*]{} [@KMV98].
Readers might wonder why the black hole solutions with such various kinds of horizon topology exist even in four dimensions. The black hole topology theorem proved by Hawking claims that the topology of the horizon is ${S}^{2}$ [@Hawk72]. But the assumptions of this theorem are that the spacetime is asymptotically flat and that certain energy conditions hold, which are both incompatible with the negative cosmological constant. In the presence of negative $\Lambda$, the black hole physics becomes far richer than the case of $\Lambda=0$. On the other hand, there are some theorems that restrict black hole solutions in spacetimes with $\Lambda<0$. Anderson [*et al.*]{} proved that the Schwarzschild$-$anti-de Sitter black hole is the unique static solution for asymptotically anti$-$de Sitter vacuum spacetimes with $\Lambda<0$ [@ACD02]. Galloway [*et al.*]{} proved some theorems that restrict the black hole solutions in $\Lambda<0$ spacetimes [@GSW03]. However, we expect that there would be a great possibility of the existence of unknown solution series.
In this paper, we consider the existence of the series of black hole solutions which describes a continuous change from the Schwarzschild$-$anti-de Sitter black hole to the cylindrical or planar one. Our expectation is easily understood by looking at the recent study of black strings and black holes in higher-dimensional Kaluza-Klein spacetimes. Gregory and Laflamme analyzed the stability of the black string in higher-dimensional spacetime [@GL93]. They showed that the black string is unstable for a perturbation along the string, if the wavelength is sufficiently large. It was also shown that there is a static perturbation, which was a strong implication for the existence of a new sequence of static solutions. This static perturbation was investigated in detail by Gubser [@Gub02], and subseqently Wiseman [@Wise03] and Kudoh and Wiseman [@KW03] numerically solved the sequence that connects the black string to the black hole in higher-dimensional Kaluza-Klein spacetime. We can expect that a similar situation would occur in four-dimensional $\Lambda<0$ spacetimes.
Motivated by this expectation, we analyze the static perturbation of the Schwarzschild$-$anti-de Sitter spacetime as a starting point. The existence of such solutions is the manifestation of the existence of the new solution series. We consider the axisymmetric, even-parity perturbations in the Regge-Wheeler formalism [@RW57]. Each component of the metric perturbation is represented by a product of a radial function and Legendre’s polynomial $P_l(\cos\theta)$. In the case of $\Lambda=0$, the solution diverges either at the horizon or at infinity [@RW57]. If $\Lambda<0$, however, it is possible to construct a solution which does not diverge everywhere from the horizon to infinity for all multipole modes corresponding to $l=2,3,...$ . The horizon of the perturbed solution is not geometrically spherical. Hence we call this solution a deformed black hole hereafter. Because some radial functions of the perturbation components asymptote to constant values at spacelike infinity, our solution is not asymptotically anti$-$de Sitter: a two-surface at spacelike infinity is also deformed. This is consistent with the uniqueness theorem of Anderson [*et al.*]{} However, our solution still describes the weakly asymptotically anti$-$de Sitter spacetime, in the sense of the Ashtekar-Magnon definition [@AM84]. Our result also does not contradict the theorems derived by Galloway [*et al.*]{}
To understand the physical implication of the black hole deformation, we would like to discuss some features of this solution such as the mass, the horizon area, and the first law of black hole thermodynamics. Because some of the metric coefficients diverge at infinity and to take a limit to infinity is a delicate problem, there are ambiguities in defining the mass of the spacetime with $\Lambda<0$. Several mass definitions in the asymptotically anti$-$de Sitter spacetimes have been proposed (see [@CN02] and references therein), and some of them are applicable also to the weakly asymptotically anti$-$de Sitter spacetimes. Among these, we use two well-known mass definitions proposed by Abbott and Deser [@AD82] and by Ashtekar and Magnon [@AM84], which can be easily applied to deformed black holes. The first-order perturbation deforms the Schwarzschild$-$anti-de Sitter black hole without changing the two masses and the horizon area. Hence we should consider the $l=0$ mode of the second-order perturbation, which is generated by the terms of a product of two first-order perturbation components in the second-order equation. The two definitions give totally different results for the mass derived from the second-order perturbation. The Abbott-Deser mass diverges to minus infinity, while the Ashtekar-Magnon mass gives a finite value. As we will discuss in detail later, this result would be due to the fact that the Abbott-Deser mass is not gauge invariant at second order. Since the definition of the Ashtekar-Magnon mass is covariant, it would provide a real amount of energy contained in the spacetime with a deformed black hole. Therefore we expect that quasistatic deformation of the Schwarzschild$-$anti-de Sitter black hole occurs with a finite change in the total energy and thus an investigation of the thermodynamic first law of the deformed black holes with this mass definition is meaningful.
One can easily find that the Schwarzschild$-$anti-de Sitter black holes obey the thermodynamic laws like asymptotically flat black holes. The first law was extended to the static black holes with unusual topology by Vanzo [@Van97]. He used the mass defined as the on-shell value of the Hamiltonian, which gives the same value as the Abbott-Deser mass, and showed that the first law holds with this mass definition. It was explicitly shown that the first law with the usual form also holds for the Kerr$-$Newman$-$anti-de Sitter black holes [@CCK00], and there are some approaches to the proof of the first law for the asymptotically anti$-$de Sitter, stationary black hole spacetimes under more general assumptions [@Sil02; @Bar03]. We will analyze the first law of the deformed black holes as follows. There are two characteristic scales for these spacetimes: i.e., the Schwarzschild radius $R_S\equiv 2m$ and the anti$-$de Sitter radius $R_A\equiv \sqrt{-3/\Lambda}$. We analytically consider the case that the Schwarzschild radius is much smaller than the anti$-$de Sitter radius (i.e., $\alpha\equiv R_S/R_A\ll 1$) and the case that the Schwarzschild radius is much larger than the anti$-$de Sitter radius (i.e., $\alpha\gg 1$). Then we numerically solve the $\alpha\sim 1$ cases. We will explicitly construct the solutions for the first-order perturbation and calculate the horizon area and the Ashtekar-Magnon mass. Our calculation shows that the first law in the usual form approximately holds for the $\alpha\gg 1$ case, and it does not hold for the other cases. In other words, the first law has a correction term which can be interpreted as the work term necessary for the deformation of the two boundary surfaces of the spacetime: i.e., the horizon and two-surface at spacelike infinity. If we compare the horizon area of the Schwarzschild$-$anti-de Sitter black hole and that of the deformed black hole under the same mass, the latter becomes larger. Therefore deformation of the Schwarzschild$-$anti-de Sitter black hole will be claimed to be a process compatible with the usual area law.
The outline of the paper is as follows. In Sec. II, we derive the equations for a static, axisymmetric perturbation of Schwarzschild$-$anti-de Sitter spacetime. We show that there exists a solution for the first-order perturbation which does not diverge everywhere, and that this spacetime is the weakly asymptotically anti$-$de Sitter spacetime. Then we derive the second-order equation for the $l=0$ mode and the general formula for the horizon area of the deformed black hole. In Sec. III, we derive the general formulas of the Abbott-Deser mass and the Ashtekar-Magnon mass for the deformed black holes. We discuss the property of these two definitions and show that the Abbott-Deser mass is not gauge invariant. In Sec. IV, we explicitly construct the solutions for the first-order perturbation in two cases $\alpha\ll 1$ and $\alpha\gg 1$. Then we analyze the first law of the deformed black holes. In Sec. V, we numerically calculate for the $\alpha\sim 1$ case and discuss the dependence of the first law on the value of $\alpha$. In Sec. VI, we summarize our results and discuss their physical implications.
Static, axisymmetric perturbation
=================================
In this section, we derive the static, axisymmetric perturbation of the Schwarzschild$-$anti-de Sitter black hole. The background metric is given in the Schwartzschild-like coordinates $(t, r, \theta, \phi)$ as follows: $$d\hat{s}^2=-e^{2\nu_0}dt^2+e^{2\mu_0}dr^2+r^2
\left(d\theta^2+\sin^2\theta d\phi^2\right),
\label{SAD1}$$ $$e^{2\nu_0}=e^{-2\mu_0}= 1 - \frac{2m}{r} - \frac{1}{3} \Lambda r^{2}.
\label{SAD2}$$ As we see from the investigations in [@RW57] and [@GNPP00], the metric of a spherically symmetric spacetime with first- and second-order even-parity static perturbations can be written in diagonal form $$d\hat{s}^2=-e^{2\nu}dt^2+e^{2\mu}dr^2+e^{2\psi}r^2
\left(d\theta^2+\sin^2\theta d\phi^2\right),
\label{metric_deformed}$$ where $\nu$, $\mu$, and $\psi$ are expanded with a small deformation parameter $\epsilon$ as follows: $$\begin{aligned}
&\nu=\nu_0+\epsilon \nu_1+\epsilon^2 \nu_2,
\label{expand_nu}\\
&\mu=\mu_0+\epsilon \mu_1+\epsilon^2 \mu_2,
\label{expand_mu}\\
&\psi=\epsilon \psi_1+\epsilon^2 \psi_2.
\label{expand_psi}\end{aligned}$$ Here all first- and second- order functions are given by the sum of the products of a radial function and Legendre’s polynomial $P_l(\cos\theta)$, after the appropriate gauge transformations. We use this Regge-Wheeler-like gauge for an investigation of the deformed black holes.
First-order perturbation
------------------------
We give the first-order perturbation as $$\begin{aligned}
&\nu_1=-\mu_1=-H^{(1)}(r) P_{l}(\cos\theta),
\label{function_nu1_mu1}\\
&\psi_1= K^{(1)}(r) P_{l}(\cos\theta),
\label{function_psi1}\end{aligned}$$ where the first equality in Eq. is imposed by the difference of $\theta\theta$, $\phi\phi$ components of the Einstein equation $R_{\mu\nu}=\Lambda g_{\mu\nu}$. We restrict our attention to $l\ge 2$, because $l=0$ and $l=1$ modes are absorbed to the coordinate transformation and the changing of the mass. The first-order equations become $$r^2e^{2\nu_0}H^{(1)}_{,rr}
+ 2r(re^{2\nu_0})_{,r} H^{(1)}_{,r}
- r^2(e^{2\nu_0})_{,r}K^{(1)}_{,r}
- \left[2 \Lambda r^2 + l(l+1)\right] H^{(1)} = 0,
\label{Rtt}$$ $$\begin{gathered}
r^2e^{2\nu_0}\left(H^{(1)}_{,rr}-2K^{(1)}_{,rr}\right)
+ 2r\left( re^{2\nu_0}\right)_{,r} H^{(1)}_{,r} \\
- r^2\left[ (e^{2\nu_0})_{,r}+4r^{-1}e^{2\nu_0}\right] K^{(1)}_{,r}
- \left[2 \Lambda r^2- l(l+1)\right] H^{(1)} = 0,
\label{Rrr}\end{gathered}$$ $$\begin{aligned}
e^{2\nu_0}\left(H^{(1)}_{,r} - K^{(1)}_{,r}\right)
+ \left(e^{2\nu_0}\right)_{,r}H^{(1)} = 0,
\label{Rr theta}\end{aligned}$$ $$\begin{gathered}
r^2e^{2\nu_0} K^{(1)}_{,rr}
- 2re^{2\nu_0}H^{(1)}_{,r}
+ r^2\left[ (e^{2\nu_0})_{,r}+4r^{-1}e^{2\nu_0}\right] K^{(1)}_{,r} \\
- 2\left( re^{2\nu_0} \right)_{,r} H^{(1)}
- (l^2+ l -2)K^{(1)} = 0,
\label{R phi phi}\end{gathered}$$ where ${,r}$ denotes the derivative with respect to $r$. These equations come from $tt, rr, r\theta$ components and the sum of the $\theta\theta, \phi\phi$ components of the Einstein equation, respectively. Although there are four equations for two functions $H^{(1)}$ and $K^{(1)}$, these equations do not overdetermine $H^{(1)}$ and $K^{(1)}$ because we can derive Eq. and the derivative of Eq. using Eq. , Eq. , and their derivatives. Eliminating $K^{(1)}$ from Eqs. and , we obtain the equation for the quantity $M\equiv r^2e^{2\nu_0}H^{(1)} $: $$\begin{aligned}
M_{,rr} - 2\left(r^{-1}+{\nu_0}_{,r}\right) M_{,r}
-e^{-2\nu_0}\left[ 2\Lambda +r^{-2}(l^2+l-2)\right] M = 0.
\label{H1}\end{aligned}$$ Eliminating $K^{(1)}_{,rr}$ and $K^{(1)}_{,r}$ from Eqs. , , , and , we find that $K^{(1)}$ is expressed in terms of $H^{(1)}$ as $$K^{(1)} = \frac{1}{(l^2+l-2)}
\left\{
r^2(e^{2\nu_0})_{,r}H^{(1)}_{,r}
+ \left[
r^2(2{\nu_0}_{,r})\left(e^{2\nu_0}\right)_{,r}
+l^2+l-2e^{2\nu_0}
\right]H^{(1)}
\right\} .
\label{K1}$$
Now we show the existence of solutions which do not diverge in the range from the horizon to spatial infinity. The two independent solutions for $H^{(1)}$ of Eq. are $(r-r_h)$ and $1/(r-r_h)$ in the neighborhood of the horizon, while $1/r$ and $1/r^2$ for large $r$. Here we have introduced the horizon radius $r_h$, which is the maximum solution of $e^{2\nu_0}(r_h)=0$. Thus, if we choose $(r-r_h)$ near the horizon, the solution $H^{(1)}$ becomes regular everywhere because it behaves like $\sim 1/r$ at large $r$. The behavior of $H^{(1)}$ at infinity can be written as $$H^{(1)}=\frac{a_1}{r}+\frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots,
\label{H1asymptotic}$$ where $a_3, a_4, ...$ are expressed with $a_1$ and $a_2$ like $$\begin{aligned}
&a_3=-\frac{3(l^2+l-4)}{2\Lambda}a_1,
\label{a3}\\
&a_4=-\frac{1}{2\Lambda}\left[18ma_1+(l^2+l-6)a_2\right],
\label{a4}\end{aligned}$$ because $H^{(1)}$ contains only two integral constants. The ratio of $a_1$ and $a_2$ is determined by imposing the regularity at the horizon. Using Eq. , we see that the behavior of $K^{(1)}$ at infinity becomes $$K^{(1)}=c_0+\frac{c_1}{r}+\frac{c_2}{r^2}+\cdots,
\label{K1asymptotic}$$ where the coefficients can be determined using Eq. as follows: $$\begin{aligned}
& c_0=\frac{2\Lambda a_2}{3(l^2+l-2)},
\label{c0}\\
& c_1=-a_1,
\label{c1}\\
& c_2=0.
\label{c2}\end{aligned}$$ Thus, $K^{(1)}$ is also finite at $r=\infty$. Similarly, if we write the behavior of $H^{(1)}$ near the horizon as $$H^{(1)}={\tilde{a}_1}(r-r_h)+\tilde{a}_2{(r-r_h)^2}+\cdots,
\label{H1_horizon}$$ the behavior of $K^{(1)}$ near the horizon becomes $$K^{(1)}=\tilde{c}_0+\tilde{c}_1{(r-r_h)}+\cdots,
\label{K1_horizon}$$ where the coefficients can be determined using Eq. such as $$\begin{aligned}
& \tilde{c}_0=\frac{4(3m-r_h)\tilde{a}_1}{l^2+l-2},
\label{tilde_c0}\\
& \tilde{c}_1=2\tilde{a}_1.\end{aligned}$$ Thus two functions $H^{(1)}$ and $K^{(1)}$ remain finite everywhere from the horizon to infinity, and hence we have established the existence of the new solution series of $\Lambda<0$ spacetimes.
Now we examine some properties of this solution. The induced metric on a two-surface $t={\rm const}$ and $r=r_h$ becomes $$d\hat{s}^2=\left[1+2\epsilon \tilde{c}_0 P_l(\cos\theta)\right]r_h^2
(d\theta^2+\sin^2\theta d\phi^2), \label{metric_horizon}$$ which implies that the geometry of the horizon in (say) the $l=2$ case is prolate for $\tilde{c}_0>0$ and oblate for $\tilde{c}_0<0$. Thus the horizon geometry deviates from the geometrically spherical surface and this solution represents the deformed black hole. To see the structure of spacelike infinity, we consider a conformal transformation $ds^2=\Omega^2 d\hat{s}^2$ where we choose $\Omega=r^{-1}$. The induced metric of $r={\rm const}$ surface at infinity of the conformally transformed spacetime becomes $$ds^2=\frac13\Lambda dt^2+\left[1+2\epsilon {c}_0 P_l(\cos\theta)\right]
(d\theta^2+\sin^2\theta d\phi^2), \label{eq:infinity}$$ which means that the conformal boundary of this spacetime is also deformed. This indicates that this spacetime is not asymptotically anti$-$de Sitter in the sense of the Ashtekar-Magnon definition [@AM84]. According to their definition, the spacetime is asymptotically anti$-$de Sitter if the spacetime satisfies the Einstein equation with $\Lambda<0$ and with an energy momentum tensor that satisfies an appropriate falloff condition, and its conformal boundary $\mathcal{I}$ is topologically $S^2\times R$, and the conformal group of $\mathcal{I}$ is the anti$-$de Sitter group. The last condition is equivalent to that $\mathcal{I}$ admits a global chart $(t, \theta, \phi)$ such that the metric on $\mathcal{I}$ is conformally related to the metric $$ds_0^2=\frac13\Lambda dt^2+d\theta^2+\sin^2\theta d\phi^2.$$ This is clearly inconsistent with Eq. . However, the spacetime of the deformed black hole is weakly asymptotically anti$-$de Sitter [@AM84]: this is defined by the above conditions except the last condition. The deformed black hole spacetime has a deformed surface at spacelike infinity, and thus the existence of this solution is not contradictory to the uniqueness theorem of the asymptotically anti$-$de Sitter black hole [@ACD02].
To solve analytically $H^{(1)}$ and $K^{(1)}$ for general $\alpha$ is rather difficult. We will explicitly construct the analytic solutions in two simple cases $\alpha\equiv 2m\sqrt{-\Lambda/3}\ll 1$ and $\alpha\gg 1$ in Sec. IV and the numerical solutions in the $\alpha\sim 1$ cases in Sec. V. Here, we show that $H^{(1)}$ and $K^{(1)}$ do not change their sign for $r_h<r<\infty$. Using Eq. , $K^{(1)}$ is given by $$K^{(1)}=H^{(1)}+\int_{r_h}^{r}{2\nu_0}_{,r} H^{(1)}dr+\tilde{c}_0.
\label{K1_positive}$$ We consider the $\tilde{a}_1>0$ case (hence $\tilde{c}_0>0$). If $H^{(1)}$ changes its sign from positive to negative at $r=r_c$, $K^{(1)}(r_c)$ should be positive from Eq. and $H^{(1)}_{,r}(r_c)$ should be negative. However, this contradicts Eq. . Hence $H^{(1)}$ is always positive. Using Eq. again, we see that $K^{(1)}$ also always takes a positive value. The proof for the case $\tilde{a}_1<0$ is similar. Thus $H^{(1)}$ and $K^{(1)}$ have the same sign and do not change their sign. This means that there is a natural relation between the geometry of the horizon and that of the conformal boundary of this spacetime. Because $\tilde{c}_0$ and $c_0$ have the same sign, the conformal boundary is prolate if the horizon is prolate and vice versa. Moreover, using Eq. again, we see that $|K^{(1)}-H^{(1)}|$ is a monotonically increasing function of $r$. Because $H^{(1)}(r_h)=H^{(1)}(\infty)=0$, we find $|K^{(1)}(r_h)|<|K^{(1)}(\infty)|$ and hence $|\tilde{c}_0|<|c_0|$. Thus the conformal boundary at infinity is more deformed compared to the horizon.
As we see from Eq. , the first-order perturbation does not change the area of the horizon. It also does not affect the mass of the spacetime, as we will see in the next section. Hence to see the change in the mass and horizon area due to the deformation of the black hole, we should consider the $l=0$ mode of the second-order perturbation.
$l=0$ mode of second-order perturbation
---------------------------------------
We consider the second-order perturbation which is generated by the $l\ge 2$ mode of the first-order perturbation of the black hole. The second-order perturbation is given as follows: $$\begin{aligned}
&\nu_2=-{H}_0^{(2)}(r)
+\sum_{n\neq 0}H_{n}^{(2)}(r)P_{n}(\cos\theta),
\label{function_nu2}\\
&\mu_2=\tilde{H}_0^{(2)}(r)
+\sum_{n\neq 0}\tilde{H}_{n}^{(2)}(r)P_{n}(\cos\theta),
\label{function_mu2}\\
&\psi_2=K_0^{(2)}(r)
+\sum_{n\neq 0}K_{n}^{(2)}(r)P_{n}(\cos\theta).
\label{function_psi2}\end{aligned}$$ Because there remains the degree of freedom to choose the $r$ coordinate, we can impose $H^{(2)}\equiv H_0^{(2)}=\tilde{H}_0^{(2)}$ only for the $l=0$ mode. We will treat only the functions of the $l=0$ mode, $H^{(2)}$ and $K^{(2)}\equiv K_0^{(2)}$, for the second-order perturbation.
The second-order Einstein equations of the $l=0$ mode are $$\begin{gathered}
r^2e^{2\nu_0}H^{(2)}_{,rr}
+ 2r(re^{2\nu_0})_{,r}H^{(2)}_{,r}
- r^2(e^{2\nu_0})_{,r}K^{(2)}_{,r}
- 2 \Lambda r^2 H^{(2)} \\
= \frac{2}{2l +1}
\left\{
e^{2\nu_0}r^2
\left[ \left( H^{(1)}_{,r} \right)^{2} - H^{(1)}_{,r} K^{(1)}_{,r}\right]
+ \left[ \Lambda r^2+ l \left( l + 1 \right) \right]
\left( H^{(1)} \right)^{2}
- l \left( l + 1 \right) H^{(1)} K^{(1)}
\right\},
\label{R2tt}\end{gathered}$$ $$\begin{gathered}
r^2e^{2\nu_0}\left(H^{(2)}_{,rr}-2K^{(2)}_{,rr}\right)
+ 2r(re^{2\nu_0})_{,r} H^{(2)}_{,r}
-r^2\left[(e^{2\nu_0})_{,r}+4r^{-1}e^{2\nu_0}\right]K^{(2)}_{,r}
- 2 \Lambda r^2 H^{(2)} \\
= \frac{2}{2l +1}
\Big\{
e^{2\nu_0}r^2
\left[ \left( H^{(1)}_{,r} \right)^{2}
+ \left( K^{(1)}_{,r} \right)^{2}
- H^{(1)}_{,r} K^{(1)}_{,r} \right]
+ \left[ \Lambda r^2 - l \left( l + 1 \right) \right]
\left( H^{(1)} \right)^{2}\\
+ l \left( l + 1 \right) H^{(1)} K^{(1)}
\Big\},
\label{R2rr}\end{gathered}$$ $$\begin{gathered}
r^2e^{2\nu_0}K^{(2)}_{,rr}
- 2re^{2\nu_0}H^{(2)}_{,r}
+ r^2\left[ (e^{2\nu_0})_{,r}+4r^{-1}e^{2\nu_0}\right] K^{(2)}_{,r}
- 2(re^{2\nu_0})_{,r}H^{(2)}
+ 2 K^{(2)} \\
= \frac{-1}{2l +1}
\Big\{
2r^2e^{2\nu_0}
\left[ \left( K^{(1)}_{,r} \right)^{2} -
H^{(1)}_{,r} K^{(1)}_{,r} \right]
+ \left( 2 \Lambda r^2 +l^2 +l -2 \right)
\left( H^{(1)} \right)^{2} \\
- 2\left( l^2+l-2\right) H^{(1)} K^{(1)}
+ 2\left( l^2+l-1 \right) \left( K^{(1)} \right)^{2}
\Big\},
\label{R2phiphi+R2thetatheta}\end{gathered}$$ where we have used the formulas $$\begin{aligned}
\langle \left[P_l(\cos\theta)\right]^2\rangle&={1}/{(2l+1)},\\
\langle
\left[d{P}_l(\cos\theta)/d\theta\right]^2
\rangle &={l(l+1)}/{(2l+1)},\end{aligned}$$ where $$\langle f\rangle\equiv\frac12\int_0^\pi f(\theta)\sin\theta d\theta.
\label{theta_average}$$ These equations come from $tt, rr$ components and the sum of the $\theta\theta, \phi\phi$ components of the Einstein equation, respectively. The $r\theta$ component and the difference of the $\theta\theta, \phi\phi$ components provide no conditions for the $l=0$ mode. Similarly to the case of the first-order perturbation, these equations do not overdetermine $H^{(2)}$ and $K^{(2)}$.
The homogeneous solution for Eqs. , , and is given by $$\begin{aligned}
&K^{(2)}_{hom}=C_1+C_2/r,\\
&H^{(2)}_{hom}=e^{-2\nu_0}
\left[C_1-{(e^{2\nu_0})_{,r}C_2}/{2}+{C_3}/{r}\right].\end{aligned}$$ This is not physical because these perturbations are absorbed to the coordinate transformation and changing the mass like $$\begin{aligned}
&\bar{t}=(1-\epsilon^2C_1)t,\\
&\bar{r}=(1+\epsilon^2C_1)r+\epsilon^2C_2,\\
&\bar{m}=(1+3\epsilon^2C_1)m+\epsilon^2C_3. \end{aligned}$$ Moreover, $H^{(1)}_{hom}$ diverges on the horizon. Hence, in order to obtain a solution that does not diverge for $r_h\le r\le \infty$, we should impose the homogeneous solution to be zero. This means that $K^{(2)}$ behaves like $$K^{(2)}=\frac{d_2}{r^2}+\frac{d_3}{r^3}+\cdots,
\label{K2asymptotic}$$ for large $r$, where $d_2, d_3,...$ are determined like $$\begin{aligned}
&d_2=- \frac{a_{1}^{2}}{2 \left( 2l + 1 \right)} ,\\
&d_3= \frac{l(l+1) a_{1} c_{0}}{(2l+1)\Lambda } .
\label{asymptotic_d3}\end{aligned}$$ The fact that $K^{(2)}=O(1/r^2)$ at large $r$ provides a boundary condition for solving $K^{(2)}$ for $r_h<r<\infty$. The solution is written as $$K^{(2)} = \frac{-1}{(2l+1)}
\int_{r}^{\infty} \frac{1}{{r^{\prime\prime}}^{2}}
\int_{r^{\prime\prime}}^{\infty}
\biggl[ {r^\prime}^{2} \left( K^{(1)}_{,r^\prime} \right)^{2}
- \frac{2l(l+1)}{e^{2\nu_0}} H^{(1)} \left(
H^{(1)} - K^{(1)} \right) \biggl]
dr^{\prime} dr^{\prime\prime}.
\label{K2}$$ We also see that $H^{(2)}$ behaves like $$H^{(2)}=\frac{b_2}{r^2}+\frac{b_3}{r^3}+\cdots.
\label{H2asymptotic}$$ Although $b_2$ is determined like $$b_2=\frac{1}{2 \left( 2l + 1 \right)}
\left(
a_1^2-\frac{6c_0^2}{\Lambda}(l^2+l-1)
\right),$$ it is not possible to determine $b_3$ in terms of $a_1$ and $c_0$, because we cannot impose $C_3$ of the homogeneous solution to be zero at infinity. It is determined only by explicitly solving $H^{(2)}$ with an appropriate boundary condition at the horizon $r=r_h$ which we introduce later.
Near the horizon, the behavior of $K^{(2)}$ is written as $$K^{(2)}=\tilde{d}_0+\tilde{d_1}{(r-r_h)}+\cdots,
\label{K2_horizon}$$ where the coefficients are determined only by explicitly calculating Eq. : $$\begin{aligned}
&\tilde{d}_0=
\frac{1}{(2l+1)}\left(
-\frac{c_0a_1}{r_h}
+\frac12\left(c_0^2-\tilde{c}_0^2\right)
+\int_{r_h}^{\infty}
\frac{(r-r_h)l(l+1)}{r_hre^{2\nu_0}}
H^{(1)}\left(2H^{(1)}-K^{(1)}\right)
dr
\right), \label{tilded0}
\\
& \tilde{d}_1=
\frac{1}{(2l+1)}
\left(
\frac{c_0a_1}{r_h^2}-2\tilde{c}_0\tilde{a}_1
-\int_{r_h}^{\infty}\frac{l(l+1)}{r_h^2e^{2\nu_0}}H^{(1)}
\left(2H^{(1)}-K^{(1)}\right)dr
\right).\label{tilded1}\end{aligned}$$ Once $K^{(2)}$ is calculated, we find that $H^{(2)}$ behaves like $$H^{(2)}=\tilde{b}_0+\tilde{b_1}{(r-r_h)}+\cdots,
\label{H2_horizon}$$ near the horizon, where the coefficients are determined by Eqs. and like $$\begin{aligned}
&\tilde{b}_0=\frac{r_h}{2(3m-r_h)}
\left(
(3m-r_h)\tilde{d}_1+\tilde{d}_0+\frac{l^2+l-1}{2l+1}\tilde{c}_0^2
\right),
\label{tildeb0}\\
&\tilde{b}_1=\frac12\tilde{d}_1+\frac{\Lambda r_h^2}{2(3m-r_h)}\tilde{b}_0.\end{aligned}$$ The fact $H^{(2)}(r_h)=\tilde{b}_0$ gives a boundary condition for solving $H^{(2)}$ for $r_h\le r\le\infty$. The solution of $H^{(2)}$ satisfying this boundary condition is written as $$\begin{gathered}
H^{(2)} = \frac{1}{2(2l+1)re^{2\nu_0}}
\int_{r_{h}}^{r} \bigg\{
(2l+1)
\left[
2K^{(2)}
+
\left({r^{\prime}}^2e^{2\nu_0}\right)_{,r^{\prime}}
K^{(2)}_{,r^\prime}
\right]
- {r^\prime}^2e^{2\nu_0}
K^{(1)}_{,r^\prime}
\left(2H^{(1)}_{,r^\prime} -K^{(1)}_{,r^\prime}\right)
\\
+ \left( 2\Lambda {r^\prime}^{2} + 3l^2+3l-2 \right)
\left( H^{(1)} \right)^{2}
- 2(l^2+l-1) K^{(1)}\left(2H^{(1)} -K^{(1)}\right)
\bigg\} dr^\prime.
\label{H2}\end{gathered}$$
Now we calculate the horizon area of the deformed black hole. The induced metric on the horizon $t={\rm const}$ and $r=r_h$ becomes $$d\hat{s}^2=\left(
1+2\psi_1+2\psi_1^2+2\psi_2
\right)\big|_{r=r_h}
r_h^2\left(d\theta^2+\sin^2\theta d\phi^2\right).$$ Substituting Eqs. and and then using Eqs. and , the change in the horizon area $\delta A$ is easily calculated: $$\begin{aligned}
\frac{\delta A}{A} =
2 \epsilon^{2} \left( \frac{\tilde{c}_0^2}{2l + 1}
+ \tilde{d}_0\right),
\label{A}\end{aligned}$$ where $A\equiv\pi r_h^2$ is the area of the Schwarzschild$-$anti-de Sitter black hole. Because $\tilde{d}_0$ is given in Eq. , we can calculate $\delta A$ without explicitly constructing solutions of $H^{(2)}$ and $K^{(2)}$. This quantity $\delta A/A$ provides an indicator for the strength of nonlinear effect near the horizon.
We also consider the area of the $t={\rm const}$ surface of the conformal boundary $\mathcal{I}$ in the case where we choose the conformal factor $\Omega=r^{-1}$, because it becomes the measure for the strength of nonlinear effect near infinity. We write this area and the increase in this area $S$ and $\delta S$, respectively. This is given by $$\begin{aligned}
\frac{\delta S}{S} =
2 \epsilon^{2} \left( \frac{c_0^2}{2l + 1} \right).
\label{S}\end{aligned}$$ Because this is proportional to $c_0^2$ and is positive definite, this quantity provides a natural factor for normalizing the thermodynamical quantities, such as $\delta A/A$ and $\delta m/m$.
Mass of the deformed black hole
===============================
In this section, we calculate the mass of the deformed black hole. As we mentioned in the Introduction, we consider two definitions of mass: the Abbott-Deser mass [@AD82] and the Ashtekar-Magnon mass [@AM84]. We will find that these two definitions of mass give totally different results and will discuss the reason for this difference.
Abbott-Deser mass
-----------------
Abbott and Deser found a conserved quantity for the spacetimes with a cosmological constant [@AD82]. They divided the metric tensor $\hat{g}_{\mu\nu}$ into two parts: $$\hat{g}_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu},$$ where $\bar{g}_{\mu\nu}$ is the solution of the Einstein equation which has the Killing vector field ${\bar{\xi}}_{\mu}$, and $h_{\mu\nu}$ represents a deviation from the background metric $\bar{g}_{\mu\nu}$. The conserved quantity associated with the Killing vector field $\bar{\xi}_{\mu}$ is $$E(\bar{\xi})=\frac{1}{8\pi}\int
\sqrt{-\bar{g}} d^2x\left(
\bar{\xi}_{\nu}D_{\beta}K^{\mu\alpha\nu\beta}
-K^{\mu\beta\nu\alpha}D_{\beta}\bar{\xi}_{\nu}
\right)
D_{\mu}tD_{\alpha}r,$$ where $\bar{g}$ is the determinant of $\bar{g}_{\mu\nu}$, $D_\mu$ denotes the covariant derivative with respect to the background metric $\bar{g}_{\mu\nu}$, $t$ and $r$ are the usual time and radial coordinates, and $K^{\mu\alpha\nu\beta}$ is the superpotential defined by $$K^{\mu\alpha\nu\beta}=
\frac12
\left(
\bar{g}^{\mu\beta}H^{\nu\alpha}+\bar{g}^{\nu\alpha}H^{\mu\beta}
-\bar{g}^{\mu\nu}H^{\alpha\beta}-\bar{g}^{\alpha\beta}H^{\mu\nu}
\right),$$ where $$H^{\mu\nu}=h^{\mu\nu}-\bar{g}^{\mu\nu}h^{\alpha}_{\alpha}/2.$$ The indices are moving with respect to $\bar{g}_{\mu\nu}$, and the integral is taken on the two-sphere at infinity $r=\infty$. If $\bar{\xi}_{\mu}$ is the usual (past-directed) timelike Killing vector, $E(\bar{\xi})$ becomes the total mass generated by $h_{\mu\nu}$ which is called the Abbott-Deser mass $M_{AD}$.
Now we calculate the Abbott-Deser mass of the deformed black hole. We adopt the Schwarzschild$-$anti-de Sitter spacetime given by Eqs. and as the background spacetime. By a straightforward calculation, we find that the Abbott-Deser mass of the spacetime of Eq. becomes $$M_{AD}=\lim_{r\to\infty}\frac{r^2}{8\pi}\int
\left[
e^{2\nu_0}
\left(
\frac1r\left(e^{2(\mu-\mu_0)}-e^{2\psi}\right)
-(e^{2\psi})_{,r}
\right)
+\frac12(e^{2\nu_0})_{,r}(e^{2\psi}-1)
\right]d\Omega,
\label{AD_mass}$$ where $d\Omega\equiv \sin\theta d\theta d\phi$ and we used $\bar{\xi}^{\mu}=(-1, 0, 0, 0)$. By expanding this formula in $\epsilon$ using Eqs. and , $M_{AD}$ is written like $$M_{AD}=\epsilon M^{(1)}_{AD}+\epsilon^2 M^{(2)}_{AD}+\cdots.$$ Because $\mu_1$ and $\psi_1$ are proportional to $P_l(\cos\theta)$, the integration in Eq. immediately leads to $M^{(1)}_{AD}=0$ for $l\ge 2$. For the second-order Abbott-Deser mass $M^{(2)}_{AD}$, we find, by substituting Eqs. , , , and and then using Eqs. , , , and , $$M^{(2)}_{AD}=\lim_{r\to\infty}
\left(\frac{2\Lambda c_0a_1}{3(2l+1)}r^2+\cdots\right)=-\infty.
\label{AD_result}$$ Thus the Abbott-Deser mass diverges to minus infinity at second order.
One interpretation for this divergence of the Abbott-Deser mass is that the deformed black hole is a far-lower-energy state compared to the Schwarzschild$-$anti-de Sitter black hole and the Schwarzschild$-$anti-de Sitter black hole would rapidly deform toward the lower-energy state. However, we would like to point out that such an interpretation may not be correct. To see this, we consider the coordinate transformation $r\to r+\epsilon a$ of the Schwarzschild$-$anti-de Sitter metric. The resulting metric is given by Eq. with $$\begin{aligned}
&e^{-2\nu}=e^{2\mu}=e^{2\mu_0}
+\left(e^{2\mu_0}\right)_{,r}a\epsilon
+\left(e^{2\mu_0}\right)_{,rr}a^2\epsilon^2/2+\cdots,\\
&e^{2\psi}=1+{2a}\epsilon/r+{a^2}\epsilon^2/r^2.\end{aligned}$$ Substituting these formulas to Eq. , we find $M^{(1)}_{AD}=0$ and $$M^{(2)}_{AD}=\lim_{r\to\infty}\left(-\frac{5\Lambda a^2}{6}r+\cdots\right)
=\infty.
\label{AD_coordinate}$$ Thus the Abbott-Deser mass for the perturbation generated by the coordinate transformation also diverges at second order. This indicates that the Abbott-Deser mass is not gauge invariant at second order and hence its divergence may be spurious. In fact, we can obtain a finite mass for the deformed black hole, if we use the covariant definition of mass given by Ashtekar and Magnon.
Ashtekar-Magnon mass
--------------------
Ashtekar and Magnon constructed a conserved quantity of a weakly asymptotically anti$-$de Sitter spacetime $(\hat{M}, \hat{g}_{\mu\nu})$ [@AM84]. They considered the conformally transformed spacetime $(M, g_{\mu\nu})$ where $g_{\mu\nu}=\Omega^2\hat{g}_{\mu\nu}$ and the situation that $M$ has a boundary $\mathcal{I}$ whose topology is $R\times S^2$. Their conserved quantity is defined on the spacelike $S^2$ surface (denoted by $\mathcal{C}$) on the conformal boundary $\mathcal{I}$ as follows: $$Q_{\xi}\equiv -\frac{1}{8\pi}\left(-\frac{3}{\Lambda}\right)^{3/2}
\int_{\mathcal{C}}\Omega^{-1}C_{\alpha\mu\beta\nu}
\xi^{\alpha} n^{\mu}N^{\beta}n^{\nu}dS,$$ where $n_{\mu}=\nabla_{\mu}\Omega$, $N^{\beta}$ is the timelike unit normal on $\mathcal{C}$, $\xi^{\alpha}$ is the conformal Killing vector field on $\mathcal{I}$, $dS$ is the proper area element of $\mathcal{C}$, and $C_{\alpha\mu\beta\nu}$ denotes the Weyl tensor of $M$. All index moving and covariant derivatives are with respect to the metric of the conformally transformed spacetime $(M, g_{\mu\nu})$. This definition is conformally invariant with the same choice of the coordinate components of $\xi^{\mu}$. If $\xi^\mu$ is the (future-directed) timelike Killing vector field, this conserved quantity is the Ashtekar-Magnon mass $M_{AM}$ of the spacetime. For the axisymmetric spacetime with metric , we find by a straightforward calculation that the Ashtekar-Magnon mass becomes $$\begin{gathered}
M_{AM}=\lim_{r\to\infty}-\sqrt{\frac{-3}{16\Lambda}}
\int_{0}^{\pi}
\xi^t r^2e^{\nu}
\left(
e^{-2\mu}\left[\nu_{,rr}+\nu_{,r}(\nu_{,r}-\mu_{,r})\right]
+\frac{e^{-2\psi}}{r^2}\mu_{,\theta}\nu_{,\theta}+\frac{\Lambda}{3}
\right)
e^{2\psi}\sin\theta d\theta.
\label{AM_mass}\end{gathered}$$ If we calculate this quantity for the Schwarzschild$-$anti-de Sitter spacetime, we obtain $M_{AM}=m$ choosing $\xi^t=1$. We can easily show that this quantity does not change under the coordinate transformation $r\to r+\epsilon a$.
In the above definition, how to choose the norm of $\xi^{\mu}$ has not been specified. Here we would like to discuss this criterion using one concrete example, because the mass calculation of the deformed black hole crucially depends on the choice of the norm of $\xi^{\mu}$. If we make a coordinate transformation $r\to ar$ and $t\to bt$ to the Schwarzschild$-$anti-de Sitter spacetime, the resulting metric becomes Eq. with $$\begin{aligned}
b^{-2}e^{2\nu}=a^2e^{-2\mu}&=1-2m/ar-\Lambda a^2r^2/3,\\
e^{2\psi}&=a^2.\end{aligned}$$ Choosing $\Omega=1/r$, the metric of the conformal boundary $\mathcal{I}$ becomes $$ds^2=\left(\Lambda a^2b^2/3\right)dt^2
+a^2\left(d\theta^2+\sin^2\theta d\phi^2\right).$$ Calculating Eq. for this metric, we obtain $M_{AM}=\xi^t bm$. Because the mass should be invariant under the coordinate transformation, this means that we should choose $\xi^t=b^{-1}$. One of the natural general criteria that recover this choice is as follows: if there is a conformal transformation such that the norm of the Killing vector field $\xi^{\mu}$ becomes constant on $\mathcal{I}$, we should choose $\xi^{\mu}$ that satisfy $$\left({3/}{\Lambda}\right)\xi^\mu\xi_\mu={S}/{4\pi},
\label{choosing_norm}$$ for the calculation of the Ashtekar-Magnon mass, where $S$ denotes the proper area of the $t={\rm const}$ surface on $\mathcal{I}$. We adopt this criterion in calculating the Ashtekar-Magnon mass of the deformed black hole.
Now we calculate the Ashtekar-Magnon mass of the deformed black hole. From the above discussion, we use $\xi^t$ in the calculation of Eq. as follows: $$\xi^t=\lim_{r\to\infty}\sqrt{\langle e^{2\psi}\rangle}
=\lim_{r\to\infty}1+
\epsilon^2\left(
\langle\psi_2\rangle+\langle\psi_1^2\rangle
\right)+O(\epsilon^4),$$ where the definition of $\langle f\rangle$ is given in Eq. . Substituting this formula and expanding Eq. using Eqs. , , and , $M_{AM}$ can be written like $$M_{AM}=M^{(0)}_{AM}+\epsilon M^{(1)}_{AM}+\epsilon^2 M^{(2)}_{AM}+\cdots.$$ We immediately obtain $M^{(0)}_{AM}=m$ and $M^{(1)}_{AM}=0$ for $l\ge 2$. Substituting Eqs. , , , , and and then using Eqs. , , , and for $M^{(2)}_{AM}$, we obtain, after a rather lengthy calculation, $$M^{(2)}_{AM}
=\frac13\Lambda\left(\frac{2\left(a_1a_2-a_3c_0\right)}{2l+1}-b_3\right)
+\frac{2a_1c_0}{2l+1}
+\frac{3mc_0^2}{2l+1}.
\label{AM_result_1}$$ This is finite in contrast to the Abbott-Deser mass. Because the Ashtekar-Magnon mass is a covariant definition of mass which is welldefined even in the weakly asymptotically anti$-$de Sitter spacetimes, we can regard it as a real amount of energy contained in the spactimes with deformed black holes. The validity of the Ashtekar-Magnon mass is also supported by the expectation that the continuous change of mass should result from the continuous deformed black hole series. Hence we consider that there is a possibility of the quasistatic deformation of the spacetime boundaries in weakly asymptotically anti$-$de Sitter spacetimes and we investigate the thermodynamic law of the deformed black holes with this mass definition in the following two sections.
To simplify Eq. , we consider the method of expressing $b_3$ with $H^{(1)}$ and $K^{(1)}$. By summing Eqs. and and then rewriting the right-hand side using Eqs. and , we find $$\begin{gathered}
\left(r^2e^{2\nu_0}H^{(2)}_{,r}\right)_{,r}
+2mH^{(2)}_{,r}
-\frac23\Lambda\left(r^3H^{(2)}\right)_{,r}
-\left(r^2e^{2\nu_0}K^{(2)}_{,r}\right)_{,r}\\
=\frac{1}{2l+1}
\left[
\left(r^2e^{2\nu_0}H^{(1)}H^{(1)}_{,r}\right)_{,r}
-l(l+1)\left(H^{(1)}\right)^2
\right].\end{gathered}$$ Integrating this equation from the horizon to large $r$ and substituting Eqs. , , and , we obtain $$\frac{\Lambda}{3}b_3=\frac{5l(l+1)-6}{2(2l+1)}a_1c_0
+2(3m-r_h)\tilde{b}_0
-\frac{l(l+1)}{2l+1}\int_{r_h}^{\infty}\left(H^{(1)}\right)^2dr.$$ Substituting this formula, Eqs. and into Eq. , we finally find $$M^{(2)}_{AM}=
\frac{-(l^2+l+2)}{2(2l+1)}a_1c_0
-2(3m-r_h)\tilde{b}_0
+\frac{l(l+1)}{2l+1}\int_{r_h}^{\infty}\left(H^{(1)}\right)^2dr
+\frac{3mc_0^2}{2l+1},
\label{AM_result_2}$$ where $\tilde{b}_0$ can be written in terms of $\tilde{d}_0$ and $\tilde{d}_1$ using Eq. , which in turn are given in Eqs. and . Hence we can express $M^{(2)}_{AM}$ in terms of the quantities of only the first-order perturbation.
Approximate calculation for $\alpha\gg 1$ and $\alpha\ll 1$
===========================================================
As we mentioned in the Secs. I and II, to solve analytically the equations of the perturbation is rather difficult. However, we can construct the solution approximately in the two simple cases: one is the case that the anti$-$de Sitter radius is much larger than the Schwarzschild radius $\alpha \ll 1$, and the other is that the anti$-$de Sitter radius is much smaller than the Schwarzschild radius $\alpha \gg 1$. In this section, we will show the solution for these two cases and calculate the horizon area and the Ashtekar-Magnon mass. Using these results, we can discuss the properties of the first law of the black hole thermodynamics for the deformed black holes.
$\alpha \ll 1$ case
-------------------
Setting a new coordinate $x\equiv r/2m$, the function $e^{2\nu_0}$ becomes $$e^{2\nu_0}= 1 - {1}/{x} + \left( \alpha x \right)^{2}.$$ In the case of $\alpha\ll 1$, the horizon location is $x=x_h\simeq 1$. In the region $1\le x\lesssim \alpha^{-2/3}$, the order of the second term is larger than $O(\alpha^{2/3})$ and the order of the third term is smaller than $O(\alpha^{2/3})$. Hence the sum of the first and second terms is much larger than the third term: the spacetime is Schwarzshild-like. In the region $\alpha^{-2/3}\lesssim x\le \infty$, the order of the second term is smaller than $O(\alpha^{2/3})$ and the order of the third term is greater than $O(\alpha^{2/3})$. Because the sum of the first and third terms is much greater than the second term, the spacetime is similar to the anti$-$de Sitter spacetime. Hence, we can use the matching method: we construct solutions for the Schwarzschild spacetime and the anti$-$de Sitter spacetime in the regions $1\le x\lesssim \alpha^{-2/3}$ and $\alpha^{-2/3}\lesssim x\le \infty$, respectively, and then match these two solutions in the overlapping region $x \sim \alpha^{-2/3}$.
The general solutions of $H^{(1)}$ for Schwarzschild spacetime are given in [@RW57]. The solutions that satisfy the boundary condition on the horizon can be expressed in terms of associated Legendre’s polynomials, $H^{(1)}_{in}=P_l^2(2x-1)$. To construct a solution $H^{(1)}$ in the anti$-$de Sitter regime, we introduce a new coordinate $y\equiv \sqrt{-\Lambda/3}r$. Setting $Y\equiv y^2$ and $M=Y^{(l+2)/2}\tilde{M}$, Eq. in the anti$-$de Sitter regime becomes $$Y(1+Y)\tilde{M}_{,YY}+
\left[
(l+1/2)Y+(l+3/2)
\right]\tilde{M}_{,Y}
+\left[l(l-1)/4\right]\tilde{M}=0.$$ Because this equation is related to the hypergeometric equation, one of the solutions becomes $\tilde{M} =
{}_2F_1 \left( l/2 ,\ (l-1)/2; \ l + {3}/{2} ; \ -Y \right),$ and the corresponding solution is $$H^{(1)}_{out}=\frac{y^l}{y^2+1}
{}_2F_1 \left( l/2 ,\ (l-1)/2; \ l + {3}/{2} ; \ -y^2 \right),
\label{H1_out}$$ which behaves like $H^{(1)}_{out}\simeq y^l$ in the matching region. Since the behavior of $H^{(1)}_{in}$ is proportional to $x^l$ in the overlapping region, we can match these two solutions. The solution of $H^{(1)}_{in}$ that smoothly continues to Eq. can be written as $$H^{(1)}_{in}=-
\left(\frac{\alpha}{2}\right)^l
\frac{(l-2)!}{(2l-1)!!}P_l^2(2x-1).
\label{H1_in}$$ Although the convergence region of Eq. is $0\le y\le 1$, we can make an analytic continuation to the region $1\le y\le \infty$ using well-known techniques. Hence we have constructed the solution of $H^{(1)}$ in the $\alpha\ll 1$ case. Using Eq. , we can also write down the solutions $K^{(1)}_{in}$ and $K^{(1)}_{out}$ in terms of associated Legendre’s polynomials and the hypergeometric functions, respectively, although we do not show them explicitly here. Using this solution of $K^{(1)}$ or observing the behavior of $H^{(1)}_{in}$ and $H^{(1)}_{out}$ near the horizon and infinity and then using Eqs. and , we find $$\frac{\tilde{c}_0}{c_0}=
\frac{\Gamma((l-1)/2)\Gamma((l+3)/2)\Gamma(l+3)}{4\Gamma(l+3/2)\Gamma(l+1/2)}
\left(\frac{\alpha}{4}\right)^l.$$ Hence, for $\alpha\ll 1$, the deformation of the horizon is much smaller than that of spacelike infinity, and this tendency is enhanced for larger $l$.
Now that we have constructed a solution of the first-order perturbation, we would like to calculate the mass and horizon area. However, this calculation for general $l$ requires integration of the products of the hypergeometric functions, and the analytic calculation is rather difficult. Hence we consider analytically only the $l=2$ case, where the function in Eq. reduces to the elementary functions. For $l>2$, we numerically evaluate the relation of the area and mass. Before doing this, we would like to remark on the general properties of the second-order solution in the $\alpha\ll1$ case. As we can see from Eqs. and , the order of the second-order perturbation becomes $O(\alpha^{-1}\epsilon^2)$, which is much larger than $\epsilon^2$. This indicates that the nonlinear effect rapidly increases with an increase in $\epsilon$. This perturbative analysis is only reliable for a sufficiently small $\epsilon$ that satisfies $\epsilon < \alpha$.
Now we calculate the horizon area and the mass for the $l=2$ case. The solutions of $H^{(1)}_{out}$ and $K^{(1)}_{out}$ corresponding to Eq. can be rewritten as $$\begin{aligned}
& H^{(1)}_{out}=\frac{5}{8y^3\left(y^2+1\right)}
\left[
-y\left(5y^2+3\right)+3\left(y^2+1\right)^2\arctan y
\right],\\
& K^{(1)}_{out}=\frac{5}{2}-\frac{15}{8y^3}
\left[
y+\left(y^2-1\right)\arctan y
\right].\end{aligned}$$ Using Eqs. , , and , we find $$\begin{aligned}
&\frac{\delta A}{A} \simeq
\frac{3\pi}{8}
\left(12\log2 -11\right)\alpha^{-1}\epsilon^2,\\
&\frac{\delta m}{m}\simeq
\frac{3\pi}{4}
\left(3\log2 -4\right)\alpha^{-1}\epsilon^2,\\
&\frac{\delta S}{S}\simeq
\frac{5}{2}\epsilon^2,\end{aligned}$$ where $\delta m\equiv M^{(2)}_{AM}\epsilon^2$. Note that both $\delta A$ and $\delta m$ are negative. Because $(\delta A/A)/(\delta S/S)=O(\alpha^{-1})$, the back reaction to the horizon area is far larger than that to the spacelike area at infinity. This result implies $$\frac{(\delta m/\delta A)}{(\kappa/8\pi)}=
\frac{12\log 2-16}{12\log 2-11}\simeq 2.86,
\label{thermodynamic_alpha_ll_1_coeffecient}$$ which indicates that the first law in the usual form does not hold for deformed black holes. In other words, the first law for the deformation of the Schwarzschild$-$anti-de Sitter black holes has a correction term such that $$\begin{aligned}
\delta m = \frac{\kappa}{8 \pi} \delta A + \delta W,
\label{thermodynamic_alpha_ll_1}\end{aligned}$$ where $\delta W$ can be expressed in terms of the coefficients of $H^{(1)}$ and $K^{(1)}$ as $\delta W=-a_1c_0\epsilon^2/5$ in the $\alpha\ll 1$ case. Because $c_0$ is related to the deformation of spacelike infinity, this term would be related to the work which is necessary for the deformation of infinity. In the next section, we will discuss the origin of this work term in more detail.
Because the work term $\delta W$ in Eq. is negative, the change in the horizon area $\delta A$ of the deformed black hole and that of the Schwarzschild$-$anti-de Sitter black hole $\delta {A_0}$ for the same $\delta m$ (i.e., $\kappa\delta A_0/8\pi=\delta m$) satisfy the relation $\delta A_0<\delta A$. This implies that the area of the deformed black hole is larger than that of the Schwarzschild$-$anti-de Sitter black hole if compared under the same Ashtekar-Magnon mass. Interestingly, we can claim that the deformation of the Schwarzschild$-$anti-de Sitter black hole is a process consistent with the area theorem. Although the area theorem has not been proved in the spacetime with $\Lambda<0$ and there are some counterexamples for this theorem such as the black holes in Brans-Dicke theory [@Kan96], we expect that this result indicates the importance of the solution sequence of the deformed black hole.
$l$ 2 3 4 5 6 7 8 9
------------------------------------ --------- --------- --------- --------- --------- --------- --------- ---------
$\alpha (\delta A/A)/(\delta S/S)$ $-1.26$ $-1.86$ $-2.45$ $-3.02$ $-3.60$ $-4.17$ $-4.73$ $-5.30$
$\alpha (\delta m/m)/(\delta S/S)$ $-1.81$ $-5.18$ $-11.2$ $-20.5$ $-34.0$ $-52.4$ $-76.4$ $-107.$
$\alpha (\delta W/m)/(\delta S/S)$ $-1.18$ $-4.24$ $-9.94$ $-19.0$ $-32.2$ $-50.3$ $-74.0$ $-104.$
: The value of $\alpha (\delta A/A)/(\delta S/S)$, $\alpha (\delta m/m)/(\delta S/S)$, and $\alpha (\delta W/m)/(\delta S/S)$ for $l=2,...,9$. The work term is negative for all $l$ and it becomes large as $l$ increases.
Now we discuss the $l>2$ cases. Calculating the integrals in Eqs. and numerically, we evaluate the factor $\alpha(\delta A/A)/(\delta S/S)$, $\alpha(\delta m/m)/(\delta S/S)$, and $\alpha(\delta W/m)/(\delta S/S)$. In all cases, $\delta A$, $\delta m$, and $\delta W$ are negative. The results are shown in Table I. The work term estimated by $(\delta W/m)/(\delta S/S)$ increases for larger $l$ and the difference from the ordinary thermodynamical relation becomes large. A higher-multipole deformation requires a larger work term. Interestingly, we have found that the relation $$\delta W=-\frac{(l-1)(l+2)}{4(2l+1)}a_1c_0\epsilon^2
\label{work_a1c0}$$ holds with accuracy $10^{-6}$. This indicates that this would be the exact value and that our numerical calculation is accurate. Similarly to the $l=2$ case, the area of the deformed black hole is larger than that of the Schwarzschild$-$anti-de Sitter black hole if compared under the same mass.
$\alpha \gg 1$ case
-------------------
In the case of $\alpha\gg 1$, we have the relation $x_h\simeq \alpha^{-2/3}\ll 1$ for the location of the black hole $x=x_h$. Near the horizon, the first term of $e^{2\nu_0}=1-1/x+(\alpha x)^2$ is $\alpha^{-2/3}$ times smaller compared to the other two terms. In the region $x\gtrsim 1$, the third term $(\alpha x)^2$ in $e^{2\nu_0}$ is more than $\alpha^2$ times larger than the other two terms. Hence, for all $x_h\le x\le\infty$, the first term is small compared to the sum of the other two terms and we neglect it in the following analysis. In this approximation, we should neglect terms whose order is $\alpha^{-2/3}$ times the leading order.
Setting $z\equiv r/r_h$ and $Z\equiv z^3$, Eq. is written as $$\begin{aligned}
3Z \left( Z - 1 \right) {M}_{,ZZ}
-\left( 2Z+1 \right) {M}_{,Z} +
{2} M = 0,\end{aligned}$$ under the above approximation. The solution satisfying the boundary condition becomes $M=2Z-3Z^{2/3}+1$, and the corresponding solution $H^{(1)}$ becomes $$\begin{aligned}
H^{(1)}
= - \frac{1}{z} + \frac{3z}{z^{2} + z + 1}+O(\alpha^{-2/3}).\end{aligned}$$ Using Eq. , we derive $$\begin{aligned}
K^{(1)} &=& \frac{6 \alpha^{2/3}}{l^2+l-2} + O(1).\end{aligned}$$ In summary, the solutions of $H^{(1)}$ and $K^{(1)}$ are $$\begin{aligned}
&H^{(1)}=O(\alpha^{-2/3}),
\label{H1_alpha_gg_1}\\
&K^{(1)}=\left(2l+1\right)^{1/2}+O(\alpha^{-2/3}),
\label{K1_alpha_gg_1}\end{aligned}$$ where we used the degree of freedom to choose the amplitude of $K^{(1)}$. It is a remarkable fact that $K^{(1)}$ is almost constant and hence $|\tilde{c}_0|\simeq |c_0|$. Although we mentioned in Sec. II that the conformal boundary at infinity is more deformed compared to the horizon, the deformations of the two are similar if $|\Lambda|$ is large. This feature does not depend on $l$. These results are in contrast to the $\alpha\ll 1$ case.
Substituting Eqs. and into Eqs. , , and , we immediately find $$\begin{aligned}
\delta A/A&=\left[{2}+O(\alpha^{-2/3})\right]\epsilon^2,\\
\delta S/S&=\left[{2}+O(\alpha^{-2/3})\right]\epsilon^2,\\
\delta m/m&=\left[{3}+O(\alpha^{-2/3})\right]\epsilon^2. \end{aligned}$$ In this case, the back reaction to the horizon area is similar to that to the spacelike area at infinity. This result leads to $$\begin{aligned}
\delta m \simeq \left({\kappa}/{8 \pi}\right) \delta A,
\label{thermodynamic_alpha_gg_1}\end{aligned}$$ which means that the thermodynamic law of the deformed black holes is almost the same as that of the Schwarzschild$-$anti-de Sitter black holes if $|\Lambda|$ is sufficiently large. In contrast to the $\alpha\ll 1$ case, the work term is small in the $\alpha\gg 1$ case. Hence we find that there is a correlation between the value of the work term and the difference of the deformation of the horizon and the spacelike surface at infinity: the absolute value of the work term decreases as $|\tilde{c}_0/c_0|$ approaches unity.
Numerical calculation for $\alpha\sim 1$
========================================
In this section, we numerically investigate the deformed black hole for $\alpha\sim 1$ to complete the perturbative analysis. Equations , , , and can be reduced to two first-order differential equations for $(rH^{(1)})$ and $K^{(1)}$, which asymptote to constant values for large $r$. We solved these equations using the Runge-Kutta method from the horizon $r=r_h$ to the cutoff value $r=r_c$. We selected the grid number and the cutoff value $r=r_c$ as follows. Because there are two characteristic length scales $R_A$ and $R_S$, we set $10^2$ grids within the smaller length scale and solved in a range which is $10^2$ times as long as the larger length scale. Hence, the cutoff value is $r_c=r_h+10^2\times{\rm max}[R_A, R_S]$ and the total grid number becomes $10^4\times{\rm max}[\alpha, \alpha^{-1}]$. Beyond the cutoff $r=r_c$, we approximate $H^{(1)}$ and $K^{(1)}$ with the formulas $H^{(1)}\simeq a_1/r+a_2/r^2$ and $K^{(1)}\simeq c_0+c_1/r$, from which we determine the value of $c_0$ and $a_1$ using Eq. , Eq. , and the numerical values of $(rH^{(1)})$ and $K^{(1)}$ at $r=r_c$, and evaluate the integrals in Eqs. and beyond the cutoff $r>r_c$. The numerical error is about $0.1$%, which is estimated by using several different grid numbers and the cutoff values. In the $\alpha=10^{-3}$ case, our numerical results coincide with the values in Table I with $0.1$% accuracy.
Now we show the numerical results. Figure 1 shows the behavior of the ratio $(\tilde{c}_0/c_0)$ of the deformation of the two spacetime boundaries—i.e., the horizon and the two-surface at spacelike infinity—as a function of $\alpha$. The asymptotic behaviors for $\alpha\gg 1$ and $\alpha\ll 1$ derived in Sec. IV are also shown. We see that the matching method gives a fairy good approximation for $\alpha\lesssim 10^{-1}$. The behavior of $1-(\delta A/A)/(\delta S/S)$ and $3/2-(\delta m/m)/(\delta S/S)$ are shown in Figs. 2 and 3, respectively. We see that both $(\delta A/A)/(\delta S/S)$ and $(\delta m/m)/(\delta S/S)$ are proportional to $\alpha^{-1}$ for $\alpha\ll 1$, which is consistent with the analysis of Sec. IV. In the region $\alpha\gg 1$, $1-(\delta A/A)/(\delta S/S)$ and $3/2-(\delta m/m)/(\delta S/S)$ asymptote to zero and these values are proportional to $\alpha^{-2/3}$. This is also consistent with the results in Sec. IV.
Figure 4 shows the behavior of the work term $(\delta W/m)/(\delta S/S)$. We see that the value of $\log_{10}[-(\delta W/m)/(\delta S/S)]$ does not diverge and hence $\delta W$ always takes a negative value. This implies that the area of the deformed black hole is larger than that of the Schwarzschild$-$anti-de Sitter black hole if compared under the same Ashtekar-Magnon mass for arbitrary $\alpha$. In the region $\alpha\ll 1$, the value of $(\delta W/m)/(\delta S/S)$ is proportional to $\alpha^{-1}$, which coincides with the results of Sec. IV. This behavior of $(\delta W/m)/(\delta S/S)$ indicates that quasistatic deformation requires a larger absolute value of the work for smaller $\alpha$ and it is consistent with the fact that the Schwarzschild spacetime does not allow the quasistatic deformation. The value of $(\delta W/m)/(\delta S/S)$ is proportional to $\alpha^{-4/3}$ in the $\alpha\gg 1$ region, although both $1-(\delta A/A)/(\delta S/S)$ and $3/2-(\delta m/m)/(\delta S/S)$ are proportional to $\alpha^{-2/3}$. This is probably because the terms which are proportional to $\alpha^{-2/3}$ in $\delta A/A$ and $\delta m/m$ cancel each other in the calculation of the work term, although we have not proceeded this analysis. The work term rapidly decreases in the region $\alpha\gg 1$. For all $\alpha$, the absolute value of $(\delta W/m)/(\delta S/S)$ becomes large as $l$ increases. The higher-multipole deformation requires a larger work term.
Figure 5 shows the value of $\delta W/a_1c_0\epsilon^2$, which was introduced with Eq. in the approximate analysis for $\alpha\ll 1$ in Sec. IV. The value of $\delta W/a_1c_0\epsilon^2$ is almost constant in the region where the matching method provides a good approximation. Although this quantity slightly changes with an increase of $\alpha$, it seems to asymptote to some nonzero constant value. This is supported by the following consideration. For $\alpha\gg 1$, we see that $K^{(1)}\sim 1$ and $H^{(1)}\sim \alpha^{-2/3}/z$ from Eqs. and . Using $z=r/r_h$ and $r_h\simeq 2m\alpha^{-2/3}$, we have $a_1\sim\alpha^{-4/3}m$ and thus $a_1c_0\epsilon^2\sim \alpha^{-4/3}m\epsilon^2$. This leads to $-a_1c_0\epsilon^2\sim\delta W$ for $\alpha\gg 1$. Hence the value of $-\delta W/a_1c_0\epsilon^2$ is always $O(1)$ and the relation, Eq. , approximately holds for all $\alpha$.
We can construct another quantity which has the same order as $\delta W$ as follows: $$-\frac{\delta W}{r_h}
\sim \left(1-\frac{r_h}{2m}\right)^{-1/2}
\left({c}_0^2-\tilde{c}_0^2\right).
\label{work_deform}$$ According to this formula, the work term normalized by the horizon radius depends only on two factors. One is related to the horizon radius, which shows that the absolute value of the work term becomes larger for smaller $\alpha$. The other factor depends on the values of $\tilde{c}_0$ and $c_0$, which are related to the deformation of the horizon and spacelike infinity, respectively. Because the deformation of the two boundary surfaces becomes similar $|\tilde{c}_0/c_0|\simeq 1$ for larger $\alpha$, the absolute value of the work term decreases with an increase of $\alpha$. Conversely, it becomes large with a decrease in $\alpha$, which corresponds to a decrease in $|\tilde{c}_0/c_0|$. Hence we interpret that the origin of the work term is related to the difference of the deformation between the two boundary surfaces of the spacetime. The work would be necessary for the deformation of both horizon and spacelike infinity, and their difference would appear in the first law as the work term.
Summary and Discussion
======================
In this paper, we analyzed the static deformation of the Schwarzschild$-$anti-de Sitter black holes using perturbative techniques. We showed that there exists a regular solution for the first-order perturbation. The resulting spacetime contains the deformed black hole whose horizon deviates from the geometrically spherically symmetric surface. The spacelike infinity of this spacetime is deformed simultaneously. Hence this spacetime is not asymptotically anti$-$de Sitter, although this is still weakly asymptotically anti$-$de Sitter in the sense of the Ashtekar-Magnon definition [@AM84]. In the $\alpha\ll 1$ case, the deformation of the horizon is about $\alpha^l$ times smaller than the deformation of spacelike infinity, while the deformations of the two are similar in the $\alpha\gg 1$ case.
We considered the $l=0$ mode of the second-order perturbation and calculated the change in the horizon area and the mass. In the mass calculation, we used two definitions of the mass: the Abbott-Deser mass and the Ashtekar-Magnon mass. The Abbott-Deser mass for the deformed black hole diverges to minus infinity. If this result is realistic, we are forced to conclude that the Schwarzschild$-$anti-de Sitter black hole would rapidly deform toward the lower-energy state. But this definition is not gauge invariant at second order and we consider that this result may be spurious. Because the Ashtekar-Magnon mass is a covariant definition of the mass from which we obtained finite results, we expect that it represents a real amount of energy although one assumption is imposed in Eq. in choosing the norm of $\xi^t$. Our results indicate that the quasistatic deformation of the spacetime boundaries may occur with a finite change in the total energy in the weakly asymptotically anti$-$de Sitter spacetimes. Of course only with the analysis in this paper, we cannot rigorously conclude that such a process actually occurs. But our expectation that the spacetime boundaries are flexible is also supported by the fact that no one has proved that spacelike infinity of the weakly asymptotically anti$-$de Sitter spacetime should be rigid; i.e., it has a global timelike conformal Killing vector field. There might exist many weakly asymptotically anti$-$de Sitter solutions whose geometrical configuration at spacelike infinity temporally evolves.
We studied the thermodynamic first law of the deformed black holes using the Ashtekar-Magnon mass. In the $\alpha\gg 1$ case, the first law in the usual form is approximately recovered, while in the $\alpha\ll 1$ case, the first law does not hold: the contribution of the work term $\delta W$ becomes important. In Sec. V, we numerically calculated this work term in the range $10^{-3}\le\alpha\le 10^3$ for $l=2,...,6$ and confirmed that it is always negative in this regime. Let us discuss the implications of this first law from the viewpoint of black hole thermodynamics. Although the ratio of $\delta m/m$ and $\delta A/A$ is fixed for each $\alpha$ in Secs. IV and V, we can consider these parameters to be independent if we further take account of the thermodynamical relation of the background Schwarzschild$-$anti-de Sitter spacetime, $\delta m_0=(\kappa/8\pi)\delta A_0$. The first law for the deformation process is given by $\delta m=(\kappa/8\pi)\delta A+\delta W$ with $\delta W<0$ for two independent parameters $\delta m$ and $\delta A$. If the deformation process occurs without changing the horizon area (i.e., $\delta A=0$ and $\delta m=\delta W$), the black hole mass decreases (i.e., $\delta m<0$) and the black hole evolves towards the lower-energy state. Hence the deformation may work as a process of energy extraction from a spherical black hole. On the other hand, if there is a deformation process in which the black hole mass does not change, the negative work term implies that the horizon area of the deformed black hole increases in this process. This indicates that quasistatic deformation can be a process consistent with the area theorem. Although whether the area theorem holds for these spacetimes is quite uncertain, we expect that this is an indication for the importance of the solution series of the deformed black holes. The value of $(\delta W/m)/(\delta S/S)$ is given in Fig. 4 as a function of $\alpha$. It is proportional to $\alpha^{-1}$ for $\alpha\ll 1$. This indicates that the deformation requires the larger absolute value of the work for smaller $\alpha$. This is consistent with the fact that the Schwarzschild spacetime does not allow a continuous static deformation. For $\alpha\gg 1$, the absolute value of $(\delta W/m)/(\delta S/S)$ is proportional to $\alpha^{-4/3}$ and rapidly decreases with the increase in $\alpha$. The higher-multipole deformation requires a larger absolute value of the work term. Because the work term satisfies the relation for any $\alpha$, this term is closely related to the difference of the deformation between the two boundaries of this spacetime: i.e., the horizon and spacelike two-surface at infinity.
Here, we discuss the reason why the work term is negative using the Hartle-Hawking formula [@HH72] (see also [@Carter79] for a review). Using the Raychaudhuri equation, Hartle and Hawking derived the following formula for an increase in the horizon area in the quasistationary evolution of a black hole: $$\frac{\kappa}{8\pi}\delta A
=\oint dA\int_{t_0}^{t_1}\left(\frac{\sigma^2}{16\pi}+T_{ab}l^al^b\right)dt,
\label{Hartle-Hawking}$$ where $t_0$ and $t_1$ denote the time for the initial state and final state, respectively, $\sigma$ denotes the shear scalar of the null geodesic congruence of the horizon, $T_{ab}$ is the energy-momentum tensor of the matter field that crosses the horizon, and $l^a$ is the Killing vector field on the horizon. In the asymptotically flat case, this formula is equivalent to the thermodynamic first law. If matter crosses the horizon and the flow of gravitational wave energy can be ignored, the second term of the integral in Eq. becomes the change of mass $\delta M$ and the first term becomes zero because $\sigma$ has the same order as $\delta A$ and thus $\sigma^2$ is much smaller than $\delta A$. (Here we do not consider any change of angular momentum.) If there is no matter and the gravitational wave energy is absorbed into the black hole, the first term has the same order as $\delta A$ in this case and gives $\delta M$. Hence the first law can be derived in general. The first term of the integral in Eq. has an analogy with the entropy generated by a surface shear viscosity with magnitude $\eta_\nu=1/16\pi$ of the ordinary viscous fluid. Now we discuss the application of this Hartle-Hawking formula to the deformed black hole. If some energy flux crosses the Schwarzschild$-$anti-de Sitter horizon to induce a quasistatic deformation of the spacetime, a part of the first term of the integral in Eq. would contribute to the work term $\delta W$ necessary for the deformation, while the remaining part (in addition to the second term) would contribute to the mass term $\delta M$. The important fact is that the shear $\sigma$ is $O(\epsilon)$ in this case, and hence all $\delta A$, $\delta M$, and $\sigma^2$ have the same order $O(\epsilon^2)$. The first term of the integral in Eq. cannot be ignored, and introducing an unknown positive parameter $\beta$, the work term and mass term may be given by $$\begin{aligned}
\delta W
&=-\oint dA\int_{t_0}^{t_1}(\beta{\sigma^2}/{16\pi})dt,\\
\delta M
&=\oint dA\int_{t_0}^{t_1}
\left(\frac{(1-\beta)\sigma^2}{16\pi}+T_{ab}l^al^b\right)dt,\end{aligned}$$ which requires the work term to be negative. Although we have not investigated the validity of the relations in terms of the Ashtekar-Magnon mass, this discussion provides a natural interpretation of the reason why the work term obtained in this paper becomes negative.
Our remaining problems are as follows. As for the problems of the solution sequence of deformed black holes, we should construct solutions beyond the perturbative region. This would probably require numerical calculations. Because spacelike infinity is also deformed, we should impose the structure of spacelike infinity in the calculation. This arbitrariness in choosing the infinity structure would lead to great degrees of freedom of the solution series of deformed black holes. The condition for the existence of a solution due to the choice of infinity structure is of interest. Next, we should analyze the stability of spacetimes with deformed black holes. This will require an analysis of the quasinormal frequency of the deformed black holes. At the same time, we would like to analyze the perturbation from the Schwarzschild$-$anti-de Sitter black hole which gives the time-dependent geometrical configuration at spacelike infinity. With this analysis, what spacetime is an attractor of the weakly asymptotically anti$-$de Sitter spacetimes might become clear. Concerning the application of these spacetimes with deformed black holes, these solutions might contribute to the brane world scenario. Although our analysis is restricted to four-dimensional cases, it is natural to expect that similar solutions exist in higher dimensions. Hence, by appropriate cutting and pasting, these spacetimes might provide interesting models of the brane world scenario. Finally, we would like to analyze the implication for the AdS/CFT correspondence of these solutions. In the usual Schwarzschild$-$anti-de Sitter black holes, the quasinormal frequencies have a relation to the correlation function of the field on the boundary. By analyzing this correspondence in the spacetime with deformed black holes, these spacetimes might shed new light on the AdS/CFT correspondence. These are the problems we would like to investigate as the next step.
The work of H.Y. is supported in part by a grant-in-aid from Nagoya University 21st Century COE Program (ORIUM).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In spite of all [**no-go**]{} theorems (e.g., von Neumann, Kochen and Specker,..., Bell,...) we constructed a realist basis of quantum mechanics. In our model both classical and quantum spaces b are rough images of the fundamental [**prespace.**]{} Quantum mechanics cannot be reduced to classical one. Both classical and quantum representations induce reductions of prespace information.'
author:
- |
Andrei Khrennikov[^1]\
MSI, University of Växjö, S-35195, Sweden
title: Classical and quantum spaces as rough images of the fundamental prespace
---
[**1. Introduction.**]{} In preprints \[1\] [^2] there was constructed a contextual quantum representation of the Kolmogorovian model. That mathematical construction can be used as a [**realist basis of quantum mechanics**]{} (QM). Existence of such a realist “underground" of QM was the question of the great debate since first days of QM, see, e.g., \[2\], \[3\] for detail. It should be reminded that A. Einstein strongly supported the idea that such a realist underground of QM could finally be found and W. Heisenberg and N. Bohr claimed that it would be impossible. In this note we present main ideas of \[1\] without to go in rather technical mathematical details of contextual representation of the Kolmogorovian model in a Hilbert space.
It should be underlined from the very beginning that we do not discuss a reduction of quantum physics to classical one.
[**2. Prespace, classical space and quantum space.**]{} In my model both quantum and classical states are rough images of contexts – complexes of physical conditions. In the mathematical model \[1\], cf. \[4\], contexts are described as sets of fundamental parameters. We call the space of fundamental parameters [**prespace**]{} and denote it $\Omega.$ Contexts are representated by a family of subsets of $\Omega.$ The prespace $\Omega$ is underground of the classical space $X_{\rm{cl}}={\bf R}^3$ as well as the quantum (Hilbert) space $X_{\rm{q}}=H.$ QM gives essentially richer picture of the prespace $\Omega:$ the QM-representation of $\Omega$-contexts generates essentially larger class of images than the classical representation. In particular, it is impossible to reduce quantum picture of the prespace $\Omega$ to the classical one.
This is a very delicate point of considerations. Dynamics in the prespace $\Omega$ is a deterministic dynamics. But it is not “classical dynamics” since the latter takes place not in the prespace $\Omega$ but in the classical space $X_{\rm{cl}}.$ This is not a question of the mathematical realization of the prespace $\Omega$ and the classical space $X_{\rm{cl}}.$ It may be that the prespace $\Omega$ can also be described as $\Omega= {\bf R}^m$ (or even as $\Omega= {\bf R}^3$ – so in the same way as $X_{\rm{cl}}={\bf R}^3,$ cf. section 8). The crucial point is that $X_{\rm{cl}}$ is created via the huge [**reduction of information**]{} in the process of transition from the prespace contexts to points of $X_{\rm{cl}}.$ Each classical point $x \in X_{\rm{cl}}$ is the image of a domain $B_x$ of the prespace $\Omega$, see \[1\], and this domain can contain huge (may be even infinite) number of prepoints $\omega =\omega^x.$ By our model there exist mappings: $$\Omega \to X_{\rm{cl}}, \; \; \; \Omega \to X_{\rm{q}}, \; \; \;\mbox{and}\; \; \;
X_{\rm{q}}\to X_{\rm{cl}}$$ Thus we also obtain the map: $$\Omega \to X_{\rm{q}}\to X_{\rm{cl}}$$ But there is no pathway: $$\Omega \to X_{\rm{cl}}\to X_{\rm{q}}$$ We underline (see further considerations) that the correspondence principle is based on the map $\Omega \to X_{\rm{q}}$ and not at all on the map $X_{\rm{q}}\to X_{\rm{cl}}.$
[**3. Fundamental incompatible preobservables.**]{} Contextual probabilistic representation \[1\] of $\Omega$-contexts in the quantum space $X_{\rm{q}}=H$ is based on a fixed pair of incompatible preobservables ([**reference observables**]{}): $$\label{RO}
b, a: \Omega \to {\bf R}$$ In our model \[1\] preobservables are functions $d: \Omega \to {\bf R}.$ Denote the set of all preobservables ${\cal O} (\Omega).$ We interpret observables $d \in {\cal O} (\Omega)$ as [**realist observables:**]{} by fixing a prepoint $\omega \in \Omega$ we are able to fix the value $d= d(\omega).$ We are not able to measure an arbitrary $d \in {\cal O} (\Omega).$ But reference preobservables (\[RO\]) and functions of those preobservables, $f(b), f(a)$ can be measured \[1\]. We denote the space of quantum observables by the symbol ${\cal O}(H).$ In mathematical models ${\cal O}(\Omega)$ and ${\cal O}(H)$ are represented by spaces of (Kolmogorovian) random variables and self-adjoint operators, respectively.
[**4. Position-momentum picture of prespace.**]{} In principle a contextual probabilistic representation of the prespace $\Omega$ in the quantum space $X_{\rm{q}}=H$ can be based on any pair of incompatible preobservables. However, it seems that we (human beings) can use only the special pair of reference preobservables: $$(q,p)= (\rm{position, momentum}).$$ Thus modern quantum as well as classical physics are the [*position-momentum pictures*]{} of the prespace. All classical and quantum observables are functions of position and momentum observables. In quantum case we use functions $\hat{d}= u(\hat{q}, \hat{p})$ of operators of the position $\hat{q}$ and the momentum $\hat{p}.$
By choosing another pair of reference preobservables we obtain another quantum picture of the prespace $\Omega.$ However, it seems that at the present time we are not able to measure preobservables $d \in {\cal O} (\Omega)$ distinct from functions $f(q)$ and $g(p).$
[**5. Nonequivalence of quantum pictures of the prespace.**]{} It should be underlined that quantum pictures of the prespace $\Omega$ based on two different pairs of incompatible observables $(b,a)$ and $(v,w)$ are in general nonequivalent. Of course, the same mathematical formalism – the Hilbert space formalism – can be used for any quantum picture of the $\Omega.$[^3] But we should pay attention to physical structures of representations. So we should not forget about the $(q,p)$-origin of QM (as a physical theory and not as only a mathematical formalism), see also \[5\], \[6\].
[**6. No “no-go?”**]{} Existence of a realist underground model of QM looks very surprising in the view of various no-go theorems, e.g., von Neumann \[7\], Kochen-Specker \[8\], Bell \[9\],... But all those no-go theorems suffered of the absence of physical justification for the list of assumptions on the correspondence between a realist prequantum model and QM. J. Bell performed the brilliant analysis of assumptions on the “real-quantum" correspondence which were assumed (very often indirectly) in previous no-go theorems, see \[9\]. We should agree with Bell that von Neumann, Kochen and Specker and many others wanted too much for the “real-quantum" correspondence. Thus despite all pre-Bellian no-go theorems J. Bell was sure that it is possible to construct a realist basis of QM. However, J. Bell also wanted too much for the “real-quantum" correspondence, see,e.g., \[10\], \[6\] for analysis of Bell’s assumptions. As a consequence, he came to the conclusion that every realist prequantum model should be [**nonlocal.**]{}
[**7. Correspondence between preobservables and quantum observables.**]{} Correspondence-maps $$W: {\cal O}(\Omega) \to {\cal O}(H)$$ between realist preobservables and quantum observables which were considered by von Neumann, Kochen and Specker, ..., Bell,... were too straightforward. Neither von Neumann and Kochen-Specker nor Bell had physical arguments to present a list of “natural features” of such a correspondence $W.$ I neither have physical arguments. But I have strong probabilistic arguments. There exists a unique quantum representation of a Kolmogorovian model and this representation automatically induces a map $W$ which have very special properties \[1\]. Neither von Neumann and Kochen-Specker nor Bell maps have such properties.
In our realist model the map $W$ is defined only on a proper domain $D_W$ in ${\cal O}(\Omega),$ namely $$D_W={\cal O} (b,a)=\{d(\omega) = f(b(\omega)) + g(a(\omega))\},$$ where $(b,a)$ is the pair of reference preobservables determining the quantum picture of the $\Omega.$ And in general the map $W$ does not preserve conditional probability distributions. Here we consider two conditionings: contextual conditioning in the $\Omega$ and quantum state coditioning in the $H.$
But (!) conditional averages are preserved by the map $W:$ $$E(d/C)= (\hat{d}\phi_C, \phi_C),$$ where $\hat{d}= W(d), d\in D_W, $ and $\phi_C\in H$ is the image of a prespace context $C.$
It is very important that quantum Hamiltonians belong to the $W$-image of the set of preobservables. The operator $$\hat{{\cal H}}= \frac{\hat{a}^2}{2} + V(\hat{b})$$ is the image of the energy preobservable $${\cal H}(\omega)= \frac{a(\omega)^2}{2} + V(b(\omega)).$$ But as we have already underlined the $W$ does not preserve probability distributions. So $\hat{{\cal H}}$ and ${\cal H}(\omega)$ have different probability distributions. But they have the same average.
[**8. Quantization and the correspondence principle.**]{} As we noticed, it was proved \[1\] that quantum Hamiltonian $\hat{{\cal H}}$ has the same contextual average as the prespace Hamiltonian ${\cal H}:$ $$\label{CPR}
E({\cal H}/C)= (\hat{{\cal H}}\phi_C, \phi_C).$$ We can speculate that this coincidence of averages is the real basis of quantization rules. By (\[CPR\]) to obtain the correct average of the energy preobservable ${\cal H}(\omega)$ we should put quantum images of the reference preobservables into the prespace Hamiltonian: $$\label{CPR1}
b\to \hat{b}=W(b), \; a\to \hat{a}=W(a)$$ and, in particular, for the (position, momentum) quantization $$\label{CPR2}
q_\Omega\to \hat{q}=W(q_\Omega), \; p_\Omega\to \hat{p}=W(q_\Omega)$$
We should sharply distinguish the prespace position and momentum, $q_\Omega, p_\Omega,$ and classical space position and momentum $q_{X_{\rm{cl}}}, p_{X_{\rm{cl}}}.$ In our model (in the opposite to the very common opinion) $\hat{q}\not=W(q_{X_{\rm{cl}}})$ and $\hat{p}\not=W(p_{X_{\rm{cl}}}).$
By our model the root of the quantization rule is the equality (\[CPR\]) and Hamiltonian dynamics in the prespace $\Omega$ and not Hamiltonian dynamics in the classical space $X_{\rm{cl}}.$ [^4]
[**Conclusion.**]{} [*In spite of all [**no-go**]{} theorems the realist model of QM exists.*]{}
I would like to thank A. Plotnitsky for numerous discussions on Heisenberg-Bohr interpretation of QM and L. Accardi, L. Ballentine, S. Gudder for discussions on the role of conditional probabilities in QM which were extremely important for the creation of a contextual representation of a Kolmogorovian model in a Hilbert space \[1\], cf. \[4\].
[**References**]{}
1\. A. Khrennikov, [*Contextual approach to quantum mechanics and the theory of the fundamental prespace.*]{} quant-ph/0306003.
2\. A. Plotnitsky, Quantum atomicity and quantum information: Bohr, Heisenberg, and quantum mechanics as an information theory. Proc. Int. Conf. [*Quantum Theory: Reconsideration of Foundations.*]{} Ed. A. Khrennikov, Ser. Math. Modelling, [**2**]{}, 309-342, Växjö Univ. Press, 2002\
(http://www.msi.vxu.se/forskn/quantum.pdf).
3\. A. Khrennikov, [Växjö interpretation of quantum mechanics.]{} Proc. Int. Conf. [*Quantum Theory: Reconsideration of Foundations.*]{} Ed. A. Khrennikov, Ser. Math. Modelling, [**2**]{}, 164-169, Växjö Univ. Press, 2002 (http://www.msi.vxu.se/forskn/quantum.pdf).
4\. A. Yu. Khrennikov, Ensemble fluctuations and the origin of quantum probabilistic rule. [*J. Math. Phys.*]{}, [**43**]{}, N. 2, 789-802 (2002).
5\. L. De Broglie, [*The current interpretation of quantum mechanics. A critical study.*]{} Elsevier Publ., Amsterdam (1964).
6\. A. Yu. Khrennikov, I. V. Volovich, Local Realism, Contextualism and Loopholes in Bell‘s Experiments. Proc. Int. Conf. [*Foundations of Probability and Physics-2.*]{} Ed. A. Khrennikov, Ser. Math. Modelling, [**5**]{}, 325-344, Växjö Univ. Press, 2003. (quant-ph/0212127).
7\. J. von Neumann, [*Mathematical foundations of quantum mechanics.*]{} Princeton Univ. Press, Princeton, N.J. (1955).
8\. S. Kochen and E. Specker, [*J. Math. Mech.*]{}, [**17**]{}, 59-87 (1967).
9\. J. S. Bell, [*Speakable and unspeakable in quantum mechanics.*]{} Cambridge Univ. Press (1987).
10\. Khrennikov A.Yu., [*Interpretations of Probability.*]{} VSP Int. Sc. Publishers, Utrecht/Tokyo (1999).
[^1]: International Center for Mathematical Modeling in Physics and Cognitive Sciences, [email protected]; supported by EU-Network “QP and Applications.”
[^2]: Plenary talk and a topic of the round table at the International Conference “Quantum Theory: Reconsideration of Foundations-2”, June-2003, Växjö, Sweden.
[^3]: But Hilbert spaces $H^{b/a}$ and $H^{v/w}$ corresponding to pictures based on $(b,a)$ and $(v,w)$ can be different.
[^4]: It may be that dynamics in $\Omega$ and $X_{\rm{cl}}$ are mathematically described in the same way. But we should distinguish Hamiltonian prespace dynamics and classical Hamiltonian dynamics.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Xin Chen
title: Prospects for Higgs CP property measurements at the LHC
---
Introduction {#intro}
============
After the Higgs was discovered in 2012 [@higgs1; @higgs2], understanding its properties, and looking for any possible deviations of its properties from the SM prediction, becomes a very important task of LHC. This includes measuring the its spin (scalar or tensor), its mass which has important implications for the vacuum stability and cosmology, its flavor couplings to look for Beyond Standard Model (BSM) signatures of flavor changing effect, its exotic decay modes such as $H\to\text{invisible}$, and its CP property which is related to the matter-antimatter imbalance in the universe. If the Higgs is in a pure CP eigen state, we will find if it is CP even or odd. On the other hand, if it is in a CP mixture, we will measure the CP mixing angle. This is a more exciting scenario, as it gives rise to an extra CP violation source, and the current known source (a single complex phase in CKM) is too small to explain the matter-antimatter imbalance.
CP test in the bosonic decays of the Higgs {#boson_decay}
==========================================
The CP symmetry of the Higgs coupling to vector bosons are parametrized with an effective Lagrangian by the ATLAS collaboration as [@EFT1; @EFT1b] $$\begin{aligned}
\begin{array}{ll}
\mathcal{L}_0^V = & \left\{ c_\alpha \kappa_{SM} \left[ \frac{1}{2} g_{HZZ} Z_\mu Z^\mu + g_{HWW}W_\mu^+ W^{-\mu} \right] -\frac{1}{4} \frac{1}{\Lambda} \left[ c_\alpha \kappa_{HZZ} Z_{\mu\nu} Z^{\mu\nu} + s_\alpha \kappa_{AZZ} Z_{\mu\nu} \tilde{Z}^{\mu\nu} \right] \right. \\
& \left. -\frac{1}{2} \frac{1}{\Lambda} \left[ c_\alpha \kappa_{HWW} W_{\mu\nu}^+ W^{-\mu\nu} + s_\alpha \kappa_{AWW} W^+_{\mu\nu} W^{-\mu\nu} \right] \right\} X_0 ,
\end{array}
\label{eq:eq1}\end{aligned}$$ where $\alpha$ is the CP mixing angle, $c_\alpha = \cos{\alpha}$, $s_\alpha = \sin{\alpha}$, $\Lambda$ is scale cut-off for dim-6 operators, $\tilde{V}_{\mu\nu}(V=W/Z)$ is the dual tensor of $V_{\mu\nu}$, and $X_0$ is the neutral resonance such as the Higgs. The terms with $1/\Lambda$ coefficients are BSM dim-6 operators, with the $c_\alpha$ ($s_\alpha$) terms representing the BSM CP even (odd) contributions. Based on Eq. \[eq:eq1\], the pure CP states of Higgs can be parametrized as in Tab. \[tab:tab1\]. CMS collaboration used a similar expression, and the part with $Z$ boson terms only are [@EFT2] $$\mathcal{L}_{HVV} \sim a_1 \frac{m_Z^2}{2} HZ^\mu Z_\mu - \frac{\kappa_1}{\Lambda_1^2} m_Z^2 H Z_\mu \square Z^\mu - \frac{1}{2} a_2 H Z^{\mu\nu}Z_{\mu\nu} - \frac{1}{2} a_3 HZ^{\mu\nu}\tilde{Z}_{\mu\nu},
\label{eq:eq2}$$
--------- ---------------------------- --------------- ---------------- ---------------- ----------
$J_P$ Model
$\kappa_{SM}$ $\kappa_{HVV}$ $\kappa_{AVV}$ $\alpha$
$0^+$ Standard Model Higgs boson 1 0 0 0
$0_h^+$ BSM spin-0 CP-even 0 1 0 0
$0^-$ BSM spin-0 CP-odd 0 0 1 $\pi/2$
--------- ---------------------------- --------------- ---------------- ---------------- ----------
: The definitions of $HVV$ coupling coefficients for pure Higgs CP states (ATLAS) [@EFT1]. []{data-label="tab:tab1"}
Both ATLAS and CMS have tested the CP states of the Higgs assuming it is in a pure CP state. Both experiments used a Matrix Element (ME) method to test the $0^+$ state against $0^-$. ATLAS combined three channels in the test: $H\to WW\to e\nu \mu\nu$ 0-jet category, $H\to\gamma\gamma$ and $H\to ZZ\to 4l$. The results are as shown in Fig. \[fig:fig1\]. CMS combined $H\to WW, ZZ, Z\gamma, \gamma\gamma$ channels and for the pure CP test [@EFT2]. The CP odd $0^-$ model is excluded at CL better than $99.9\%$ by both experiments.
![ The test results of Higgs with different spin and CP states at ATLAS [@EFT1]. The measurements are indicated by the black solid bars. []{data-label="fig:fig1"}](fig_01.png){width="6.cm"}
When the Higgs is in a CP-mixed state, both experiments have made constraints on the coefficients of different couplings in Eq. \[eq:eq1\] and \[eq:eq2\], as shown in Tab. \[tab:tab2\] and Fig. \[fig:fig2\]. One can conclude that the non-SM tensor couplings are consistent with zero for both experiments.
----------------------------------------------------- ---------- ---------- ------------------------------------ ------------------------------------
Coupling ratio
Combined Expected Observed Expected Observed
$\tilde{\kappa}_{HVV}/\kappa_{SM}$ 0.0 -0.48 $(-\infty,-0.55]\cup[4.80,\infty)$ $(-\infty,-0.73]\cup[0.63,\infty)$
$(\tilde{\kappa}_{AVV}/\kappa_{SM})\cdot\tan\alpha$ 0.0 -0.68 $(-\infty,-2.33]\cup[2.30,\infty)$ $(-\infty,-2.18]\cup[0.83,\infty)$
----------------------------------------------------- ---------- ---------- ------------------------------------ ------------------------------------
: The best-fit and excluded ranges for the coefficients of the BSM CP-even and CP-odd terms in Eq. \[eq:eq1\] at ATLAS, with $H\to ZZ^*$ and $H\to WW^*$ combined [@EFT1].[]{data-label="tab:tab2"}
![ The bounds on the coefficients of the different BSM terms for $H\to ZZ^*$, $H\to WW^*$, $H\to Z\gamma$ and $H\to \gamma\gamma$ decays at CMS [@EFT2]. []{data-label="fig:fig2"}](fig_03.png){width="12.cm"}
Another important way to search for Higgs CP violation is in the Vector-Boson-Fusion (VBF) production of the Higgs. One of the most promising channel for this search is $H\to\tau\tau$ decay. The combined Run-1 sensitivity of ATLAS and CMS has already exceeded $5\sigma$ [@Htt]. This channel is special in that it is sensitive to CP in both $HVV$ and $Hf\bar{f}$ couplings. In the 2HDM model, the rate of $H\to\tau\tau$ is often enhanced, and mixed CP for Higgs can be accommodated.
Omitting the BSM CP-even terms, the $HVV$ coupling can be written as [@OO] $$\mathcal{L}_{eff} = \mathcal{L}_{SM} + \tilde{g}_{HAA}H\tilde{A}_{\mu\nu}A^{\mu\nu} + \tilde{g}_{HAZ}H\tilde{A}_{\mu\nu}Z^{\mu\nu} + \tilde{g}_{HZZ}H\tilde{Z}_{\mu\nu}Z^{\mu\nu} + \tilde{g}_{HWW}H\tilde{W}^+_{\mu\nu}W^{-\mu\nu}.
\label{eq:eq3}$$ Since the vector bosons in the process are not directly observable, it is easier to treat the CP-odd terms as one by the assumption in Eq. \[eq:eq4\] about the relations of the coefficients: $$\tilde{g}_{HAA} = \tilde{g}_{HZZ} = \frac{1}{2} \tilde{g}_{HWW} = \frac{g}{2m_W} \tilde{d}, ~~\text{and}~~\tilde{g}_{HAZ}=0.
\label{eq:eq4}$$ Traditionally, the signed $\Delta\phi$ between the two tagging in the VBF process are used as the CP sensitive variable [@VBF]. ATLAS used a different variable, the Optimal Observable (OO), which is expected to perform better. It is based on the ME of $$\mathcal{M} = \mathcal{M}_{SM} + \tilde{d} \cdot \mathcal{M} _{CP-odd}, ~~\text{and}~~
\left|\mathcal{M}\right|^2 = \left|\mathcal{M}_{SM}\right|^2 + \tilde{d} \cdot 2 Re\left( \mathcal{M}_{SM}^*\mathcal{M}_{CP-odd} \right) + \tilde{d}^2 \left| \mathcal{M}_{CP-odd} \right|^2,
\label{eq:eq5}$$ and the OO is defined as $$OO = \frac{2 Re \left(\mathcal{M}_{SM}^* \mathcal{M}_{CP-odd} \right)}{\left| \mathcal{M}_{SM} \right|^2}.
\label{eq:eq5b}$$
With all 4-momenta of the final state particles (the Higgs and two tagging jets) measured, the LO ME of SM and CP-odd can be calculated from HAWK [@hawk], and then the OO can be calculated per event. As Fig. \[fig:fig3\] shows, of there is no CP violation, the mean of the OO distribution should be zero. For positive (negative) CP-odd component (determined by $\tilde{d}$), its mean will be shifted to positive (negative) values. This method can be applied to other decays such as $H\to\gamma\gamma$.
![ The distributions of OO for the SM ($\tilde{d}=0$) and two other values of $\tilde{d}$ in the VBF $H\to\tau\tau$ signal at ATLAS [@OO]. []{data-label="fig:fig3"}](fig_04.png){width="6.cm"}
To obtain a pure signal sample, a cut is first made on the Multi-Variate-Analysis (MVA) output score. A signal to background ratio of about 0.3 is achieved by this cut. Next, a likelihood fit to the OO distribution is performed to find the best value for $\tilde{d}$. Figure \[fig:fig4\] shows the OO distribution in the $H\to\tau\tau\to ll+4\nu$ subchannel, and the increase of the negative-log-likelihood $\Delta$NLL with respect to its minimum in the likelihood scan. Each point in the plot indicates a $\Delta$NLL calculated with a particular hypothesis template and the data. The $68\%$ CL interval is found by the intersection points of the $\Delta$NLL curve and the horizontal line at $\Delta\text{NLL}=0.5$. The result is $-0.11<\tilde{d}<0.05$. While the result shows that the CP-odd contribution is consistent with zero, it is about 10 times tighter than the one shown in Tab. \[tab:tab2\] for the $HVV$ coupling.
![ The distributions of the OO in the $H\to\tau\tau\to ll+4\nu$ subchannel (left), and the $\Delta$NLL for different $\tilde{d}$ hypotheses (right) at ATLAS [@OO].[]{data-label="fig:fig4"}](fig_05.png "fig:"){width="5.7cm"} ![ The distributions of the OO in the $H\to\tau\tau\to ll+4\nu$ subchannel (left), and the $\Delta$NLL for different $\tilde{d}$ hypotheses (right) at ATLAS [@OO].[]{data-label="fig:fig4"}](fig_06.png "fig:"){width="6cm"}
In the HL-LHC phase, the Missing Transverse Energy and the di-tau mass resolution will degrade as the pileup increases. The signal-background discrimination increases with the tracking coverage, as the VBF forward jets are better separated from the pileup jets with the association between the jets and the primary vertex, shown in Fig. \[fig:fig5\]. Assuming zero theory uncertainty, the uncertainty on the signal strength $\mu$ at 8-18% (2-5%) can be achieved with the ATLAS $\tau_l \tau_h$ alone [@HL1] (CMS $\tau\tau$ inclusive [@HL2]). The projected uncertainty of $\mu$ for different tracking extensions at ATLAS is listed in Tab. \[tab:tab3\]. Thus, we would expect a corresponding increase in the precision for the Higgs CP measurement in the VBF channel due to both increased data and extended tracking coverage.
![ The jet-PV association efficiency as a function of the VBF quark $\eta$ in the $H\to\tau\tau$ channel at CMS. The improvement is evident with the tracking volume extension [@HL2]. []{data-label="fig:fig5"}](fig_07.png){width="5.0cm"}
forward pile-up jet rejection $50\%$ $75\%$ $90\%$
------------------------------- -------- -------- --------
forward tracker coverage
Run-I tracking volume
$\eta<3.0$
$\eta<3.5$
$\eta<4.0$
: The projected uncertainty on the signal strength for different scenarios of tracking coverage and forward pileup jet rejection at HL-LHC for ATLAS [@HL1].[]{data-label="tab:tab3"}
CP test in the fermionic decays of the Higgs {#fermion_decay}
============================================
The CP-odd Yukawa coupling can enter the Lagrangian as a dimension-4 operator as in Eq. \[eq:eq6\], thus the $Hf\bar{f}$, especially the $H\tau\tau$ coupling, is a sensitive probe of CP at tree-level rather than the loop level as with the dimension-6 operators in the $HVV$ coupling. $$\mathcal{L} = -g_\tau \left( \cos\phi\bar{\tau}\tau + i\sin\phi\bar{\tau}\gamma_5\tau \right)h,
\label{eq:eq6}$$ where $\phi$ is the mixing angle between the CP even and odd terms. The CP of $H\tau\tau$ coupling can be distinguished by the transverse tau spin correlations, as the decay width is proportional to $$\Gamma(H,A\to\tau^-\tau^+) \sim 1 - s_z^{\tau-}s_z^{\tau+} \pm s_T^{\tau-}s_T^{\tau+} ,
\label{eq:eq7}$$ where $s_{z,T}$ are the tau spin components in the longitudinal and transverse directions with respect to the tau momentum. As a result, the angle between the planes spanned by the tau and its charged track is sensitive to the CP. For example, in the $\tau\to\pi\nu$ decay, one can look at the angle between tau decay planes to extract the CP mixing angle $\phi$: $$\frac{d\Gamma\left(H\to\tau\tau\to\pi^+\pi^-+2\nu \right)}{d\phi_{CP}} \sim 1-\frac{\pi^2}{16}\cos\left(\phi_{CP}-2\phi \right),
\label{eq:eq8}$$ where $\phi_{CP}$ is the angle between the tau decay planes in the ditau rest frame. Using the $H\to\tau\tau$ decay to measure the CP is experimentally challenging because the neutrinos are not reconstructed. There are two main methods to extract the CP [@CP1; @CP2; @CP3; @CP4; @CP5]. One is by using the impact parameters to approximately reconstruct the tau decay plane from its leading track. This is best done for the $\tau\to\pi\nu$ decay, and the analyzing power is compromised for the other tau decays. The other is by using the $\tau\to\rho\nu\to\pi^\pm\pi^0\nu$ decay. The tau decay plane can be approximately reconstructed by the track and the neutral pion. However, the relative energy of $\pi^\pm$ and $\pi^0$ need to be classified in order to maximize the analyzing power. In order to use the methods, the tau decay modes (or its substructure) need to be well differenciated. In ATLAS [@sub1], after the initial tau reconstruction and identification, the hadronic tau candidate is further analyzed with a particle flow (PF) algorithm to better reconstruct and differenciate the neutral pions. The energy in the calorimeter deposited by $\pi^\pm$ is then subtracted, and the remaining energy cluster is reconstructed as $\pi^0$ and identified with a Boosted Decision Tree (BDT) method. The final decay mode classification is done with another BDT. The reconstruction and identification efficiencies for a single tau at ATLAS is listed in Tab. \[tab:tab4\]. The efficiency for separating the different decay modes and the amount of crosstalk is visualized in Fig. \[fig:fig7\]. In general, there is a non-negligible fraction of 2 (1) $\pi^0$ reconstructed as 1 (0) $\pi^0$. In Fig. \[fig:fig8\], the tau energy resolution with the PF is compared with the old baseline method, and the reconstructed tau mass and relative abundance of different modes are compared with data. In conclusion, with the substructure, a factor of 2 improvement of tau energy with respect to the calo-based method at the low $p_T$ (the resolution of the neutral $\pi^0$ is about 16%), and a factor of 5 improvement in the angular resolution can be achieved. For the neutral $\pi^0$, a precision of 0.006 (0.012) can be reached for it $\eta$ ($\phi$) measurement.
Decay mode $\mathcal{B}$ \[%\] $\mathcal{A}\cdot\epsilon_{reco}$ \[%\] $\epsilon_{ID}$ \[%\]
--------------------- --------------------- ----------------------------------------- -----------------------
$h^\pm$ 11.5 32 75
$h^\pm\pi^0$ 30.0 33 55
$h^\pm\geq 2\pi^0$ 10.6 43 40
$3h^\pm$ 9.5 38 70
$3h^\pm\geq 1\pi^0$ 5.1 38 46
: The Branching Fractions, acceptance, reconstruction and identification efficiencies for a single tau in different decay modes at ATLAS [@sub1]. []{data-label="tab:tab4"}
![ The efficiencies for the generated tau decay modes to be reconstructed as different measured modes (left), and the fractions of different reconstructed modes coming from different generated modes indicating the purity of the reconstruction (right), at ATLAS [@sub1]. []{data-label="fig:fig7"}](fig_10.png "fig:"){width="4.7cm"} ![ The efficiencies for the generated tau decay modes to be reconstructed as different measured modes (left), and the fractions of different reconstructed modes coming from different generated modes indicating the purity of the reconstruction (right), at ATLAS [@sub1]. []{data-label="fig:fig7"}](fig_11.png "fig:"){width="4.7cm"}
![ The resolution of the reconstructed tau energy (left), the mass (middle) and modes (right) of the reconstructed taus with substructure at ATLAS [@sub1]. []{data-label="fig:fig8"}](fig_12.png "fig:"){width="4cm"} ![ The resolution of the reconstructed tau energy (left), the mass (middle) and modes (right) of the reconstructed taus with substructure at ATLAS [@sub1]. []{data-label="fig:fig8"}](fig_13.png "fig:"){width="4cm"} ![ The resolution of the reconstructed tau energy (left), the mass (middle) and modes (right) of the reconstructed taus with substructure at ATLAS [@sub1]. []{data-label="fig:fig8"}](fig_14.png "fig:"){width="4cm"}
CMS also uses the PF constituents (charged and neutral particles) to reconstruct taus [@sub2; @sub3]. Discriminants are then applied to reject jets and leptons (with a MVA based tau identification). Multiple $\tau_h$ hypotheses for each jet are constructed by a combinatorial approach. The one passing all cuts and with the highest $p_T$ is selected as the decay mode for the tau candidate. The mode reconstruction efficiency, and tau mass with Run-1 and Run-2 data are shown in Fig. \[fig:fig9\]. Compared with ATLAS, CMS currently has less decay modes classified, but the crosstalk among different modes is also smaller than ATLAS. Good agreement with Run-1 and Run-2 data is achieved, although the $W$+jets and QCD background are visibly higher in the latter.
![ The efficiencies for the generated tau decay modes to be reconstructed as different measured modes (left), the tau mass distributions with Run-1 (middle) and Run-2 (right) data, at CMS [@sub2; @sub3]. []{data-label="fig:fig9"}](fig_15.png "fig:"){width="4cm"} ![ The efficiencies for the generated tau decay modes to be reconstructed as different measured modes (left), the tau mass distributions with Run-1 (middle) and Run-2 (right) data, at CMS [@sub2; @sub3]. []{data-label="fig:fig9"}](fig_16.png "fig:"){width="4.0cm"} ![ The efficiencies for the generated tau decay modes to be reconstructed as different measured modes (left), the tau mass distributions with Run-1 (middle) and Run-2 (right) data, at CMS [@sub2; @sub3]. []{data-label="fig:fig9"}](fig_17.png "fig:"){width="3.8cm"}
Both ATLAS and CMS used the embedding technique to estimate the $Z\to\tau\tau$ background with reduced systematics. This is done by replacing muons from the $Z\to\mu\mu$ data by simulated taus and decaying the taus, and then embedding the simulated tau decay products into the original real data event. No spin correlation information is conserved by this procedure. The Tauola and TauSpinner packages [@tsp] could be used to ensure the correct $Z$ boson polarization and to restore the di-tau spin correlation.
Conclusion
==========
Testing the CP nature of Higgs is one of the important tasks after its discovery. With the diboson channels ($H\to\gamma\gamma, ZZ, WW$), and assuming pure CP state for the Higgs, the CP even state is favored by data. The tensor structure of the $HVV$ coupling can also be probed for mixed CP scenarios. No significant CP mixing effect is observed and limits are set on the CP-odd terms in the effective Lagrangian. The $H\to\tau\tau$ channel is ideal for probing CP through both the $HVV$ coupling through VBF, and the $H\tau\tau$ Yukawa coupling. Combined sensitivity of ATLAS and CMS already exceeded $5\sigma$ with the Run-1 data. The former has results with VBF $H\to\tau\tau$ based on the Run-1 data. The CP-mixing parameter in the VBF production is currently consistent with zero. The CP mixing in $H\to\tau\tau$ decay can be large in theory, but experimentally very challenging. Both ATLAS and CMS have refined the tau reconstruction with substructure information, which is essential for the CP study in the tau decays. Looking forward, the CP test precision will improve with the HL-LHC data, in which just a few percents uncertainty on the $H\to\tau\tau$ signal strength is expected.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The dynamics of accretion onto a schwarzschild black hole is studied using Paczynski-Wiita pseudo newtonian potential. Steady state solution of the flow equations is obtained using thin disc approximation, including the effect of bremsstrahlung cooling in the energy equation. The topology of transonic solutions are got and the conditions for shock formation (Rankine-Hugoniot conditions) are checked. Shock and no-shock regions in the parameter space of accretion rate and specific angular momentum are identified. We motivate the time-dependent numerical simulation of the flow, as a candidate for explaining the quasi periodic oscillations observed in black hole candidates.'
author:
- 'Sivakumar G. Manickam and Sandip K. Chakrabarti'
title: 'Accretion onto a Black Hole in the Presence of Bremsstrahlung Cooling - Parameter Space Study'
---
Introduction
============
The parameter space of shock formation in adiabatic transonic accretion discs was first discussed in Chakrabarti(1989). Though time-dependent numerical simulation of transonic accretion disc in the presence of cooling was dealt in Molteni, Sponholz & Chakrabarti (1996, hereafter MSC96), the detailed study of parameter space was reserved for the future. In this paper we describe the procedure for obtaining the flow topology in the presence of bremsstrahlung cooling and obtain the dependence of topology and shock formation on the parameters, accretion rate and specific angular momentum.
Flow equations
==============
The equations which govern the flow are the basic conservation equations of mass, momentum and energy. The pseudo newtonian potential of Paczynski-Wiita (1980) is used, which mimics the gravitational field around a schwarzschild black hole to a sufficient accuracy. The effect of bremsstrahlung cooling (Lang 1980) in electron-proton plasma is included in the energy equation. ${\bf v}(v_r, v_{\phi}, v_z), \rho$ and $p$ are the velocity, density and pressure respectively.
Continuity equation:
$${{{\partial \rho}\over{\partial t}} +{ \nabla .{({\rho}{\bf v})}} = 0 }$$
Euler equation:
$${{\partial \bf v} \over {\partial t}} + (\bf v. \nabla ) \bf v = - {\nabla p \over \rho} - \nabla g$$
where $ g = -{GM \over (r-r_g)} $ is the Paczynski-Wiita potential, $G$ is the gravitational constant, M is the mass of the black hole, $r_g={2GM \over c^2}$ is the schwarzschild radius and $c$ is the velocity of light.
Energy equation:
$$\nabla . ( \rho \epsilon \bf v) + \Lambda - \Gamma = - {\partial \over {\partial t}} (\rho \epsilon)$$
where $ \epsilon = {1 \over 2} {v_r}^2 + U + {p \over \rho} + g + {1 \over 2} {v_ \phi}^2 $ is the specific energy, $U={p \over \rho (\gamma-1)}$ is the thermal energy, $\gamma$ is the adiabatic index, $ \Lambda = 1.43 \times 10^{-27} {\rho ^2 \over m_p ^2} T^{1/2} g_f $ is the expression for bremsstrahlung cooling, $T$ is the temperature, $m_p$ is the mass of proton, $g_f$ is the gaunt factor and $\Gamma$ is the heating term.
Thin disc approximation:
We use cylindrical polar coordinates $(r,\phi,z)$ for the axisymmetric flow. Because of axisymmetry and thinness of the disc the following assumptions are made.
$ \rho(\phi^0, z^0) $ i.e. $\rho$ is not a function of $\phi$ and $z$
$ {v_r}(\phi^0, z^0) $
$ {v_ \phi}(\phi^0, z^0) $
$ {v_z}(r^0, \phi^0, z^0) $
$ p(r) $ i.e. $p$ is a function of r only
$ g(r) $
$ \epsilon(\phi^0) $
By the help of these assumptions it is possible to write the steady state equations in the following form.
$${{\psi ^ \prime - a^2 /r +j{g_f}{2 \over 3} {\rho \over v {m_p}^2} T^{1/2}} \over a^2 /v -v}
= {dv \over dr} \ ; \ \psi = -{GM \over (r-r_g)} + {1 \over 2} {v_ \phi}^2$$
$${1 \over \rho} {d \rho \over dr} + {1 \over v} {dv \over dr} + {1 \over r} = 0$$
$$v{dv \over dr} + {2a \over \gamma}{da \over dr}+{a^2 \over \rho \gamma}{d\rho \over dr}+\psi ^ \prime=0$$
where $^\prime$ denotes the derivative with respect to r, $j=1.43 \times 10^{-27}$ in cgs units and the polytropic relation $p=K \rho ^ \gamma$ is used to obtain the sound speed $a=\sqrt
{\partial p \over
\partial \rho}=\sqrt{\gamma p \over \rho}$. These equations are solved using fourth-order Runge-Kutta method (Press et al. 1992).
Sonic point analysis
====================
The flow can pass through a point where denominator in the expression for ${dv \over dr}$ becomes zero. If it happens that the numerator also becomes zero, then ${dv \over dr}$ can be finite. We call such a point a critical point. In this problem the critical point is same as sonic point. At critical point, using l’ Hospital’s rule, we get (for $\gamma={5\over 3}$),
$${dv \over dr} = {-B \pm {\sqrt{(B^2-4AC)}} \over 2A}$$
$$A = {8 \over 3}$$
$$B = -{1\over3}{v\over r}-{4\over 3} \zeta \rho a^{2\alpha-2} (1+{\alpha\over 3}) +{5\over 3}{\psi^\prime \over v}$$
$$C = {5\over 3}{\psi ^ \prime \over r}+ \psi ^ {\prime \prime} -{2\over 3}\zeta {a^{2\alpha-1}\over r}
\rho (1-\alpha)-{10\over9}\alpha\rho\zeta a^{2\alpha-3}
\psi ^\prime$$
$$\zeta = jg_f{1\over m_p^2} ({\mu m_p \over \gamma k})^\alpha$$
where $\alpha=0.5$, $\mu=0.5$ and $k$ is the boltzmann constant. Now starting from the critical point we obtain the transonic solution topologies.
Numerical scheme
================
Fluid dynamical problems are inherently sensitive to the boundary conditions. We formulate the problem as follows. The variables which are functions of space and time are ${\bf v}, \rho, p$ or ${\bf v}, \rho, a$. The parameters are accretion rate, specific angular momentum($\lambda$), $\gamma$, cooling process and viscosity. We consider inviscous flow, set $\gamma = 5/3$ and the mass of the black hole $M$ is chosen as $10^8 M_{\odot}$. So the free parameters are accretion rate and $\lambda$. The location of the critical points, $r_{c1}, r_{c2}$, is the freedom we have, subjected to the constraint that shock conditions should be satisfied. The natural constraint of stability is likely to decide the uniqueness of the solution. The algorithm for obtaining the numerical solution is,
1\. Choose accretion rate and $\lambda$
2\. Choose $r_{c1}$ and $r_{c2}$ range and find corresponding $a_{c1}$ and $a_{c2}$
3\. Obtain solution topology by using fourth-order Runge-Kutta method
4\. Check the shock conditions
The typical components of the topology are shown in fig.1, where mach number is plotted as a function of the radial distance. Fig.2 and 3 shows the solution topologies for chosen parameter values. Rankine-Hugoniot shock conditions (Landau & Lifshitz 1984) basically ensure that mass, momentum and energy are conserved in spite of a discontinuity. For infinitesimally thin and non-dissipative shock, the shock conditions take the form,
$$p_1 + \rho _1 {v_1}^2 = p_2 + \rho _2 {v_1}^2, \ \ \ \ \ {1 \over 2} {v_1}^2 + {{a_1}^2 \over \gamma -1} = {1 \over 2} {v_2}^2 + {{a_2}^2 \over \gamma -1}$$
where the suffixes 1 and 2 refer to pre-shock and post-shock quantities.
Parameter space
===============
We use the following procedure to obtain the parameter space. The accretion rate in eddington units, is varied in the range (.0001, 500) and $\lambda$ in the units of ${2GM \over c}$, is varied in the range $(1.6,2.5)$. It is suggested that (Chakrabarti 1996) flows with $\lambda$ in the range between that of marginally stable and marginally bound orbit would form steady accretion discs, when self-gravity of the disc is neglected. For a chosen accretion rate and $\lambda$ we scan the r-axis from 1.5$r_g$ to 1000$r_g$ to find if it can be a critical point. The critical point can be of X or Alp or plA or lA or x type as shown in fig.1. X looks like the English alphabet X, Alp like Greek alphabet alpha which opens towards infinity, plA is reflected Alp which opens towards inner boundary, lA is plA which doesn’t reach the inner boundary when $\lambda$ is high and x is X which doesn’t reach the inner boundary when accretion rate is high. For certain parameter values there is an intermediate range of r which cannot be a critical point (denoted by - in topology column of Table 1). Table 1 shows the parameter space. We obtain supersonic branches for outer critical points and subsonic branches for inner critical points. When shock conditions are satisfied the flow makes a transition from supersonic to subsonic branch and reaches the inner boundary (chosen as 1.5$r_g$) supersonically. When shock conditions are not satisfied accretion is still possible if X type critical point exists.
Discussion and conclusions
==========================
Of all the different possible branches of accretion for chosen parameter value, the real flow is likely to choose the branch which is most stable, as perturbations are always present in a real situation. If the assumption of steady flow is relaxed the flow might still choose a steady solution branch if the time scale of change is large. Time-dependent numerical simulation of the flow (MSC96) shows that flow solution oscillates about the steady state solution when certain resonance condition is met. The ’perturbations’ which occur in a numerical code and physical perturbations should be related, to increase the faith in numerical results. Such numerical studies will be pursued in the future. The oscillation of shock location (MSC96) would mean the size of post-shock region, which is the source for hard photons (Chakrabarti & Titarchuk 1995), is also oscillating. These would result in quasi periodic oscillations as reported in Chakrabarti & Manickam (2000).
We thank Indian Space Research Organization for funding the project, Quasi-Periodic Oscillations in Black Hole Candidates, of which this work forms a part.
Chakrabarti, S. K. 1989, , 347, 365
Chakrabarti, S. K. 1996, Physics Reports, 266, No. 5 & 6, 229
Chakrabarti, S. K. & Manickam, S. G. 2000 , 531, L41
Chakrabarti, S. K. & Titarchuk, L. 1995 , 455, 623
Landau, L. D. & Lifshitz, E. M. 1984, Fluid Mechanics, second edition, Maxwell Macmillan International Editions
Lang, K. R. 1980, Astrophysical Formulae, second edition, Springer-Verlag
Molteni, D., Sponholz, H. & Chakrabarti, S. K. 1996, , 457, 805, (MSC96)
Paczynski, B. & Wiita, P. J. 1980, A&A, 88, 23
Press, W. H., Teukolsky, S. A., Vellerling, W. T. & Flannery, B. P. 1992, Numerical Recipes in Fortran - The Art of Scientific Computing, second edition, Cambridge University Press
[clcc]{}
.0001 & 1.6 & X &\
.0001 & 1.65 & X plA - Alp X & noshock\
.0001 & 1.7 & X plA - Alp X & shk\
.0001 & 1.9 & X plA - Alp X & shk2\
.0001 & 2.0 & X - Alp & noshock\
.001 & 1.6 & X &\
.001 & 1.65 & X plA - Alp X & shk\
.001 & 1.9 & X plA - Alp X & shk2\
.001 & 2.0 & X - Alp & noshock\
.01 & 1.6 & X &\
.01 & 1.65 & X plA - Alp X & shk\
.01 & 1.9 & X plA - Alp X & shk2\
.01 & 2.0 & X lA - Alp & noshock\
.1 & 1.6 & X &\
.1 & 1.65 & X plA - Alp X & shk\
.1 & 1.9 & X plA - Alp X & shk\
.1 & 2.0 & X - Alp & noshock\
1.0 & 1.6 & X &\
1.0 & 1.65 & X plA - Alp X & shk\
1.0 & 1.7 & X plA - Alp X &\
1.0 & 1.8 & X plA - Alp X &\
1.0 & 1.9 & X - Alp X & shk2\
1.0 & 2.0 & X - Alp & noshock\
1.0 & 2.1 & X - Alp &\
1.0 & 2.3 & X lA - Alp &\
1.0 & 2.4 & lA - Alp &\
1.0 & 2.5 & lA - Alp &\
2.0 & 1.6 & X &\
2.0 & 1.65 & X plA - Alp X & shk\
2.0 & 1.8 & X plA - Alp X &\
2.0 & 1.9 & X lA - Alp X & shk1\
2.0 & 2.0 & X lA - Alp & noshock\
2.0 & 2.3 & X lA - Alp &\
2.0 & 2.4 & x lA - Alp &\
5.0 & 1.6 & X &\
5.0 & 1.65 & X plA - Alp X x & shk\
5.0 & 1.7 & X plA - Alp X &\
5.0 & 1.8 & X plA - Alp X &\
5.0 & 1.9 & X lA - Alp X x & shk\
5.0 & 2.0 & X lA - Alp X x & noshock\
5.0 & 2.3 & X x lA - Alp x &\
20\. & 1.65 & X plA - Alp X x &\
20. & 2.4 & X x lA - Alp x &\
30\. & 1.65 & X plA - Alp X x &\
50\. & 1.65 & X x &\
50. & 1.7 & X plA - Alp X x & shk\
50. & 1.8 & X plA - Alp X x & shk1\
50. & 1.9 & X plA lA - Alp X x & shk\
50. & 2.0 & X x lA - Alp X x & noshock\
50. & 2.4 & X x lA - Alp x &\
500\. & 1.6 & X x &\
500. & 1.8 & X x &\
500. & 2.4 & x &\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report on observations of the stellar populations in twelve fields spanning the region between the Magellanic Clouds, made with the Mosaic-II camera on the 4-meter telescope at the Cerro-Tololo Inter-American Observatory. The two main goals of the observations are to characterize the young stellar population (which presumably formed [*in situ*]{} in the Bridge and therefore represents the nearest stellar population formed from tidal debris), and to search for an older stellar component (which would have been stripped from either Cloud as stars, by the same tidal forces which formed the gaseous Bridge). We determine the star-formation history of the young inter-Cloud population, which provides a constraint on the timing of the gravitational interaction which formed the Bridge. We do not detect an older stellar population belonging to the Bridge in any of our fields, implying that the material that was stripped from the Clouds to form the Magellanic Bridge was very nearly a pure gas.'
author:
- Jason Harris
title: 'The Magellanic Bridge: The Nearest Purely Tidal Stellar Population'
---
= ==1=1=0pt =2=2=0pt
Introduction {#sec:intro}
============
Interactions are known to be an important driver of galaxy evolution, but a detailed understanding of their influence remains elusive. The Magellanic Clouds are a particularly compelling target for investigating the effects of minor so-called “harassment” interactions, due to their proximity to the Milky Way, their close association over at least the past several Gyr, and their abundant gas reservoirs, which allow for ongoing star formation. The strongest evidence that their interaction has played an important role in driving the evolution of the Clouds lies in the extra-tidal features of the Magellanic Stream and Magellanic Bridge. Unlike the Magellanic Stream, which appears to be a pure-gas feature [@gr98], there is a known stellar population associated with the Magellanic Bridge, and by measuring the ages, chemical abundances, and kinematics of these stars, we can obtain strong constraints on the evolution of the dynamical event which formed the Bridge, and study in detail how star formation proceeds in the wake of such an event.
The Magellanic Bridge was first reported in observations by [@hkm63], and a young stellar component (the “inter-Cloud population”) was discovered by [@ikd85], who estimated the age of the stars at about $10^8$ yr. [@db98] provided the most comprehensive study of the inter-Cloud population to date; they observed five fields in the western Bridge, and found stars as young as 10–25 Myr in both clusters and in a diffuse field component up to $9^\circ$ from the SMC. The young inter-Cloud population probably formed [*in situ*]{} in the Bridge, in the wake of the Bridge-forming event, making it the nearest example of a stellar population whose formation was unambiguously triggered by a tidal interaction. Yet surprisingly, no detailed analysis of the star formation history of these stars has been performed to date, and we do not even know the full extent of the population, since no fields along the ridgeline of the gas have been observed east of the midpoint between the Clouds.
Furthermore, no study to date has specifically searched for an older component of the inter-Cloud population, which would represent a population of stars that was stripped from the Clouds during the Bridge-forming event. Tidal forces during an interaction should affect both gas and stars, so the inter-Cloud population should have an old component, other things being equal.
The present study will address these open questions regarding the inter-Cloud population in the Magellanic Bridge. We present our observations and data reduction in Section \[sec:obsdata\]. In Section \[sec:youngstars\], we trace the eastward extent of the young inter-Cloud population, and in Section \[sec:oldstars\], we search for tidally-stripped stars in our Bridge fields. We briefly examine the outer structure of the LMC using our four fields nearest that galaxy in Section \[sec:expdisk\]. Finally, we present the star-formation history of the young inter-Cloud population in Section \[sec:sfh\], and summarize the results in Section \[sec:summary\].
Observations and Data Reduction {#sec:obsdata}
===============================
The Observations
----------------
The data were obtained on the nights of January 4 and 5 2006 (UT), at the Cerro-Tololo Inter-American Observatory (CTIO) 4-meter telescope. We used the Mosaic-II camera, which images a 36$^\prime\times$36$^\prime$ field onto a 8k$\times$8k CCD detector array, to obtain short and long exposures in Washington $C$, Harris $R$, and Cousins $I$ filters at twelve field positions spanning the inter-Cloud region, and at one offset field at a similar Galactic latitude, but to the west of the SMC (see Table \[tab:exposures\] and Figure \[fig:fields\]).
The field positions were selected to uniformly sample the inter-Cloud region, approximately following the ridgeline of the gas which forms the Magellanic Bridge [@put00]. In addition, we selected two fields to lie off the main ridgeline, but still in regions of abundant emission (fields mb03 and mb14). Our fields include the eastern half of the inter-Cloud region, where there are very few known star clusters, and where @db98’s fields mostly lie far from the ridgeline.
The exposure times listed in Table \[tab:exposures\] were chosen in order to detect stars as faint as the ancient main-sequence turn-off with S/N$>10$ in all three filters. At each field position and for each of the $C$, $R$, and $I$ filters, we obtained a pair of long exposures for cosmic-ray rejection, and a short exposure to record the photometry of brighter stars that are saturated in the long exposures. Seeing was stable during both nights; the FWHM varied between 0.7$^{\prime\prime}$ and 1.0$^{\prime\prime}$. We observed standard star fields several times per night for photometric calibration (see Table \[tab:standards\]), twilight flats were obtained during evening and morning twilight of both nights, and bias frames were obtained in the afternoon prior to both nights.
Data Reduction {#sec:reduction}
--------------
The data reduction followed the procedure documented by the NOAO Deep Wide Field Survey team [@jan03], and utilized the [mscred]{} package in IRAF[^1]. Before reducing the data, we obtained an updated world coordinate system database for the CTIO 4-meter telescope from the CTIO website, dated from May 2004, and we also obtained an updated crosstalk-correction parameter file.
We first processed the bias frames using [ccdproc]{} and [ zerocombine]{}. We then used [ccdproc]{} to perform overscan correction and trimming, bias subtraction, bad pixel masking, amplifier merging and crosstalk correction on all data images.
We ran [objmasks]{} on each twilight-sky exposure to mask out any detected stars in the images, and then combined the exposures using [sflatcombine]{}. This yielded normalized $C$, $R$, and $I$ twilight-sky flats for each observing night. We then divided each data frame by the appropriate normalized flat, using [ccdproc]{} with the [sflatcor]{} parameter activated. To further flatten the data images, we also constructed night-sky flats from the collection of data images, after running [objmasks]{} on each image to mask out detected objects. We then ran [ccdproc]{} once again to apply the night-sky flats.
We empirically determined the world coordinate system (WCS) for each exposure using [msccmatch]{}, which uses an astrometric database to identify stars in the image and determine the WCS from their known coordinates. We chose to use the Guide Star Catalog, version 2 for the astrometric database, which we found yielded fit residuals that were smaller by about a factor of two compared to the USNO-A2 catalog. [msccmatch]{} provides an interactive interface with which the fit can be improved by eliminating outlying mismatched points. We were able to get the fit residuals in each image below 0.25 arcsec.
Next, we used [mscstack]{} to combine each pair of long exposures into a single, averaged image. We had initially activated the cosmic-ray rejection feature of [mscstack]{}, but we found that if the seeing changed between the two exposures, the stellar flux in the averaged image would be significantly clipped by the rejection algorithm. Instead, we stacked the images without rejection, and used [craverage]{} to detect cosmic-ray hits in the combined image and replace the affected pixels with the average of the surrounding pixel values. We also ran [mscstack]{} on the short-exposure images, despite the fact that they did not have multiple exposures to combine, because [mscstack]{} also performs a pixel-value replacement for the pixels in the bad pixel masks.
The final step in the reduction pipeline is to run [mscpixarea]{}, which corrects each image for the variable pixel scale across the field. We then use [mscsplit]{} to separate the eight CCD images per field into separate FITS files. We choose to analyze the CCD images separately, in order to properly account for slight differences between the CCDs.
Instrumental Photometry
-----------------------
We used the [daophot]{} package in IRAF to perform stellar photometry on our images, using the method of point-spread function (PSF) fitting. For each CCD image, we ran [daofind]{} to identify peaks, and [phot]{} to obtain preliminary relative photometry of the detected sources.
Next, we used [pstselect]{} to select fifty bright, relatively isolated sources to form a high-quality sample from which an empirical PSF model will be built. Candidate PSF stars are rejected if they contain saturated pixels, or if their centers are within ten pixels of any pixel flagged as bad by our data reduction procedure.
The PSF fitting is performed in a fully automated way, without user intervention. However, we do examine residual images after PSF subtraction to ensure that the automatic model parameters are correct. All available forms for the PSF are explored to find the best-fitting model. Next, [group]{} and [nstar]{} are run to identify and photometer stars that are close neighbors of the PSF sample. [ Substar]{} is used to subtract these close neighbors from the image. We then repeat the PSF fitting, this time using the neighbor-subtracted image to produce a more accurate model.
Once the second-pass PSF has been determined, we run [allstar]{} to perform iterative PSF photometry of all detected sources in the image. Then we run [daofind]{} on an image in which all known objects have been subtracted using the PSF model, in order to find fainter stars in the image. Then [allstar]{} is run on the subtracted image, using this new list of fainter stars[^2]. The two allstar photometry lists are combined into a single photometry table for the frame, and the IRAF task [wcsctran]{} is used to convert the stars’ X,Y pixel coordinates to right ascension and declination, using the world coordinate solution we determined during the data reduction procedure.
The above photometry procedure results in some spurious detections due to two artifacts: bleed columns, and scattered-light halos around extremely bright stars. The bleed columns are flagged as bad pixels by the data reduction procedure, but this does not prevent [ daofind]{} from identifying sources along the bleed columns, nor does it prevent [allstar]{} from trying to photometer these false sources. We therefore remove objects from the photometry table that are within 10 pixels of a flagged bad pixel. Extremely bright stars have large scattered-light halos in the images, resulting in circular concentrations of false detections centered on these stars due to the elevated flux levels. To clean the photometry tables of sources detected in the wings of extremely bright stars, we reject all sources with anomalously high estimated sky values. In the absence of these extremely bright stars, the sky levels are quite stable, making the sky levels an efficient way to identify spurious objects in the wings of bright objects.
The [allstar]{} program provides very accurate relative photometry, but because the PSF models are uncertain at large radii, the allstar magnitudes are normalized to a relatively small aperture size of 4 pixels. We therefore need to apply an aperture correction to convert the allstar magnitudes to true instrumental magnitudes that represent the total flux recorded from each star. To determine the aperture correction, we select bright, isolated objects from our sample and perform concentric-aperture photometry using [phot]{} with a series of aperture sizes, up to 17 pixels. The per-star aperture correction is simply the difference between the star’s allstar magnitude and its large-aperture magnitude: $(m_{als} - m_{ap17})$. However, since individual objects suffer from measurement errors and contamination from neighboring objects, we need to statistically determine the characteristic aperture correction for the entire frame. The distribution of per-star aperture-correction values is typically a Gaussian with an asymmetric tail to negative values. The Gaussian spread is due to measurement uncertainties, and the asymmetric tail is due to flux contamination, which is never completely mitigated by our selection of isolated objects.
An accurate determination of the aperture correction’s value and uncertainty requires that we attempt to isolate the underlying Gaussian shape from the asymmetric skew caused by contaminating flux. To do this, we first determine the approximate position of the distribution’s peak, and then fit a Gaussian function to the points to the positive side of this peak value, thereby ignoring the negative half that may suffer from flux contamination. We adopt the central value of the fitted Gaussian as the frame’s aperture correction, and its width as the uncertainty in the frame’s aperture correction. The instrumental magnitude of each star is simply its allstar magnitude plus the frame’s aperture correction; we also add the aperture correction uncertainty in quadrature to each star’s photometric uncertainty.
Standard Star Observations and Photometric Calibration
------------------------------------------------------
The final step in our determination of the stellar photometry is to place the instrumental magnitudes we have measured onto a standard photometric system, using standard-star observations. Standard star fields were imaged several times on both of the observing nights. One standard field (SA 101) was observed at two separate visits on each night in order to measure the effect of atmospheric extinction.
The standard star observations are presented in Table \[tab:standards\]. We selected well-known standard fields, first measured by [@lan73], expanded for wide-field CCD instruments by [@ste00], and calibrated for the Washington $C$ filter by [@gei96]. The standard fields were reduced using the pipeline procedure described in Section \[sec:reduction\]. We then identified sources in each standard field with [daofind]{}, and performed concentric-aperture photometry on all sources with [ phot]{} in IRAF’s [daophot]{} package.
We use the measured aperture photometry of the observed standard stars, together with their total photometry as published by [@ste00] and [@gei96], to solve the following photometric calibration equation for each filter and each CCD in the Mosaic-II array, and independently for the two nights of observing:
$$M = m + A + B*(R-I) - C*X$$
where $M$ is the published total magnitude, $m$ is the observed instrumental aperture magnitude, $A$ is the photometric zeropoint, $B$ is the color term, $R-I$ is the star’s true color, $C$ is the atmospheric extinction term, and $X$ is the airmass. Once we have determined $A$, $B$ and $C$ for each CCD and filter, we will be able to convert the observed photometry of any star to its total photometry, given its observed color and the airmass at which it was observed.
We first determine $C$, the atmospheric extinction term, by examining the photometry from SA 101, the standard field that was observed at different airmasses during each night. The $C$ parameter is independent of the color term and zeropoint, so we simply need to fit a linear regression through the observed stars’ magnitudes as a function of the observed airmass. The slope of the linear regression is $C$, the atmospheric extinction term. Note that we need not restrict ourselves to the actual standard stars for this step; since we only need the relative photometry to determine the extinction term, all of the stars observed in field SA 101 can be employed.
Having determined $C$, we proceed to simultaneously determine the zeropoint and color term. The observed magnitudes of all standard stars are first corrected for atmospheric extinction ($m_x = m -
C*X$); we then construct the quantity $M-m_x$, the difference between the the published total magnitude of the star and its extinction-corrected instrumental magnitude, and fit a linear regression through the $M-m_x$ values as a function of the published $R-I$ colors. The slope of this regression is the color term $B$, and its zeropoint is $A$, the photometric zeropoint correction. Table \[tab:photcalib\] presents the photometric calibration parameters for each CCD and filter, and for each of the two observing nights.
Note that in Table \[tab:photcalib\], the parameters for the Washington $C$ filter are the same in all eight CCDs. The reason for this is that there are too few standard stars calibrated for the $C$ filter to support an independent determination for each CCD (see Table \[tab:standards\]), so we were forced to determine an average $C$ calibration for the entire mosaic.
As noted in Table \[tab:photcalib\], there were too few $R$ and $I$ standards present in some of the CCDs to support an independent determination of their color terms and zeropoints. For these cases, we perform a bootstrap estimate of the parameter values from those published at the CTIO website[^3]. We determine the mean offset between our determined values of the photometric calibration parameters, and those published by CTIO, for the CCDs that we were able to analyze. We then apply this mean offset to the published values of the remaining CCDs, as an estimate of what we would have measured if we had observed enough standard stars in those CCDs. For the uncertainty in these bootstrapped parameters, we simply adopt the standard deviation of the mean offset between the observed and published values.
We simultaneously perform a positional match of the sources in the $C$, $R$ and $I$ photometry lists, and apply the above photometric calibration to produce catalogs with total $CRI$ photometry. We then match objects between the $CRI$ catalogs from the short and long exposures of each field. For the positional matching, we use a maximum match radius of 0.5$^{\prime\prime}$; to be retained in the catalog, an object must be detected in the $R$ band, with a matching detection in either $C$ or $I$. For objects which are matched between the short and long exposures, we adopt the weighted mean photometry in the final catalog; objects present only in the short or long catalog are included as well. The final calibrated composite photometry catalogs for the twelve observed fields in the Magellanic Bridge (plus the observed offset field) are presented in Table \[tab:catalog\], and rendered as pairs of Hess diagrams in Figure \[fig:cmds\]. A Hess diagram is a pixelized color-magnitude diagram (CMD) in which each pixel value is proportional to the number of stars in the region covered by that pixel. The CMD of the offset field is presented separately in Figure \[fig:offset-cmd\].
Statistical Subtraction of Foreground/Background Contamination {#sec:statsub}
--------------------------------------------------------------
It is clear from comparing Figures \[fig:cmds\] and \[fig:offset-cmd\] that the stellar populations in many of the fields are dominated by foreground Galactic (and background extragalactic) contamination. In order to study the underlying inter-Cloud populations, we need to first perform a statistical subtraction of the contaminant foreground/background population. In doing so, we will assume that the population observed in the offset field is representative of the contaminant population in each Bridge field (a reasonable assumption, given the similar Galactic latitude of the offset field).
We proceed by first determining a scaling factor for normalizing the number of objects in the offset field to the number of contaminant objects in each Bridge field. This is necessary to account for variations in the effective area covered by each field (which arise from masking out regions contaminated by bad pixels, very bright stars and bleed trails). The normalization factor is simply the ratio of object counts in the target and offset fields, for a selected subregion of each CMD that is expected to contain only contaminant objects. For the $C-R$ CMD, the normalization region is defined by the criteria $C-R > 2.4$, $R < 22.4 - (C-R)$, and $R > 23.8 -
2*(C-R)$; for the $R-I$ CMD the criteria are $R-I > 1.0$ and $18 < I <
20$. The normalization regions are outlined with dashed lines in Figure \[fig:offset-cmd\]. In the $C-R$ CMD, the objects in the normalization region constitute 12% of the total number of objects, while in the $R-I$ CMD, the fraction is 14%. The normalization factor computed for each target field ranges between 0.65 and 0.95. We multiply the offset field’s Hess diagrams by these scaling factors, and subtract them from the target field’s Hess diagrams. The resultant statistically-cleaned Hess diagrams for each field are shown in Figure \[fig:clean-cmds\].
The statistical subtraction was generally successful in removing a component from each field’s CMD that is consistent with the contaminant population in the offset field. However, there are some artifacts present that bear explanation. Specifically, the faint end of many of the $R-I$ CMDs appear to show an oversubtracted contaminant population. This is simply due to the fact that the offset field’s $R-I$ CMD has a fainter detection limit than that in most of the target fields.
Analysis {#sec:analysis}
========
Extent of the Young Inter-Cloud Population {#sec:youngstars}
------------------------------------------
Young (age $<1$ Gyr) stars provide an unambiguous tracer of the inter-Cloud population, because no foreground or background contaminants are expected to share the bright, blue region of the CMD with these stars. From previous work on the young inter-Cloud population by [@db98], we expected to observe a population of young stars near the SMC, coincident with the young cluster population cataloged by [@bs95]. However, we did not know how far the young population would extend toward the LMC along the ridgeline. We isolate stellar populations younger than 1 Gyr in the $C-R$ CMD, by selecting those stars with $R<20$ mag and $C-R<0$ mag (see dashed lines in Figure \[fig:cmds\]). Stars matching these criteria are absent in all of our fields east of mb09, which corresponds roughly to the eastern extent of the @bs95 clusters. Interestingly, field mb09 is also near the position along the Bridge where the gas surface density drops to the critical threshold for star formation of 3–4 $M_\odot pc^{-2}$ [@ken89], which corresponds to the $5\times10^{21} cm^{-2}$ seen throughout the eastern Bridge in Figure 4a of [@bru05]. West of field mb09, the gas density is sustained at a level three times higher, and this is where star formation has been active in the Bridge. It would seem that the same star formation threshold observed for disk galaxies holds for this tidal debris environment as well.
Searching for Tidally-Stripped Stars in the Inter-Cloud Region {#sec:oldstars}
--------------------------------------------------------------
It is perhaps not surprising that the young inter-Cloud population is confined to those regions where the gas density is relatively high, if we accept the hypothesis that these stars formed [*in situ*]{}, following the formation of the gaseous Bridge by a recent gravitational encounter between the Clouds. However, the tidal forces that presumably formed the Bridge should have stripped stars and gas with equal efficiency, so we expect to observe a population of such tidally-stripped stars in the inter-Cloud region. [@yn03] conducted detailed numerical modeling of the stars and gas in the SMC, as it orbits both the LMC and Milky Way, in an attempt to reproduce the broad physical parameters of the Magellanic system. In their best-fitting model, there is an abundant population of stars in the inter-Cloud region which formed in the SMC, and had been ejected into the inter-Cloud region by a tidal interaction with the LMC. The tidally-stripped stars should have a similar age distribution to the stars in the galaxy from which they were stripped (at least for ages prior to the Bridge-forming event when their histories diverged). Since the stellar populations in both Magellanic Clouds exhibit a prominent red giant branch and a “red clump” horizontal branch, these bright features serve as ideal tracers of a putative stellar population that had been stripped from either of the Clouds during the Bridge-forming event.
While some of the fields in Figure \[fig:clean-cmds\] do show red giant branch and red clump features, these older populations appear to be confined to the fields nearest the SMC (fields mb02 and mb03) or the LMC (fields mb16–mb20). Furthermore, the surface density of these tracer populations increases sharply as the galactocentric separation of the field decreases, consistent with populations that are bound to the LMC and SMC. In Section \[sec:expdisk\], we will demonstrate that the red giant populations in fields mb16–mb20 are consistent with a plausible exponential disk distribution centered on the LMC. For now, we simply conclude that the red giant populations in these six fields near the SMC and LMC are very likely composed of stars bound to each respective galaxy, and are not indicative of a tidally-stripped stellar population in the Magellanic Bridge.
The Hess diagrams of the remaining six fields (mb06–mb14) show no red features that can be associated with an old inter-Cloud population. However, the strength of this non-detection is limited by the presence of the contaminant population. To enhance our sensitivity to a potentially sparse old stellar population, we construct a composite pair of Hess diagrams from these six “true Bridge” fields, and perform a new statistical contaminant subtraction on the composite population (see left panels of Figure \[fig:fakergb\]). Even in this composite Hess diagram which covers more than two square degrees of the inter-Cloud region, there is no detectable trace of an underlying red giant branch or red clump feature.
We can place an upper limit on the surface density of red giant branch stars in these six “true Bridge” by adding an artificial old stellar population at the distance of the Magellanic system ($m-M=18.7$ mag, intermediate between the two Clouds) to the composite inter-Cloud population. The artificial old stellar population is drawn from a theoretical isochrone [@gir02] with $Z=0.002$ and $log(age)=10.0$, to which we add photometric errors consistent with the data. We modulate the number of artificial stars added until a red giant branch is marginally detectable (Figure \[fig:fakergb\]). We conclude from this exercise that there are fewer than 1000 red giant branch stars at the distance of the Bridge and brighter than $R=23$ mag in the observed composite population. By applying a stellar mass function [@kro01], we can convert the upper limit on the number of observed red giants to an upper limit on the total stellar mass present in a putative old stellar population. However, the conversion factor depends on the assumed age of the stars, because the fraction of the total stellar population that is brighter than $R=23$ mag varies with age. For a 10 Gyr population, the upper mass limit is 14800 $M_\odot$, and for a 2.5 Gyr population, the upper mass limit is 5300 $M_\odot$. Thus, the stellar surface mass density in these six “true Bridge” fields is $\le0.009\ M_\odot\ pc^{-2}$; this is more than 400 times smaller than the average surface mass density of in the Magellanic Bridge [4 $M_\odot\ pc^{-2}$, converted from the characteristic column density in the Bridge, @bru05]. There does not appear to be any trace of a tidally-stripped stellar population in the Magellanic Bridge, at least in these six fields along the ridgeline.
One potential caveat in this analysis is that we have assumed that the putative tidally stripped stellar population would be spatially coincident with the gaseous Bridge. This need not be the case; if ram-pressure from the Milky Way halo has played a significant role in the evolution of the Magellanic system [@mas05], then it is possible that the gaseous Bridge is now displaced from the region occupied by tidally-stripped stars between the Clouds. We investigate this possibility using data from the 2-Micron All-Sky Survey [2MASS, @skr06].
Using the Gator web-based database query service[^4] at the NASA/IPAC Infrared Science Archive, we obtained near-infrared $JHK$ photometry from the 2MASS All-Sky Point Source Catalog, in two regions (shown as dashed boxes in Figure \[fig:fields\]). The first 2MASS region (the “full-bridge region”) was selected to cover all plausible locations where a tidally-stripped inter-Cloud population might exist. We selected a range in right ascension between 2.5$^h$ and 3.5$^h$, because these limits are bracketed by fields mb06 and mb14, which define the edges of the “pure bridge” section of our sample, uncontaminated by LMC or SMC stars. We selected a very large range in declination, from $-77^\circ$ to $-69^\circ$, to cover all plausible trajectories of a putative tidally-stripped stellar population. The second 2MASS region (the “SW-LMC region”) was selected as a comparison field that is known to contain an old stellar population at the distance of the Magellanic system. This region spans 4.2$^h$ to 5$^h$ in right ascension, and $-75^\circ$ to $-74^\circ$ in declination. It is coincident with our fields mb18, mb19 and mb20, in which we have observed an old stellar population associated with the LMC (Section \[sec:oldstars\]). While the full-bridge region covers a solid angle ten times larger than that of the SW-LMC region (35 square degrees and 3.2 square degrees, respectively), the 2MASS catalog contains about the same number of stars in both regions (94000 stars in the full-bridge region, and 92000 stars in the SW-LMC region), due to the larger stellar surface density of the SW-LMC region.
The 2MASS $J-K$ CMDs for these two regions are shown in Figure \[fig:2mass-cmds\]. In the SW-LMC region, there is an abundant population of red objects consisting of old stars associated with the the LMC. Following [@nw00], we identify the various subpopulations of these red objects. The bulk of the population extends in a narrow diagonal sequence from $J-K$=1 mag,$K$=14 mag to $J-K$=1.25 mag,$K$=11 mag. Along this sequence, there is a sharp drop in the density of stars around $K$=12.3 mag; this is the tip of the red giant branch. The stars in this sequence brighter than $K$=12.3 mag are oxygen-rich asymptotic giants, while the stars which extend redward of $J-K$=1.25 mag are carbon-rich asymptotic giants. In the full-bridge region’s $J-K$ CMD, there is a small number of stars whose photometry is consistent with these features (notably the $\sim6$ red objects around $K=11$ mag which may be Carbon stars at the Magellanic distance), but considering the very large solid angle covered by the full-bridge region, we do not regard these objects as a significant detection of an old inter-Cloud population. We can place an upper limit on the number of red giants at the distance of the Magellanic system that can remain undetected in the 2MASS CMD, using the same synthetic population analysis described above for our optical CMDs. A red giant population containing 150 stars brighter than $K=14$ mag is easily detectable when added to the 2MASS CMD, which implies that any old inter-Cloud stellar population that may be present has a total stellar mass no greater than $2\times10^6\
M_\odot$. This is 1% of the total mass in the Magellanic Bridge [@bru05]. However, the area covered by our full-Bridge region is about three times smaller than the area used to define the Bridge by @bru05, so the true limit from the 2MASS data is closer to 3%. Thus, even accounting for the possibility that a putative tidally-stripped stellar population may be displaced from the gaseous Magellanic Bridge, we can still conclude that the Bridge material was more than 97% gas when the Bridge was formed.
The Outer Disk of the Large Magellanic Cloud {#sec:expdisk}
--------------------------------------------
In fields mb16–mb20 we observe old stellar populations that we conclude are bound members of the LMC, based on the sharp increase in their surface density with decreasing angular separation from the LMC. [@gal04] and others have found that the LMC’s stellar radial profile follows an exponential disk to projected radii beyond 7 kpc, with no sign of a break which might indicate the onset of a kinematic halo. Fields mb16–mb20 have projected separations from the LMC of between 5 kpc and 8.5 kpc; however, when the orientation of the LMC disk [@vdm01] is taken into account, the in-disk galactocentric distances of these fields are between 6 kpc and 10.5 kpc. We use the number of stars in the red clump feature as a proxy for the stellar surface density, and plot the surface density profile in Figure \[fig:expdisk\]. The solid curve represents the best-fit exponential-disk model, with a scale length of $\alpha=0.98$ kpc, and the dotted and dashed curves are exponential disk models fit by previous authors, as noted in the figure caption. The surface density profile of the red clump stars in fields mb16–mb20 are generally consistent with previous measurements of the LMC’s outer exponential disk, but the fact that the profile is somewhat steeper in this southwestern quadrant is interesting. A comprehensive survey of the stellar populations in the outer LMC is currently underway; we will therefore postpone further discussion of the LMC’s structure until this survey is completed, when more definitive conclusions can be made.
Characterizing the Purely Tidal Stellar Population in the Inter-Cloud Region {#sec:sfh}
----------------------------------------------------------------------------
We have determined that the stars in the inter-Cloud region appear to be exclusively composed of a stellar population that formed [*in situ*]{}, in the wake of the Bridge-forming event (modulo some contribution from stars still bound to the LMC and to the SMC, in the observed Bridge fields nearest those galaxies). This isolation of a tidally-triggered stellar population provides an important opportunity to examine the nature and evolution of star formation processes in tidal debris. We measure the age distribution of the inter-Cloud population to determine when the star formation occurred, and how long it lasted. Since these stars presumably formed in the wake of the Bridge-forming event, these measurements provide an important constraint on the timing of that event. We will also look for spatial structure in the age distribution, which may provide insights into how star formation proceeds when triggered by a gravitational interaction.
Previous studies of the inter-Cloud population have estimated the age of the youngest stars present, through simple isochrone fitting [e.g., @db98]. Here we will perform a more detailed analysis, using the StarFISH star formation history fitter [@hz01]. This analysis is motivated by the clear presence of composite stellar populations in some of our fields, and by our goal to constrain the duration of star-formation activity in the Bridge.
StarFISH constructs a library of synthetic CMDs, each of which represents a model of what the photometric observations would yield, if the observed stellar population had a single age and a single metallicity. The model photometry is derived from theoretical isochrones; in this case we chose the latest Padua isochrones [@gir02]. In order to accurately predict the observed photometric distribution in the CMDs, the models include a distance modulus, a distribution of extinction values, and a detailed model of the photometric errors. The distance modulus was simply chosen to be that of the SMC, 18.9 mag, because the young stellar populations in the Bridge are near the SMC on the sky. The distribution of extinction values is drawn from regions near the eastern edge of the MCPS SMC extinction map [@zar02]. For the photometric errors, we employ an analytic model that reproduces the error statistics in the observed fields. While we usually advocate for an empirical model based on artificial stars tests, these tests are only strictly necessary when the data images are crowded. In the present case, even in our field with the highest stellar surface density (mb20), we have detected roughly 109000 stars in $8192^2$ pixels, corresponding to a mean separation between objects of almost 14 pixels.
The StarFISH model library provides synthetic CMDs for the range of ages and metallicities thought to be present in the observed population; in the present case we constructed synthetic CMDs for 16 age bins spanning ages 10 Myr to 12 Gyr, spaced uniformly in $log(age)$, and for three metallicity bins, $Z=0.001$, $Z=0.002$, and $Z=0.004$. The best-fit SFH is found by determining the combination of amplitudes modulating these synthetic CMDs which produces a composite model CMD that most closely matches the observed CMD. To take advantage of the full $CRI$ photometric data set in determining the SFH, the fit is actually performed on the CMD pair: $C-R$ vs. $R$ and $R-I$ vs. $I$.
Because the contaminant population dominates many of our observed fields, it is important to account for contaminants in the SFH fit. We could have run StarFISH on the statistically-cleaned data set, but we instead chose to use the observed data set, and simply include the contaminant offset-field population as an additional amplitude in the model, in addition to the normal set of synthetic CMDs. The code will then select a multiplicative amplitude factor that optimally accounts for the contaminant population, just as it does for each of the synthetic CMDs.
The star formation histories of our twelve observed Magellanic Bridge fields are presented in Figure \[fig:sfh\], and the results are consistent with the qualitative analysis of the CMDs presented in Sections \[sec:youngstars\] and \[sec:oldstars\]. Recent star formation has occurred only in the fields west of mb11, and old stellar populations are confined to the six fields nearest the SMC (mb02 and mb03) and LMC (mb16–mb20). In fields mb11, mb13 and mb14 there is no trace of a stellar population associated with the Magellanic system.
Our StarFISH analysis shows that star formation in the Bridge began around 200–300 Myr ago, and this measurement provides an important constraint on the timing of the Bridge-forming event. In field mb02, the field nearest the SMC, we see a prolonged star formation episode spanning ages 80–300 Myr, and in mb03 we see a slightly shorter episode spanning ages 100–200 Myr. In the other fields in which a young stellar population is present (mb06–mb09), the star formation rates are much lower, making ages and durations more difficult to determine reliably. To boost the signal, we construct a composite population from these three fields, and determine the SFH of the composite population (see Figure \[fig:comp-sfh\]). In these more easterly fields, we see evidence for two distinct episodes of star formation, 160 Myr and 40 Myr ago. It is interesting that while star formation was active throughout the western Bridge 100–200 Myr ago, the more recent episode 40 Myr ago was apparently confined to regions further from the SMC. We note that [@db98] also found that the youngest inter-Cloud populations are to be found in fields eastward of the SMC Wing.
The StarFISH solutions indicate that no significant star formation occurred in the Bridge more recently than 40 Myr ago. This conclusion can be confirmed by direct inspection of the CMDs in Figure \[fig:cmds\]: in no region do we see a significant number of main sequence stars brighter than $R=15$ mag, corresponding to the main-sequence turn off position of a 40 Myr isochrone at the distance of the Magellanic system.
The conclusion that star formation in the Bridge largely ceased around 40 Myr seems to be at odds with a variety of previous research that finds evidence of stellar populations much younger than this. [@db98] used main-sequence isochrone fitting to conclude that the western Bridge contains stars as young as 10–25 Myr. [@mea86] reported the discovery of DEM 171, a large circular H$\alpha$ filament in the Bridge which is likely photoionized by one or more massive O stars. [@bs95] found that some of the clusters in their Bridge catalog have associated emission nebulae, again implying the presence of massive stars. [@miz06] detected cold molecular clouds in the Bridge, which demonstrates at least the potential for ongoing star formation.
This apparent contradiction can be partly reconciled by understanding that we are not claiming there are absolutely no stars in the Bridge younger than 40 Myr; we find that the star-formation rate dropped off around 40 Myr ago, and has remained consistent with zero since then. We also note that our field selection covers the HI ridgeline of the Magellanic Bridge uniformly (see Figure \[fig:fields\]), [ *except*]{} for the segment around $RA=2^h$, where much of the evidence for more recent star formation is to be found. These explanations do not reconcile our result with the conclusions of [@db98], however. They reported the widespread presence of stars aged 10–25 Myr in a number of fields east of $RA=2^h$. This is based on an analysis of their Figure 7 (lower panel), in which theoretical isochrones are overplotted on a composite CMD from four of their observed fields. While the observed main sequence does appear to follow the shape of the 10 Myr isochrone, it is clearly truncated around $M_V=-3$ mag, whereas a 10 Myr population should have a main sequence that extends up to $M_V=-5$ mag. A main sequence turn-off at $M_V=-3$ mag is consistent with the 40 Myr age that we have found for the youngest bulk population in the Bridge.
Summary {#sec:summary}
=======
We have observed stellar populations in twelve fields uniformly spanning the region between the Magellanic Clouds. Our fields were selected to follow the ridgeline of the gas that forms the Magellanic Bridge, in order to look for stars that formed [*in situ*]{} in the Bridge from gas that had already been removed from one of the Clouds, and also for stars that were stripped from either of the Clouds by the same tidal forces that presumably stripped the gas.
We observed the previously known young stellar population in the western half of the inter-Cloud region, most recently characterized by [@db98], and extend on previous analyses in two key ways. First, we determine that the eastward extent of these stars is truncated around $\alpha=3^h$, corresponding also to the eastward extent of the star clusters cataloged by [@bs95], and to the point at which the surface density falls below the critical threshold for star formation as determined by [@ken89]. Second, we use the StarFISH program to determine quantitative star formation histories of the young inter-Cloud population, finding that star formation in the Bridge commenced about 200–300 Myr ago, and continued over an extended interval, until about 40 Myr ago.
We found no evidence for a population of tidally-stripped stars in the inter-Cloud region, and our non-detection allows us to conclude that the material stripped from the Clouds into the Bridge was very nearly a pure gas, with an upper limit on the mass fraction in stars of less than $10^{-4}$ if coincidence with the gaseous Bridge is assumed, and 0.03 otherwise. This can potentially be understood if the pre-collision SMC had an extended envelope of gas, surrounding a more tightly bound stellar component. In this case, a weak tidal interaction might unbind the gas envelope while leaving the stellar component undisturbed. It is known that some dwarf galaxies have gas extending beyond 2–3 times the radii occupied by their stellar populations [@sh96], so perhaps this scenario is plausible. In fact, the recent numerical simulation of the tidal history of the SMC by cite[yn03]{} included such an extended gas envelope, in order to produce a pure-gas Magellanic Stream. Nevertheless, the inter-Cloud region in their best model contains an abundant population of tidally-stripped stars. In addition, Figure 4a of [@bru05] shows that the gas in the Magellanic Bridge appears to be contiguous with the higher-density gas in the central regions of the SMC, which are currently abundantly populated with stars. If the gaseous Bridge formed via the tidal extraction of this high-density gas from the central regions of the SMC, then the question remains: where are the stars in the Bridge that should have felt these same tidal forces?
Future observations may be able to address this question. While some kinematic measurements exist for a handful of stars in the inter-Cloud region [@mau87; @kid97], radial-velocity kinematics of a truly representative sample of the young inter-Cloud population would help us to better understand the dynamical evolution of the Magellanic Bridge. A much deeper understanding of the SMC’s complex three-dimensional structure and kinematics would certainly help as well. These measurements (along with our current understanding of the orbital motions of the Clouds and the Milky Way) could then be used to motivate new detailed numerical simulations specifically targeting the formation of the Magellanic Bridge as a pure-gas feature. We may then better understand how tidal features are formed during minor harassment interactions, and what role such interactions play in driving the evolution of the participant galaxies.
I am very grateful for extended discussions with Edward Olszewski on the interpretation of the data presented here, and I would also like to thank Kurtis Williams, Abhijit Saha, Knut Olsen, Tim Abbot and Armin Rest for their assistance with the reduction of the Mosaic-II images. Finally, I would like to gratefully acknowledge the constructive comments made by the anonymous referee. These comments prompted the 2MASS analysis, and substantially improved the paper. JH is supported by NASA through Hubble Fellowship grant HF-01160.01-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.
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[^1]: IRAF is the Image Reduction and Analysis Facility, distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^2]: Alternatively, we could have performed the second-pass allstar on the original image using a concatenated list of all detected objects. We tested both methods on an image from mb20, the most crowded field. The differences in the photometry are consistent with the uncertainties estimated by allstar
[^3]: http://www.ctio.noao.edu/mosaic/ZeroPoints.html
[^4]: http://irsa.ipac.caltech.edu/applications/Gator
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this note we show that the exceptional algebraic set of a discrete group in $PSL(3,\C)$ should be a finite union of: complex lines, copies of the Veronese curve or copies of the cubic $xy^2-z^3$.'
address:
- ' UCIM, Av. Universidad s/n. Col. Lomas de Chamilpa, C.P. 62210, Cuernavaca, Morelos, México.'
- ' IIT UACJ, Av. del Charro no. 450 Nte. Col. Partido Romero CP 32310, Ciudad Juárez, Chihuahua, México.'
author:
- Angel Cano
- Luis Loeza
title: 'Exceptional Algebraic sets for discrete groups of $PSL(3,\Bbb{C})$'
---
Introduction {#introduction .unnumbered}
============
Complex Kleinian groups first appeared in mathematics with the works of Henri Poincaré, as a way to qualitatively study the solutions of ordinary differential equations of order two, one can say that the success of Poincaré was because he managed to establish a dictionary between differential equations and group actions. Subsequently this theory achieved a new boom with the introduction of quasi-conformal maps and the discovering of bridges between hyperbolic three-manifolds this theory. At the beginning of the 90’s A. Verjovsky and J. Seade began studying, see [@SV], the discrete groups of projective transformations that act in projective spaces, as a proposal to establish a dictionary between actions of discrete groups and the theory of foliations, partial or ordinary differential equations. The purpose of this note is to understand those groups whose dynamic could be described in terms of a group acting in a algebraic curve in the complex-projective plane,[*i. e.*]{} groups leaving an invariant algebraic curve, analogous results but in the case of iteration holomorphic maps in $\P^n$ has been studied extensively, see [@BCS; @CLi; @FS].
Recall that in the classical case, see [@Gr], a set in $\P^1$ is said to be exceptional for the action of a group $G\subset \PSL(2,\C)$, discrete or not, if it is invariant under the action of $G$ and is finite. In analogy with this, we will say that $S\subset P^2$ is an [*exceptional algebraic set*]{} for the action of a group, in our case discrete, if it is invariant and a complex algebraic curve, compare with the definition of algebraically mixing in [@SV]. As we will see, the geometry of the curve as well as the group is very restricted. More precisely in this article we show:
\[t:main1\] Let $G\subset PSL(\C)$ discrete and $S$ a complex algebraic surface invariant under $G$, if $S_0$ is a irreducible component of $S$, then $S_0$ is a complex line, the Veronese curve or the projective curve induced by the polynomial $p(x,y,z)=xy^2-z^3$.
\[t:main2\] Let $G\subset PSL(\C)$ discrete and non-virtually commutative. If $S$ is a complex algebraic surface invariant under $G$, then either $S$ is finite union of lines lines or the Veronese curve. Moreover, if the number of lines is least four, then the largest number of lines in general position is three.
\[t:main3\] Let $G\subset PSL(3,\C)$ be a discrete virtually cyclic group and $S$ a complex algebraic surface invariant under $G$, then $S$ is:
1. A finite union of lines. If the number of lines is least four, then the lines are concurrent.
2. A finite union of copies of the Veronese curve and a finite union (possibly empty) on tangent and secant lines. Moreover the number of tangents does not exceed two and the number of secants is at most one.
3. A finite union of copies of the cubic induced by the polynomial $xy^2-z^3$ and finite union (possibly empty) on tangent and secant lines. Moreover the number of tangents does not exceed two and the number of secants is at most one.
\[c:main\] Let $G\subset PSL(3,\C)$ be a discrete group with an algebraic exceptional set, then either $G$ is virtually affine or is the representation of a Kleinian group of Möbius transformations though the irreducible representation of $Mob(\hat \C)$ into $PSL(3,\C)$.
The paper is organized as follows: Section \[s:nb\] reviews some elementary well-known facts on the subject that are used in the sequel. In Section \[s:examples\], we provide a collection of examples which depicts all the possible ways to constructing groups with exceptional algebraic surfaces as well as the respective surfaces. In Section \[GDC\] we show that every irreducible component of an exceptional algebraic set is a line, a copy of the Veronese curve of a cubic with a cusp as a singularity, the proof of this fact relies strongly in the use of the Plücker Formulas as well as in the Hurwitz theorems for curves, most of the material on algebraic curves used in this note is well known but the interested reader could see [@Fischer; @miranda; @sha] for full details. Finally in section \[s:main\] we prove the main theorem of this article.
Preliminaries {#s:nb}
==============
In this section we establish some elementary facts that we use in the sequel.
Projective Geometry
-------------------
The complex projective plane $\mathbb{P}^2_{\mathbb{C}}$ is the quotient space $(\mathbb{C}^{3}\setminus
\{{\bf 0}\})/\mathbb{C}^*
,$ where $\mathbb{C}^*$ acts on $\mathbb{C}^3\setminus\{{\bf 0}\}$ by the usual scalar multiplication. Let $[\mbox{
}]:\mathbb{C}^{3}\setminus\{{\bf 0}\}\rightarrow
\mathbb{P}^{2}_{\mathbb{C}}$ be the quotient map. A set $\ell\subset \mathbb{P}^2_{\mathbb{C}}$ is said to be a complex line if $[\ell]^{-1}\cup \{{\bf 0}\}$ is a complex linear subspace of dimension $2$. Given $p,q\in
\mathbb{P}^2_{\mathbb{C}}$ distinct points, there exists a unique complex line passing through $p$ and $q$, such line is denoted by $\overleftrightarrow{p,q}$. The set of all complex lines in $\P^2$, denoted $Gr(\P^2)$, equipped with the topology of the Hausdorff convergence, actually is diffeomorphic to $\P^2$ and it is its projective dual $ \P^{*2} \cong Gr(\P^2)$.
Consider the action of $\mathbb{Z}_{3}$ (viewed as the cubic roots of the unity) on $\SL(3,\mathbb{C})$ given by the usual scalar multiplication. Then $$\PSL(3,\mathbb{C})=\SL(3,\mathbb{C})/\mathbb{Z}_{3}\,,$$ is a Lie group whose elements are called projective transformations. Let $[[\mbox{ }]]:\SL(3,\mathbb{C})\rightarrow \PSL(3,\mathbb{C})$ be the quotient map, $g\in \PSL(3,\mathbb{C})$ and ${\bf g}\in
\GL(3,\mathbb{C})$. We say that ${\bf g}$ is a lift of $g$ if there exists a cubic root $\alpha$ of $Det({\bf g})$ such that $[\alpha^{-1} {\bf g}]=g$, by abuse of notation in the following we will use $[\mbox{ }] $ instead $[[\mbox{ }]] $ . We use the notation $(g_{ij})$ to denote elements in $\SL(3,\Bbb{C})$. One can show that $ \PSL(3,\mathbb{C})$ acts transitively, effectively and by biholomorphisms on $\mathbb{P}^2_{\mathbb{C}}$ by $[{\bf g }]([w])=[{\bf g }(w)]$, where $w\in
\mathbb{C}^3\setminus\{{\bf 0}\}$ and ${\bf g }\in \GL(3,\mathbb{C})$.\
If $g$ is an element in $\PSL(3,\C)$ and ${\bf g}$ is a lift of $g$, we say [@CNS] that:
- $g$ is a elliptic if ${\bf g }$ is diagonalizable with unitary eigenvalues.
- $g$ is parabolic if ${\bf g }$ is non-diagonalizable with unitary eigenvalues.
- $g$ is loxodromic if ${\bf g }$ has some non-unitary eigenvalue.
Recall a subgroup is weakly semi-controllable, if it have a global fixed point $p \in \P^2$; hence for each line $\mathcal L$ in $\P^2 \setminus \{p\}$ one has a canonical holomorphic projection map $\pi$ from $\P^2 \setminus \{p\}$ into $\mathcal L \cong \P^1$. This defines a group morphism: $$\begin{matrix}
\Pi = \Pi_{p,\ell,G} : G \rightarrow Bihol(\ell)\cong \PSL(2,\C)\\
\Pi(g)(x) = \pi(g(x))
\end{matrix}$$ which essentially is independent of all choices, see [@CNS] for details.
Examples {#s:examples}
========
In this section we present examples of curves invariant under Lie groups later we will see that any irreducible curve is one of the curves presented here.
\[e:line\]\[Lines\] The group of affine projective transformation leaves invariant a complex line
\[e:ver\]\[Veronese Curves\] Recall the Veronese embedding is given by $$\begin{array}{l}
\psi:\Bbb{P}^1_\Bbb{C}\rightarrow \Bbb{P}^2_\Bbb{C}\\
\psi([z,w])=[z^2,2zw, w^2].
\end{array}$$
And the unique irreducible representation of $PSL(2,\Bbb{C})$ into $PSL(3,\Bbb{C})$ is given by Let us consider $\iota: \PSL(2,\Bbb{C})\rightarrow \PSL(3,\Bbb{C})$ given by $$\iota\left(\frac{az+b}{cz+d}\right )=\left [
\begin{array}{lll}
a^2&ab&b^2\\
2ac&ad+bc&2bd\\
c^2&dc&d^2\\
\end{array}
\right ].$$ Then $\iota PSL(2,\Bbb{C})$ leaves invariant $Ver=\psi(\P^1)$.
\[e:cubic\]\[Cubic with a cusp\] Consider the homogeneous cubic polynomial $p(x,y,z)=xy^2-z^3$ then the projective curve induced by $p$ is a cubic with a cusp in \[1:0:0\] and a inflection point in \[0:1:0\]. Moreover, the cubic and the lines $\overleftrightarrow{e_1,e_2}$, $\overleftrightarrow{e_3,e_2}$ and $\overleftrightarrow{e_1,e_3}$ are invariant under the Lie group $$\C_p^*=
\left \{
\begin{pmatrix}
a^{-5}&0 &0\\
0 &a^{4}& 0\\
0 &0& a\\
\end{pmatrix}: a\in \C^*
\right \}$$
\[Pencil of lines\] Consider the Lie group given by $$\C^2_\infty=
\left \{
\begin{pmatrix}
1 & a &b \\
0 & 1 & 0\\
0 &0& 1\\
\end{pmatrix}:a,b\in \C
\right \}.$$ Then $\C^2_\infty$ leaves invariant any pencil lines passing trough $[e_1]$.
Geometry and dynamic of the invariant curves {#GDC}
============================================
\[l:fp\] Let $G\subset PSL(3,\C)$ be a discrete group and $S$ an algebraic reducible complex curve invariant under $G$. If $S$ is not a line and $\gamma\in \Gamma$ has infinite order then $S$ contains a fixed point of $\gamma$ and the action of $\gamma$ restricted to $S$ has infinite order. Moreover, if $S$ is non singular, then $S$ has genus 0.
Since $G$ is discrete there is an element $g\in G$ with infinite order. Once again since $G$, thus $g$ is either loxodromic or parabolic. We must consider two cases:\
Case 1.- $g$ is loxodromic. Since $g$ is loxodromic there is a point $p$ an a line $\ell$ such that: $p\notin \ell$, $p\cup \ell$ is invariant under $g$ and $(g^n)_{n\in \Bbb{N}}$ converges uniformly on compact sets of $\P^2-\ell$ to $p$.\
Claim 1.- $p\in S$. Since $S$ is not a line by Bézout Theorem we know that $S\cap \ell$ is a finite set, thus we can find a point $q\in S-\ell$. Since $S$ is closed and invariant under $g$, we conclude $p\in S$.\
Case 2.- $g$ is parabolic. Since $g$ is parabolic there is a point $p$ an a line $\ell$ such that: $p\in \ell$, $p$, and $ \ell$ are invariant under $g$ and $(g^n)_{n\in \Bbb{N}}$ converges uniformly on compact sets of $\P^2-\ell$ to $p$. In similar way we can show that $p\in S$.\
Finally observe that for any point $q$ in $S-\ell$ the set $\{g^n q\}$ is infinite, which show the claim.\
Let us show the final part of the lemma. Let us assume that $\pi_1(S)$ is non-trivial, then there exist a non trivial class in $\pi_1(S)$, say $[h]$. Let us assume without lost of generality that $G$ contains a loxodromic element in other case the proof will be similar. As before there is a point $p$ an a line $\ell$ such that: $p\notin \ell$, $p\cup \ell$ is invariant under $g$ and $(g^n)_{n\in \Bbb{N}}$ converges uniformly on compact sets of $\P^2-\ell$ to $p$. Recall that $[h]$ can be assume as the homotopy class of a path based in $p$, since $S\cap \ell$ is finite we can assume that $h$ is a loop based on $p$ that does not contains points in $\ell$. For $N$ large we have that $g^N(h)$ is contained in a simply connected neighborhood of $p$ that is $[g^N h]$ is trivial, thus $g^{N}_{\#}: \pi_1(S)\rightarrow \pi_1(S) $ is not a group isomorphism, which is a contradiction, since $g^N$ is a homeomorphism.
As an immediate consequence of the proof of the previous lemma we have:
If $M\subset \P^2$ an embedded k-manifold with $k\geq 2$ invariant under a discrete group $G\subset PSL(3,\C)$, then $S$ is simply connected
\[l:genus\] Let $G\subset PSL(3,\Bbb{C})$ be a discrete group and $S$ a $G$-invariant complex irreducible algebraic curve. Then the dual group $G^*$ leaves invariant the dual algebraic curve $S^*$. Moreover, if $S$ has singularities then $S^*$ does.
Let $p\in S^*$ then there is a sequence of points $(p_n)\in S^*$ such that each $p_n$ converges to $p$ and for each $n$ we have $p_n^*$ is a tangent line to $S$ at a non-singular point. Since the action of $G$ is by biholomorphism of $\P^2$ we deduce for each $\gamma\in\Gamma$ we have $ \gamma( p_n^*)$ is a tangent line to $S$ at a non-singular point. Trivially $\gamma( p_n^*)$ converges to $\gamma( p^*)$.
Let us prove the other part of the Lemma. let us assume that $S^*$ is non-singular, then by the Riemann-Hurwitz formula we have $$0=(n-1)(n-2)$$ where $n$ is the degree of $S^*$. Thus $S^*$ is a line or $S$ is a quadratic, since $S^{**}=S$, see page 74 in [@Fischer], we deduce that $S^*$ is a quadratic. By the Plücker class formula we have the degree of $S$ is given by $n(n-1)=2$. Since every quadratic in $P^2$ is projectively equivalent to the Veronese curve and the Veronese curve is non-singular we deduce $S$ is non-singular, which is a contradiction.
\[l:cs\] Let $G\subset PSL(3,\Bbb{C})$ be a discrete group and $S$ a $G$-invariant complex irreducible algebraic curve. If $S$ has singularities then $S$ is a cubic with one node and one inflection point.
It’s well known that $S$ has a finite number of singularities. Since $\Gamma$ acts on $S$ by biholomorphisms of $\P^2$ we conclude that $\Gamma$ takes singularities of $S$ into singularities of $S$, thus $\Gamma_0=\bigcap_{p\in Sing (S)}Isot(\Gamma_0,p)$, here $Sing(S)$ denotes the singular set of $S$, is a finite index subgroup of $\Gamma$.\
Claim 1.- The genus of $S$ is $0$. Let $\widetilde{S}$ be the desingularization of $S$, then there is a birrational equivalence $f:\widetilde{S}\rightarrow {S}$. Now, let $\gamma_0 \in \Gamma$ be an element with infinite order, then there is $m\in \Bbb{N}$ such that $\gamma_1=\gamma_0^n\in \Gamma_1$. since $f$ is a birrational equivalence we can construct $\widetilde {\gamma}_{1}:\widetilde{S}-f^{-1}Sing (S) \rightarrow \widetilde{S}-f^{-1}Sing (S)$ a biholomorphism such that the following diagram commutes:
$$\label{e:diag}
\xymatrix{\widetilde{S}-f^{-1}Sing (S)\ar[r]^{\widetilde {\gamma}_{1}}\ar[d]^{f}& \widetilde{S}-f^{-1}Sing (S)\ar[d]^{f}\\
S-Sing(S)\ar[r]^{\gamma_1} & S-Sing(S)}$$
Since each arrow in the previous diagram is a biholomorphism we deduce that $\widetilde {\gamma}_{1}$ admits an holomorphic extension to $\widetilde{S}$. Observe that by Lemma \[l:fp\], $\gamma_1$ has a fixed point in $S$ and its action on $S$ has infinite order, then by diagram \[e:diag\] we deduce that $\widetilde {\gamma}_{1}$ has infinite order and at least one fixed point. Recall that a Riemann surface whose group of biholomorphisms is infinite should have genus 1 or 0, see Theorem in 3.9 in [@miranda], and in the case of Riemann surfaces of genus 1 the subgroup of biholomorphism sharing a fixed point should be finite, see Proposition 1.12 in [@miranda], which concludes the proof of the claim.\
Claim 2.- The curve $S$ has at most two singularities. Moreover, the singular set is either a node or at most simple cusp. Let $f$, $\widetilde{S}$, $\gamma_1$, $\widetilde \gamma_{1}$ as in claim 1. Thus $\widetilde \gamma_{1}$ fixes each point in $f^{-1}(Sing(S))$. Since $\widetilde S$ has genus 0, we deduce $\widetilde \gamma_{1}$ can conjugated to a Möbius transformation. Because $\widetilde \gamma_{1}$ has infinite order, we conclude $f^{-1}(Sing(S_j))$ contains at most two points. If $p$ is a singular point in $S$ we have $f^{-1}{p}$ is either one point or two points. If it is one point, $p$ should be a cusp a in the remaining case $p$ a simple node. Now it is clear that either $S$ has one simple node or a at most two cusp.\
Claim 3.- $S$ has degree 3 and the singular set is either a simple node or a single cusp. Given that $Sing(S)$ contains only cusp and simple nodes and the genus is 0, by applying Clebsch’s genus formula, see page 179 in [@Fischer], to $S$ and we get: $$0=genus(S)=(n-1)(n-2)-2(d+s)$$ where $n$ is the degree, $d$ is the number of nodes and $s$ is the number of cusp in $S$. In our case the previous equation implies following possibilities: $$\left\{
\begin{array}{ll}
d=1 & s=0 \\
d=0 & s=1\\
d=0 & s=2
\end{array}
\right.$$ Substituting this values in the Clebsch’s genus formula we get $n=3$ and also we conclude the case $d=0, s=2$ is not possible.\
On the hand, by Lemma \[l:genus\] the curve $S^*$ is singular and has degree three, thus by Plücker class formula, see page 89 in [@Fischer], we obtain: $$3=deg(S^*)=deg(S)(deg(S)-1)-2d-3s=6-2d-3s$$ which is only possible when $d=0$ and $s=1$, that is the singular set consist of a single cusp. To conclude the proof we need to use Plücker inflection point formula, recall this formula is given by, see page 89 in [@Fischer]: $s^*=3deg(S)(deg(S)-2)-6d-8s$, where $s^*$ is the number on inflection points in $S$.
\[l:cc\] Let $G\subset PSL(3,\Bbb{C})$ be a discrete group and $S$ be an irreducible singular curve invariant under $G$. Then there is a a projective transformation $\gamma\in PSL(3,\Bbb{C})$ such that $gS$ is the curve induce by the polynomial $p(x,y,z)=xy^2-z^3$ and $\gamma G\gamma^{-1}\subset \Bbb{C}^*_p$, see example \[e:cubic\].
By Lemma \[l:cs\] we have $S$ is a cubic with a cusp, then there is a projective transformation $\gamma$ such that $\gamma S$ is the Curve induced by the polynomial $p(x,y,z)=xy^2-z^3$, see cite [@sha]. Since $G$ acts by biholomorphism of $\Bbb{P}^2_\Bbb{C}$ the group $gGg^{-1}$ leaves invariant the singular point and the inflection point of $gS$. On the other hand, a straightforward computation shows $\overleftrightarrow{e_2,e_3}$ is the tangent line to $gS$ at $[e_2]$ and $\overleftrightarrow{e_1,e_3}$ is the unique tangent line to $gS$ at $[e_1]$, once again since the action of $G$ on $\Bbb{P}^2_\Bbb{C}$ is biholomorphisms we conclude that $\overleftrightarrow{e_2,e_3}$ and $\overleftrightarrow{e_1,e_3}$ are G-invariant, thus $[e_3]$ is fixed by $gGg^{-1}$. Therefore each element in $gGg^{-1}$ has a diagonal lift. Let $g\in gGg^{-1}$ and $(g_{ij})\in SL(3,\Bbb{C})$ be a lift of $g$, considering $[1:1:1]\in gS$ we conclude: $$g_{11}g_{22}^2-g_{33}^3=0$$ Using the fact $g_{11}g_{22}g_{33}=1$, we deduce $g_{22}=g_{33}^4$ and $g_{11}=g_{33}^{-5}$, which concludes the proof.
In the case of discrete groups preserving the Veronese group the Kulkarni limit set is the set of lines tangent to the Veronese group at points to the curve at points of the usual limit set of the action restricted to the curve, see [@CL]. For groups preserving the Curve the limit set is not hard to check that its Kulkarni limit set consist of two lines. In the case of affine groups the description of the limit set is not so simple so we will omit here, we refer to the interested reader to [@BCNS].
Proof of the main Theorems {#s:main}
==========================
Since $S$ is an algebraic surface, we know $S$ is a finite union of irreducible curves, say $S=\bigcup_{j=1}^n S_j$, then $G_0=\bigcap Isot(G, s_j)$ is a finite index subgroup of $G$, leaving invariant each $S_j$. If $S_j$ has singularities then by Lemmas \[l:cs\] and \[l:cc\] we have that $S_j$ is projectively equivalent to the curve induced by $xy^2-z^3$ and the group $G_0$ is virtually cyclic. If $S_j$ is non- singular by the Riemann Hurwitz theorem and Lemma \[l:fp\] we deduce that $S_j$ is either a line or a copy of the Veronese curve, this shows Theorem \[t:main1\].
In order to proof Theorem \[t:main2\]. Observe, the previous argument also shows that if $G$ is non-virtually cyclic, then each connected component of $S$ is either a line or a copy of the Veronese curve. Let us assume that $S$ contains at least to irreducible component, say $S_0$ and $S_1$, also let us assume that $S_0$ is projectively equivalent to the Veronese curve, then $S_0\cap S_1$ is finite, non-empty and $G_0$ invariant. On the other hand, since $G_0$ leaves invariant $S_0$ and $S_0$ is biholomorphicaly equivalent to the sphere Lemma \[l:fp\] ensures that $G_0$ is virtually commutative, which is a contradiction. Thus if $S$ contains a Veronese curve the curve is exactly the Veronese curve, if each reducible component is a line then by proposition 5.15 in [@BCN] the largest number of lines in general position in $S$ is 3.
Now the proofs of Theorem \[t:main3\] and Corollary \[c:main\] are trivial, so we omit it here.$\square$\
It would be interesting to understand how this result changes in the higher dimensional setting as well as in the case when we consider smooth manifold or real algebraic manifold as the invariant sets.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank to the UCIM UNAM and their people for its hospitality and kindness during the writing of this paper. We also grateful with J. F. Estrada and J. J. Zacarías for fruitful conversations.
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} |
---
abstract: 'The analysis of stellar populations has, by and large, been developed for two limiting cases: spatially-resolved stellar populations in the color-magnitude diagram, and integrated light observations of distant systems. In between these two extremes lies the semi-resolved regime, which encompasses a rich and relatively unexplored realm of observational phenomena. Here we develop the concept of pixel color magnitude diagrams (pCMDs) as a powerful technique for analyzing stellar populations in the semi-resolved regime. pCMDs show the distribution of imaging data in the plane of pixel luminosity vs. pixel color. A key feature of pCMDs is that they are sensitive to all stars, including both the evolved giants and the unevolved main sequence stars. An important variable in this regime is the mean number of stars per pixel, $\npix$. Simulated pCMDs demonstrate a strong sensitivity to the star formation history (SFH) and allow one to break degeneracies between age, metallicity and dust based on two filter data for values of $\npix$ up to at least $10^4$. We extract pCMDs from [*Hubble Space Telescope (HST)*]{} optical imaging of M31 and derive non-parametric SFHs from $10^6$ yr to $10^{10}$ yr for both the crowded disk and bulge regions (where $\npix\approx30-10^3$). From analyzing a small region of the disk we find a non-parametric SFH that is smooth and consistent with an exponential decay timescale of 4 Gyr. The bulge SFH is also smooth and consistent with a 2 Gyr decay timescale. pCMDs will likely play an important role in maximizing the science returns from next generation ground and space-based facilities.'
author:
- 'Charlie Conroy & Pieter G. van Dokkum'
title: 'Pixel Color Magnitude Diagrams for Semi-Resolved Stellar Populations: The Star Formation History of Regions within the Disk and Bulge of M31'
---
Introduction {#s:intro}
============
Analysis of the stellar populations of both star clusters and galaxies has provided the foundation for much of our understanding of the formation and evolution of galaxies. Interpreting observations of complex stellar populations relies on two key ingredients - stellar evolution models and stellar spectral libraries. The task of comparing observations to models reduces fundamentally to the task of searching for linear combinations of these two key ingredients (usually cast as the star formation history, SFH, and metallicity, Z), often with the inclusion of additional ingredients such as dust attenuation and emission.
Despite these underlying common ingredients, the analysis of stellar populations in practice is almost universally treated in one of two limiting cases. In the first case, fully resolved stellar populations are available, where one is able to robustly measure fluxes of each star in the system (down to some limit). One then models these observations in the color magnitude diagram (CMD) by fitting stellar evolution models to the data [e.g., @Dolphin02; @Tolstoy09]. In the second case, fully unresolved stellar populations are assumed, e.g., when observing distant galaxies. In this case all of the stars add together to contribute to an integrated spectrum which is then modeled with stellar population synthesis techniques [e.g., @Walcher11; @Conroy13b]. Poisson fluctuations are generally negligible and one can assume a fully populated initial mass function (IMF).
A useful way to describe these two extremes within a common framework is by considering the mean number of stars per pixel, $\npix$.[^1] In the case of fully resolved stellar populations this number is very small, e.g., $\lesssim10^{-2}$. In contrast, distant galaxies are typically in the regime where $10^6\lesssim\npix\lesssim10^{12}$. The intermediate, ‘semi-resolved’ regime where $1\lesssim\npix\lesssim10^6$, is relatively unexplored territory. Examples at the upper end of this regime includes surface brightness fluctuations [SBF; @Tonry88], fluctuation spectroscopy [@vanDokkum14b], pixel-level time variability due to the finite number of long period variables per pixel [@Conroy15a], and ‘dis-integrated’ light analysis of stellar halos [@Mould12]. At the lower end of this regime, @Beerman12 demonstrated that strong constraints on the ages of low mass star clusters could be obtained by analyzing the integrated light of the unevolved main sequence stars after isolating and removing the rare luminous evolved stars.
Even with the exquisite angular resolution and PSF stability delivered by [*Hubble Space Telescope (HST)*]{} imaging, the main body of every galaxy beyond the nearest dwarfs and out to at least the Virgo cluster are in the semi-resolved regime. For example, [*HST*]{} imaging of M31, obtained by the Panchromatic Hubble Andromeda Treasury (PHAT) Survey is crowding-limited well above the oldest main sequence turnoff point across the entire disk of M31 [@Dalcanton12]. Producing a complete photometric catalog down to the oldest main sequence turnoff point is considered the gold standard in resolved star analysis because the turnoff point is the most reliable age indicator, and hence resolving the oldest main sequence turnoff point enables the most precise and accurate SFHs for the entire age range. In spite of this crowding, strong constraints on the detailed SFH can still be obtained in such regions by modeling the upper main sequence and evolved giants that are above the crowding limit [e.g., @Lewis15; @Williams15]. The situation in the bulge of M31 is much worse - the data are so crowded in the optical and NIR that point source photometry of even the brightest giants is difficult to interpret without extensive simulations to map the completeness and bias resulting from crowding [@Dalcanton12; @Williams14]. This is not surprising - the mean number of stars per pixel in the bulge is $\sim10^2-10^3$ at [*HST*]{} resolution.
The goal of this paper is to develop a new analysis framework that enables seamless transition from the fully resolved to fully unresolved regimes. The basic idea is to forego point source photometry and instead construct pixel CMDs (pCMDs) directly from the imaging data. This approach requires generating models in the image plane, which entails creating complex stellar populations pixel-by-pixel and then convolving the model image with the PSF. In many other respects the analysis procedure is similar to modeling resolved stellar populations, in which one converts the photometric catalog into a Hess diagram in CMD space.
We expect that pCMDs will be an important tool for analyzing future data. On the near horizon are an array of new facilities, instruments, and observatories, including the [*James Webb Space Telescope (JWST)*]{}, WFIRST, Euclid, the Large Synoptic Survey Telescope (LSST), and three 30m class ground-based telescopes (ELTs). While these facilities will revolutionize many aspects of extragalactic astrophysics, it is important to underline the fact that almost none of these facilities will deliver significantly better spatial resolution than [*HST*]{}. The only possibility for real gain is with the ELTs, provided that they can deliver diffraction-limited imaging over wide fields with a highly stable PSF [see e.g., @Olsen03; @Greggio12; @Schreiber14 for the expected gains in the limit of a perfectly known and stable PSF]. Even in this case the regime of semi-resolved populations will simply be pushed to somewhat larger distances, such as the Virgo and Coma clusters. In order to maximize returns from these new facilities, we must therefore develop new tools specifically for the semi-resolved universe.
This paper is organized as follows. In Section \[s:crowd\] we provide a brief review of the crowding limit. Section \[s:models\] describes the modeling of pCMDs and in Section \[s:fit\] we describe our approach to fitting models to data in pCMD space. Section \[s:data\] contains a comparison between models and observations of M31. A discussion and summary are provided in Sections \[s:disc\] and \[s:sum\]. Where necessary we adopt the AB magnitude zeropoint system [@Oke83].
The Crowding Limit {#s:crowd}
==================
Crowding of sources will limit the flux at which one can reliably estimate photometry. This crowding, or confusion limit, has been studied extensively in both the radio and optical astronomy communities [e.g., @Scheuer57; @Condon74; @Renzini98; @Hogg01; @Olsen03]. @Olsen03 provided detailed simulations of the crowding limit for realistic stellar populations as a function of many of the controlling parameters (age, metallicity, IMF, wavelength). Careful analysis of real data requires measuring the crowding limit across the field by injecting artificial stars into the image [e.g., @Dalcanton12; @Williams14]. In this paper we are only interested in an approximate crowding limit in order to provide perspective on the delineation between the resolved and semi-resolved regime. For this purpose we adopt the rule of thumb which states that photometry becomes confusion-limited when the background flux level is equal to that which would be produced if light from the source were spread out over 30 resolution elements [@Hogg01]. In our case we assume that one resolution element is equal to 10 [*HST*]{} ACS pixels, as 10 pixels contain $\approx60$% of the total flux. We have compared this simple estimate of the crowding limit to the models presented in @Olsen03 and find that the simple rule of thumb reproduces the predictions from the detailed models generally to within 0.5 mag, which, for the purposes of guiding the eye, is sufficient.
The $\npix$ parameter depends on the distance, surface brightness (which is proportional to the stellar surface density), mass-to-light ratio, and the IMF. Figure \[fig:npix\] shows the relation between $\npix$, distance, and surface brightness. We have assumed a mass-to-light ratio of 4.0 and a mass-to-number ratio of 2.0 in order to convert luminosities into numbers, and a pixel resolution element of $0.05\arcsec$ (i.e., the pixel scale of ACS). Also shown in this figure is the approximate crowding limit for three important stellar phases: the main sequence turn-off at 13 Gyr, the red clump, and the tip of the red giant branch (RGB). Above these limits in $\npix$ the phases are crowding limited, which means that stars in this phase, and fainter, cannot be reliably separated from the background. Distances to a variety of stellar systems including Milky Way satellite galaxies and globular clusters (GCs), M31, and the Virgo and Coma clusters are included at the bottom of the figure. As mentioned in the Introduction, most of the next-generation facilities, including WFIRST and JWST, will not deliver significant improvements in spatial resolution compared to [*HST*]{}, and so we can expect the landscape depicted in Figure \[fig:npix\] to remain largely unchanged for the next $10-20$ years.
![Relation between the mean number of stars per pixel, $\npix$, distance, and surface brightness for a pixel scale of 0.05 and a resolution element of 10 pixels. Also shown are the approximate crowding limits for three key stellar phases: the oldest main sequence turnoff (MS TO), the red clump, and the tip of the RGB. Approximate locations of a variety of objects are included at the bottom of the figure. The intersection between the dashed and solid lines marks the region above which that particular stellar phase is crowding limited. For example, at the distance of M31, regions with a surface brightness of 25 mag arcsec$^{-2}$ fully resolve the RGB tip, barely resolve the red clump, and do not resolve the MS TO. []{data-label="fig:npix"}](f1.eps){width="47.00000%"}
Pixel Color Magnitude Diagrams {#s:models}
==============================
{width="90.00000%"} {width="90.00000%"} {width="90.00000%"}
Methods
-------
pCMDs are constructed by simply measuring magnitudes within a pixel and plotting those pixel magnitudes in a color magnitude diagram. Obviously this requires the pixels from different images, bands, etc. to be registered to the exact same reference frame and to have the same pixel scale. The modeling of stellar populations in pCMD space is in principle very straightforward. Our goal is to model the image plane, and from that image construct pCMDs. In order to model the image plane we must create stellar populations [*at each pixel*]{} and then convolve the model image with the PSF. At each pixel we draw stars according to weights specified by the product of the IMF, the star formation history (SFH), and the mean number of stars per pixel. The flux of each star can be reddened according to a reddening law. The stars within a pixel are summed together to produce the final spectral energy distribution of the pixel.
In practice we adopt a Salpeter IMF [@Salpeter55] with a lower-mass cutoff of $0.08{M_{\odot}}$ and isochrones from the MIST project [@Choi16]. The SFH is set by weights supplied in 7 age bins and can either be specified non-parametrically or via parametric relations (a constant model or an exponential model with a decay timescale denoted by $\tau_{\rm SF}$). Note that the SFH extends from $10^6$ yr to $10^{10}$ yr. We allow for a single reddening parameter, $E(B-V)$, that is applied to all stars equally, with an $R_V=3.1$ reddening law from @Schlafly11. Bandpass filters are adopted from the [*HST*]{} ACS camera and for brevity we refer to the [ *HST*]{} filters with the following abbreviations: $\gband=$F475W and $\iband=$F814W. We adopt a single PSF derived from ACS F814W imaging when convolving images (the F475W and F814W PSFs are very similar and so this simplification is unlikely to impact our comparison to observations in later sections).
In most respects the modeling process is similar to modeling resolved stellar populations. However, there are key differences: first, since we are modeling the image plane, we must take into account the PSF in the modeling procedure. Second, computational considerations require us to model a finite image plane, which implies that the model contains a non-trivial stochastic noise component (we return to this point in Section \[s:fit\]). Third, the key variable $\npix$ must be modeled. In reality one might need to consider a distribution function for $\npix$ rather than a single value. A simplifying feature of modeling in pCMD space is that one need not explicitly consider stellar binarity since we are not attempting to resolve individual stars. For these reasons the number of free parameters is not necessarily much larger, although the modeling of populations in pCMD space is substantially more computationally intensive.
![Sensitivity of pCMDs to the star formation history. Solar metallicity models with $\npix=10^2$ are shown for four SFHs: a 10 Gyr single-age model, $\tau$-model SFHs with $\tau_{\rm SF}=2$ and 5 Gyr, and a constant SFH. The pCMDS are displayed as Hess diagrams with a logarithmic color mapping. Isochrones at 0.01 and 10 Gyr are shown in grey to guide the eye. There is clear sensitivity to the different age components in these SFHs in the mean color of the faint pixels, the relative numbers of RGB and upper main sequence stars, and the distribution of stars along the upper main sequence.[]{data-label="fig:varysfh1"}](f3.eps){width="45.00000%"}
Information Content of pCMDs
----------------------------
In this section we illustrate the information content of pCMDs as a function of $\npix$, SFH, and metallicity.
We begin with Figure \[fig:overview\], which shows the image plane and pCMDs as a function of $\npix$ in the $\gband$ and $\iband$ filters. We show pCMDs both with and without convolution with the PSF in order to illustrate the critical role played by the PSF. Models were generated for solar metallicity 10 Gyr single-age stellar populations with zero reddening. The distribution of points in the pCMD is represented by a Hess diagram with a logarithmic color mapping. Some of the discrete features evident in the top right panel are the result of the finitely-sampled isochrone, PSF and image-plane (by $\npix\gtrsim10^2$ these numerical issues become negligibly small).
One sees clearly from this figure that the morphology in the pCMD varies smoothly from $\npix=1$ to $\npix=10^4$, indicating that the information content in these diagrams also varies smoothly. The approximate crowding limit is shown in the PSF-convolved pCMDs in order to underline the rich morphology that lies in the regime that is not included in resolved stellar population analysis. We also show in the top right panel the stars that are recoverable as resolved sources above the crowding limit. By $\npix\ge10^2$ there are no stars in the entire image above the (approximate) crowding limit.
![Same as Figure \[fig:varysfh1\], now for $\npix=10^4$.[]{data-label="fig:varysfh3"}](f4.eps){width="45.00000%"}
Figures \[fig:varysfh1\] and \[fig:varysfh3\] show the effect of different SFHs in pCMD space for two values of $\npix$. In each figure we compare a single age population at 10 Gyr to two exponentially-declining ($\tau$-model) SFHs with timescales of $\tau_{\rm SF}=2$ and 5 Gyr, and a constant SFH. All models are reddening-free and at solar metallicity. Several important trends are evident. First, there is clearly a varying balance between luminous red and blue pixels, due to the varying influence of upper main sequence and RGB stars. Second, the distribution of stars along the upper main sequence is clearly changing with the SFH. Third, the mean color of the faintest pixels and the faint limit of the faintest pixels also varies with SFH. Also notice that some of the finger-like features extending vertically toward brighter fluxes are the result of one or a few rare bright stars. In pixel space a single bright star will occupy many pixels owing to the effect of the PSF. These pixels will vary in brightness at approximately constant color (depending in detail on the level of similarity of the PSFs of the two filters). In many cases those finger-like structures would be recoverable as resolved sources above the crowding limit.
Figure \[fig:mpix\_z\] shows pCMDs as a function of metallicity. Models were generated for a 10 Gyr single-age solar metallicity population with zero reddening. Also shown is a reddening vector assuming the reddening law of @Schlafly11 for a reddening of $E(B-V)=0.3$ and $R_V=3.1$. As is well-known, an increase in dust or metallicity will result in redder colors. In unresolved data it is generally impossible to separate the effects of dust, metallicity, and age with a single color. However, with semi-resolved data there is clearly much more information. Notice that while the pCMDs become redder overall with increasing metallicity, the morphology of the data in pCMD space also changes, becoming more horizontally-aligned with increasing metallicity. We therefore expect that pCMDs will enable the disentangling of metallicity and dust effects even when only two imaging bands are available. Though not shown, we have made similar diagrams for $\iband-H_{160}$ colors and find an even stronger sensitivity to metallicity, suggesting that optical-NIR colors may be better suited for jointly estimating metallicities and reddening.
![pCMDs as a function of metallicity. Each panel shows a solar metallicity 10 Gyr single-age model displayed as a Hess diagram (color mapping is logarithmic). Also shown is a reddening vector for a standard $R_V=3.1$ reddening law and $E(B-V)=0.3$. While both metallicity and reddening result in overall redder colors, the morphology of the pCMD contains a significant amount of metallicity information. For example the pCMDs transition from being more vertically-aligned to more horizontally-aligned with increasing metallicity. It should therefore be possible to separately constrain metallicity and reddening with only two band imaging. []{data-label="fig:mpix_z"}](f5.eps){width="45.00000%"}
Fitting Data in PCMD Space {#s:fit}
==========================
In this section we describe our approach to fitting data in pCMD space and demonstrate its effectiveness by testing against mock observations.
{width="80.00000%"}
Fitting Technique
-----------------
The basic framework for fitting observations in pCMD space is as follows. One must simulate an image plane (actually two, one for each band) of some dimension for a given input SFH, reddening, and $\npix$, convolve with the appropriate PSF, add observational errors, bin into a Hess diagram in pCMD space, compute the likelihood of this model given the data, and iterate. We implement this in practice in the following manner. The image plane is simulated at a resolution of $256^2$ pixels. The SFH has a free, non-parametric form discretized into 7 age bins with the following boundaries: $(6.0,7.0,8.0,8.5,9.0,9.5,10.0,10.2)$ in units of log age (yr). We assume that the SFR is constant within each bin. The mass in each bin, $M_i$, constitutes 7 free parameters ($\npix$ is derived from the integral of the SFH). The reddening is taken to be a single parameter: log $E(B-V)$. The final free parameter is the metallicity, \[Z/H\]. We fit for a single metallicity by interpolating within the isochrone tables (which include the bolometric corrections). There are thus 9 free parameters in total. The priors on the parameters are flat in log space over the boundaries: \[Z/H\]$=(-1.1,0.5)$, log $E(B-V)=(-6.0,0.0)$, log $M_i/M_{\rm tot}=(-10.0,0.0)$. The lower limit on the metallicity was simply a practical consideration given that we are interested in this paper in metal-rich regions within M31.
The PSF is implemented by dividing the image into 16 subregions. Within each subregion the image is convolved with a PSF that has been shifted by 1/4 of a pixel. We do this because we have assumed in the model that the stellar populations reside at the center of each pixel, whereas in reality the stars can of course fall anywhere in the image plane, not only at pixel centers.
Observational (photon counting) errors are applied to the model by converting the pixel fluxes into counts by specifying a distance to the object of interest and an exposure time. At each pixel we then draw from a Poisson distribution with a mean given by the mean number of counts in each pixel.
The default MIST isochrones are very densely sampled in mass at each age [see @Dotter16 for details]. In order to ease the computational burden in the fitting we have made use of isochrones that are sampled with $5\times$ fewer equivalent evolutionary points, resulting in $\sim100-200$ points per isochrone. In testing we have found that this reduction in isochrone points has a very minor effect on the resulting pCMD, mostly affecting the very rare and luminous stars, which in any event do not receive significant weight in the fit.
Errors on both the data and the model Hess diagrams are computed directly from the number of pixels at each point (assuming Poisson statistics). Both the model and data Hess diagrams are normalized to unity.
We begin the fitting procedure by fitting only for $\npix$ and a smooth exponentially-declining SFH specified by $\tau_{\rm SF}$. We fix the reddening to zero and the metallicity to solar. This two parameter fit provides a good starting position for the main fitting routine, which employs the Markov chain Monte Carlo (MCMC) technique `emcee` [@Foreman-Mackey13]. We use 256 walkers and find that the solution is typically well-converged after $10^3$ iterations.
Fitting pCMD data presents several unique challenges not encountered when fitting resolved CMD data. Chief among them is the Poisson nature of drawing stars and populating them in a finite image. For $256^2$ image pixels and $N_{\rm iso}\approx 3000$ isochrone points this requires $\sim10^8$ Poisson draws per likelihood call. This is computationally very expensive. Moreover, the stochastic nature of the model implies that the exact same set of parameters will result in a somewhat different model, and hence a different $\chi^2$ value. Such a model would require more sophisticated search techniques than standard MCMC. This latter issue could be mitigated by simulating an image of sufficiently large number of pixels, but in our testing even $1024^2$ pixels was not sufficient to overcome the Poisson noise. In order to circumvent these two issues (the expense of drawing many Poisson numbers and the stochastic nature of the model), we decided to make the following approximation. Rather than drawing random numbers for the computation of each Poisson draw, we created a fixed set of $256^2\times N_{\rm iso}$ random numbers at the beginning of the program. This ensures that each isochrone point has a unique random number at each image pixel, and it also guarantees that the model is deterministic and is computationally faster by about a factor of 3. We then fit each dataset 10 times with a different random number seed and combine the 10 posteriors in an attempt to account for the uncertainties in the model induced by stochastic effects (note that when we generate mock data, we use a random number seed different the 10 used in the fitting).
We make two additional simplifications. We assume that isochrone points with a mass $<0.7{M_{\odot}}$ or a mean number (which is the product of the IMF and SFH weights) $\langle N\rangle>10^3$ contain no pixel-to-pixel variation and hence are not discretely drawn. For the rest, we draw from a Poisson distribution if $\langle N\rangle<100$ and a Gaussian distribution otherwise. With these simplifications each likelihood call takes $\approx1$ s.
As this is the first attempt to fit observations in pCMD space, we have taken a somewhat simplified approach to fitting the data. Areas for future improvement include the following: 1) exploring techniques for rapidly generating models that do not suffer from stochastic effects, e.g., by simulating images of much larger numbers of pixels; 2) allowing for more metallicity components, either in the form of a metallicity distribution function at a fixed age or the freedom for each age component to have its own metallicity; 3) a dust model characterized by more than a single $E(B-V)$ value, e.g., allowing for a distribution function of reddening values as in @Dalcanton15; 4) fitting not just $\npix$ but $P(\npix)$, i.e., allowing for the fact that a given physical region will in nearly all cases have a distribution of $\npix$ values; 5) a more detailed modeling of the PSF, including e.g., its spatial variation across the image. All of these improvements are straightforward to implement although most will significantly increase the computational expense of each likelihood evaluation.
Tests With Mock Data
--------------------
Figure \[fig:fitmock\] shows the results of several tests of SFH recovery with mock data. The mock data were constructed assuming a solar metallicity population with zero reddening. In each panel the resulting best-fit cumulative SFH is shown as a solid line with the grey bands marking the $1\sigma$ uncertainties. The input SFH is also shown as a dashed line. While we focus here on the SFH recovery, we are simultaneously also fitting for metallicity and dust content. The metallicities are recovered to within 0.05 dex, the reddening is constrained to be $<0.01$, and the overall normalization, $\npix$ is recovered to within 0.1 dex.
In the top panels we fit mock pCMDs generated with a constant SFH for two values of $\npix$. The upper left panel shows a model with $\npix=10^2$. The recovered SFH agrees very well with the input value, with statistical uncertainties of less than a factor of two for all age bins except the youngest where the uncertainties are a factor of three. In the upper right panel we show a model with $\npix=10^4$ and $100\times$ fewer pixels. In other words, the total mass of the two populations in the top panels are the same. One can think of the top panels as being of the same underlying system with the right panel observed at a distance $10\times$ greater than the left panel. As a consequence, the information content is lower in the right panel compared to the left and the uncertainties are therefore larger. Nonetheless, even at $\npix=10^4$ one can reliably recover the full SFH to remarkably high precision.
The bottom left panel shows the result for an old stellar population with an exponential decay time of $\tau_{\rm SF}=1$ Gyr and $\npix=10^2$. Here again the best-fit SFH agrees very well with the input model.
The final test is shown in the bottom right panel of Figure \[fig:fitmock\]. Here we take a constant SFH and increase the mass in the second youngest bin by a factor of 5. In other words, the system has an underlying constant SFH with a recent, large burst of star formation. Here again the recovery is excellent, indicating that we can recovery fairly detailed structure in the SFH of crowding-limited systems by analyzing their pCMDs.
Overall we find these tests very encouraging as they imply that we can reliably infer SFHs in a variety of astrophysically interesting regimes including old and young stellar populations and populations with bursty SFHs. It is also encouraging that the best-fit solutions and $1\sigma$ uncertainties encompass the input model in essentially all age bins for all of our tests.
Comparison to Observations {#s:data}
==========================
Having laid out the basic idea behind pCMDs and demonstrated that one can [*quantitatively*]{} recover SFHs by fitting models to data in pCMD space, we now turn to a comparison with observations. For this purpose we utilize [*HST*]{} observations of M31 obtained through the PHAT survey [@Dalcanton12]. Specifically, we use the brick-level drizzled mosaics available in the MAST archive. We adopt a distance modulus to M31 of 24.47 [@McConnachie05]. In order to simulate the effect of photon noise we adopt an exposure time in the $\gband$ and $\iband$ bands of 3620s and 3235s, respectively [@Dalcanton12].
The Bulge of M31
----------------
We first consider the old stellar population in the bulge of M31. We selected two regions from Brick 1 spanning a factor of $\approx10$ in $\npix$. The first region lies $\approx200$ pc from the center of M31 and has $\npix\sim10^3$ [note that this is well beyond the inner $\sim10$ pc where a young cluster of blue stars resides; @Lauer12]. The second region lies 1 kpc from the center and has $\npix\sim10^2$. Regions were selected over a relatively narrow range in radius from the center in order to identify groups of pixels that would have a relatively narrow distribution of $\npix$. The first and second regions include 253,000 and 79,000 pixels, respectively. No attempt was made to remove artifacts, clearly resolved bright stars, or other features. Note that the crowding limit is so severe in the bulge of M31 that it is only possible to reliably photometer the most luminous giants based on optical-NIR [*HST*]{} data [@Dalcanton12; @Williams14].
The left panels of Figure \[fig:bulge\] show a comparison of pCMDs between these two regions of the bulge of M31 and the best-fit models. The effect of the PSF and observational (photon counting) uncertainties are included in the models. Overall the best-fit models do a good job of reproducing the features in the observed pCMDs. In detail the observations appear to span a slightly wider range in colors at a fixed luminosity. This may be pointing to the fact that our models are too simplistic. For example, allowing for a metallicity distribution function would result in a broader range of colors at fixed luminosity. We defer these complications to future work.
The right panel of Figure \[fig:bulge\] shows the derived cumulative SFHs for these two regions after fitting the observed pCMDs to models. Also shown are simple SFHs to guide the eye. The derived SFHs are steeply declining with time and are broadly consistent with a $\tau-$model SFH with $\tau_{\rm SF}\approx2$ Gyr. It is also interesting to note that the recovered SFHs are not declining as fast as possible, i.e., a $\tau_{\rm SF}=1$ Gyr model appears to be ruled out by the data. This is noteworthy in light of the integrated light analysis presented in @Dong15. These authors modelled FUV-NIR photometry in the bulge of M31 and presented evidence for a young stellar population with an age of $\sim600-800$ Myr and a mass fraction of $\sim1-2$% in the inner 700 pc. We show their derived age and mass fraction for the young component in the right panel of Figure \[fig:bulge\] for their $50-55$bin, which corresponds closely to our 0.2 kpc region. Their result agrees very well with our derived SFH for this region.
The existence of hot stars associated with old stellar populations (including blue stragglers hot horizontal branch stars, and post-AGB stars) has long complicated the measurement of low levels of SF in such systems, as these hot evolved stars can, in certain circumstances, masquerade as young main sequence stars. Such stars, if present in significant numbers in the bulge of M31, could also bias our derived SFHs high at young and intermediate ages. The most conservative interpretation of our results is that they represent upper limits, as some of the bluest pixels could be due to the hot evolved stars not currently included in our models. We will explore the effect of hot evolved stars on the derived SFHs from pCMDs if future work.
The best-fit metallicities are close to solar and the reddening values are $E(B-V)\lesssim0.01$. There is some age-dust degeneracy. In light of Figure \[fig:mpix\_z\] it would be preferable to use an optical-NIR color in order to more strongly separate these two variables. Nonetheless, the modest degeneracy between age and metallicity does not have a significant impact on the recovery of the SFH.
The Star-Forming Disk of M31
----------------------------
We now consider the stellar population in the star forming disk of M31. For this comparison we selected a single $\approx100$ pc $\times$ 100 pc region from Brick 6 data. The region corresponds to one of the 9000 regions analyzed by @Lewis15, who used the resolved stars in this region to constrain the SFH over the past $500$ Myr. The resulting pCMD is shown in the upper left panel of Figure \[fig:disk\]. We compare these observed pCMD to three models: the best-fit model, and two simple models. The latter two models are solar metallicity, reddening-free, and have a constant SFH and a $\tau-$model SFH with $\tau_{\rm SF}=2$ Gyr. These simple models were generated with $\npix=10^{1.5}$, which is very close to the best-fit $\npix$ value.
The right panel of Figure \[fig:disk\] shows the derived cumulative SFH for this region. We also show several simple SFHs in order to guide the eye. The derived SFH agrees remarkably well with a smooth exponential model with $\tau_{\rm SF}=4$ Gyr. The best-fit metallicity is close to solar and the reddening is low, $E(B-V)\sim0.01$.
The region that we have analyzed was chosen so that a direct comparison could be made with the resolved star analysis in @Lewis15. The results are shown in Figure \[fig:fitcompare\]. The top panel shows an $\iband-$band image of a $40\times90$ pc subregion of the full $100\times100$ pc region used in the analysis[^2]. The middle panels show the resolved star CMD and the pixel CMD for the full $100\times100$ pc region. @Lewis15 fit the region of the resolved CMD blueward of the dotted line in order to focus on the main sequence, which is easier to model than the evolved giants. There are 1800 stars in the fitted region of the diagram. The sharp cutoff at $\iband\approx2$ corresponds to the 50% completeness limit of the resolved star catalog [@Lewis15].
The derived SFHs from these two approaches are compared in the bottom panel of Figure \[fig:fitcompare\]. The resolved star SFH from Lewis et al. is based on modeling the main sequence and hence is limited to the most recent $\approx500$ Myr; the main sequence turnoff is below the crowding limit for older stellar populations [see @Williams15 for SFHs derived to older ages in M31 based on modeling the evolved giants]. The $1\sigma$ uncertainties on the resolved star SFHs are statistical only; at the youngest age they are dominated by Poisson uncertainties in the number of stars above the crowding limit. The resolved star SFH was re-computed specifically for this comparison for the exact same age bins as the pixel CMDs (courtesy of A. Lewis), and the results were multiplied by 1.5 to convert from a @Kroupa01 IMF to our adopted Salpeter IMF.
The overall agreement is encouraging. Three of the four age bins agree very well within the errors. The third age bin differs more significantly. In addition to the overall differences in techniques, there are a variety of issues that could drive these differences. First and foremost, we adopt a different set of stellar evolution models than those in @Lewis15, who adopt the Padova models. It would be interesting to repeat both analysis with a common set of stellar evolution models, as different isochrone tables can induce non-trivial systematics [e.g., @Dolphin12; @Weisz14]. Second, @Lewis15 adopt a more sophisticated dust model than employed herein. They consider a model that allows for a distribution of dust that includes foreground extinction and differential extinction; the extinction PDF is a step function between the foreground and differential extinction values. For this particular region they find a foreground extinction of $E(B-V)=0.1$ (assuming $R_V=3.1$) and a differential extinction of $E(B-V)=0.23$. Our best-fit reddening is very low by comparison: $E(B-V)\sim0.01$. We do not yet know what is responsible for driving our reddening values so low, but it could be related to our simplified treatment of reddening (one value applied to all stars equally). In future work we will explore the more sophisticated dust model employed in @Lewis15. However, we have tested the impact of our low derived reddening values on the inferred SFH. We re-ran the fitting with a lower limit on the $E(B-V)$ prior of 0.08. The resulting SFRs agree with those shown for our default model in \[fig:fitcompare\] within $1\sigma$ even though the best-fit $E(B-V)$ is closer to 0.1. This test suggests that the derived SFHs are not overly sensitive to the best-fit reddening. Another potential systematic uncertainty is our use of a PSF that does not take into account the effects of the drizzling process that was used to create the final mosaic images.
As a further test of the pCMD-based results, we have used our best-fit parameters to predict the integrated fluxes within this region from the FUV through NIR. For the F475W, F814W, F110W, and F160W filters our predicted fluxes agree to within 3%, 8%, 13%, and 15%, respectively. We have also synthesized FUV and NUV fluxes from GALEX images and find that our predicted fluxes are about a factor of 3 too bright. If we adopt a reddening of $E(B-V)=0.13$ then the model and observed fluxes are in much better agreement, providing additional support to the conclusion that our best-fit reddening values are too low.
Finally, we note that the metallicities derived in the two techniques agree well and both return metallicities within $\approx0.05$ dex of solar.
We summarize this comparison by concluding that the broad agreement between the resolved star and pCMD-based fitting for the SFH and metallicity is encouraging, but that our model for dust is likely too simplistic. Preliminary tests suggest that this shortcoming does not unduly bias our derived SFH but further tests are needed.
{width="94.00000%"}
Discussion {#s:disc}
==========
We envision pCMDs being employed in at least two related but distinct regimes. First, in the regime where resolved photometry is not possible even for the brightest stars, pCMDs offers a unique opportunity to extract stellar population information. We presented one example here for the bulge of M31. Other examples include more distant systems (e.g., massive galaxies in the Virgo cluster). As we have remarked earlier, the pCMD concept is in some sense a generalization of the SBF technique, which has been used both as a distance indicator and as a probe of stellar populations [e.g., @Tonry01; @Cantiello05; @Blakeslee10]. By analogy with the SBF technique, one could imagine including distance as an additional free parameter when fitting pCMDs. Distance results in an overall vertical shift in the pCMD, and no other parameter induces a similar vertical shift in the pCMD. For this reason we anticipate that pCMDs will deliver strong constraints on the distance.
Second, in the regime in which bright stars are resolvable but the oldest main sequence turnoff point remains below the crowding limit, as is the case in the optical-NIR throughout the disk of M31, pCMDs offer a valuable complementary tool that can be used in combination with classic resolved star techniques. In this regime the key point is that for a given age, if the crowding limit is above the turnoff point then there is additional age information in between the crowding limit and the turnoff point that is not used in traditional resolved star analysis but is contained in pCMDs. For example, @Williams07 used the surface brightness below the magnitude limit of their resolved star data to place additional constraints on the SFH of intracluster stars in the Virgo cluster. As an example of the possible synergy between resolved star and pCMD analyses, the dust model from the resolved star analysis as a prior on the dust model in the pCMD analysis. Ultimately, one could imagine jointly fitting the resolved CMD and pCMD data within the same model, thereby providing the strongest possible constraints on physical parameters of the system.
We emphasize that pCMDs provide a much less obstructed view of the main sequence than is available from standard integrated light observations. For example, for an old stellar population viewed in integrated light, the main sequence comprises $50-60$% of the light at $0.5-0.6\mu m$, and $\approx30$% at $0.8\mu m$ [e.g., @Conroy13b]. We computed similar numbers for our pCMDs as a function of pixel luminosity, for the case of a $\tau_{\rm SF}=1$ Gyr SFH and $\npix=10^2$. At the faintest pixels ($\iband\approx2$), the main sequence contributes 80% of the flux at $0.8\mu m$. The key point is that the pCMDs are sensitive both to the evolved giants and to the sea of main sequence stars, and with pCMDs we can separately extract the age and metallicity-sensitive information from these components. Moreover, by combining pCMDs with spectroscopy at the pixel level one can hope to extract even more information [e.g., @Mould12; @vanDokkum14].
The majority of ground and space-based facilities coming online in the next two decades will not deliver substantially increased angular resolution compared to current facilities, and as a consequence they will not improve upon the crowding limit offered by [*HST*]{}. The ELTs do offer the hopes of dramatically increased angular resolution. In this case many more galaxies will be in the semi-resolved regime (e.g., the central regions of galaxies in the Virgo cluster). In light of this, we believe that one must embrace crowding-limited data and develop tools for extracting information in that regime. pCMDs offer one such example in this direction.
This is the first attempt to fit observations in pCMD space, and, as a consequence, a number of simplifying assumptions have been made. Due to computational limitations we have made several shortcuts; we are optimistic that solutions to these limitations can be found in the near future. Our underlying model is relatively simple, containing only 9 parameters (7 for the SFH, and one each for metallicity and reddening). Clearly the reddening model is too simplistic and it should be straightforward to expand this component into something that rivals the resolved star analysis in sophistication [see e.g., @Dalcanton15 for the current state of the art]. Likewise for the metallicity, in principle it is straightforward to add more components, but in this case the computational burden increases linearly with each added component. Greater care needs to be taken in handling the (spatially variable) PSF. The relative simplicity of the model also suggests that our quoted uncertainties are likely lower limits. Nonetheless, there are no obvious “show stoppers” in the modeling of pCMDs, and so we encourage their use for interpreting crowding-limited data.
Summary {#s:sum}
=======
We have presented the concept of pixel color magnitude diagrams (pCMDs) as a powerful tool for analyzing stellar populations in the crowding-limited regime. We constructed stellar population models and highlighted the main dependencies on the key parameters. We also compared the model pCMDs to [*HST*]{} imaging of M31 obtained through the PHAT survey. We now summarize our main results.
- A key parameter governing the behavior of pCDMs is the mean number of stars per pixel, $\npix$. This parameter governs how mottled or smooth the image appears, and when combined with the measured flux, can provide a strong constraint on the distance to the system. This is not surprising as the “magnitude” part of pCMDs is essentially equivalent to the information contained in SBFs.
- pCMDs show strong sensitivity to the underlying SFH with the ability to resolve bursts of star formation and place strong constraints on old stellar populations even when the data are strongly crowding-limited. Our simulations have demonstrated that one can recover at least 7 age components non-parametrically with only two filter data.
- In addition to the SFH, the metallicity and dust content can also be reliably separated with pCMDs, especially with optical-NIR colors. While both metallicity and dust result in redder colors, the detailed structure of the data in pCMD space offers strong, separable constraints on both the metallicity and the overall reddening. Again, all of this can be measured with only two filter data.
- We have developed machinery to fit model pCMDs to observations. We then constructed pCMDs from [*HST*]{} imaging of M31 in several small regions in the bulge and disk. We derived non-parametric SFHs in these regions extending from $10^6$ yr to $10^{10}$ yr. These are the strongest constraints to-date on the full SFH in the bulge of M31.
- We also compared our SFHs to those derived from fitting the resolved star CMD in one disk field and found overall good agreement for ages $10^{6.5}-10^{8.7}$ yr, where the resolved star results were available. The comparison revealed notable disagreement in the derived reddening, suggesting that there are important systematic uncertainties in our current approach to fitting pCMDs. Going forward, the combination of resolved star and pCMD data should provide the strongest possible constraints on the full SFH.
We thank Julianne Dalcanton, Ben Johnson, Alexia Lewis, Phil Rosenfield and Dan Weisz for helpful discussions throughout the development of this work and for comments on an earlier draft, Jieun Choi for providing custom-made isochrones, Alexia Lewis for providing the custom-made PHAT SFHs and resolved star photometry, and Dan Foreman-Mackey for advise on statistical matters. C.C. acknowledges support from NASA grant NNX13AI46G, NSF grant AST-1313280, and the Packard Foundation. The computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.
[^1]: For simplicity we characterize the pCMDs in terms of $\npix$ but in reality the key variable is the number of stars per resolution element.
[^2]: These are projected distances; the de-projected regions are approximately $100\times450$ pc.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Based on the CR formalism of algebraically special spacetimes by Hill, Lewandowski and Nurowski, we derive a nonlinear system of two real ODEs, of which the general solution determines a twisting type II (or more special) vacuum spacetime with two Killing vectors (commuting or not) and at most seven real parameters in addition to the cosmological constant $\Lambda$. To demonstrate a broad range of interesting spacetimes that these ODEs can capture, special solutions of various Petrov types are presented and described as they appear in this approach. They include Kerr-NUT, Kerr and Debney/Demiański’s type II, Lun’s type II and III (subclasses of Held-Robinson), MacCallum and Siklos’ type III ($\Lambda<0$), and the type N solutions ($\Lambda\neq 0$) we found in an earlier paper, along with a new class of type II solutions as a nontrivial limit of Kerr and Debney’s type II solutions. Also, we discuss a situation in which the two ODEs can be reduced to one. However, constructing the general solution still remains an open problem.'
address: 'Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131 USA'
author:
- Xuefeng Zhang and Daniel Finley
title: 'CR Structures and Twisting Vacuum Spacetimes with Two Killing Vectors and Cosmological Constant: Type II and More Special'
---
Introduction
============
All algebraically special Einstein spaces—vacuum but possibly with a nonzero cosmological constant $\Lambda$—possess a repeated principal null direction, which generates a foliation of the spacetime by a 3-parameter congruence of shearfree and null geodesics [@Goldberg62; @Adamo12]. This 3-dimensional parameter space can be identified with a 3-dimensional real manifold described by the theory of CR structures with one complex and one real coordinate. CR structures were first introduced by Poincaré and extensively studied by E. Cartan [@Cartan32I; @Cartan32II]. Good sources of background on the relationship between spacetimes and CR structures can be found, for instance, in the thesis of Nurowski [@Nurowski93]. Recently, Hill, Lewandowski and Nurowski [@Hill08] generalized earlier work of [@Lewandowski90; @Nurowski93] to provide a new formulation of twisting algebraically special spacetimes with cosmological constant. It allows a classification of algebraically special spacetimes according to Cartan’s classification of 3-dimensional CR structures.
There is indeed a method to determine the equivalence of two solutions of Einstein’s equations without having to construct explicit coordinate transformations that map one into the other, the idea of which was originated in work of Cartan and pushed forward by Brans [@Brans65], Karlhede [@Karlhede80], and Skea [@Skea00a; @Skea00b]. Cartan also created a method for determining equivalence of two CR manifolds, which is much simpler than the method mentioned above for 4-dimensional manifolds. Because of the correspondence between the two, it allows a considerably simpler approach to determine the equivalence of two twisting algebraically special spacetimes, or, more usefully in this paper, the lack of such an equivalence, thereby guaranteeing that two solutions are distinct. In addition to this important reason for using CR structures, it also provides a different formulation of Einstein’s equations, which exhibits certain invariant features that are desirable for calculations, as compared to other formulations. It is favorable to have this invariant approach to study the decomposition of Einstein’s equations into some manageable form, and to have one that prefers non-zero values of the twist, since only a very limited number of such spacetimes with non-zero twists, in Petrov types II, III, and N, are actually available for study.
In the theory of exact solutions, Einstein’s equations are usually solved under assumptions of the existence of some symmetry group [@Stephani03], i.e., Killing vectors. For instance, Kerr and Debney [@Kerr70] have determined all diverging (twisting or not) algebraically special vacuums ($\Lambda=0$) with three or more Killing vectors. The case with two Killing vectors, however, is still not solved completely. We intend to address this problem in this paper with the extension to include a nonzero cosmological constant.
Building on the work of Hill, Lewandowski, and Nurowski [@Hill08], we first present, in Section 2, the twisting type II vacuum metric formulated according to CR geometry, together with our calculated Weyl scalars. Then in Section 3, we establish the transformation from the CR formalism to the canonical frame that is widely used, e.g., in [@Stephani03]. In Section 4, we generalize the ansatz that was found for twisting type N solutions in [@Zhang12a], thereby reducing the field equations to only two coupled real ODEs for two unknown functions of a single variable. From Section 5 to 8, we show, by using the transformation in Section 3, that a large variety of previously-known, twisting solutions of types II, III, as well as D ([@Stephani03], Chapters 29 and 38), with at least two Killing vectors, correspond to special solutions of these ODEs. Moreover, we study a special case when the number of ODEs can be reduced to one, which generalizes our previous results on type N. Though the general solution is yet to be found, we believe that this extension of ODEs is quite worthwhile and should provide a start for future steps forward in the study of twisting exact solutions.
CR structures and the field equations
=====================================
A CR structure[^1] is a 3-dimensional real manifold $M$ equipped with an equivalence class of pairs of 1-forms $\lambda$ (real) and $\mu$ (complex) satisfying $$\lambda \wedge \mu \wedge \bar\mu \neq 0.$$ Another pair $(\lambda',\mu')$ is equivalent to $(\lambda,\mu)$, iff there exist functions $f\neq 0$ (real) and $h\neq 0$, $g$ (complex) on $M$ such that $$%\label{CRETrfm}
\lambda' = f\lambda, \qquad \mu' = h\mu + g\lambda, \qquad
\bar\mu' = \bar h \bar\mu + \bar g \lambda.$$ For our purpose, we further assume [@Hill08; @Cartan32I] $$\begin{aligned}
\mu = \rmd \zeta, \qquad \bar\mu = \rmd \bar\zeta, \label{dmu} \\
\rmd\lambda = \rmi\mu \wedge \bar\mu
+ (c\mu + \bar c \bar\mu )\wedge \lambda, \label{dlambda}\end{aligned}$$ where $\zeta$ and $c$ are some complex-valued functions on $M$. Taking the closure of , we obtain a reality condition on the derivatives of $c$: $$\partial\bar c = \bar\partial c,$$ The same function $c$ also appears in the commutation relations of the dual basis of vector fields: $$\label{partials}
\eqalign{
\left(\partial_0, \partial, \bar\partial \right)
\hbox{~~dual to~~} \left(\lambda, \mu, \bar\mu \right), \nonumber \\
\left[\partial, \bar\partial\, \right] = -\rmi \partial_0, \qquad
\left[\partial_0, \partial \right] = c \partial_0, \qquad
\left[\partial_0, \bar\partial\, \right] = \bar c \partial_0.}$$ There is then the following theorem telling us how to construct an algebraically special spacetime on the basis of $M$.
**Theorem 1.** [@Hill08; @Nurwoski08] The CR structure (\[dmu\]-\[partials\]) on $M$ can be lifted to a spacetime $\mathcal{M}=M\times \mathbb{R}$ equipped with the metric $$\label{metric}
\eqalign{
\mathbf{g} = 2 \left( \theta^1\theta^2 + \theta^3\theta^4 \right),
\qquad \theta^1 = \mathcal{P}\,\mu = \bar\theta^2, \\
\theta^3 = \mathcal{P}\,\lambda,
\qquad \theta^4 = \mathcal{P} \left(\rmd r + \mathcal{W}\mu
+ \bar\mathcal{W} \bar\mu + \mathcal{H}\lambda \right), }$$ where $\mathcal{P}\neq 0$, $\mathcal{H}$ (real) and $\mathcal{W}$ (complex) are arbitrary functions on $\mathcal{M}$. The spacetime admits a geodesic, shearfree and twisting null congruence along the vector field $\partial_r$ ($r\in \mathbb{R}$), of which the 3-parameter leaf spaces ($r=\mathrm{const.}$) have the same CR structure as $M$. It further satisfies the Einstein equation $\textrm{Ric}(\mathbf{g})=\Lambda \mathbf{g} $, iff the metric components can be written as $$\begin{aligned}
\mathcal{P} = \frac{p}{\cos(\case{r}{2})}, \qquad \mathcal{W} = \rmi\,a\,(\rme^{-\rmi r} + 1), \\
\mathcal{H} = \frac{n}{p^4}\, \rme^{2\rmi r} + \frac{\bar{n}}{p^4}\, \rme^{-2\rmi r}
+ q\,\rme^{\rmi r} + \bar q \,\rme^{-\rmi r} + h, \\
a = c + 2 \partial \log p, \\
q = \frac{3n+\bar{n}}{p^4} + \frac{2}{3} \Lambda p^2 + \frac{2 \partial p\, \bar\partial p
- p\, \left(\partial \bar\partial p + \bar\partial \partial p \right)}{2 p^2} - \frac{\rmi}{2}\,
\partial_0 \log p - \bar\partial c, \\
h = 3\frac{n+\bar{n}}{p^4} + 2\Lambda p^2 + \frac{2 \partial p\, \bar\partial p
- p\, \left(\partial \bar\partial p + \bar\partial \partial p \right)}{p^2} - 2 \bar\partial c,\end{aligned}$$ where $c$, $n$ (complex) and $p$ (real), all functions on $M$ (independent of $r$), satisfy the following set of equations: $$\begin{aligned}
\partial \bar c = \bar\partial c, \label{ccb} \\
\left[ \partial \bar\partial + \bar\partial \partial + \bar c
\partial + c \bar\partial + \frac{1}{2} c \bar c + \frac{3}{4}
\left(\partial \bar c + \bar\partial c \right) \right] p =
\frac{n+\bar{n}}{p^3} + \frac{2}{3} \Lambda p^3, \label{NurowskiEqn} \\
\partial n + 3c\,n = 0, \label{pncn} \\
R_{33} = 0. \label{R330}\end{aligned}$$ Here the Ricci tensor component $R_{33}$ as well as the Weyl scalars $\Psi_2$, $\Psi_3$ and $\Psi_4$ are given by $$\begin{aligned}
R_{33} &= \Bigg\{ \frac{8}{p^4}\, (\partial + 2 c)
\! \left[ p^2\! \left(\partial\bar{I} - 2\Lambda (2\bar\partial \log p + \bar c) p^2 \right) \right]
\nonumber \\
& + \frac{16}{p}\, \Lambda \Big[ \Big(\partial \bar\partial +
\bar\partial \partial + \bar c \partial
+ c \bar\partial + \frac{1}{2} c \bar c + \frac{3}{4}
(\partial \bar c + \bar\partial c ) \Big) p -
\frac{n+\bar{n}}{p^3} - \frac{2}{3} \Lambda p^3 \Big]
\nonumber \\
& + \frac{16\rmi}{p^3}\, \partial_0\! \Big(\frac{n}{p^3}\Big)
\Bigg\} \cos^4\! \left(\frac{r}{2} \right), \label{R33}\end{aligned}$$ $$\label{Psi2}
\Psi_2 = \frac{n}{2p^6}\, (\rme^{\rmi r} + 1)^3,$$ $$\begin{aligned}
\Psi_3 =& \Bigg\{ \frac{2\rmi}{p^2} \left[\partial\bar{I}
- 2\Lambda (2\bar\partial \log p + \bar c) p^2 \right] \label{Psi3} \\
% - 12\rme^{\rmi r} \sin(r) % Two different ways to write down the factors
& + 6\rmi \left(2 \bar\partial \log p + \bar{c} \right)
\frac{n}{p^6}\, (\rme^{2\rmi r} - 1)
% - 8\rmi \rme^{\rmi r/2} \cos\!\left(\frac{r}{2}\right)
- 4\rmi \bar\partial \Big(\frac{n}{p^6}\Big) (\rme^{\rmi r} + 1)
\Bigg\} \rme^{\rmi r/2} \cos^3\! \left(\frac{r}{2}\right),
\nonumber\end{aligned}$$ $$\fl \eqalign{
\Psi_4 = 2 \rme^{-\rmi r/2} \cos^3 \left(\frac{r}{2}\right)
\Bigg\{\! -(2\bar\partial \log p + \bar c) \left[\partial \bar{I}
- 2\Lambda (2\bar\partial \log p + \bar c) p^2\right]
\frac{\rme^{2\rmi r} - 1}{p^2}
\\
+ (\bar\partial + 2\bar{c}) \left[\partial\bar{I} - 2\Lambda (2\bar\partial \log p + \bar c) p^2\right]
\frac{\rme^{\rmi r} + 1}{p^2} + \frac{\rmi}{p^2} \partial_0 \bar{I}
\\
+ \frac{2}{3} \Lambda \left[(\bar\partial + \bar c)
(2\bar\partial \log p + \bar{c}) + 2(2\bar\partial \log p + \bar{c})^2 \right]
-3(2\bar\partial \log p + \bar{c})^2 \frac{n}{p^6}\, \rme^{4\rmi r}
\\
+ \left[ (\bar\partial - 2\bar\partial \log p)(2\bar\partial \log p + \bar{c})\, \frac{n}{p^6}
+ 3(2\bar\partial \log p + \bar{c}) \bar\partial \Big(\frac{n}{p^6}\Big) \right] \rme^{3\rmi r}
\\
+ \left[ 3\left( 2\bar\partial^2 \log p - 16 (\bar\partial \log p)^2
+ 2\bar{c} \bar\partial \log p + \bar{c}^2 \right) \frac{n}{p^6}
+ \frac{(16\bar\partial \log p + \bar{c}) \bar\partial n - \bar\partial^2 n}{p^6} \right] \rme^{2\rmi r}
\\
+ \left[ 3\left( 2 \bar\partial^2 \log p - 8(\bar\partial \log p)^2 + 8\bar{c} \bar\partial \log p
- \bar\partial \bar{c} - \bar{c}^2\right) \frac{n}{p^6}
+ \frac{7(2\bar\partial \log p - \bar{c}) \bar\partial n - 2\bar\partial^2 n}{p^6} \right] \rme^{\rmi r}
\\
+ 2\left( \bar\partial^2 \log p - 2(\bar\partial \log p)^2 + 5\bar{c} \bar\partial \log p
- \bar\partial \bar{c} - 3\bar{c}^2\right) \frac{n}{p^6}
+ \frac{(4\bar\partial \log p - 5\bar{c}) \bar\partial n - \bar\partial^2 n}{p^6}
\Bigg\}, }$$ with the function $I$ defined by $$I = \partial \left(\partial \log p + c \right) + \left(\partial \log p + c \right)^2.$$
Following the same procedure as [@Hill08], which uses Cartan’s structure equations to calculate the curvature tensor, we present our calculated $R_{33}$, $\Psi_3$ and $\Psi_4$ above with nonzero $\Lambda$ and $n$, as a complement to [@Hill08]. Moreover, to facilitate future calculations, we have arranged the expression of $R_{33}$ so that its second square bracket can be immediately removed by the field equation , whereas the terms $\partial\bar{I} - 2\Lambda (2\bar\partial \log p + \bar c) p^2$ are made prominent as they also appear in $\Psi_3$ and $\Psi_4$.
To solve the field equations (\[ccb\]-\[R330\]) in practice, one needs to introduce a real coordinate system $(x, y ,u)$ on $M$ such that $$\!\!\! \begin{array}{ll}
\zeta = x + \rmi y, \qquad & \partial_\zeta = \frac{1}{2} \left(\partial_x - \rmi \partial_y \right), \\
\partial = \partial_\zeta - L \partial_u,
\qquad & \partial_0 = \rmi(\bar\partial L - \partial\bar{L}) \partial_u,
\end{array} \qquad
\lambda = \frac{\rmd u + L \rmd\zeta + \bar{L} \rmd\bar\zeta}{\rmi(\bar\partial L - \partial\bar{L})}, \label{lambda}$$ with $L=L(\zeta, \bar\zeta, u)$ a complex-valued function [@Hanges88] satisfying $$\bar\partial L - \partial\bar{L} \neq 0,$$ which is needed for a nonzero twist (cf. ). In addition, $L$ relates to the function $c$ by $$c = -\partial \ln (\bar\partial L - \partial\bar{L}) - \partial_u L, \label{cL}$$ as imposed by the commutation relations . Hence generally, the system (\[ccb\]-\[R330\]) are in fact PDEs for the unknown functions $L$, $n$ and $p$ of the coordinate variables $(\zeta,\bar\zeta,u)$.
For other possible coordinate choices, the metric (\[dmu\]-\[cL\]) admits the following coordinate freedom ([@Zhang12b], see Section 2.6): $$\label{CoTfm}
r'=r, \qquad \zeta' = f(\zeta),
\qquad u'=F(\zeta,\bar\zeta,u), \qquad \partial_u F \neq 0,$$ with $f(\zeta)$ holomorphic and $F$ a real-valued function, which generates the transformation laws $$\begin{aligned}
\mu' = f'\mu, \qquad \lambda' = f'\bar{f}' \lambda,
\qquad f' \equiv \rmd f/\rmd \zeta, \\
\partial' = \frac{1}{f'} \partial,
\qquad \partial_0' = \frac{1}{f'\bar{f}'} \partial_0,
\qquad \left[\partial', \bar\partial' \, \right] = -\rmi \partial_0',
\qquad \left[\partial_0', \partial' \right] = c' \partial_0', \\
c' = \frac{1}{f'} c + \frac{f''}{(f')^2},
\qquad p' = \frac{1}{|f'|} p,
\qquad n' = \frac{1}{(f'\bar{f}')^3} n, \label{cpnpcpn} \\
L' = -\frac{1}{f'} (\partial_\zeta F - L \partial_u F) = -\frac{1}{f'} \partial F,
\quad \bar\partial'L' - \partial'\bar{L}'
= \frac{\partial_u F}{f'\bar{f}'} (\bar\partial L - \partial \bar{L}).\end{aligned}$$ Note that the function $F$ does not appear above on the level of $c$, $p$ and $n$, as well as $\partial$ and $\partial_0$, the fact of which indicates an invariant feature of the CR formalism. As expected, the field equations for the new $p'$, $c'$ and $n'$ take on the same form of (\[ccb\]-\[R330\]) with $(\partial_0,\partial,\bar\partial)$ simply replaced by $(\partial_0',\partial',\bar\partial')$. These transformation properties will be used to simplify our metrics.
Transformations to the canonical frame {#sec:canonical}
======================================
Given the algebraically special twisting metric form (\[dmu\]-\[cL\]) formulated according to CR geometry, it is important to know how it is related to other pre-existing formalisms that have been extensively studied in the past. Here we quote from [@Stephani03] (p. 439–441) a most commonly used one by Kerr, Debney *et al.* [@Kerr63; @Debney69; @Robinson69]. For simplicity, we only consider $\Lambda=0$ and follow closely the notation of [@Stephani03] with sub- or superscript $s$ added to avoid confusion.
**Theorem 2.** A spacetime admits a geodesic, shearfree and twisting null congruence along the vector field $\partial_{r_s}$ and satisfies the Einstein equation $\textrm{Ric}(\mathbf{g})=0$, iff the metric can be written as $$\label{metricStph}
\eqalign{
\mathbf{g} = 2 (\mathbf{\omega}^1 \omega^2 - \omega^3 \omega^4),
\qquad \omega^1 = -\frac{\rmd\zeta}{P_s \bar\rho_s} = \bar\omega^2, \\
\omega^3 = \rmd u + L\rmd \zeta + \bar{L} \rmd \bar\zeta,
\qquad \omega^4 = \rmd r_s + W_s \rmd \zeta + \bar{W}_s \rmd \bar\zeta + H_s \omega^3,}$$ with metric components $$\label{compStph}
\eqalign{
\rho_s^{-1} = -(r_s + \rmi \Sigma_s),
\qquad \frac{2\rmi \Sigma_s}{P_s^2} = \bar\partial L - \partial \bar{L} \neq 0, \\
W_s = \rho_s^{-1} \partial_u L + \rmi \partial\Sigma_s, \qquad \partial = \partial_\zeta - L\partial_u, \\
H_s = \frac{1}{2}K_s - r_s\partial_u \log P_s - \frac{m_s r_s + M_s \Sigma}{r_s^2 + \Sigma^2}, \\
K_s = 2P_s^2 \mathrm{Re} \left[\partial \left(\bar\partial \log P_s - \partial_u \bar{L} \right)\right]}$$ such that the functions $m_s$, $M_s$, $P_s$ (real) and $L$ (complex), all only dependent on the coordinates $(\zeta, \bar\zeta,
u)$, satisfy a system of PDEs: $$\begin{aligned}
P_s^{-3} M_s = \mathrm{Im}\ \partial\partial \bar\partial \bar\partial V_s,
\qquad P_s = \partial_u V_s, \label{VsPs}\\
\partial (m_s + \rmi M_s) = 3 (m_s + \rmi M_s) \partial_u L, \\
\partial_u \left[ P_s^{-3}(m_s + \rmi M_s)\right]
= P_s [\partial + 2(\partial \log P_s - \partial_u L)] \partial I_s,
\label{Rs330}\end{aligned}$$ where the function $I_s$ is defined by $$\label{Is}
I_s = \bar\partial \left(\bar\partial \log P_s - \partial_u \bar{L} \right)
+ \left(\bar\partial \log P_s - \partial_u \bar{L} \right)^2 = P_s^{-1} \partial_u \bar\partial \bar\partial V_s.$$ Additionally, the Weyl scalar $\Psi^s_2$ is given by $$\Psi^s_2 = (m_s + \rmi M_s) \rho_s^3.$$
In this metric form, the coordinates $(\zeta, \bar\zeta, u)$ and the function $L$ have been chosen identically with those introduced in ; hence, each is not given a sub- or superscript $s$. Taking $\Lambda=0$ and by a tedious but straightforward calculation, one can show that the metrics (\[metricStph\]-\[Is\]) and (\[dmu\]-\[cL\]) are equivalent to each other by the transformation [@Zhang12b] $$\begin{aligned}
& P_s = \frac{2 p}{\rmi(\bar\partial L - \partial \bar{L})}, \label{Psp} \\
& r_s = -\frac{2 p^2}{\rmi(\bar\partial L - \partial \bar{L})} \tan\left(\frac{r}{2}\right),
\qquad |r|<\pi, \label{rsr} \\
& m_s = \frac{16(n - \bar{n})}{(\bar\partial L - \partial \bar{L})^3},
\qquad M_s = \frac{16(n + \bar{n})}{\rmi(\bar\partial L - \partial \bar{L})^3}, \label{mMsn}\end{aligned}$$ with the inverse $$\begin{aligned}
p = \frac{\rmi}{2} (\bar\partial L - \partial \bar{L}) P_s, \label{pPs} \\
r = 2\arctan\left( -\frac{2}{\rmi (\bar\partial L - \partial \bar{L}) P_s^2}\, r_s \right),
\label{rrs} \\
n = \frac{1}{32}(m_s + \rmi M_s)(\bar\partial L - \partial \bar{L})^3. \label{nmMs}\end{aligned}$$ In particular, the field equation can be transformed into with $\Lambda=0$, despite their drastically different appearances. Also, one gets $I_s=\bar{I}$ when substituting into the definition . For more details about these transformations, one may see Section 2.6 of [@Zhang12b]. The relation and together with will be used later to translate known solutions of the canonical field equations (\[VsPs\]-\[Rs330\]) to solutions of (\[ccb\]-\[R330\]).
Reductions to ODEs
==================
Now we go back to the metric (\[dmu\]-\[cL\]). Following the same idea as [@Zhang12a] for solving the field equations, we assume that the unknowns $p$, $c$ and $n$ have no $u$-dependence, i.e., $\partial_0 p = \partial_0 c =
\partial_0 n = 0$. This assumption avoids the involvement of the function $L$ inside the operator $\partial$ since now we have, e.g., $\partial p = \partial_\zeta p$. Therefore the system (\[ccb\]-\[R330\]) becomes effectively PDEs for the unknowns $c$, $p$ and $n$ instead of $L$, $p$ and $n$. Once $c=c(\zeta,\bar\zeta)$ is solved, one may further determine a function $L$ without $u$-dependence from (cf. ). Altogether, this means that the resulting spacetime shall possess a Killing vector in the $u$-direction, which is, in fact, an assumption widely used in many research articles on algebraically special solutions (see, e.g., [@Stephani03] Chapter 29).
We apply the assumption and rewrite the system (\[ccb\]-\[R330\]) (likewise for $\Psi_3$ and $\Psi_4$) as $$\begin{aligned}
\partial_\zeta \bar{c} = \partial_{\bar\zeta} c, \label{ccb1} \\
2 \partial_\zeta \partial_{\bar\zeta}p + \bar{c} \partial_\zeta p
+ c \partial_{\bar\zeta} p + \frac{1}{2} c \bar{c} p + \frac{3}{2} (\partial_\zeta \bar{c}) p
= \frac{n+\bar{n}}{p^3} + \frac{2}{3}\Lambda p^3, \label{NurowskiEqn1} \\
\partial_\zeta n + 3c\,n = 0, \label{pncn1} \\
(\partial_\zeta + 2 c)\! \left[ p^2 \partial_\zeta \bar{I}
- 2\Lambda (2\partial_{\bar\zeta} \log p + \bar{c}) p^4 \right] = 0, \label{R3301}\end{aligned}$$ where in the last equation ($R_{33}=0$) we have used to simplify the expression of $R_{33}$, and the function $I$ is given by $$I = \partial_\zeta \left(\partial_\zeta \log p + c \right) + \left(\partial_\zeta \log p + c \right)^2.$$ This is the set of PDEs we aim to solve for the unknowns $p(\zeta,\bar\zeta)$, $c(\zeta,\bar\zeta)$ and $n(\zeta,\bar\zeta)$.
Generalizing the ansatz [@Zhang12a; @Zhang12b] we found from the classical symmetries [@Krasilshchik99] of the type N case of (\[ccb1\]-\[R3301\]) with $n=0$, we assume the following forms for the unknowns: $$\label{pcnz}
\fl p(\zeta, \bar\zeta) = \frac{F_1(z)}{\sqrt{A \bar{A}}}, \quad
c(\zeta, \bar\zeta) = \frac{\partial_\zeta A + \rmi F_2(z) + C_1}{A}, \quad
n(\zeta, \bar\zeta) = \frac{F_3(z) + \rmi F_4(z)}{(A\bar{A})^3}$$ with a new real variable $$z = -\rmi \left( \int \frac{1}{A} \rmd \zeta -
\int \frac{1}{\bar A} \rmd \bar{\zeta} \right)
= \mathrm{Im} \int \frac{2}{A} \rmd \zeta.$$ Here the function $A=A(\zeta)$ is an arbitrary function of $\zeta$ that is sufficiently smooth, and the constant $C_1$ and the undetermined functions $F_{1-4}(z)$ are all real-valued. The constraint equation has been taken into account in the form of $c(\zeta,\bar\zeta)$ so that it is satisfied.
Inserting the ansatz into (\[NurowskiEqn1\]-\[R3301\]), we obtain a remarkable reduction to a system of four compatible real ODEs for $F_{1-4}(z)$ only, with all other dependence on $A, \bar{A}\neq 0$ factored out: $$\begin{aligned}
0 = -F_1'' + F_2 F_1' + \frac{1}{3}\Lambda F_1^3
+ \frac{1}{4} (3 F_2' - F_2^2 - C_1^2) F_1 + \frac{F_3}{F_1^3}, \label{NurowskiEqnz} \\
0 = F_3' - 3(F_2 F_3 + C_1 F_4), \label{F3ODE} \\
0 = F_4' + 3(C_1 F_3 - F_2 F_4), \label{F4ODE} \\
0 = (H' - 2F_2 H)' - 2 F_2(H' - 2F_2 H) + 4 C_1^2 H, \label{R330z}\end{aligned}$$ where the function $H(z)$ is defined by $$H = F_1'' F_1 - (F_1')^2 - \Lambda F_1^4 - F_2' F_1^2.$$ The two inner equations are derived from the real and imaginary parts of , respectively. Note that they are linear in $F_3$ and $F_4$. Thus if $F_2$ is given, one can solve them by $$\label{F34F2}
\eqalign{
F_3 = \exp\left(3\! \int\! F_2\rmd z \right)
[B_1 \sin(3C_1 z) - B_2 \cos(3C_1 z)], \\
F_4 = \exp\left(3\! \int\! F_2\rmd z \right)
[B_2 \sin(3C_1 z) + B_1 \cos(3C_1 z)], }$$ where $B_{1,2}$ are real constants. With $F_3$ expressed in terms of $F_2$, we are left with only two nonlinear equations for $F_1$ and $F_2$: $$\label{2ODEs}
\eqalign{
0 =& -F_1'' + F_2 F_1' + \frac{1}{3}\Lambda F_1^3 + \frac{1}{4} (3 F_2' - F_2^2 - C_1^2) F_1 \\
&+ \frac{\exp\left(3\! \int\! F_2\rmd z \right)}{F_1^3}\, [B_1 \sin(3C_1 z) - B_2 \cos(3C_1 z)], \\
0 =& (H' - 2F_2 H)' - 2 F_2(H' - 2F_2 H) + 4 C_1^2 H, }$$ which can be easily converted to a set of ODEs if one introduces, e.g., $F_2=K'(z)$ to remove the integral in the first equation. The system with , or alternatively (\[NurowskiEqnz\]-\[R330z\]), constitutes the main result of this paper.
Before moving on to solve (\[NurowskiEqnz\]-\[R330z\]) in the next few sections, we should make a few general remarks concerning the metric (\[dmu\]-\[cL\]) equipped with the ansatz . First, despite the appearance of a free function $A(\zeta)\neq 0$ in the ansatz, its different choices do not generate new metrics. In fact, using the coordinate change $\zeta'=\int\! \frac{2}{A(\zeta)} \rmd\zeta$ permitted by and, accordingly, the transformation law with $f'=2/A(\zeta)$, one can always replace a function $A$ by a constant $A=2$. This is also consistent with the fact that the local CR structure determined by the function $c$ in is independent of the choice of $A(\zeta)$ [@Zhang12a]. Hence for simplicity, we can just set $A(\zeta)=\bar{A}(\bar\zeta)=2$ without loss of generality, and hence $z=\mathrm{Im}\zeta=y$. With this choice of $A$, the function $L$ and the 1-form $\lambda$ can be determined from and as $$\begin{aligned}
L = -\rme^{-C_1 x} \int \exp\left(\int F_2 \rmd z \right) \rmd z, \label{Lxz} \\
\lambda = \frac{\rme^{C_1 x} \rmd u
- 2 \left[\int \exp\left(\int F_2 \rmd z \right) \rmd z \right] \rmd x}
{\exp\left(\int F_2 \rmd z \right)}. \label{lambdaz}\end{aligned}$$ Once again, though there exist other $L$’s satisfying , one can always use the remaining coordinate freedom $u'=F(\zeta,\bar\zeta,u)$ (cf. ) to convert them to the $u$-independent real expression [@Zhang12a]. Given such $L$ and $\lambda$, the class of metrics determined by (\[pcnz\]-\[R330z\]) admits at least two Killing vectors $$\label{KV}
X_1 = \partial_u, \qquad X_2 = \partial_x - C_1 u\, \partial_u,$$ with the commutation relation $$[X_1, X_2] = -C_1 X_1.$$ These vectors, verifiable by direct calculation, are both inherited from the symmetries of the underlying CR structures [@Nurowski88; @Zhang12b].
Type D solutions: Kerr-NUT
==========================
All type D vacuum solutions, twisting or not, are known [@Kinnersley69; @Plebanski76], which include perhaps the most famous algebraically special solutions such as Kerr’s rotating black-hole solution. Hence it is worthwhile to consider whether our equations (\[NurowskiEqnz\]-\[R330z\]) can capture some of these physically important solutions. But first we should comment that the CR formalism (\[dmu\]-\[cL\]) used here is constructed on just one single shearfree null congruence aligned with a multiple principal null direction , which is unique in type II, III and N spacetimes. However, a type D spacetime possesses two such congruences, each along one of the two doubly degenerate principal null directions, and consequently one cannot treat them at the same time in the CR formalism. This suggests that the CR formalism may not provide the most convenient approach for finding type D solutions, as compared to other approaches that are specially designed to make use of both congruences.
A spacetime is of type D iff it satisfies the conditions $$\label{CTD}
3\Psi_3\Psi_4 - 2\Psi_2^2 = 0, \qquad \Psi_2\neq 0.$$ This equality with the ansatz applied gives rise to a number of lengthy ODEs (as one may sense by looking at the expression of $\Psi_4$) from the coefficients of various powers of $\rme^{\rmi r}$ required to vanish. This fairly complicated situation (except when $\Psi_3=\Psi_4=0$, see \[App:TD\]) needs a specialized paper to elaborate; hence it is not further discussed here (also because there is no new type D solution to be found). Instead, by applying the results of Section \[sec:canonical\], we will simply show that the Kerr-NUT solution can be retrieved as a special solution of (\[NurowskiEqnz\]-\[R330z\]) through the ansatz .
In the canonical frame (\[metricStph\]-\[Is\]), the Kerr-NUT solution [@Stephani03] (p. 453) is given by $$\eqalign{
P_s = 1 + \frac{\zeta \bar\zeta}{2},
\qquad L = -\frac{\rmi}{\zeta P_s^2} [ 2M + (M + a)\zeta \bar\zeta ], \\
m_s = m, \qquad M_s = M, \qquad \Lambda = 0,}$$ where $m$, $M$ and $a$, each a real constant, are called the mass, the NUT parameter and the Kerr parameter, respectively. Then inserting them into , and , we obtain the following solution to (\[ccb1\]-\[R3301\]): $$\begin{aligned}
p = -\frac{4(M-a)+2(M+a)\zeta\bar\zeta}{(2+\zeta\bar\zeta)^2}, \\
c = \frac{2\bar\zeta [2(M-2a)+(M+a)\zeta\bar\zeta]}
{[2(M-a)+(M+a)\zeta\bar\zeta](2+\zeta\bar\zeta)}, \\
n = -\frac{16\rmi [2(M-a)+(M+a)\zeta\bar\zeta]^3 (m+\rmi M)}{(2+\zeta\bar\zeta)^9}.\end{aligned}$$ which, as expected, satisfies the condition for type D with nonzero $\Psi_3$ and $\Psi_4$. Without a dependence on $u$, these expressions can be cast into the form of our ansatz by $$\begin{aligned}
A(\zeta) = -\rmi \zeta, \qquad z = \log(\zeta \bar\zeta), \qquad C_1=0,\end{aligned}$$ such that $$\label{KerrNUT}
\eqalign{
\fl F_1 = -\frac{2[2(M-a)+(M+a)\rme^z]\rme^{z/2}}{(2+\rme^z)^2},
\qquad F_2 = \frac{4(M-a)+8a\rme^z-(M+a)\rme^{2z}}
{[2(M-a)+(M+a)\rme^z](2+\rme^z)}, \\
\fl F_3 = \frac{16M[2(M-a)+(M+a)\rme^z]^3\rme^{3z}}{(2+\rme^z)^9},
\qquad F_4 =
-\frac{16m[2(M-a)+(M+a)\rme^z]^3\rme^{3z}}{(2+\rme^z)^9}.}$$ One can verify that they are indeed a solution to (\[NurowskiEqnz\]-\[R330z\]) with $\Lambda=0$. Note that the function $A(\zeta)$ above will still be serving as a free function, as long as $F_{1-4}$ are obtained. The Kerr-NUT solution is contained in the Demiański solution as a special case (cf. ). For other examples of type D solutions, see \[App:TDCS\] and with $B_2=0$.
Type N solutions with nonzero cosmological constant
===================================================
The type N solutions require $$\Psi_2=\Psi_3=0, \qquad \Psi_4\neq 0.$$ These conditions lead to the following special case of (\[NurowskiEqnz\]-\[R330z\]) with $F_3 = F_4 = 0$: $$\label{TN}
\eqalign{
0 = -F_1^{\prime \prime} + F_2 F_1' + \case{1}{3}\Lambda F_1^3
+ \case{1}{4} (3 F_2' - F_2^2 - C_1^2) F_1, \nonumber \\
\Psi_3 \propto H' - 2F_2 H - 2\rmi C_1 H = 0,}$$ plus one inequality $$\begin{aligned}
\Psi_4 &=& -\frac{4\Lambda}{3\bar{A}^2 F_1^2}\,
\Big[ 2 F_1 F_1'' + 6 (F_1')^2 - 10 (F_2 + \rmi C_1) F_1 F_1' \nonumber \\
& & - (F_2' - 3 F_2^2 - 6 \rmi C_1 F_2 + 3 C_1^2) F_1^2 \Big]\,
\rme^{-\rmi r/2} \cos^3\! \left(\frac{r}{2}\right) \neq 0,\end{aligned}$$ which imposes $\Lambda\neq 0$ for type N. To better see that the system is included in (\[NurowskiEqnz\]-\[R330z\]), one can rewrite equation as $$0 = (H' - 2F_2 H - 2\rmi C_1 H)'
- 2(F_2' - \rmi C_1)(H' - 2F_2 H - 2\rmi C_1 H).$$ In fact, the field equation $R_{33}=0$ can always be removed by $\Psi_2=\Psi_3=0$ for general type N vacuums (cf. (\[R33\]-\[Psi3\])). Notice that the second equation of is complex; we have two cases for solutions.
*Case 1:* $H=0$, $\Lambda\neq 0$. This simpler case has been investigated in [@Zhang12a] (see [@Zhang12b] for more details). The equations for this case read $$\label{TN1}
\eqalign{
0 = -F_1^{\prime \prime} + F_2 F_1' + \case{1}{3}\Lambda F_1^3
+ \case{1}{4} (3 F_2' - F_2^2 - C_1^2) F_1, \\
0 = H = -F_1^{\prime \prime} F_1 + (F_1')^2 + \Lambda F_1^4 + F_2' F_1^2.}$$ By introducing a real function $J=J(z)$ and $$\label{F12J}
F_1 = \pm \sqrt{J'}, \qquad F_2 = \frac{J^{\prime\prime}}{2 J'} - \Lambda J, \qquad J'>0,$$ we can reduce the first equation of to $$\label{JEQ}
J''' = \frac{(J^{\prime\prime})^2}{2J'} - 2 \Lambda J J^{\prime\prime} -
\frac{10}{3} \Lambda (J')^2 - 2 (\Lambda^2 J^2 + C_1^2) J',$$ while the second equation is automatically satisfied. Since this ODE has no explicit dependence on the variable $z$, we can immediately lower its order by the transformation $J'=P(J)>0$ such that $$\label{PEQ}
P'' = - \frac{(P' + 2 \Lambda J)^2}{2P} - \frac{2 C_1^2}{P} - \frac{10}{3} \Lambda,
\qquad \Lambda\neq 0,$$ which, in the case of $C_1=0$, can be further reduced to an Abel ODE [@Polyanin95] of the first kind $$\label{Abel}
f' = \frac{4}{t} \left(t+\frac{3}{2}\right)\left(t+\frac{1}{3}\right) f^3
+ \frac{5}{t} \left(t+\frac{2}{5}\right) f^2 + \frac{1}{2t}\ f,$$ by $J = \exp(\int\! f(t) \rmd t)/\Lambda$, $P(J) = t \exp(2 \int\!
f(t) \rmd t)/\Lambda$. Unfortunately, this Abel ODE has not been identified as a known solvable type.
Various aspects of the equation were examined in [@Zhang12a], including the weak Painlevé property [@Conte08] and constructions of various special and series solutions. All degenerate solutions of type O ($\Psi_4=0$, conformally flat) were found. The only known type N solution with $
\Lambda\neq 0$ in closed forms was first discovered by Leroy [@Leroy70] and presented in the CR formalism by Nurowski [@Nurwoski08]. It corresponds to $$\label{Leroy}
\eqalign{
P(J) = -\frac{1}{3}\Lambda J^2 - \frac{3 C_1^2}{4\Lambda} >0, \\
F_1 = \pm \frac{\sqrt{3} C_1}{2 s \sin(\frac{1}{2} C_1 (z + C_0))},
\qquad F_2 = -\frac{2 C_1}{\tan(\frac{1}{2} C_1 (z + C_0))},}$$ with $\Lambda=-s^2<0$ and $C_0$ a real constant (removable by a translation $z+C_0\rightarrow z$). Nonetheless, the equation does also admit type N solutions with $\Lambda>0$ and two additional parameters besides $\Lambda$ and $C_1$.
*Case 2:* $H\neq 0$, $C_1=0$, $\Lambda\neq 0$. The system becomes $$\label{TN2}
\eqalign{
0 = -F_1^{\prime \prime} + F_2 F_1' + \case{1}{3}\Lambda F_1^3
+ \case{1}{4} (3 F_2' - F_2^2) F_1, \\
0 = H' - 2F_2 H.}$$ Very little is known about the solutions of this system except for one of type O [@Zhang12b] given by $$\label{hyqN}
F_1 = \pm \frac{\sqrt{6}}{2s(z+C_0)}, \qquad F_2 = -\frac{2}{z+C_0},
\qquad H = \frac{3}{4 s^2 (z+C_0)^4},$$ with $\Lambda = -s^2<0$ and $C_0$ a real constant. Particularly, this solution has the hyperquadric CR structure (the most symmetrical one) [@Jacobowitz90].
Type III solutions
==================
Similar to the case of type N, the equations for type III ($\Psi_2=0$, $\Psi_3\neq 0$; $F_3 = F_4 = 0$) are given by $$\label{TIII}
\eqalign{
0 = -F_1'' + F_2 F_1' + \case{1}{3}\Lambda F_1^3
+ \case{1}{4} (3 F_2' - F_2^2 - C_1^2) F_1, \\
0 = (H' - 2F_2 H)' - 2 F_2(H' - 2F_2 H) + 4 C_1^2 H,}$$ which are subject to $$\Psi_3 \propto H' - 2F_2 H - 2\rmi C_1 H \neq 0.$$ Using the first equation of , we can lower the order of the second ODE, such that the resulting set of equations contains derivatives up to the second-order in $F_1$ and the third-order in $F_2$. Hence the general solution carries another five real parameters in addition to $\Lambda$ and $C_1$. Considering that one of these parameters is simply the translation $z \rightarrow z+C_0$ (no explicit dependence on $z$ in (\[NurowskiEqnz\]-\[R330z\])), and thus removable, one can see that the final type III metric determined by has at most six parameters including $\Lambda$ and $C_1$ (see the conclusions).
Two classes of twisting type III vacuum solutions are known, respectively for $\Lambda=0$ and $\Lambda<0$. The one with $\Lambda=0$ is due to Held [@Held74] and Robinson [@Robinson75], which generalizes the non-twisting Robinsion-Trautman type III vacuum solution and generally admits only one Killing vector $\partial_u$. The subclasses with two Killing vectors (commuting or not) were found by Lun [@Lun78]. It can be shown that Lun’s case I type III metric corresponds to the following solution of : $$\label{Lun3}
\eqalign{
\Lambda =0, \qquad C_1 = 0, \\
F_1 = \frac{\sqrt{3}}{2 z^3} (E_1 z^{2+\sqrt{13}/2} + E_2 z^{2-\sqrt{13}/2}), \\
F_2 = -\frac{(5-\sqrt{13}) E_1 z^{2+\sqrt{13}/2} + (5+\sqrt{13}) E_2 z^{2-\sqrt{13}/2}}
{2z (E_1 z^{2+\sqrt{13}/2} + E_2 z^{2-\sqrt{13}/2})}. }$$ Likewise, his case II type III metric puts forth a second solution of : $$\label{Lun31}
\eqalign{
\Lambda =0, \qquad C_1 = \frac{1}{4}, \\
F_1 = \frac{\sqrt{3}}{4\cos(\frac{z}{2})} (E_1 G + E_2 G^{-1}), \\
F_2 = \frac{\sqrt{13}\, (E_1 G - E_2 G^{-1})}{4\cos(\frac{z}{2})\, (E_1 G + E_2 G^{-1})}
+ \frac{5}{4}\tan(\case{z}{2}), \\
G(z) = \left( \frac{\sin(\case{z}{2})+1}
{\cos(\case{z}{2})} \right)^{\sqrt{13}/2}. }$$ In both cases, $E_{1,2}$ are real constants. More details will be given in the next section as degenerate cases of the related type II solutions and .
The other known type III solution requires $\Lambda<0$ and is due to MacCallum and Siklos [@Siklos81] (see also [@Stephani03] p. 201). As a solution of , it is given by $$\label{Siklos}
F_1 = \pm \frac{\sqrt{39}}{4s(z+C_0)}, \qquad F_2 = -\frac{5}{2(z+C_0)},
\qquad C_1=0,$$ with $\Lambda=-s^2$, a real constant $C_0$ and $$\Psi_3 \propto H'-2F_2 H = -\frac{585}{256\Lambda(z+C_0)^5}.$$
Type II solutions
=================
Based on the structure of (\[NurowskiEqnz\]-\[R330z\]), we can consider the type II solution ($\Psi_2\neq 0$, $3\Psi_2 \Psi_4 -
2\Psi_3^2\neq 0$) according to three different cases.
*Case 1:* $F_3=0$, $F_4\neq 0 \Rightarrow C_1=0$. The equations (\[NurowskiEqnz\]-\[R330z\]) are reduced to $$\begin{aligned}
0 = -F_1'' + F_2 F_1' + \case{1}{3}\Lambda F_1^3
+ \case{1}{4} (3 F_2' - F_2^2) F_1, \\
0 = (H' - 2F_2 H)' - 2 F_2(H' - 2F_2 H), \\
F_4 = B_1 \exp\left(3\! \int\! F_2\rmd z \right),\end{aligned}$$ with $B_1\neq 0$ a real constant. Since the first two equations above are identical to with $C_1=0$ (also cf. and ), one can generate this kind of type II solutions directly from existing type N and III solutions, i.e., with $C_1=0$, and , which works as if one is adding a “mass source” to them. A similar idea can be found in [@Stephani03] p. 447. In addition, for type D solutions with $F_3=0$ and $C_1=0$, see \[App:TDCS\].
*Case 2:* $F_3\neq 0$, $C_1=0$. The associated equations are given by $$\begin{aligned}
0 = -F_1'' + F_2 F_1' + \case{1}{3}\Lambda F_1^3
+ \case{1}{4} (3 F_2' - F_2^2) F_1 + \frac{F_3}{F_1^3}, \\
0 = (H' - 2F_2 H)' - 2 F_2(H' - 2F_2 H), \\
F_2 = \frac{F_3'}{3F_3}, \qquad
F_4 = B_1 \exp\left(3\! \int\! F_2\rmd z \right) = B_1 F_3,\end{aligned}$$ with $B_1$ a real constant. Certainly one may use the third equation above to turn the first two into ODEs for $F_1$ and $F_3$ only.
Lun’s case I solution [@Lun78; @McIntosh87] with four parameters can be shown to belong to this case. It reads, in the canonical frame (\[metricStph\]-\[Is\]), $$\begin{aligned}
P_s = \sqrt{\case{2}{3}}\, (\zeta + \bar\zeta)^{3/2}, \qquad \zeta = x + \rmi y, \\
L = -\frac{3\rmi}{16} x^{-3/2} \left[(3+\sqrt{13}) E_1 x^{\sqrt{13}/2}
+ (3-\sqrt{13}) E_2 x^{-\sqrt{13}/2} \right] + \frac{3\rmi M}{32 x^3}, \\
m_s = m, \qquad M_s = M, \qquad \Lambda = 0,\end{aligned}$$ or, in the CR formalism as a solution of (\[ccb1\]-\[R3301\]) or (\[NurowskiEqnz\]-\[R330z\]), $$\label{Lun2}
\eqalign{
A(\zeta) = -2\rmi, \qquad z = x, \qquad C_1 = 0, \\
p = \frac{1}{2} F_1(z) = \frac{\sqrt{3}}{16 z^3}\, G(z), \\
c = -\frac{1}{2} F_2(z) \\
\ \ =(z G)^{-1} \left[(5-\sqrt{13}) E_1 z^{2+\sqrt{13}/2}
+ (5+\sqrt{13}) E_2 z^{2-\sqrt{13}/2} + 6M z^{1/2} \right], \\
n = \frac{1}{64} \left(F_3(z) + \rmi F_4(z)\right)
= \frac{27\, \rmi}{2^{20} z^{27/2}}\, G^3 (m+\rmi M), \\
G(z) = 4\left( E_1 z^{2+\sqrt{13}/2} + E_2 z^{2-\sqrt{13}/2} \right) + 3M z^{1/2}, }$$ with $m$, $M$ and $E_{1,2}$ real constants. When $m = M = 0$, the solution degenerates to the type III solution .
The Kerr and Debney/Demiański’s four-parameter solution [@Stephani03] (see p. 449) also falls under this case. It is given by $$\eqalign{
P_s = 1 + \frac{\zeta \bar\zeta}{2},
\qquad L = -\rmi P_s^2 \left[ 2M/\zeta + (M+a)\bar\zeta
+ \case{1}{4} b \bar\zeta \log(\bar\zeta/\sqrt{2}\,) \right], \\
m_s = m, \qquad M_s = M, \qquad \Lambda = 0,}$$ and corresponds to $$\label{Demianski}
\eqalign{
A(\zeta) = -\rmi\zeta, \qquad z = \log(\zeta \bar\zeta),
\qquad C_1=0, \\
F_1 = -\frac{[16(M-a)+8(M+a)\rme^z + b G_1]
\rme^{z/2}}{4(2+\rme^z)^2}, \\
F_2 = \frac{8[4(M-a)+8a\rme^z-(M+a)\rme^{2z}] + b G_2}
{8[2(M-a)+(M+a)\rme^z](2+\rme^z) + b G_1 (2+\rme^z)}, \\
F_3 = \frac{M[16(M-a)+8(M+a)\rme^z + b G_1]^3\rme^{3z}}{32(2+\rme^z)^9}, \\
F_4 = -\frac{m[16(M-a)+8(M+a)\rme^z + b G_1]^3\rme^{3z}}{32(2+\rme^z)^9}, \\
G_1(z) = (2+\ln 2-z)(2-\rme^z) - 8, \\
G_2(z) = (3+\ln 2-z)(4+\rme^{2z}) - 8(\ln 2-z)\rme^z - 24,}$$ which is a solution of (\[NurowskiEqnz\]-\[R330z\]). Here $m$, $M$, $a$ and $b$ are four real parameters. Clearly, the Kerr-NUT solution is a special case with $b=0$.
Besides these known solutions, we have obtained an additional one (see the derivation in Case 3 below) which turns out to be a limiting case ($C_1\rightarrow 0$) of the solution : $$\label{new}
\eqalign{
C_1 = 0, \qquad \Lambda = 0, \\
F_1 = -2B_2 z^2 + C_2 z + C_3,
\qquad F_2 = \frac{4B_2 z - C_2}{2B_2 z^2 - C_2 z - C_3}, \\
F_3 = -B_2 F_1^3, \qquad F_4 = B_1 F_1^3.}$$ Here $C_{2,3}$ and $B_{1,2}$ are real constants. Its comparisons with Lun’s and Demiański’s solutions will be discussed in \[App:Cmp\]. Particularly when $B_2=0$, the solution becomes type D.
*Case 3:* $F_3\neq 0$, $C_1\neq 0 \Rightarrow F_4\neq 0$. This corresponds to the most general case for solutions. As one may check, Lun’s case II four-parameter solution [@Lun78; @McIntosh87], which is given by $$\begin{aligned}
P_s = \sqrt{\case{2}{3}}\, (\zeta + \bar\zeta)^{3/2},
\qquad \zeta = x + \rmi y, \qquad w=y/x, \\
L = x^{-3/2} \Big\{ \case{1}{6} [
E_1 (w+(w^2+1)^{1/2})^{\sqrt{13}/2} (w-\case{\sqrt{13}}{2}(w^2+1)^{1/2}) \\
+ E_2 (w+(w^2+1)^{1/2})^{-\sqrt{13}/2} (w+\case{\sqrt{13}}{2}(w^2+1)^{1/2}) \Big] \\
+ \case{3}{160} [(m+\rmi M)(1+\rmi w)^{3/2} (2-3\rmi w)
+ (m-\rmi M)(1-\rmi w)^{3/2} (2+3\rmi w)] \Big\}, \\
m_s - \rmi M_s = (m+\rmi M) x^{3/2} (1+\rmi w)^{3/2}, \qquad \Lambda = 0,\end{aligned}$$ can be converted to a solution of (\[NurowskiEqnz\]-\[R330z\]): $$\label{Lun21}
\fl \eqalign{
A(\zeta)=\zeta, \qquad \zeta=|\zeta|\,\rme^{\rmi z/2}, \qquad C_1 = \frac{1}{4}, \\
F_1 = \frac{\sqrt{3} (E_1 G + E_2 G^{-1})}{4\cos(\case{z}{2})}
+ \frac{3\sqrt{3}\,\sin(\case{z}{2})
\left(M \sin(\case{z}{4}) - m \cos(\case{z}{4}) \right) }
{16\cos^{5/2}(\case{z}{2})}, \\
F_2 = \frac{4\cos^{3/2}(\case{z}{2}) [ 5\sin(\case{z}{2}) (E_1 G + E_2 G^{-1})
+ \sqrt{13} (E_1 G - E_2 G^{-1}) ] - 3 (M G_1 - m G_2) }
{16 \cos^{5/2}(\case{z}{2}) (E_1 G + E_2 G^{-1})
+ 6 \sin(z) \left(M \sin(\case{z}{4}) - m \cos(\case{z}{4}) \right)}, \\
F_3 = \frac{3\sqrt{3} F_1^3 \left( M \cos(\case{3z}{4}) + m \sin(\case{3z}{4}) \right)}
{2^8 \cos^{9/2}(\case{z}{2})}, \qquad
F_4 = \frac{3\sqrt{3} F_1^3 \left( -M \sin(\case{3z}{4}) + m \cos(\case{3z}{4}) \right)}
{2^8 \cos^{9/2}(\case{z}{2})}, \\
G(z) = \left( \frac{\sin(\case{z}{2})+1}{\cos(\case{z}{2})} \right)^{\sqrt{13}/2},
\qquad G_1(z) = \case{5}{4} \sin(\case{5z}{4})
- \case{7}{4} \sin(\case{3z}{4}) - 5\sin(\case{z}{4}), \\
G_2(z) = \case{5}{4} \cos(\case{5z}{4})
+ \case{7}{4} \cos(\case{3z}{4}) - 5\cos(\case{z}{4}), }$$ with $E_{1,2}$, $M$ and $m$ real constants. In the case of $M=m=0$, the solution reduces to the type III solution .
Besides Lun’s example, we have also considered the special case of $H=0$ for the system (\[NurowskiEqnz\]-\[R330z\]), which turns out to be fully soluble when $\Lambda=0$. The derivation follows closely the type N *Case 1* (cf. ), and utilizes the same ansatz that makes the function $H(z)$ vanish (hence, satisfied). More specifically, we have $$\label{newF}
\eqalign{
F_1 = \pm \sqrt{J'}, \qquad F_2 = \frac{J''}{2J'} - \Lambda J
= \frac{F_1'}{F_1} - \Lambda J, \qquad J'>0, \\
F_3 = F_1^3 \exp\left(-3\Lambda\! \int\! J\rmd z \right) [B_1 \sin(3C_1 z) - B_2 \cos(3C_1 z)], \\
F_4 = F_1^3 \exp\left(-3\Lambda\! \int\! J\rmd z \right) [B_2 \sin(3C_1 z) + B_1 \cos(3C_1 z)], }$$ with the last two equations derived from . Therefore when $\Lambda$ vanishes, the equation can be reduced to a linear ODE for $F_1(z)$ alone: $$F_1'' = -C_1^2 F_1 + 4[B_1 \sin(3C_1 z) - B_2 \cos(3C_1 z)],
\qquad \Lambda = 0,$$ which has the general solution $$\label{newKD}
\fl F_1 = C_2 \cos(C_1 (z+C_0))
- \frac{1}{2C_1^2} [B_1 \sin(3C_1 z) - B_2 \cos(3C_1 z)],
\qquad C_1\neq 0,$$ or, if $C_1$ vanishes (cf. ), $$\label{new1}
F_1 = -2B_2 z^2 + C_3 z + C_4, \qquad C_1=0,$$ where $C_{0-4}$ and $B_{1,2}$ are real constants. The conditions $H=0$ and $\Lambda=0$ exclude type III and N as special cases. The solution corresponding to with $C_1=-\case{1}{2}$ coincides with a special case of Kerr and Debney’s type II solution [@Stephani03] (p. 608), and hence is not new (see \[App:KD\] for more details). Also, by setting $C_0=0$, $C_2=-\frac{B_2}{2C_1^2}+C_4$ and $B_1=-\frac{2}{3}C_1 C_3$, one can obtain from $\eref{newKD}$ in the limit $C_1\rightarrow
0$, whereas the two Killing vectors become commuting. For a generally non-vanishing $\Lambda$, we obtain from an equation for $J(z)$: $$\label{JEQII}
\eqalign{
J''' &= \frac{(J^{\prime\prime})^2}{2J'} - 2 \Lambda J J^{\prime\prime} -
\frac{10}{3} \Lambda (J')^2 - 2 (\Lambda^2 J^2 + C_1^2) J' \\
& + 8 (J')^{1/2} \exp\left(-3\Lambda\! \int\! J\rmd z \right)
[B_1 \sin(3C_1 z) - B_2 \cos(3C_1 z)],}$$ which generalizes the equation for type II. By introducing $J=K'$ (or $-3\Lambda J=K'/K$, etc.), one can transform this equation to a fourth-order ODE for $K(z)$. However, we do not have any type II solution with $\Lambda\neq 0$ for this ODE.
Conclusions
===========
With the real coordinates $\{x, z, u, r\}$ and $A(\zeta)=2$ in the ansatz , we present here our new class of vacuum twisting type II metrics admitting two Killing vectors: $$\mathbf{g} = \frac{F_1^2(z)}{2 \cos^2(\frac{r}{2})}
\left[ \rmd\zeta \rmd\bar\zeta + \lambda \left(\rmd r
+ \mathcal{W} \rmd\zeta + \bar\mathcal{W} \rmd\bar\zeta + \mathcal{H} \lambda \right) \right],$$ with $\zeta = x + \rmi z$ and $$\begin{aligned}
\lambda = \frac{\rme^{C_1 x} \rmd u
- 2 \left[\int \exp\left(\int F_2 \rmd z \right) \rmd z \right] \rmd x}
{\exp\left(\int F_2 \rmd z \right)}, \\
\mathcal{W} = \frac{1}{2} \left( \frac{2F_1'}{F_1} - F_2 + \rmi C_1 \right) (\rme^{-\rmi r}+1), \\
\fl \mathcal{H} = -\frac{1}{2} \left[ \left(\frac{F_1'}{F_1}\right)^\prime
- \Lambda F_1^2 - F_2' - \frac{2( F_3(\cos r + 1) + F_4\sin r )}{F_1^4}
\right]\! (\cos r + 1) - \frac{1}{6} \Lambda F_1^2 \cos r,\end{aligned}$$ where $C_1$ is an arbitrary real parameter and the real functions $F_{1-4}(z)$ satisfy $$\label{FODEs}
\eqalign{
0 = -F_1'' + F_2 F_1' + \frac{1}{3}\Lambda F_1^3
+ \frac{1}{4} (3 F_2' - F_2^2 - C_1^2) F_1 + \frac{F_3}{F_1^3}, \\
0 = (H' - 2F_2 H)' - 2 F_2(H' - 2F_2 H) + 4 C_1^2 H, \\
F_3 = \exp\left(3\! \int\! F_2\rmd z \right)
[B_1 \sin(3C_1 z) - B_2 \cos(3C_1 z)], \\
F_4 = \exp\left(3\! \int\! F_2\rmd z \right)
[B_2 \sin(3C_1 z) + B_1 \cos(3C_1 z)],}$$ with the function $H(z)$ defined by $$H = F_1^{\prime \prime} F_1 - (F_1')^2 - \Lambda F_1^4 - F_2' F_1^2.$$ One can determine $F_{1,2}$ from the first two equations of , which are in fact a pair of ODEs for $F_1$ and $K=\int
F_2\rmd z$. This allows the metric to have at most eight real parameters including $\Lambda$ and $C_1$ (see discussion in Section 7), and its two Killing vectors are $$\partial_u, \qquad \partial_x - C_1 u\, \partial_u.$$ Most of the previously known twisting vacuum solutions with two Killing vectors, as presented in Chapters 29 and 38 of [@Stephani03], have been shown to belong to this class, the only exception being the general case of Kerr and Debney’s type II solution (see \[App:KD\]). Additionally, for $H=0$, the system can be reduced to a single ODE , or even be fully integrated when $\Lambda=0$. This leads to the discovery of a limiting solution (type II with two commuting Killing vectors) of a special case of Kerr and Debney’s type II solution, which we believe has not been discussed or published before. Despite all these special solutions with maximally four parameters, the general solution of is still quite unknown. We believe that this problem poses a major challenge.
Altogether, we hope that this work may provide a platform for all types of twisting algebraically special solutions to be studied in a connected and unified manner, given the history that many of those known solutions were derived by quite different approaches or special assumptions. For future research, this new class of metrics may be further examined to study important issues such as the cosmic no-hair conjecture, the asymptotic stability of the Kerr solution [@Natorf12], and the formation of rotating black holes that might be described by certain solutions of (see, e.g., [@Bicak95; @Oliveira04] and references therein).
Type D solutions: $\Psi_3=\Psi_4=0$ {#App:TD}
===================================
For simplicity, we consider type D solutions with $\Psi_2\neq 0$ and $\Psi_3=\Psi_4=0$. In order to acquire $\Psi_3=0$ with $\Psi_3$ depending on $r$, we need, at least, for the coefficient of $\rme^{2\rmi r}$ to vanish in , $$\label{TD}
0 = 2\bar\partial \log p + \bar{c},$$ which, through the ansatz , adds two more equations to the system: $$\label{TD1}
C_1=0, \qquad F_1' = \case{1}{2} F_2 F_1.$$ Thus we can simplify the original (\[NurowskiEqnz\]-\[R330z\]) to $$\begin{aligned}
F_2' = -\frac{4}{3}\Lambda F_1^2 - \frac{4F_3}{F_1^4}, \\
F_3' = 3F_2 F_3, \\
F_4' = 3F_2 F_4, \label{TD4}\end{aligned}$$ with the last equation being automatically satisfied by the equations presented above. The system (\[TD1\]-\[TD4\]) can be fully solved with the general solution $$\label{TDsol}
\eqalign{
F_1 = C_3 \sec\! \Big(\frac{z + C_0}{C_2}\Big),
\qquad F_2 = \frac{2}{C_2} \tan\! \Big(\frac{z + C_0}{C_2}\Big), \\
\fl F_3 = -\frac{C_3^4(3 + 2\Lambda C_2^2 C_3^2)}{6C_2^2}
\sec^6\! \Big(\frac{z + C_0}{C_2}\Big),
\qquad F_4 = C_4 \sec^6\! \Big(\frac{z + C_0}{C_2}\Big).}$$ Remarkably, though we have only started with one extra condition , it is enough for the solution to be of type D, i.e., that we have $$\Psi_2 = \frac{-C_3^4(3+2\Lambda C_2^2 C_3^2)+ 6\rmi C_2^2 C_4}
{12 C_2^2 C_3^6} (\rme^{\rmi r} + 1)^3,
\qquad \Psi_3 = \Psi_4 = 0.$$
With $A(\zeta)=2$ and , the resulting metric can be written as $$\mathbf{g} = \frac{C_4^2}{2 \cos^2(\frac{r}{2}) \cos^2(\frac{z + C_0}{C_2})}
\left[ \rmd\zeta \rmd\bar\zeta + \lambda \left(\rmd r + \mathcal{H} \lambda \right) \right]$$ with $\zeta = x + \rmi z$ and $$\begin{aligned}
\lambda = \cos^2\! \Big(\frac{z + C_0}{C_2}\Big) \left[ \rmd u
- 2 C_2 \tan\! \Big(\frac{z + C_0}{C_2}\Big) \rmd x \right], \\
\mathcal{H} = -\frac{1}{\cos^2(\frac{z + C_0}{C_2})}
\left[ \cos^2\! \left(\frac{r}{2}\right) \!
\left(\frac{\cos r}{C_2^2} + \frac{2 C_4 \sin r}{C_3^4} \right)
+ \frac{1}{6} C_3^2\Lambda (\cos 2r + 2\cos r) \right].\end{aligned}$$ One can immediately remove the parameter $C_0$ by $z + C_0
\rightarrow z$. Besides the cosmological constant $\Lambda$, the metric contains three real parameters $C_2$, $C_3$ and $C_4$.
Type D solutions from classical symmetries {#App:TDCS}
==========================================
Here we list a number of special solutions one may encounter when searching for group-invariant solutions from classical symmetries [@Krasilshchik99] of the system (\[NurowskiEqnz\]-\[R330z\]) or its various special cases. Incidentally, all these solutions turn out to be of type D, even though the condition $2\Psi_2^2=3\Psi_3\Psi_4$ is never used in their derivation. As expected for type D solutions, they all have $C_1=0$, which means that the two Killing vectors are commuting. In what follows, $C_{2-4}$ are real constants.
We start with solutions with $F_3=0$. For $\Lambda = 0$, we have $$\begin{aligned}
F_1 = C_3 \rme^{C_2 z}, \qquad F_2 = 2 C_2,
\qquad F_3 = 0, \qquad F_4 = C_4 \rme^{6C_2 z}, \\
F_1 = \frac{C_2}{z^2}, \qquad F_2 = -\frac{3}{z},
\qquad F_3 = 0, \qquad F_4 = \frac{C_3}{z^9}.\end{aligned}$$ For $\Lambda = -s^2<0$, we have ([@Zhang12a] and cf. ) $$\begin{aligned}
F_1 = \frac{\sqrt{6}}{3s z}, \qquad F_2 = -\frac{5}{3z},
\qquad F_3 = 0, \qquad F_4 = \frac{C_2}{z^5}, \\
F_1 = \frac{\sqrt{6}}{2s z}, \qquad F_2 = -\frac{2}{z},
\qquad F_3 = 0, \qquad F_4 = \frac{C_2}{z^6}.\end{aligned}$$ In case of $F_4=0$, they all become type O solutions of or .
For solutions that admit non-vanishing $F_3$ and $\Lambda$, we have $$\begin{aligned}
F_1 = C_2\neq 0, \qquad F_2 = 0,
\qquad F_3 = -\case{1}{3} C_2^6\Lambda, \qquad F_4 = C_3, \\
F_1 = \frac{C_2}{z}, \qquad F_2 = -\frac{2}{z},
\qquad F_3 = -\frac{3C_2^4 + 2C_2^6 \Lambda}{6z^6},
\qquad F_4 = \frac{C_3}{z^6},\end{aligned}$$ Note that the first solution above is given by constants, which is a consequence of the translational invariance ($z\rightarrow z+C_0$) of the system (\[NurowskiEqnz\]-\[R330z\]). Lastly for $\Lambda=0$, we obtain $$F_1 = C_3 \rme^{C_2 z}, \qquad F_2 = \case{4}{3}C_2,
\qquad F_3 = \case{1}{9} C_3^4 C_2^2 \rme^{4C_2 z},
\qquad F_4 = C_4 \rme^{4C_2 z}.$$
Comparisons of type II solutions with $C_1=0$ {#App:Cmp}
=============================================
Here we compare the three type II solutions from *Case 2*, i.e., , and , and show that is different from the other two. Generally, to see that two twisting type II vacuum metrics are different, i.e., not being related by a coordinate transformation, it is sufficient to show that their CR structures along the shearfree null congruences are not equivalent [@Hill08]. This can be decided by evaluating the six Cartan invariants [@Nurowski93; @Zhang12a], which are the same only for two equivalent CR structures. Among these invariants, the first one, in Cartan’s original notation, is given by $$\eqalign{
\alpha(\zeta,\bar\zeta) = -\frac{5 \bar{r} \partial_\zeta r + r \partial_\zeta \bar{r} + 8 c r\bar{r}}
{8 \sqrt{\bar{r}} \cdot \sqrt[8]{(r\bar{r})^{7}}}, \\
\bar{r} = \case{1}{6}\left(\partial_{\zeta} l + 2 c l\right), \qquad
l = -\partial_\zeta \partial_{\bar\zeta} c - c \partial_{\bar\zeta} c,}$$ which only relies on the function $c=c(\zeta,\bar\zeta)$ (hence $F_2(z)$ from the ansatz ). The use of $\alpha$ alone will be adequate for our comparison.
For simplicity, we consider the special case of with $C_2=C_3=0$, i.e., $$F_1 = -2B_2 z^2, \qquad F_2 = \frac{2}{z},
\qquad F_3 = -B_2 F_1^3, \qquad F_4 = B_1 F_1^3.$$ This solution is still of type II but has a constant invariant $\alpha$ given by $$\alpha^2 = -\frac{25}{14} \sqrt{21}.$$ Yet another case with $\alpha$ being constant is when $B_2=0$ in , in which case the solution is of type D and $$\alpha^2 = -\frac{16}{5} \sqrt{10}.$$ This same quantity $\alpha$ calculated from Lun’s solution , however, is generally a function of $z$ and only becomes a constant when two of the three free parameters $E_{1,2}$ and $M$ vanish, i.e., that we have $$\begin{aligned}
\alpha^2 = \frac{\sqrt{15}}{10} \ \ \mathrm{for}\ \ E_1=E_2=0, \\
\alpha^2 = \sqrt{2} (4-\sqrt{13}) \ \ \mathrm{for}\ \ E_1=M=0, \\
\alpha^2 = \sqrt{2} (4+\sqrt{13}) \ \ \mathrm{for}\ \ E_2=M=0,\end{aligned}$$ none of which is equal to those of . As for Demiański’s solution , one can see (using Maple) that its invariant $\alpha$ is never a constant (even when $b=0$ for type D; particularly for the NUT solution with $b=a=0$, $\alpha$ is not defined due to $\bar{r}=0$, and the corresponding CR structure is hyperquadric [@Lewandowski90]) within the full range of the parameters $M$, $a$ and $b$. Therefore we conclude that the solution is different from and .
Kerr and Debney’s type II solution {#App:KD}
==================================
The solution by Kerr and Debney [@Kerr70] (see also [@Stephani03] p. 608) admits two non-commuting Killing vectors and reads $$\label{KDII}
\eqalign{
P_s = 1, \qquad L = A_1 \bar\zeta^2 \zeta^{1+\sigma}
+ A_2 \bar\zeta \zeta^{\sigma/3}, \\
\rmi M_s - m_s = 2A_1(1+\sigma)\zeta^\sigma, \qquad \Lambda=0,}$$ with $\mathrm{Re}\,\sigma = -3$ and $A_{1,2}$ complex constants. The special case with $\sigma = -3$ can be captured by the ansatz , and it corresponds to ($E_{1-4}$ real) $$\eqalign{
A(\zeta)=\zeta, \qquad \zeta=|\zeta|\,\rme^{\rmi z/2}, \qquad C_1 = -\frac{1}{2}, \\
F_1 = E_3\sin(\case{z}{2}) - E_4\cos(\case{z}{2})
+ 2E_1\sin(\case{3z}{2}) - 2E_2\cos(\case{3z}{2}), \\
F_2 = F_1'/F_1, \qquad
F_3 = -F_1^3 \left( E_1 \sin(\case{3z}{2}) - E_2 \cos(\case{3z}{2}) \right), \\
F_4 = F_1^3 \left( E_2 \sin(\case{3z}{2}) + E_1 \cos(\case{3z}{2}) \right), \\
A_1 = E_1 + \rmi E_2, \qquad A_2 = E_3 + \rmi E_4. }$$ Modulo some redefinition of parameters, this is the same solution as and with $C_1=-\case{1}{2}$. It is not clear how to make such a conversion for the general case of .
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[^1]: All our considerations are local.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Let $K$ be the function field of a $p$-adic curve, $G$ a semisimple simply connected group over $K$ and $X$ a $G$-torsor over $K$. A conjecture of Colliot-Thélène, Parimala and Suresh predicts that if for every discrete valuation $v$ of $K$, $X$ has a point over the completion $K_v$, then $X$ has a $K$-rational point. The main result of this paper is the proof of this conjecture for groups of some classical types. In particular, we prove the conjecture when $G$ is of one of the following types: (1) ${}^2A_n^*$, i.e. $G=\mathbf{SU}(h)$ is the special unitary group of some hermitian form $h$ over a pair $(D\,,\,\tau)$, where $D$ is a central division algebra of square-free index over a quadratic extension $L$ of $K$ and $\tau$ is an involution of the second kind on $D$ such that $L^{\tau}=K$; (2) $B_n$, i.e., $G=\mathbf{Spin}(q)$ is the spinor group of quadratic form of odd dimension over $K$; (3) $D_n^*$, i.e., $G=\mathbf{Spin}(h)$ is the spinor group of a hermitian form $h$ over a quaternion $K$-algebra $D$ with an orthogonal involution. Our method actually yields a parallel local-global result over the fraction field of a $2$-dimensional, henselian, excellent local domain with finite residue field, under suitable assumption on the residue characteristic.'
author:
- Yong HU
title: '**Hasse Principle for Simply Connected Groups over Function Fields of Surfaces**'
---
[**MSC classes:**]{} 11E72, 11E57
Introduction
============
Let $K$ be a field and $G$ a smooth connected linear algebraic group over $K$. The cohomology set $H^1(K,\,G)$ classifies up to isomorphism $G$-torsors over $K$, and a class $\xi\in H^1(K,\,G)$ is trivial if and only if the corresponding $G$-torsor has a $K$-rational point. Let $\Omega_K$ denote the set of (normalized) discrete valuations (of rank 1) of the field $K$. For each $v\in\Omega_K$, let $K_v$ denote the completion of $K$ at $v$. The restriction maps $H^1(K,\,G)\to H^1(K_v,\,G)$, $v\in\Omega_K$ induce a natural map of pointed sets $$H^1(K,\,G){\longrightarrow}\prod_{v\in \Omega_K}H^1(K_v,\,G)\,.$$If the kernel of this map is trivial, we say that the *Hasse principle* with respect to $\Omega_K$ holds for $G$-torsors over $K$.
In the case of a $p$-adic function field, by which we mean the function field of an algebraic curve over a $p$-adic field (i.e., a finite extension of $\mathbb{Q}_p$), the following conjecture was made by Colliot-Thélène, Parimala and Suresh.
\[conj1p1NEW\] Let $K$ be the function field of an algebraic curve over a $p$-adic field and let $G$ be a semisimple simply connected group over $K$.
Then the kernel of the natural map $$H^1(K,\,G){\longrightarrow}\prod_{v\in\Omega_K}H^1(K_v,\,G)$$is trivial. In other words, if a $G$-torsor has points in all completions $K_v\,,\,v\in\Omega_K$, then it has a $K$-rational point.
\[para1p2NEWTEMP\] Let $K$ be a $p$-adic function field with field of constants $F$, i.e., $K$ is the function field of a smooth projective geometrically integral curve over the $p$-adic field $F$. Let $A$ be the ring of integers of $F$. It is in particular a henselian excellent local domain of dimension 1. By resolution of singularities, there exists a proper flat morphism $\mathcal{X}\to
\mathrm{Spec} A$, where $\mathcal{X}$ is a connected regular 2-dimensional scheme with function field $K$. We will say that $\mathcal{X}$ is a $p$-adic *arithmetic surface* with function field $K$, or that $\mathcal{X}\to\mathrm{Spec} A$ is a *regular proper model* of the $p$-adic function field $K$.
An analog in the context of a 2-dimensional base is as follows. Let $A$ be a henselian excellent 2-dimensional local domain with *finite* residue field $k$ and let $K$ be the field of fractions of $A$. Again by resolution of singularities, there exists a proper birational morphism $\mathcal{X}\to \mathrm{Spec} A$, where $\mathcal{X}$ is a connected regular 2-dimensional scheme with function field $K$. We will say that $\mathrm{Spec} A$ is a *local henselian surface* with function field $K$ and that $\mathcal{X}\to\mathrm{Spec} A$ is a *regular proper model* of $\mathrm{Spec}
A$.
Experts have also been interested in the following analog of Conjecture$\;$\[conj1p1NEW\]:
\[ques1p2NEW\] Let $K$ be the function field of a local henselian surface $\mathrm{Spec} A$ with finite residue field and let $G$ be a semisimple simply connected group over $K$.
Does the Hasse principle with respect to $\Omega_K$ hold for $G$-torsors over $K\,?$
Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field. For most quasi-split $K$-groups, the Hasse principle may be proved by combining an injectivity property of the Rost invariant map (cf. [@CTPaSu Thm.$\;$5.3]) and results from higher dimensional class field theory of Kato and Saito.
The goal of this paper is to prove the Hasse principle for groups of several types in the non-quasisplit case. To give precise statement of our main result, we will refine the usual classification of absolutely simple simply connected groups in some cases.
\[para1p4NEWTEMP\] Let $E$ be a field and let $G$ be an absolutely simple simply connected group over $E$. We say that $G$ is of type
\(1) ${}^1A_n^*$, if $G={\mathbf{SL}}_1(A)$ is the special linear group of some central simple $E$-algebra $A$ *of square-free index*;
\(2) ${}^2A_n^*$, if $G={\mathbf{SU}}(h)$ is the special unitary group of some nonsingular hermitian form $h$ over a pair $(D,\,\tau)$, where $D$ is a central division algebra *of square-free index* over a separable quadratic field extension $L$ of $E$ and $\tau$ is an involution of the second kind on $D$ such that $L^{\tau}=E$; when the index of division algebra $D$ is odd (resp. even), we say the group $G={\mathbf{SU}}(h)$ is of type ${}^2A_n^*$ of odd (resp. even) index;
\(3) $C_n^*$, if $G={\mathbf{U}}(h)$ is the unitary group (also called symplectic group) of a nonsingular hermitian form $h$ over a pair $(D,\,\tau)$, where $D$ is *quaternion algebra* over $E$ and $\tau$ is a symplectic involution on $D$;
\(4) $D_n^*$ (in characteristic $\neq 2$), if $G={\mathbf{Spin}}(h)$ is the spin group of a nonsingular hermitian form $h$ over a pair $(D,\,\tau)$, where $D$ is *quaternion algebra* over $E$ and $\tau$ is an orthogonal involution on $D$;
\(5) $F_4^{red}$ (in characteristic different from $2,\,3$), if $G=\mathbf{Aut}_{alg}(J)$ is the group of algebra automorphisms of some *reduced* exceptional Jordan $E$-algebra $J$ of dimension 27.
Recall also that $G$ is of type
\(6) $B_n$ (in characteristic $\neq 2$), if $G={\mathbf{Spin}}(q)$ is the spin group of a nonsingular quadratic form $q$ of dimension $2n+1$ over $E$;
\(7) $G_2$ (in characteristic $\neq 2$), if $G=\mathbf{Aut}_{alg}(C)$ is the group of algebra automorphisms of a Cayley algebra $C$ over $E$.
\[para1p5NEWTEMP\] In the local henselian case, we shall exclude some possibilities for the residue characteristic. To this end, we define for any semisimple simply connected group $G$ a set $S(G)$ of prime numbers as follows (cf. [@Ser94 $\S$2.2] or [@Gille10 p.44]):
$S(G)={\{\,{2}\,\}}$, if $G$ is of type $G_2$ or of classical type $B_n,\,C_n$ or $D_n$ (trialitarian $D_4$ excluded);
$S(G)={\{\,{2\,,\,3}\,\}}$, if $G$ is of type $E_6,\,E_7,\,F_4$ or trialitarian $D_4$;
$S(G)={\{\,{2\,,\,3\,,\,5}\,\}}$, if $G$ is of type $E_8$;
$S(G)$ is the set of prime factors of the index ${\mathrm{ind}}(A)$ of $A$, if $G={\mathbf{SL}}_1(A)$ for some central simple algebra $A$;
$S(G)$ is the set of prime factors of $2.{\mathrm{ind}}(D)$, if $G={\mathbf{SU}}(h)$ for some nonsingular hermitian form $h$ over a division algebra $D$ with an involution of the second kind.
In the general case, define $S(G)=\cup S(G_i)$, where $G_i$ runs over the almost simple factors of $G$.
When $G$ is absolutely simple, let $n_G$ be the order of the Rost invariant of $G$. Except for a few cases where $n_G=1$, the set $S(G)$ coincides with the set of prime factors of $n_G$ (cf. [@GMS Appendix$\;$B] or [@KMRT $\S$31.B]).
We summarize our main results in the following two theorems.
\[thm1p3NEW\][^1] Let $K$ be the function field of a $p$-adic arithmetic surface and $G$ a semisimple simply connected group over $K$. Assume $p\neq 2$ if $G$ contains an almost simple factor of type ${}^2A_n^*$ of even index.
If every almost simple factor of $G$ is of type $${}^1A_n^*\,,\,{}^2A^*_n\,,\,B_n\,,\,C_n^*\,,\,D_n^*\,,\,F_4^{red}\,\,\text{
or }\;\;G_2\,,$$then the natural map $$H^1(K,\,G){\longrightarrow}\prod_{v\in\Omega_K}H^1(K_v\,,\,G)$$has a trivial kernel.
\[thm1p4NEWv2\] Let $K$ be the function field of a local henselien surface with finite residue field of characteristic $p$. Let $G$ be a semisimple simply connected group over $K$. Assume $p\notin S(G)$.
If every almost simple factor of $G$ is of type $${}^1A_n^*\,,\,{}^2A^*_n \text{ of odd index}\,,\,\,B_n\,,\,C_n^*\,,\,D_n^*\,,\,F_4^{red}\,\,\text{
or }\;\;G_2\,,$$then the natural map $$H^1(K,\,G){\longrightarrow}\prod_{v\in\Omega_K}H^1(K_v\,,\,G)$$has a trivial kernel.
If moreover the Hasse principle with respect to $\Omega_K$ holds for quadratic forms $q$ of rank $6$ over $K ($i.e., $q$ has a nontrivial zero over $K$ if and only if it has a nontrivial zero over every $K_v\,,\,v\in\Omega_K)$, then the same result is also true for an absolutely simple group of type ${}^2A_n^*$ of even index.
In fact, it suffices to consider only divisorial discrete valuations in the above theorems.
\[remark1p8v2\] Let $K$ be as in Theorem$\;$\[thm1p3NEW\] or \[thm1p4NEWv2\]. Assume the residue characteristic $p$ is not 2.
\(1) By [@Salt97 Thm.$\;$3.4] (the arithmetic case) and [@Hu11 Thm.$\;$3.4] (the local henselian case), a central division algebra of exponent 2 over the field $K$ is either a quaternion algebra or a biquaternion algebra. So for a group of type $C_n$, say $G={\mathbf{U}}(h)$ with $h$ a hermitian form over a symplectic pair $(D,\,\tau)$, the only case not covered by our theorems is the case where $D$ is a biquaternion algebra. Similarly, for a group of classical type $D_n$, say $D={\mathbf{Spin}}(h)$ with $h$ a hermitian form over an orthogonal pair $(D,\,\tau)$, the only remaining case is the one with $D$ a biquaternion algebra.
\(2) In Theorem$\;$\[thm1p4NEWv2\], the hypothesis on the Hasse principle for quadratic forms of rank 6 is satisfied if $K=\mathrm{Frac}(\mathcal{O}[\![t]\!])$ is the fraction field of a formal power series ring over a complete discrete valuation ring $\mathcal{O}$ (whose residue field is finite), by [@Hu10 Thm.$\;$1.2]. In the arithmetic case, this is established in [@CTPaSu Thm.$\;$3.1].
In the rest of the paper, after some preliminary reviews in Section$\;$\[sec2\], we will prove our main theorems case by case: the cases ${}^1A_n^*$, $C_n$, $F_4^{red}$ and $G_2$ in Section$\;$\[sec3\]; the cases $B_n$ and $D_n^*$ in Sections$\;$\[sec4\] and \[sec5\]; and the case ${}^2A_n^*$ in Section$\;$\[sec6\].
Our proofs use ideas from Parimala and Preeti’s paper [@PaPr]. In particular, two exact sequences of Witt groups, due to Parimala–Sridharan–Suresh and Suresh respectively, play a special role in some cases. Other important ingredients include Hasse principles for degree 3 cohomology of $\mathbb{Q}/\mathbb{Z}(2)$ coming from higher dimensional class field theory of Kato and Saito (cf. [@Ka86] and [@Sai87]), as well as the work of Merkurjev and Suslin on reduced norm criterion and norm principles ([@Suslin85], [@Mer96]). For spinor groups and groups of type ${}^2A_n^*$ of even index, we also make use of results on quadratic forms over the base field $K$ obtained in [@PaSu], [@Le10] (see also [@HHK]) in the $p$-adic case and in [@Hu11] in the local henselian case.
Some reviews and basic tools {#sec2}
============================
In this section, we briefly review some basic notions which will be used frequently and we recall some known results that are essential in the proofs to come later.
Throughout this section, let $L$ denote a field of characteristic different from 2.
Hermitian forms and Witt groups
-------------------------------
We will assume the readers have basic familiarity with the theory of involutions and hermitian forms over central simple algebras (cf. [@Schar], [@Knus], [@KMRT]). For later use, we recall in this subsection some facts on Witt groups, the “key exact sequence” of Parimala, Sridharan and Suresh and the exact sequence of Suresh. The readers are referred to [@BP1 $\S$3 and Appendix$\;$2], [@BP2 $\S$3] and [@PaPr $\S$8] for more information.
Unless otherwise stated, all hermitian forms and skew-hermitian forms (in particular all quadratic forms) in this paper are assumed to be nonsingular.
\[para2p1NEW\] Let $L$ be a field of characteristic different from 2, $A$ a central simple algebra over $L$ and $\sigma$ an involution on $A$. Let $E=L^{\sigma}$. We say that $\sigma$ is an $L/E$-*involution* on $A$. To each hermitian or skew-hermitian form $(V,\,h)$ over $(A,\,\sigma)$, one can associate an involution on ${\mathrm{End}}_A(V)$, called the *adjoint involution* on ${\mathrm{End}}_A(V)$ with respect to $h$. This is the unique involution $\sigma_h$ on ${\mathrm{End}}_A(V)$ such that $$h(x,\,f(y))=h(\sigma_h(f)(x)\,,\,y)\,,\quad\forall\;x,\,y\in
V\,,\;\;\forall\;f\in{\mathrm{End}}_A(V)\,.$$
For a fixed finitely generated right $A$-module $V$, define an equivalence relation $\sim$ on the set of hermitian or skew-hermitian forms on $V$ (with respect to the involution $\sigma$) by $$h\,\sim\, h'\;\iff\; \text{there exists }\;\lambda\in E^* \text{
such that }h=\lambda.h'\,.$$Let $\mathcal{H}^+(V)$ (resp. $\mathcal{H}^-(V)$) denote the set of equivalence classes of hermitian (resp. skew-hermitian) forms on $V$ and let $\mathcal{H}^{\pm}(V)=\mathcal{H}^+(V)\cup\mathcal{H}^-(V)$. The assignment $h\mapsto \sigma_h$ defines a map from $\mathcal{H}^{\pm }(V)$ to the set of involutions on ${\mathrm{End}}_A(V)$. If $\sigma$ is of the first kind, then the map $h\mapsto \sigma_h$ induces a bijection between $\mathcal{H}^{\pm}(V)$ and the set of involutions of the first kind on ${\mathrm{End}}_A(V)$, and the involutions $\sigma_h$ and $\sigma$ have the same type (orthogonal or symplectic) if $h$ is hermitian and they have opposite types if $h$ is skew-hermitian. If $\sigma$ is of the second kind, then the map $h\mapsto \sigma_h$ induces a bijection between $\mathcal{H}^+(V)$ and the set of $L/E$-involutions on ${\mathrm{End}}_A(V)$. (cf. [@KMRT p.43, Thm.$\;$4.2].)
If $A=L$ and $\sigma=\mathrm{id}$, a hermitian (resp. skew-hermitian) form $h$ is simply a symmetric (resp. skew-symmetric) bilinear form $b$. In this case, $b\mapsto \sigma_b$ defines a bijection between equivalence classes of nonsingular symmetric or skew-symmetric bilinear forms on $V$ modulo multiplication by a factor in $L^*$ and involutions of the first kind on ${\mathrm{End}}_L(V)$. If $q$ is the quadratic form associated to a symmetric bilinear form $b$, we also write $\sigma_q$ for the adjoint involution $\sigma_b$.
\[para2p2NEW\] Let $(A,\,\sigma)$ be a pair consisting of a central simple algebra $A$ over a field $L$ of characteristic $\neq 2$ and an involution (of any kind) $\sigma$ on $A$. The orthogonal sum of hermitian forms defines a semigroup structure on the set of isomorphism classes of hermitian forms over $(A,\,\sigma)$. The quotient of the corresponding Grothendieck group by the subgroup generated by hyperbolic forms is called the *Witt group* of $(A,\,\sigma)$ and denoted $W(A,\,\sigma)=W^1(A,\,\sigma)$. The same construction applies to skew-hermitian forms and the corresponding Witt group will be denoted $W^{-1}(A,\,\sigma)$.
If $A=L$ and $\sigma=\mathrm{id}$, then $W(A,\,\sigma)$ is the usual Witt group $W(L)$ of quadratic forms (cf. [@Lam], [@Schar]). One has a ring structure on $W(L)$ induced by the tensor product of quadratic forms. The classes of even dimensional forms form an ideal $I(L)$ of the ring $W(L)$. For each $n\ge 1$, we write $I^n(L)$ for the $n$-th power of the ideal $I(L)$. As an abelian group, $I^n(L)$ is generated by the classes of $n$-fold Pfister forms.
\[para2p3NEW\] Let $D$ be a quaternion division algebra over a field $L$ of characteristic $\neq 2$. Let $\tau_0$ be the standard (symplectic) involution on $D$. The Witt group $W(D,\,\tau_0)$ has a nice description as follows (cf. [@Schar p.352]).
If $h: V\times V\to D$ is a hermitian form over $(D,\,\tau_0)$, then the map $$q_h\,:V{\longrightarrow}L\,,\quad q_h(x):=h(x,\,x)$$defines a quadratic form on the $L$-vector space $V$, called the *trace form* of $h$. If $h$ is isomorphic to the diagonal form $\langle \lambda_1\,,\dotsc, \lambda_r\rangle$, then $q_{h}$ is isomorphic to the form $\langle \lambda_1\,,\dotsc,
\lambda_r\rangle\otimes n_D$, where $n_D$ denotes the norm form of the quaternion algebra $D$. By [@Schar p.352, Thm.$\;$10.1.7], the assignment $h\mapsto q_h$ induces an injective group homomorphism $W(D,\,\tau_0)\to W(L)$, whose image is the principal ideal of $W(L)$ generated by (the class of) the norm form $n_D$ of $D$. In particular, two hermitian forms over $(D,\,\tau_0)$ are isomorphic if and only if their trace forms are isomorphic.
\[para2p4NEW\] Let $L/E$ be a quadratic extension of fields of characteristic different from 2. The nontrivial element $\iota$ of the Galois group ${\mathrm{Gal}}(L/E)$ may be viewed as a unitary involution on the $L$-algebra $A=L$. The Witt group $W(L,\,\iota)$ can be determined as follows (cf. [@Schar pp.348–349]):
As in (\[para2p3NEW\]), to each hermitian form $h: V\times V\to L$ over $(L,\,\iota)$, one can associate a quadratic form $q_h$ on the $E$-vector space $V$, called the *trace form* of $h$, by defining $$q_h(x):=h(x,\,x)\in E\,,\,\forall\;x\in V\,.$$One can show that $h\mapsto q_h$ induces a group homomorphism $W(L,\,\iota)\to W(E)$ which identifies $W(L,\,\iota)$ with the kernel of the base change homomorphism $W(E)\to W(L)$. In particular, two hermitian forms over $(L,\,\iota)$ are isomorphic if and only if their trace forms are isomorphic. (cf. [@Schar Thm.$\;$10.1.2].)
Let $\delta\in E$ be an element such that $L=E(\sqrt{\delta})$. Then for $a\in E^*$, the trace form of $h=\langle a\rangle$ is isomorphic to $\langle a\,,\,-a\delta\rangle=a.\langle
1\,,\,-\delta\rangle$. So the image of the map $$W(L\,,\,\iota){\longrightarrow}W(E)\,;\quad h\mapsto q_h$$is the principal ideal generated by the form $\langle 1\,,\,-\delta\rangle$ (cf. [@Schar Remark$\;$10.1.3]).
\[para2p5NEW\] Let $A$ be a central simple algebra over a field $L$ of characteristic ${\mathrm{char}}(L)\neq 2$. Let $\sigma$ be an involution on $A$ and let $E=L^{\sigma}$. For any invertible element $u\in A^*$, let ${\mathrm{Int}}(u): A\to A$ denote the inner automorphism $x\mapsto u.x.u^{-1}$. If $\sigma(u)u^{-1}=\pm 1$, then ${\mathrm{Int}}(u)\circ \sigma$ is an involution on $A$ of the same kind as $\sigma$.
Conversely, let $\sigma,\,\tau$ be involutions of the same kind on $A$. If $\sigma$ and $\tau$ are of the first kind, then there is a unit $u\in A^*$, uniquely determined up to a scalar factor in $E^*$, such that $\tau={\mathrm{Int}}(u)\circ \sigma$ and $\sigma(u)=\pm u$. Moreover, the two involutions $\sigma$ and $\tau={\mathrm{Int}}(u)\circ
\sigma$ are of the same type (orthogonal or symplectic) if and only if $\sigma(u)=u$. If $\sigma$ and $\tau$ are of the second kind, then there exists a unit $u\in A^*$, uniquely determined up to a scalar factor in $E^*$, such that $\tau={\mathrm{Int}}(u)\circ \sigma$ and $\sigma(u)=u$.
Let ${\mathfrak{H}}(A,\,\sigma)={\mathfrak{H}}^1(A,\,\sigma)$ (resp. ${\mathfrak{H}}^{-1}(A,\,\sigma)$) denote the category of hermitian (resp. skew-hermitian) forms over $(A,\,\sigma)$. Let ${\varepsilon},\,{\varepsilon}'\in\{\pm 1\}$. Let $a\in A^*$ be an element such that $\sigma(a)={\varepsilon}' a$. Then the functor $$\Phi_a\,:\;\;{\mathfrak{H}}^{{\varepsilon}}(A\,,\,{\mathrm{Int}}(a^{-1})\circ \sigma){\longrightarrow}{\mathfrak{H}}^{{\varepsilon}{\varepsilon}'}(A,\,\sigma)\,;\quad (V,\,h)\longmapsto (V,\,a.h)$$is an equivalence of categories, called a *scaling*. There is also an induced isomorphism of Witt groups $$\phi_a\,:\;\;W^{{\varepsilon}}(A\,,\,{\mathrm{Int}}(a^{-1})\circ\sigma){\xrightarrow{\sim}}W^{{\varepsilon}{\varepsilon}'}(A,\,\sigma)\,.$$ In particular, if $\sigma$ and $\tau$ are involutions of the same kind and type on $A$, then there is a scaling isomorphism of Witt groups $\phi_a: W(A,\,\tau){\xrightarrow{\sim}}W(A,\,\sigma)$.
\[para2p6NEW\] Let $A$ be a central simple algebra over a field $L$ of characteristic $\neq 2$ and $\sigma$ an involution of any kind on $A$. Let $(V,\,h)$ be a hermitian form over $(A,\,\sigma)$. Let $B={\mathrm{End}}_A(V)$ and let $\sigma_h$ be the adjoint involution with respect to $h$. There is an equivalence of categories, called the *Morita equivalence*, $$\Phi_h\,:\;\;{\mathfrak{H}}(B\,,\,\sigma_h){\longrightarrow}{\mathfrak{H}}(A\,,\,\sigma)$$defined as follows (cf. [@BP1 $\S$1.4], [@Knus $\S$I.9]): For a hermitian form $(M,\,f)$ over $(B,\sigma_h)$, define a map $$h*f\,:\;\;(M\otimes_BV)\times (M\otimes_BV){\longrightarrow}A$$by $$(h*f)(m_1\otimes v_1\,,\,m_2\otimes
v_2):=h(v_1\,,\,f(m_1,\,m_2)(v_2))\,.$$One verifies that $\Phi_h(M\,,\,f):=(M\otimes_BV\,,\,h*f)$ yields a well-defined functor ${\mathfrak{H}}(B\,,\,\sigma_h)\to {\mathfrak{H}}(A\,,\,\sigma)$, which can be shown to be an equivalence (cf. [@Knus p.56, Thm.$\;$I.9.3.5]). The Morita equivalence induces an isomorphism of Witt groups: $$\phi_h\,:\;\;W({\mathrm{End}}_A(V)\,,\,\sigma_h){\xrightarrow{\sim}}W(A\,,\,\sigma)\,.$$
\[para2p7NEW\] We briefly recall the construction of the key exact sequence of Parimala, Sridharan and Suresh. The readers are referred to [@BP1 $\S$3 and Appendix$\;$2] for more details.
Let $(A,\,\sigma)$ be a central simple algebra with involution over $L$. Let $E=L^{\sigma}$. Assume there is a subfield $M\subseteq A$ which is a quadratic extension of $L$ such that $\sigma(M)=M$. Suppose $\sigma|_M=\mathrm{id}_M$ if $\sigma$ is of the first kind. Let $${\widetilde{A}}:=\{a\in A\,|\,a.m=m.a\,,\forall\; m\in M\}$$be the centralizer of $M$ in $A$. This is a central simple algebra over $M$. By [@BP1 Lemma$\;$3.1.1], there exists $\mu\in A^*$ such that $\sigma(\mu)=-\mu$ and that the restriction of ${\mathrm{Int}}(\mu)$ to $M$ is the nontrivial element of the Galois group ${\mathrm{Gal}}(M/L)$.
Set $\tau={\mathrm{Int}}(\mu)\circ\sigma$ and let $\tau_1,\,\tau_2$ be the restrictions of $\tau$ and $\sigma$ to ${\widetilde{A}}$ respectively. Then $\tau_1$ is an involution of the second kind, $\tau_2$ is of the same kind and type as $\sigma$, and $\tau$ is orthogonal (resp. symplectic) if and only if $\sigma$ is symplectic (resp. orthogonal).
One has a decomposition $A={\widetilde{A}}\oplus\mu.{\widetilde{A}}$ (as right $M$-modules). Let $\pi_1,\,\pi_2: A\to{\widetilde{A}}$ be the $M$-linear projections $$\pi_1(x+\mu y)=x\,,\quad \pi_2(x+\mu
y)=y\,,\quad\forall\;x,\,y\in{\widetilde{A}}\,.$$These induce well-defined group homomorphisms $$\pi_1\,:\;W(A,\,\tau){\longrightarrow}W({\widetilde{A}},\,\tau_1) \quad\text{and }\quad
\pi_2\,:\;W^{-1}(A,\,\tau){\longrightarrow}W({\widetilde{A}}\,,\,\tau_2)\,.$$On the other hand, let $\lambda\in M$ be an element such that $\lambda^2\in
L$ and $M=L(\lambda)$. For a hermitian form $({\widetilde{V}},\,f)$ over $({\widetilde{A}},\,\tau_1)$, define $\rho(f)$ to be the unique skew-hermitian form on $V={\widetilde{V}}\oplus {\widetilde{V}}\mu$ which extends $\lambda.f: {\widetilde{V}}\times{\widetilde{V}}\to {\widetilde{A}}$. This defines a group homomorphism $$\rho\,:\;W({\widetilde{A}},\,\tau_1){\longrightarrow}W^{-1}(A,\,\tau)\,;\quad
({\widetilde{V}},\,f)\mapsto ({\widetilde{V}}\oplus{\widetilde{V}}\mu\,,\,\rho(f))\,.$$
The sequence $$\label{eq2p7p1NEW}
W^{{\varepsilon}}(A\,,\,\tau)\overset{\pi_1}{{\longrightarrow}}
W^{{\varepsilon}}({\widetilde{A}}\,,\,\tau_1)\overset{\rho}{{\longrightarrow}}W^{-{\varepsilon}}(A\,,\,\tau)\overset{\pi_2}{{\longrightarrow}}
W^{{\varepsilon}}({\widetilde{A}}\,,\,\tau_2)$$turns out to be an exact sequence (cf. [@BP1 Appendix$\;$2]).
Since $\tau(\mu)=-\mu$, one has a scaling isomorphism (cf. (\[para2p5NEW\])) $$\phi_{\mu}^{-1}\,:\;W^{-1}(A\,,\,\tau)\overset{\sim}{{\longrightarrow}}
W(A\,,\,\sigma).$$We may thus replace $W^{-1}(A\,,\,\tau)$ in the exact sequence by $W(A,\,\sigma)$ and rewrite it as $$\label{eq2p7p2NEW}
W(A\,,\,\tau)\overset{\pi_1}{{\longrightarrow}}W({\widetilde{A}}\,,\,\tau_1)\overset{{\widetilde{\rho}}}{{\longrightarrow}}W(A\,,\,\sigma)\overset{{\widetilde{\pi}}_2}{{\longrightarrow}}W({\widetilde{A}}\,,\,\tau_2)\,$$where ${\widetilde{\rho}}=\phi_{\mu}^{-1}\circ\rho$ and ${\widetilde{\pi}}_2=\pi_2\circ\phi_{\mu}$. This exact sequence is due to Parimala, Sridharan and Suresh and is referred to as the *key exact sequence* in [@BP1].
We will only use the exact sequence in the case where $A=D$ is a quaternion algebra and $\sigma$ is an orthogonal involution. This special case was already discussed by Scharlau in [@Schar p.359].
\[para2p8v2\] Now let $D$ be a quaternion division algebra over a quadratic field extension $L$ of $E$ and let $\tau$ be a unitary $L/E$-involution on $D$ (i.e. a unitary involution such that $L^{\tau}=E$). There is a unique quaternion $E$-algebra $D_0$ contained in $D$ such that $D=D_0\otimes_EL$ and $\tau=\tau_0\otimes\iota$, where $\tau_0$ is the canonical (symplectic) involution on $D_0$ and $\iota$ is the nontrivial element of the Galois group ${\mathrm{Gal}}(L/E)$. Write $L=E(\sqrt{d})$ with $d\in E^*$. Then $D=D_0\oplus D_0\sqrt{d}$. For any hermitian form $(V,\,h)$ over $(D,\,\tau)$, we may write $$h(x,\,y)=h_1(x,\,y)+h_2(x,\,y)\sqrt{d}\quad \text{ with }\;h_i(x,\,y)\in D_0\,,\quad\text{for }\; i=1,\,2$$for any $x,\,y\in V$.
The projection $h\mapsto h_2$ defines a group homomorphism $$p_2\,:\;\;W(D,\,\tau){\longrightarrow}W^{-1}(D_0\,,\,\tau_0)\,.$$ For a hermitian form $(V_0,\,f)$ over $(D_0,\,\tau_0)$, set $$V=V_0\otimes_{D_0}D=V_0\otimes_EL=V_0\oplus V_0\sqrt{d}\,$$and let ${\widetilde{\rho}}(f)\,:\;V\times V\to D$ be the map extending $f:V_0\times V_0\to D_0$ by $\tau$-sesquilinearity. One checks that this defines a group homomorphism $${\widetilde{\rho}}\,:\;\;W(D_0,\,\tau_0){\longrightarrow}W(D,\,\tau)\,;\quad (V_0,\,f)\longmapsto (V_0\oplus V_0\sqrt{d}\,,\,{\widetilde{\rho}}(f))\,.$$ For any quadratic form $q$ over $L=E(\sqrt{d})$, there are quadratic forms $q_1,\,q_2$ over $k$ such that $q(x)=q_1(x)+q_2(x)\sqrt{d}$. We have thus group homomorphisms $$\pi_i\,:\;\;W(L){\longrightarrow}W(E)\,;\;\;q\longmapsto q_i\,,\quad i=1,\,2\,.$$ We denote by ${\widetilde{\pi}}_1\,:\;W(L)\to W(D_0,\,\tau_0)$ the composite map $$W(L)\overset{\pi_1}{{\longrightarrow}} W(E){\longrightarrow}W(D_0,\,\tau_0)$$where the map $W(E)\to W(D_0,\,\tau_0)$ is induced by base change.
Suresh (cf. [@PaPr Prop.$\;$8.1]) proved that the sequence $$W(L)\overset{{\widetilde{\pi}}_1}{{\longrightarrow}}W(D_0\,,\,\tau_0)\overset{{\widetilde{\rho}}}{{\longrightarrow}}W(D\,,\,\tau)\overset{p_2}{{\longrightarrow}}W^{-1}(D_0,\,\tau_0)$$is exact. We will refer to this sequence as Suresh’s exact sequence in the sequel.
Invariants of hermitian forms {#sec2p2}
-----------------------------
In this subsection, we recall the definitions of some invariants of hermitian forms. For more details, see [@BP1 $\S$2], [@BP2 $\S$3] and [@PaPr $\S$5, $\S$7].
\[para2p8NEW\] Let $(D,\,\sigma)$ be a central division algebra with involution over $L$. Let $E=L^{\sigma}$. Let $(V,\,h)$ be a hermitian form over $(D,\,\sigma)$. The *rank* of $(V,\,h)$, denoted ${\mathrm{rank}}(V,\,h)$ or simply ${\mathrm{rank}}(h)$, is by definition the rank of the $D$-module $V$: $${\mathrm{rank}}(h):={\mathrm{rank}}_D(V)\,.$$
\[para2p9NEW\] With notation as in (\[para2p8NEW\]), let $e_1,\dotsc, e_n$ be a basis of the $D$-module $V$ (so that ${\mathrm{rank}}(h)={\mathrm{rank}}_D(V)=n$). Let $M(h):=(h(e_i,\,e_j))$ be the matrix of the hermitian form $h$ with respect to this basis. The matrix algebra $A={\mathrm{M}}_n(D)$ has dimension $$\dim_LA=n^2\dim_LD=({\mathrm{rank}}(h).\,\deg_LD)^2\,.$$Put $$m=\sqrt{\dim_LA}={\mathrm{rank}}(h).\,\deg_LD=\frac{\dim_LV}{\deg_LD}\,.$$We define the *discriminant* ${\mathrm{disc}}(h)={\mathrm{disc}}(V,\,h)$ of the hermitian form $(V,\,h)$ by $${\mathrm{disc}}(h)=(-1)^{\frac{m(m-1)}{2}}{\mathrm{Nrd}}_A(M(h))\;\in\;\begin{cases}
E^*/E^{*2}\;\;&\; \text{ if $\sigma$ is of the first kind}\; \\
E^*/N_{L/E}(E^*)\;\;& \; \text{ if $\sigma$ is of the second kind}
\end{cases}$$If $h$ is a hermitian form over $(D,\,\sigma)$, the image of the canonical map $$H^1(E\,,\,{\mathbf{SU}}(h)){\longrightarrow}H^1(E\,,\,{\mathbf{U}}(h))$$consists of classes $[h']\in H^1(E\,,\,{\mathbf{U}}(h))$ of hermitian forms $h'$ which have the same rank and discriminant as $h$.
\[para2p10NEW\] Let $D$ be a cenral division algebra over $L$ and let $\sigma$ be an orthogonal involution on $D$. Note that the Brauer class of $D$ in the Brauer group ${\mathrm{Br}}(L)$ lies in the subgroup $${}_2{\mathrm{Br}}(L):=\{{\alpha}\in{\mathrm{Br}}(L)\,|\,2.{\alpha}=0\}\,.$$
Let $h$ be a hermitian form over $(D,\,\sigma)$. Let $$\delta\,:\;\;H^1(L\,,\,{\mathbf{SU}}(h)){\longrightarrow}H^2(L\,,\,\mu_2)={}_2{\mathrm{Br}}(L)$$be the connecting map associated to the exact sequence of algebraic groups $$1{\longrightarrow}\mu_2{\longrightarrow}{\mathbf{Spin}}(h){\longrightarrow}{\mathbf{SU}}(h){\longrightarrow}1\,.$$Let $h'$ be a hermitian form over $(D,\,\sigma)$ such that ${\mathrm{rank}}(h')={\mathrm{rank}}(h)$ and ${\mathrm{disc}}(h')={\mathrm{disc}}(h)$. Then there is an element $c(h')\in H^1(L,\,{\mathbf{SU}}(h))$ which lifts $[h']\in
H^1(L\,,\,{\mathbf{U}}(h))$. The class of $\delta(c(h'))$ in the quotient ${}_2{\mathrm{Br}}(L)/\langle [D]\rangle$ is independent of the choice of $c(h')$ (cf. [@BP1 $\S$2.1]). Following [@Bar], we define the *relative Clifford invariant* $\mathscr{C}\ell_h(h')$ by $$\mathscr{C}\ell_h(h'):=[\delta(c(h'))]\;\in\;\frac{{}_2{\mathrm{Br}}(L)}{\langle[D]\rangle}\,.$$
When $h$ has even rank $2n$ and trivial discriminant, the *Clifford invariant* $\mathscr{C}\ell(h)$ of $h$ is defined as $$\mathscr{C}\ell(h):=\mathscr{C}\ell_{H_{2n}}(h)\;\in\;\frac{{}_2{\mathrm{Br}}(L)}{\langle[D]\rangle}\,,$$where $H_{2n}$ denotes a hyperbolic hermitian form of rank $2n={\mathrm{rank}}(h)$ over $(D,\,\sigma)$. If $D=L$ and $h=q$ is a nonsingular quadratic form over $L$, then $\mathscr{C}\ell(h)$ coincides with the usual Clifford invariant of the quadratic form $q$.
\[para2p11NEW\] Let $(D,\,\sigma)$ be a central division algebra with an orthogonal involution over $L$. We denote by ${\mathbf{U}}_{2n}(D,\,\sigma)$, ${\mathbf{SU}}_{2n}(D,\,\sigma)$ and ${\mathbf{Spin}}_{2n}(A,\,\sigma)$ respectively the unitary group, the special unitary group and the spin group of the hyperbolic form over $(D,\,\sigma)$ defined by the matrix $H_{2n}={\bigl(\begin{smallmatrix} {0}& {I_n}\\ {I_n}&{0}\end{smallmatrix}\bigl)}$.
Let $h$ be a hermitian form of even rank $2n$, trivial discriminant and trivial Clifford invariant. There is an element $\xi\in H^1(L,\,{\mathbf{Spin}}_{2n}(D,\,\sigma))$ which is mapped to the class $[h]\in H^1(L,\,{\mathbf{U}}_{2n}(D,\,\sigma))$ under the composite map $$H^1(L,\,{\mathbf{Spin}}_{2n}(D,\,\sigma)){\longrightarrow}H^1(L\,,\,{\mathbf{SU}}_{2n}(D,\,\sigma)){\longrightarrow}H^1(L\,,\,{\mathbf{U}}_{2n}(D,\,\sigma))\,.$$Let $$R_{{\mathbf{Spin}}_{2n}(D,\,\sigma)}\,:\;\;
H^1(L\,,\,{\mathbf{Spin}}_{2n}(D,\,\sigma)){\longrightarrow}H^3(L\,,\,\mathbb{Q}/\mathbb{Z}(2))$$be the usual Rost invariant map of the simply connected group ${\mathbf{Spin}}_{2n}(D,\,\sigma)$ (cf. [@KMRT $\S$31.B]). It is shown in [@BP2 p.664] that the class of $R_{{\mathbf{Spin}}_{2n}(D,\,\sigma)}(\xi)$ in the quotient $$\frac{H^3(L\,,\,\mathbb{Q}/\mathbb{Z}(2))}{H^1(L\,,\,\mu_2)\cup (D)}$$ is well-defined. The *Rost invariant* ${\mathscr{R}}(h)$ of the form $h$ is defined as $${\mathscr{R}}(h):=[R_{{\mathbf{Spin}}_{2n}(D,\,\sigma)}(\xi)]\,\in\;
\frac{H^3(L\,,\,\mathbb{Q}/\mathbb{Z}(2))}{H^1(L\,,\,\mu_2)\cup (D)}\,.$$
\[para2p12NEW\] Let $(D,\,\sigma)$ be a quaternion algebra with an orthogonal involution over $L$. We will need some further analysis on the map ${\widetilde{\rho}}: W({\widetilde{D}}\,,\,\tau_1)\to
W(D,\,\sigma)$ in the exact sequence . Note that in this case ${\widetilde{D}}=M$ is a quadratic field extension of $L$ and $\tau_1$ is the nontrivial element $\iota$ of the Galois group ${\mathrm{Gal}}(M/L)$. Let ${\mathbf{U}}_{2n}(M,\,\iota)$ and ${\mathbf{SU}}_{2n}(M,\,\iota)$ denote the unitary group and the special unitary group of the hyperbolic form over $(M,\,\iota)$ defined by the matrix $H_{2n}={\begin{pmatrix} {0}& {I_n}\\ {I_n}&{0}\end{pmatrix}}$. We have $${\mathbf{U}}_{2n}(M,\,\iota)(L)=\{A\in
{\mathrm{M}}_{2n}(M)\,|\,A.H_{2n}\iota(A)^t=H_{2n}\}\,.$$Note that for $A\in{\mathrm{M}}_{2n}(M)$, $\iota(A)={\mathrm{Int}}(\mu)\circ \sigma(A)=\mu
A\mu^{-1}$ (cf. (\[para2p7NEW\])) and $$A.H_{2n}.\iota(A)^t=H_{2n}\iff
(A.H_{2n}.\iota(A)^t)^t=(H_{2n})^t\iff \iota(A).H_{2n}.A^t=H_{2n}\,.$$Therefore, for $A\in{\mathbf{U}}_{2n}(M,\,\iota)(L)$, we have $$\begin{split}
A.\mu^{-1}\lambda H_{2n}.\sigma(A)^t&=A.\mu^{-1}\lambda H_{2n}.
A^t\\
&=\mu^{-1}(\mu A\mu^{-1})\lambda H_{2n}. A^t\\
&=\mu^{-1}\lambda \iota(A).H_{2n}A^t=\mu^{-1}\lambda H_{2n}
\end{split}$$inside ${\mathrm{M}}_{2n}(D)$. So we have a natural inclusion $${\mathbf{U}}_{2n}(M,\,\iota)(L)\subseteq {\mathbf{U}}(\mu^{-1}\lambda
H_{2n})(L)=\{B\in {\mathrm{M}}_{2n}(D)\,|\, B.\mu^{-1}\lambda H_{2n}.
\sigma(B)^t=\mu^{-1}\lambda H_{2n}\}\,.$$In fact, this defines an inclusion of algebraic groups over $L$: $$\rho'\,:\;\;{\mathbf{U}}_{2n}(M,\,\iota){\longrightarrow}{\mathbf{U}}(\mu^{-1}\lambda
H_{2n})\,;\quad A\longmapsto A\,.$$ By [@KMRT p.402, Example$\;$29.19], any element $\xi$ of $H^1(L,\,{\mathbf{U}}_{2n}(M,\,\iota))$ is represented by a matrix $S\in
{\mathrm GL}_{2n}(M)$ which is symmetric with respect to the adjoint involution $\iota_{H_{2n}}$ on ${\mathrm{M}}_{2n}(M)$, and $\xi$ is the isomorphism class of the hermitian form $H_{2n}S^{-1}$. The natural map $$H^1(L\,,\,{\mathbf{U}}_{2n}(M,\,\iota)){\longrightarrow}H^1(L\,,\,{\mathbf{U}}(\mu^{-1}\lambda
H_{2n}))$$induced by the homomorphism $\rho'$ maps $\xi$ to the class of the hermitian form $\mu^{-1}\lambda H_{2n}S^{-1}$. On the other hand, by the construction of the homomorphism ${\widetilde{\rho}}: W(M,\,\iota)\to
W(D,\,\sigma)$, the form $H_{2n}S'$ over $(M,\,\iota)$ is mapped to the form $\mu^{-1}\lambda H_{2n}S^{-1}$ over $(D,\,\sigma)$. Hence the natural map $$H^1(L,\,{\mathbf{U}}_{2n}(M,\,\iota)){\longrightarrow}H^1(L\,,\,{\mathbf{U}}(\mu^{-1}\lambda
H_{2n}))$$is compatible with the restriction of $\rho$ to forms of rank $2n$.
Clearly, the inclusion $\rho': {\mathbf{U}}_{2n}(M,\,\iota)\to
{\mathbf{U}}(\mu^{-1}\lambda H_{2n})$ induces an inclusion ${\mathbf{SU}}_{2n}(M,\,\iota)\to {\mathbf{SU}}(\mu^{-1}\lambda H_{2n})$ (cf. [@BP2 p.671]). A choice of isomorphism of hermitian forms $\mu^{-1}\lambda H_{2n}\cong H_{2n}$ over $(D,\,\sigma)$ yields an injection $${\mathbf{SU}}_{2n}(M,\,\iota){\longrightarrow}{\mathbf{SU}}(H_{2n})={\mathbf{SU}}_{2n}(D,\,\sigma)\,.$$This lifts to a homomorphism $$\rho_0\,:\;\;{\mathbf{SU}}_{2n}(M,\,\iota){\longrightarrow}{\mathbf{Spin}}_{2n}(D,\,\sigma)\,.$$The composition $${\mathbf{SU}}_{2n}(M,\,\iota)\overset{\rho_0}{{\longrightarrow}}{\mathbf{Spin}}_{2n}(D,\,\sigma){\longrightarrow}{\mathbf{U}}_{2n}(D,\,\sigma)$$induces a commutative diagram $$\xymatrix{ H^1(L\,,\,{\mathbf{SU}}_{2n}(M\,,\,\iota)) \ar[dr]_{\rho'}
\ar[rr]^{\rho_0} && H^1(L\,,\,{\mathbf{Spin}}_{2n}(D\,,\,\sigma)) \ar[dl]_{} \\
& H^1(L\,,\,{\mathbf{U}}_{2n}(D\,,\,\sigma)) & }$$such that the map ${\widetilde{\rho}}\,:\;W(M\,,\,\iota)\to W(D\,,\,\sigma)$ restricted to forms of rank $2n$ and of trivial discriminant is compatible with the map $\rho'$ at the level of cohomology sets. Moreover, for any $\xi\in
H^1(L,\,{\mathbf{SU}}_{2n}(M,\,\iota))$, one has by [@BP2 Prop.$\;$3.20] $$R_{{\mathbf{Spin}}_{2n}(D\,,\,\sigma)}(\rho_0(\xi))=R_{{\mathbf{SU}}_{2n}(M\,,\,\iota)}(\xi)\,\in\;
H^3(L\,,\,\mathbb{Q}/\mathbb{Z}(2))\,,$$i.e., $\rho_0(\xi)\in H^1(L\,,\,{\mathbf{Spin}}_{2n}(D,\,\sigma))$ has the same Rost invariant as $\xi$. If $h$ is a hermitian form over $(D,\,\sigma)$ representing the class $\rho'(\xi)\in
H^1(L,\,{\mathbf{U}}_{2n}(D,\,\sigma))$, then the Rost invariant of the form $h$ is $${\mathscr{R}}(h)=[R_{{\mathbf{Spin}}_{2n}(D\,,\,\sigma)}(\rho_0(\xi))]=
[R_{{\mathbf{SU}}_{2n}(M\,,\,\iota)}(\xi)]\,\in\;\frac{H^3(L\,,\,\mathbb{Q}/\mathbb{Z}(2))}{H^1(L\,,\,\mu_2)\cup
(D)}$$by definition (cf. (\[para2p11NEW\])).
\[para3p1TEMP\] We shall also use the notion of Rost invariant of hermitian forms over an algebra with unitary involution. The definition is as follows. Let $E$ be a field of characteristic $\neq 2$, $L/E$ a quadratic field extension and $(D,\,\tau)$ a central division algebra over $L$ with a unitary $L/E$-involution. Let ${\mathbf{U}}_{2n}(D,\,\tau)$ and ${\mathbf{SU}}_{2n}(D,\,\tau)$ denote respectively the unitary group and the special unitary group of the hyperbolic form ${\bigl(\begin{smallmatrix} {0}& {I_n}\\ {I_n}&{0}\end{smallmatrix}\bigl)}$ over $(D,\,\tau)$. For a hermitian form $h$ of rank $2n$ and trivial discriminant over $(D,\,\tau)$, we may define its *Rost invariant* $\mathscr{R}(h)$ by $$\mathscr{R}(h):=[R_{{\mathbf{SU}}_{2n}(D,\,\tau)}(\xi)]\,\in\; \frac{H^3(E,\,\mathbb{Q}/\mathbb{Z}(2))}{\mathrm{Cores}_{L/E}((L^{*1})\cup (D))}\,,$$where $\xi\in H^1(E,\,{\mathbf{SU}}_{2n}(D,\,\tau))$ is any lifting of the class $[h]\in H^1(E,\,{\mathbf{U}}_{2n}(D,\,\tau))$ and $$L^{*1}=(R^1_{L/E}\mathbb{G}_m)(E)=\{ a\in L^*\,|\, N_{L/E}(a)=1 \}\,.$$Indeed, by [@PaPr Appendix, Remark$\;$B], the class $[R_{{\mathbf{SU}}_{2n}(D,\,\tau)}(\xi)]$ is independent of the choice of the lifting $\xi$, so that this Rost invariant $\mathscr{R}(h)$ is well defined. Note that if $D=D_0\otimes_EL$ for some central division algebra $D_0$ over $E$, then $$\mathrm{Cores}_{L/E}((L^{*1})\cup (D))=0$$and hence the Rost invariant of $h$ is simply the usual Rost invariant of any lifting $\xi\in H^1(E,\,{\mathbf{SU}}_{2n}(D,\,\tau))$ of the isomorphism class of $h$.
Spinor norms
------------
\[para4p1temp\] Let $E$ be a field of characteristic different from 2, $A$ a central simple algebra over $E$ and $\sigma$ an orthogonal involution on $A$. Let $h$ be a nonsingular hermitian form over $(A\,,\,\sigma)$. The exact sequence of algebraic groups $$1{\longrightarrow}\mu_2{\longrightarrow}{\mathbf{Spin}}(h){\longrightarrow}{\mathbf{SU}}(h){\longrightarrow}1\,,$$induces a connecting map $$\delta\,:\;\;{\mathbf{SU}}(h)(E){\longrightarrow}H^1(E\,,\,\mu_2)=E^*/E^{*2}$$which we call the *spinor norm* map. We will write $${\mathrm{Sn}}(h_E):=\mathrm{Im}\left(\delta\,:\;{\mathbf{SU}}(h)(E){\longrightarrow}E^*/E^{*2}\right)$$for the image of the above spinor norm map. If $A=E$, $\sigma=\mathrm{id}$ and $h=q$ is a quadratic form, the spinor norm map $\delta: {\mathbf{SO}}(q)(E)\to E^*/E^{*2}$ has an explicit description as follows (cf. [@Lam p.108]): Any element $\theta\in {\mathbf{SO}}(q)(E)$ can be written as the product of an even number of hyperplan reflections associated with anisotropic vectors $v_1,\dotsc, v_{2r}$. The spinor norm $\delta(\theta)$ is equal to the class of the product $q(v_1)\cdots q(v_{2r})$ in $E^*/E^{*2}$.
A deep theorem of Merkurjev is the following norm principle for spinor norms.
\[thm4p2temp\] With notation as in $(\ref{para4p1temp})$, assume that $\deg(A).{\mathrm{rank}}(h)$ is even and at least $4$.
Then the image ${\mathrm{Sn}}(h_E)$ of the spinor norm map is equal to the subgroup of $E^*/E^{*2}$ generated by the canonical images of the norm groups $N_{L/E}(L^*)$ over all finite field extensions $L/E$ such that $A_L$ is split and $h_L$ is isotropic.
The following corollary is immediate from the above theorem.
\[coro2p14NEW\] With notation and hypotheses as in Theorem$\;\ref{thm4p2temp}$, for any finite field extension $E'/E$, one has $$N_{E'/E}({\mathrm{Sn}}(h_{E'}))\subseteq {\mathrm{Sn}}(h_E)\,.$$
\[para2p15NEW\] With notation and hypotheses as in Theorem$\;\ref{thm4p2temp}$, the well-known norm principle for reduced norms states that the subgroup ${\mathrm{Nrd}}(A^*)\subseteq E^*$ of reduced norms is generated by the norm groups $N_{L/E}(L^*)$, where $L/E$ runs over all finite field extensions such that $A_L$ is split. So Theorem$\;$\[thm4p2temp\] implies that ${\mathrm{Sn}}(h_E)$ is contained in the canonical image of ${\mathrm{Nrd}}(A^*)$ in $E^*/E^{*2}$.
\[para4p3temp\] Let $(A,\,\sigma)$ be a central simple algebra with an orthogonal involution over a field $E$ of characteristic $\neq 2$. Let $L/E$ be a field extension which splits $A$ and let $\phi:
(A\,,\,\sigma)\otimes_EL\cong ({\mathrm{M}}_n(L)\,,\,\sigma_{q_0})$ be an isomorphism of $L$-algebras with involution, where $\sigma_{q_0}$ is the adjoint involution of a quadratic form $q_0$ of rank $n=\deg(A)$ over $L$. Let $h$ be a hermitian form over $(A\,,\,\sigma)\otimes_EL$. Then by Morita theory (cf. (\[para2p6NEW\])), $h$ corresponds via the above isomorphism $\phi$ to a quadratic form $q$ of rank $n.{\mathrm{rank}}(h)=\deg(A).{\mathrm{rank}}(h)$ over $L$. The similarity class $[q]\in W(L)$ of $q$ is uniquely determined by $h$ and is independent of the choice of $\phi$ and $q_0$. The hermitian form $h_L$ is isotropic if and only if the quadratic form $q_L$ is isotropic. So, if $\deg(A).{\mathrm{rank}}(h)$ is even and at least $4$, one has ${\mathrm{Sn}}(q_L)={\mathrm{Sn}}(h_L)$ by Theorem$\;$\[thm4p2temp\].
Some easy cases {#sec3}
===============
We shall now start the proofs of our main theorems. In a few cases, as may be already well-known to specialists, the results basically follow by combining a general injectivity result for the Rost invariant and a Hasse principle coming from higher dimensional class field theory.
\[para3p1NEW\] Recall that our base field $K$ is the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field (cf. (\[para1p2NEWTEMP\])). Namely, $K$ is either
(the case of $p$-adic arithmetic surface) the function field $F(C)$ of a smooth projective geometrically integral curve $C$ over $F$, where $F$ is a $p$-adic field with ring of integers $A$ and residue field $k$;
or
(the case of local henselian surface) the field of fractions $\mathrm{Frac}(A)$ of a 2-dimensional, henselian, excellent local domain $A$ with finite residue field $k$ of characteristic $p$.
In either case, by abuse of language we say $k$ is the residue field of $K$ and $p=\mathrm{char}(k)$ is the *residue characteristic* of $K$.
In our proofs of the main theorems, we only use local conditions at *divisorial valuations*, i.e., valuations corresponding to codimension 1 points of regular proper models (cf. (\[para1p2NEWTEMP\])). More precisely, the set $\Omega_A$ of divisorial valuations of the field $K$ is the subset of $\Omega_K$ defined as follows:
In $p$-adic arithmetic case, define $$\Omega_A=\bigcup_{\mathcal{X}\to\mathrm{Spec} A}\mathcal{X}^{(1)}\,,$$where $\mathcal{X}\to \mathrm{Spec} A$ runs over proper flat morphisms from a regular integral scheme $\mathcal{X}$ with function field $K$ and $\mathcal{X}^{(1)}$ denotes the set of codimension 1 points of $\mathcal{X}$ identified with a subset of $\Omega_K$.
In the local henselian case, define $$\Omega_A=\bigcup_{\mathcal{X}\to\mathrm{Spec} A}\mathcal{X}^{(1)}\,,$$ where $\mathcal{X}\to \mathrm{Spec} A$ runs over proper birational morphisms from a regular integral scheme $\mathcal{X}$ with function field $K$ and $\mathcal{X}^{(1)}$ denotes the set of codimension 1 points of $\mathcal{X}$ identified with a subset of $\Omega_K$.
\[para3p2NEW\] Let $L/K$ be a finite field extension. Then $L$ is a field of the same type as $K$ if $K$ is the function field of a $p$-adic arithemetic surface or a local henselian surface with finite residue field. In the $p$-adic arithmetic case, let $F'$ be the field of constants of $L$ and let $A'$ be the integral closure of $A$ in $F'$. In the local henselian case, let $A'$ be the integral closure of $A$ in $L$. Then the set $\Omega_{A'}$ of divisorial discrete valuations of $L$ is precisely the set of discrete valuations $w\in\Omega_L$ lying over valuations in $\Omega_A\subseteq \Omega_K$.
\[para3p3NEW\] By the general theory of semisimple groups (see e.g. [@KMRT p.365, Thm.$\;$26.8]), any semisimple simply connected group $G$ over $K$ is a finite product of groups of the form $R_{L/K}(G')$, where $L/K$ is a finite separable field extension, $G'$ is an absolutely simple simply connected group over $L$ and $R_{L/K}$ denotes the Weil restriction functor. For each $v\in\Omega_A$, one has $L\otimes_KK_v\cong\prod_{w\,|\,v}L_w$ and by Shapiro’s lemma, $$H^1(K,\,R_{L/K}G')\cong H^1(L\,,\,G')\quad\text{ and }\quad
H^1(K_v,\,R_{L/K}G')\cong\prod_{w\,|\,v}H^1(L_w\,,\,G')\,.$$ Therefore, to prove the Hasse principle for semisimpe simply connected groups we may reduce to the case where $G$ is an absolutely simple simply connected group.
The quasi-split case
--------------------
We recall the proof of the Hasse principle for quasi-split groups without $E_8$ factors (cf. [@CTPaSu Thm.$\;$5.4]).
The following theorem is of particular importance to us.
\[thm1p3temp\] Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Let $\Omega_A$ be the set of divisorial discrete valuations of $K\,($as defined in $(\ref{para3p1NEW}))$.
$({\mathrm{i}})$ $($Kato, [@Ka86]$)$ In the $p$-adic arithmetic case, the natural map $$H^3(K\,,\,\mathbb{Q}/\mathbb{Z}(2)){\longrightarrow}\prod_{v\in\Omega_A}H^3(K_v\,,\,\mathbb{Q}/\mathbb{Z}(2))$$is injective.
$({\mathrm{ii}})$ $($Saito, [@Sai87], cf. [@Hu11 Prop.$\;$4.1]$)$ In the local henselian case, let $n>0$ be an integer prime to $p$. Then the natural map $$H^3(K\,,\,\mu_n^{\otimes 2}){\longrightarrow}\prod_{v\in\Omega_A}H^3(K_v\,,\,\mu_n^{\otimes 2})$$is injective.
The next result is an injectivity statement for the Rost invariant of quasi-split groups.
\[thm3p5NEW\][^2] Let $E$ be a field of cohomological $2$-dimension $\le 3$ and let $G$ be an absolutely simple simply connected quasi-split group over $E$. Assume that $G$ is not of type $E_8$. Assume further the characteristic of $E$ is not $2$ if $G$ is of classical type $B_n$ or $D_n$.
Then the kernel of the Rost invariant map $R_G: H^1(E,\,G)\to
H^3(E,\,\mathbb{Q}/\mathbb{Z}(2))$ is trivial.
For a quasi-split group of type ${}^1A_n$ or $C_n$, it is well-known that $H^1(E,\,G)=1$ over an arbitrary field $E$. For exceptional groups (not of type $E_8$), the kernel of the Rost invariant is trivial over an arbitrary field by the work of Chernousov, Garibaldi and Gille (cf. [@Gille10 Thm.$\;$5.2], [@Chern03], [@Gari01] and [@Gille00]). If $G$ is of type ${}^2A_n$, $B_n$ or classical type $D_n$, the proof can be done as in [@CTPaSu Thm.$\;$5.3], by passing to a quadratic form argument.
The $p$-adic case of the following result is [@CTPaSu Thm.$\;$5.4].
\[thm3p6NEW\] Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Let $G$ be an absolutely simple simply connected quasi-split group not of type $E_8$ over $K$. Assume $p\notin S(G)$ in the local henselian case $($see $(\ref{para1p5NEWTEMP})$ for the definition of $S(G)\,)$.
Then the natural map $$H^1(K\,,\,G){\longrightarrow}\prod_{v\in\Omega_A}H^1(K_v\,,\,G)$$has a trivial kernel.
The result follows from the following commutative diagram $$\begin{CD}
H^1(K\,,\,G) @>>> \prod_{v\in\Omega_A}H^1(K_v,\,G) \\
@VVV @VVV\\
H^3(K\,,\,\mu_n^{\otimes 2}) @>>>
\prod_{v\in\Omega_A}H^3(K_v,\,\mu_n^{\otimes 2})
\end{CD}$$where the vertical maps have trivial kernel by Theorem$\;$\[thm3p5NEW\] and the bottom horizontal map is injective by Theorem$\;$\[thm1p3temp\].
Groups of type ${}^1A_n^*$
--------------------------
For groups of inner type $A_n^*$, the proof is essentially the same as the quasi-split case.
\[thm1p4temp\] Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Let $A$ a central simple $K$-algebra of square-free index $n$ and $G={\mathbf{SL}}_1(A)$. Assume $p\nmid n$ in the local henselian case.
Then the natural map $$H^1(K,\,G){\longrightarrow}\prod_{v\in\Omega_A}H^1(K_v\,,\,G)$$is injective.
A well-known theorem of Suslin ([@Suslin85 Thm.$\;$24.4]) implies that under the assumptions of the theorem, the Rost invariant map $$H^1(E\,,\,{\mathbf{SL}}_1(A))=E^*/{\mathrm{Nrd}}(A^*){\longrightarrow}H^3(E\,,\,\mu_n^{\otimes
2})\;;\quad \lambda\longmapsto (\lambda)\cup (A)$$is injective for $E=K$ or $K_v$. An argument similar to the proof of Theorem$\;$\[thm3p6NEW\] yields the result.
Groups of type $C_n^*$
----------------------
\[lemma2p1temp\] Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\neq 2$ in the local henselian case.
Then the natural map $$I^3(K){\longrightarrow}\prod_{v\in\Omega_A}I^3(K_v)$$is injective.
Consider the following commutative diagram $$\begin{CD}
I^3(K) @>>> \prod_{v\in\Omega_A}I^3(K_v)\\
@V{e_3}VV @VVV\\
H^3(K\,,\,\mathbb{Z}/2) @>>> \prod_{v\in\Omega_A}H^3(K_v\,,\,\mathbb{Z}/2)
\end{CD}$$where the vertical maps are induced by the Arason invariants. Since $\mathrm{cd}_2(K)\le 3$, we have $I^4(K)=0$. So the map $$e_3\,:\;I^3(K){\longrightarrow}H^3(K\,,\,\mathbb{Z}/2)$$is injective by [@AEJ86 Prop.$\;$3.1]. The map $$H^3(K\,,\,\mathbb{Z}/2){\longrightarrow}\prod_{v\in\Omega_A}H^3(K_v\,,\,\mathbb{Z}/2)$$is injective by Theorem$\;$\[thm1p3temp\]. The lemma then follows from the above commutative diagram.
\[thm2p2temp\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Let $D$ be a quaternion division algebra over $K$ with standard involution $\tau_0$ and $h$ a nonsingular hermitian form over $(D,\,\tau_0)$. Assume $p\neq 2$ in the local henselian case. Let $G={\mathbf{U}}(h)$ be the unitary group of the hermitian form $h$.
Then the natural map $$H^1(K\,,\,G){\longrightarrow}\prod_{v\in\Omega_A}H^1(K_v\,,\,G)$$is injective.
The pointed set $H^1(K\,,\,G)=H^1(K\,,\,{\mathbf{U}}(h))$ classifies up to isomorphism hermitian forms over $(D,\,\tau_0)$ of the same rank as $h$. Let $h_1$ and $h_2$ be hermitian forms over $(D\,,\,\tau_0)$ of the same rank as $h$. Put $h'=h_1\bot (-h_2)$. Note that $h'$ has even rank, so the class of $q_{h'}$ in the Witt group $W(K)$ lies in the subgroup $I^3(K)=I(K)\cdot I^2(K)$ (cf. (\[para2p3NEW\])). Thus $$[q_{h_1}]-[q_{h_2}]=[q_{h'}]\,\in \,I^3(K)\,.$$If $(h_1)_v\cong (h_2)_v$ for all $v\in\Omega_A$, then by Lemma$\;$\[lemma2p1temp\], $[q_{h'}]=0\in I^3(K)$. This implies that $q_{h_1}\cong q_{h_2}$ over $K$. Two hermitian forms over $(D\,,\,\tau_0)$ are isomorphic if and only if their trace forms are isomorphic as quadratic forms (cf. (\[para2p3NEW\])). So we get from the above that $h_1\cong h_2$, proving the theorem.
Groups of type $G_2$ or $F_4^{red}$
-----------------------------------
\[thm6p1temp\] Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Let $G$ be an absolutely simple simply connected group of type $G_2$ over $K$. Assume $p\neq 2$ in the local henselian case.
Then the natural map $$H^1(K\,,\,G){\longrightarrow}\prod_{v\in\Omega_A}H^1(K_v\,,\,G)$$has a trivial kernel.
The group $G$ is isomorphic to $\mathbf{Aut}_{alg}(C)$ for some Cayley algebra $C$ over $K$. Let $\xi\in H^1(K\,,\,G)$ be a locally trivial class and let $C'$ be a Cayley algebra which represents $\xi$. We have $C_{K_v}\cong C'_{K_v}$ for every $v\in\Omega_A$ by hypothesis and we want to show $C\cong C'$ over $K$. Since two Cayley algebras are isomorphic if and only if their norm forms are isomorphic and since the norm form of a Cayley algebra is a 3-fold Pfister form (cf. [@KMRT p.460]), the result follows easily from Lemma$\;$\[lemma2p1temp\].
\[thm6p2temp\] Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\nmid 6$ in the local henselian case. Let $G=\mathbf{Aut}_{alg}(J)$ be the automorphism group of a *reduced* $27$-dimensional exceptional Jordan algebra over $K$.
Then the natural map $$H^1(K\,,\,G){\longrightarrow}\prod_{v\in\Omega_A}H^1(K_v\,,\,G)$$has a trivial kernel.
Recall that (cf. [@Ser94 $\S$9]) to each exceptional Jordan algebra $J'$ of dimension 27 over a field $F$ of characteristic not 2 or 3, one can associate three invariants $$f_3(J')\in H^3(F\,,\,\mathbb{Z}/2)\,,\;\;f_5(J')\in
H^5(F\,,\,\mathbb{Z}/2)\quad\text{and }\quad g_3(J')\in H^3(F\,,\,\mathbb{Z}/3)\,.$$One has $g_3(J')=0$ if and only if $J'$ is reduced. Two reduced exceptional Jordan algebras are isomorphic if and only if their $f_3$ and $f_5$ invariants are the same.
Now our base field $K$ has cohomological 2-dimension $\mathrm{cd}_2(K)=3$. So the invariant $f_5(J')$ is always zero. Let $\xi\in H^1(K\,,\,G)$ correspond to the isomorphism class of an exceptional Jordan algebra $J'$ over $K$. Assume that $\xi$ is locally trivial in $H^1(K_v\,,\,G)$ for every $v\in\Omega_A$. By Theorem$\;$\[thm1p3temp\], we have $f_3(J)=f_3(J')$ and $g_3(J)=g_3(J')$. Since $J$ is reduced by assumption, we have $g_3(J')=0$ and hence $J'$ is reduced. Thus it follows that $J\cong
J'$ over $K$, showing that $\xi$ is trivial in $H^1(K\,,\,G)$ as desired.
Spin groups of quadratic forms {#sec4}
==============================
\[para3p1temp\] Let $E$ be a field of characteristic different from 2 and $q$ a nonsingular quadratic form of rank $\ge
3$ over $E$. Recall that ${\mathrm{Sn}}(q_E)$ denotes the image of the spinor norm map $${\mathbf{SO}}(q)(E){\longrightarrow}E^*/E^{*2}\,,\,$$i.e., the connecting map associated to the cohomology of the exact sequence $$1{\longrightarrow}\mu_2{\longrightarrow}{\mathbf{Spin}}(q){\longrightarrow}{\mathbf{SO}}(q){\longrightarrow}1\,.$$
\[prop3p2temp\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\neq 2$ in the local henselian case. Let $q$ be a nonsingular quadratic form of rank $3$ or $4$ over $K$.
Then the natural map $$\frac{K^*/K^{*2}}{{\mathrm{Sn}}(q_K)}{\longrightarrow}\prod_{v\in\Omega_A}\frac{K_v^*/K_v^{*2}}{{\mathrm{Sn}}(q_{K_v})}$$is injective.
If ${\mathrm{rank}}(q)=3$, we may assume $q=\langle 1\,,\,a\,,\,b\rangle$ after scaling. Let $D$ be the quaternion algebra $(-a\,,\,-b)_K$ over $K$. Then ${\mathrm{Sn}}(q)={\mathrm{Nrd}}(D^*)$ modulo squares. The result then follows from Theorem$\;$\[thm1p4temp\].
Assume next ${\mathrm{rank}}(q)=4$. If ${\mathrm{disc}}(q)=1$, we may assume after scaling $q=\langle 1\,,\,a\,,\,b\,,\,ab\rangle$. Put $D=(-a\,,\,-b)_K$. Then ${\mathrm{Sn}}(q)={\mathrm{Nrd}}(D^*)$ and the result follows again from Theorem$\;$\[thm1p4temp\]. If $d={\mathrm{disc}}(q)$ is nontrivial in $K^*/K^{*2}$, we may assume $q=\langle
1\,,\,a\,,\,b\,,\,abd\rangle$. Then $${\mathrm{Sn}}(q_K)={\mathrm{Nrd}}\left(D_{K(\sqrt{d})}^*\right)\cap K^*\quad\text{modulo
squares}$$by [@KMRT p.214, Coro.$\;$15.11]. The field $K(\sqrt{d})$ is a field of the same type as $K$ (cf. (\[para3p2NEW\])). Let $\Omega_{A'}$ denote the set of divisorial valuations of $K'=K(\sqrt{d})$. If ${\alpha}\in K^*$ lies in ${\mathrm{Sn}}(q_{K_v})$ for all $v\in\Omega_A$, then ${\alpha}$ is a reduced norm from $D_{K'_w}$ for all $w\in\Omega_{A'}$. By Theorem$\;$\[thm1p4temp\], ${\alpha}$ is a reduced norm from $D_{K'}=D_{K(\sqrt{d})}$. This finishes the proof.
Recall that the $u$-invariant $u(E)$ of a field $E$ of characteristic $\neq 2$ is the supremum of dimensions of anisotropic quadratic forms over $E$ (so $u(E)=\infty$, if such dimensions can be arbitrarily large).
\[prop3p3temp\] Let $E$ be a field of characteristic $\neq 2$ and $q$ a nonsingular quadratic form of rank $r$ over $E$. Assume $u(E)<2r$.
Then ${\mathrm{Sn}}(q_E)=E^*/E^{*2}$, i.e., the spinor norm map $${\mathbf{SO}}(q)(E){\longrightarrow}E^*/E^{*2}$$ is surjective.
The image ${\mathrm{Sn}}(q_E)$ of the spinor norm map consists of elements of the form $\prod^{2m}_{i}q(v_i)$, where $v_i$ are anisotropic vectors for $q$ (cf. (\[para4p1temp\])). If $q$ is isotropic over $E$, then for every ${\alpha}\in E^*$, there is a vector $v_{{\alpha}}$ such that $q(v_{{\alpha}})={\alpha}$. Let $v_1$ be a vector such that $q(v_1)=1$. Then we have ${\alpha}=q(v_{{\alpha}}).q(v_1)\in{\mathrm{Sn}}(q_E)$.
Assume next $q$ is anisotropic. For any ${\alpha}\in E^*$, the form $q\bot (-{\alpha}.q)$ is isotropic over $E$ by the assumption on the $u$-invariant. Hence there are vectors $x,\,y$ such that $q(x)-{\alpha}.q(y)=0$. Since $q$ is anisotropic, we have $\lambda:=q(y)\in E^*$ and $q(x)\in E^*$. It follows that $${\alpha}=q(x).q(y)^{-1}=\lambda^{-2}q(x).q(y)=q(x).q(\lambda^{-1}y)\,\in
\,{\mathrm{Sn}}(q_E)\,$$whence the desired result.
\[coro3p4temp\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\neq 2$ in the local henselian case. Let $q$ be a nonsingular quadratic form of rank $\ge 5$ over $K$.
Then ${\mathrm{Sn}}(q_K)=K^*/K^{*2}$, i.e., the spinor norm map $${\mathbf{SO}}(q)(K){\longrightarrow}K^*/K^{*2}$$ is surjective.
In the $p$-adic arithmetic case, we have $u(K)=8$ by [@PaSu] (if $p\neq 2$) or [@Le10] (see also [@HHK]). In the local henselian case, it is proved in [@Hu11 Thm.$\;$1.2] that $u(K)=8$. The result then follows immediately from Proposition$\;$\[prop3p3temp\].
\[thm3p5temp\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\neq 2$ in the local henselian case. Let $q$ be a nonsingular quadratic form of rank $\ge 3$ over $K$ and $G={\mathbf{Spin}}(q)$.
$({\mathrm{i}})$ The natural map $$H^1(K\,,\,G){\longrightarrow}\prod_{v\in\Omega_A}H^1(K_v\,,\,G)$$has a trivial kernel.
$({\mathrm{ii}})$ The Rost invariant $$R_G\,:\;H^1(K\,,\,G){\longrightarrow}H^3(K\,,\,\mathbb{Q}/\mathbb{Z}(2))$$has a trivial kernel if ${\mathrm{rank}}(q)\ge 5$.
Consider the exact sequence of algebraic groups $$1{\longrightarrow}\mu_2{\longrightarrow}{\mathbf{Spin}}(q)=G{\longrightarrow}{\mathbf{SO}}(q){\longrightarrow}1$$which gives rise to an exact sequence of pointed sets $$\label{eq3p5p1temp}
{\mathbf{SO}}(q)(K)\overset{\delta}{{\longrightarrow}}K^*/K^{*2}\overset{\psi}{{\longrightarrow}}
H^1(K\,,\,{\mathbf{Spin}}(q))\overset{\eta}{{\longrightarrow}} H^1(K\,,\,{\mathbf{SO}}(q))\,.$$The image of the map $\eta$ is in bijection with isomorphism classes of nonsingular quadratic forms $q'$ with the same rank, discriminant and Clifford invariant as $q$.
Let $\xi\in H^1(K\,,\,G)=H^1(K\,,\,{\mathbf{Spin}}(q))$ with $\eta(\xi)\in
H^1(K\,,\,{\mathbf{SO}}(q))$ corresponding to a quadratic form $q'$. Then in the Witt group $W(K)$ the class of $q\bot (-q')$ lies in $I^3(K)$ by Merkurjev’s theorem (cf. [@Schar p.89, Thm.$\;$2.14.3]) and its Arason invariant $e_3([q\bot (-q')])\in H^3(K\,,\,\mathbb{Z}/2)$ coincides with Rost invariant $R_G(\xi)$ of $\xi$ when ${\mathrm{rank}}(q)\ge 5$ ([@KMRT p.437]).
For (i), assume the canonical image $\xi_v$ of $\xi$ in $H^1(K_v\,,\,G)$ is trivial for every $v\in\Omega_A$. We have $$[q\bot
(-q')]_v=0\,\in\,I^3(K_v)\,,\quad\forall\;v\in\Omega_A\,.$$By Lemma$\;$\[lemma2p1temp\], we have $q\cong q'$ over $K$. This means that $\xi\in H^1(K,\,G)$ lies in the kernel of $$\eta\,:\;\;H^1(K\,,\,G){\longrightarrow}H^1(K\,,\,{\mathbf{SO}}(q))\,.$$By the exactness of the sequence , $\xi=\psi({\alpha})$ for some ${\alpha}\in
\mathrm{Coker}(\delta)=\frac{K^*/K^{*2}}{{\mathrm{Sn}}(q_K)}$. Consider now the following commutative diagram with exact rows $$\begin{CD}
1 @>>> \frac{K^*/K^{*2}}{{\mathrm{Sn}}(q_K)} @>\psi>> H^1(K\,,\,G) @>\eta>>
H^1(K\,,\,{\mathbf{SO}}(q))\\
&& @VVV @VVV @VVV\\
1 @>>> \prod_{v\in\Omega_A}\frac{K^*_v/K^{*2}_v}{{\mathrm{Sn}}(q_{K_v})}
@>\psi>> \prod_{v\in\Omega_A}H^1(K_v\,,\,G) @>\eta>>
\prod_{v\in\Omega_A}H^1(K_v\,,\,{\mathbf{SO}}(q))
\end{CD}$$The canonical image ${\alpha}_v$ of ${\alpha}$ in $\frac{K^*_v/K^{*2}_v}{{\mathrm{Sn}}(q_{K_v})}$ is trivial for all $v\in\Omega_A$. From Proposition$\;$\[prop3p2temp\] and Corollary$\;$\[coro3p4temp\], it follows that ${\alpha}=1$ and hence $\xi=\psi({\alpha})$ is trivial.
For (ii), assume the Rost invariant $R_G(\xi)$ of $\xi$ is trivial. Then the Arason invariant $e_3([q\bot (-q')])$ is zero. Since $\mathrm{cd}_2(K)\le 3$, the map $e_3: I^3(K)\to H^3(K,\,\mathbb{Z}/2)$ is injective. So we get $q\cong q'$ over $K$ and therefore $\xi=\psi({\alpha})$ for some ${\alpha}\in \frac{K^*/K^{*2}}{{\mathrm{Sn}}(q_K)}$. When the rank of $q$ is $\ge 5$, we have $K^*/K^{*2}={\mathrm{Sn}}(q_K)$ by Corollary$\;$\[coro3p4temp\]. So ${\alpha}=1$ and $\xi$ is trivial.
\[remark3p6temp\] Assertion (ii) of Theorem$\;$\[thm3p5temp\] may be compared with the following result, which was already known to experts (cf. [@CTPaSu Prop.$\;$5.2]): Let $E$ be a field of characteristic $\neq 2$ and of cohomological 2-dimension $\mathrm{cd}_2(E)\le 3$. Let $q$ be an *isotropic* quadratic form of rank $\ge 5$ over $E$. Then the Rost invariant $$H^1(E\,,\,{\mathbf{Spin}}(q)){\longrightarrow}H^3(E\,,\,\mathbb{Q}/\mathbb{Z}(2))$$for the spinor group ${\mathbf{Spin}}(q)$ has a trivial kernel.
Groups of type $D_n^*$ {#sec5}
======================
\[prop5p1NEW\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\neq 2$ in the local henselian case. Let $(D\,,\,\sigma)$ be a quaternion division algebra with an orthogonal involution over $K$ and let $h$ be a hermitian form of rank $\ge 2$ over $(D\,,\,\sigma)$.
Then the natural map $$\frac{K^*/K^{*2}}{{\mathrm{Sn}}(h_K)}{\longrightarrow}\prod_{v\in\Omega_A}\frac{K^*_v/K^{*2}_v}{{\mathrm{Sn}}(h_{K_v})}$$ is injective.
First assume ${\mathrm{rank}}(h)=2$. Put $d={\mathrm{disc}}(h)\in K^*/K^{*2}$. If $d=1\in K^*/K^{*2}$, then $h$ is isotropic and ${\mathrm{Sn}}(h)={\mathrm{Nrd}}(D^*)$ modulo squares by Merkurjev’s norm principle (Theorem$\;$\[thm4p2temp\]). The result then follows from Theorem$\;$\[thm1p4temp\]. Let us assume $d={\mathrm{disc}}(h)\in
K^*/K^{*2}$ is nontrivial. Let $(A\,,\,\tilde{\sigma})=({\mathrm{M}}_2(D)\,,\,\sigma_h)$, where $\sigma_h$ denotes the adjoint involution of $h$ on $A={\mathrm{M}}_2(D)$. The even Clifford algebra $C=C_0(A\,,\,\tilde{\sigma})$ of the pair $(A\,,\,\tilde{\sigma})$ (cf. [@KMRT $\S$8]) is a quaternion algebra over the field $K(\sqrt{d})$ and one has $${\mathrm{Sn}}(h_K)={\mathrm{Nrd}}(C^*)\cap K^*\;\pmod{K^{*2}}\,.$$(cf. [@KMRT p.94, Thm.$\;$8.10 and p.214, Coro.$\;$15.11].) As in the proof of Proposition$\;$\[prop3p2temp\], it follows from Theorem$\;$\[thm1p4temp\] that an element $\lambda\in K^*/K^{*2}$ is a spinor norm for $h_K$ if and only if it is a spinor norm for $h_{K_v}$ for all $v\in\Omega_A$.
Assume next ${\mathrm{rank}}(h)\ge 3$. Let $\lambda\in K^*$ and assume $\lambda$ is a local spinor norm for $h_{K_v}$ for every $v\in\Omega_A$. Merkurjev’s norm principle (Theorem$\;$\[thm4p2temp\]) implies that $\lambda\in
{\mathrm{Nrd}}(D^*_{K_v})$ for every $v\in\Omega_A$. Hence $\lambda\in{\mathrm{Nrd}}(D^*)$ by Theorem$\;$\[thm1p4temp\]. (Note that $K^{*2}\subseteq {\mathrm{Nrd}}(D^*)$ since $D$ is a quaternion algebra.) Let $K'/K$ be a field extension such that $D_{K'}$ is split and $\lambda=N_{K'/K}(\mu)$ for some $\mu\in (K')^*$. By Corollary$\;$\[coro2p14NEW\], $N_{K'/K}({\mathrm{Sn}}(h_{K'}))\subseteq
{\mathrm{Sn}}(h_K)$. Since $\lambda\in K^*/K^{*2}$ lies in the image of $N_{K'/K}: (K')^*/(K')^{*2}\to K^*/K^{*2}$, to show $\lambda$ is a spinor norm for $h_K$ it suffices to show that the map $$\delta'\,:\;{\mathbf{SU}}(h)(K'){\longrightarrow}(K')^*/(K')^{*2}$$is surjective. Note that $D$ splits over $K'$ by the choice of $K'$. So we see from (\[para4p3temp\]) that $\mathrm{Im}(\delta')={\mathrm{Sn}}(h_{K'})={\mathrm{Sn}}(q_{K'})$, where $q_{K'}$ is a quadratic form of rank $2.{\mathrm{rank}}(h)\ge 6$ over $K'$. Now the result follows immediately from Corollary$\;$\[coro3p4temp\].
\[prop4p5temp\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\neq 2$ in the local henselian case. Let $(D\,,\,\sigma)$ be a quaternion division algebra with an orthogonal involution over $K$. Let $h$ be a nonsingular hermitian form of even rank $\ge 2$ over $(D\,,\,\sigma)$. Assume that $h$ has trivial discriminant, trivial Clifford invariant and trivial Rost invariant $($cf. $\S\ref{sec2p2})$.
If the form $h_{K_v}$ over $(D_{K_v}\,,\,\sigma)=(D\otimes_KK_v\,,\,\sigma)$ is hyperbolic for every $v\in\Omega_A$, then the form $h$ over $(D\,,\,\sigma)$ is hyperbolic.
Let $L\subseteq D$ be a subfield which is a quadratic extension over $K$ such that $\sigma(L)=L$ and $\sigma|_L=\mathrm{id}_L$. Such an $L$ exists since $\sigma$ is an orthogonal involution. Let $\mu\in D^*$ be an element such that $\sigma(\mu)=-\mu$, ${\mathrm{Int}}(\mu)(L)=L$ and ${\mathrm{Int}}(\mu)|_L=\iota$, where $\iota$ denotes the nontrivial element of the Galois group ${\mathrm{Gal}}(L/K)$. The involution $\tau:={\mathrm{Int}}(\mu)\circ \sigma$ is a symplectic involution on $D$ (and hence coincides with the canonical involution on the quaternion algebra $D$). The “key exact sequence” of Parimala-Sridharan-Suresh (cf. ) yields the following commutative diagram with exact rows $$\xymatrix{ W(D\,,\,\tau) \ar[d]_{} \ar[r]^{\pi_1} & W(L\,,\,\iota)
\ar[d]_{} \ar[r]^{{\widetilde{\rho}}}
& W(D\,,\,\sigma) \ar[d]_{} \ar[r]^{{\widetilde{\pi}}_2} & W(L) \ar[d]_{} \\
\prod_{v\in\Omega_A}W(D_v\,,\,\tau)\ar[r]^{\pi_1} &
\prod_{v\in\Omega_A}W(L_v\,,\,\iota) \ar[r]^{{\widetilde{\rho}}} &
\prod_{v\in\Omega_A} W(D_v\,,\,\sigma) \ar[r]^{{\widetilde{\pi}}_2} &
\prod_{v\in\Omega_A}W(L_v) }$$ where for any $K$-algebra $B$ we denote $B_v=B\otimes_KK_v$ for each $v\in\Omega_A$. (Here $L_v$ need not be a field. It can be a Galois $K_v$-algebra of the form $L_{w_1}\times L_{w_2}$, where $w_1,\,w_2$ are discrete valuations of $L$ lying over $v$. But this does not affect the construction of the key exact sequence for $D_v$. Indeed, the same choice of $\mu\in D^*\subseteq D_v^*$ satisfies the condition that $\mathrm{Int}(\mu)|_{L_v}$ is the nontrivial automorphism of the $K_v$-algebra $L_v$. It is not difficult to check that the key exact sequence for $D_v$ is still well defined.)
The form ${\widetilde{\pi}}_2(h)\in W(L)$ has even rank, trivial discriminant and trivial Clifford invariant by [@BP1 Prop.$\;$3.2.2]. Hence ${\widetilde{\pi}}_2(h)\in I^3(L)\subseteq
W(L)$. Let $\Omega_{A'}$ denote the set of divisorial valuations of $L$. Then for every $w\in\Omega_{A'}$ one has ${\widetilde{\pi}}_2(h)=0$ in $W(L_w)$. By Lemma$\;$\[lemma2p1temp\], ${\widetilde{\pi}}_2(h)=0$ in $W(L)$. So by the exactness of the first row in the above diagram, there exists a hermitian form of even rank $h_0$ over $(L\,,\,\iota)$ such that ${\widetilde{\rho}}(h_0)=h\in W(D\,,\,\sigma)$.
Let ${\alpha}={\mathrm{disc}}(h_0)\in K^*/N_{L/K}(L^*)$ be the discriminant of $h_0$. One has $$\mathscr{C}\ell({\widetilde{\rho}}(h_0))=(L\,,\,{\alpha})\;\in\;\;{}_2{\mathrm{Br}}(K)/(D)$$by [@BP1 Prop.$\;$3.2.3]. Since $\mathscr{C}\ell({\widetilde{\rho}}(h_0))=\mathscr{C}\ell(h)=0$ by assumption, one has either $(L,\,{\alpha})=0$ or $(L\,,\,{\alpha})=(D)$ in ${\mathrm{Br}}(K)$. If $(L\,,\,{\alpha})=0\in{\mathrm{Br}}(K)$ then ${\alpha}$ is a norm for the extension $L/K$ so that ${\mathrm{disc}}(h_0)=1\in K^*/N_{L/K}(L^*)$. If $(L\,,\,{\alpha})=D$, writing $L=K(\sqrt{a})$ such that $D=(a\,,\,{\alpha})_K$, one has ${\mathrm{disc}}(\langle 1\,,\,-{\alpha}\rangle)={\alpha}\in
K^*/N_{L/K}(L^*)$. By the construction of the map $\pi_1$, one has $\pi_1(\langle 1\rangle)=\langle 1\,,\,-{\alpha}\rangle\in
W(L\,,\,\iota)$ (since $D=L\oplus \mu L$ with $\mu^2={\alpha}$). Replacing $h_0$ by $h_0-\pi_1(\langle 1\rangle)$, we may assume that ${\mathrm{disc}}(h_0)=1\in K^*/N_{L/K}(L^*)$. Let $2n={\mathrm{rank}}(h_0)$ and let ${\mathbf{SU}}_{2n}(L\,,\,\iota)$ denote the special unitary group of the hyperbolic form ${\bigl(\begin{smallmatrix} {0}& {I_n}\\ {I_n}&{0}\end{smallmatrix}\bigl)}$ over $(L\,,\,\iota)$. The form $h_0$, having trivial discriminant, now determines a class in $H^1(K\,,\,{\mathbf{SU}}_{2n}(L\,,\,\iota))$.
Let $H_{2n}$ be the hyperbolic form ${\bigl(\begin{smallmatrix} {0}& {I_n}\\ {I_n}&{0}\end{smallmatrix}\bigl)}$ over $(D\,,\,\sigma)$ and let ${\mathbf{U}}_{2n}(D\,,\,\sigma)$, ${\mathbf{SU}}_{2n}(D\,,\,\sigma)$ and ${\mathbf{Spin}}_{2n}(D\,,\,\sigma)$ denote respectively the unitary group, the special unitary group and the spin group of the form $H_{2n}$. By (\[para2p12NEW\]), there is a homomorphism $$\rho_0\,:\;\;{\mathbf{SU}}_{2n}(L\,,\,\iota){\longrightarrow}{\mathbf{Spin}}_{2n}(D\,,\,\sigma)\,.$$which induces a commutative diagram $$\xymatrix{ H^1(K\,,\,{\mathbf{SU}}_{2n}(L\,,\,\iota)) \ar[dr]_{\rho'}
\ar[rr]^{\rho_0} && H^1(K\,,\,{\mathbf{Spin}}_{2n}(D\,,\,\sigma)) \ar[dl]_{} \\
& H^1(K\,,\,{\mathbf{U}}_{2n}(D\,,\,\sigma)) & }$$such that the map ${\widetilde{\rho}}\,:\;W(L\,,\,\iota)\to W(D\,,\,\sigma)$ in the “key exact sequence” restricted to forms of rank $2n$ and of trivial discriminant is compatible with the map $\rho'$ at the level of cohomology sets.
By [@BP2 Prop.$\;$3.20], one has $$R_{{\mathbf{Spin}}_{2n}(D\,,\,\sigma)}(\rho_0([h_0]))=R_{{\mathbf{SU}}_{2n}(L\,,\,\iota)}([h_0])\,\in\;
H^3(K\,,\,\mathbb{Q}/\mathbb{Z}(2))\,.$$Thus by the definition of the Rost invariant ${\mathscr{R}}$ (cf. (\[para2p11NEW\])), $$0={\mathscr{R}}(h)=[R_{{\mathbf{Spin}}_{2n}(D\,,\,\sigma)}(\rho_0([h_0]))]=
[R_{{\mathbf{SU}}_{2n}(L\,,\,\iota)}([h_0])]\,\in\;\frac{H^3(K\,,\,\mathbb{Q}/\mathbb{Z}(2))}{H^1(K\,,\,\mu_2)\cup
(D)}\,.$$Therefore, there is an element $\beta\in K^*/K^{*2}=H^1(K\,,\,\mu_2)$ such that $$R_{{\mathbf{SU}}_{2n}(L\,,\,\iota)}([h_0])=(\beta)\cup (D)\,\in\;
H^3(K\,,\,\mathbb{Q}/\mathbb{Z}(2))\,.$$A direct computation shows that the element ${\widetilde{h}}_0:=\pi_1(\langle 1\,,\,-\beta\rangle)\in W(L\,,\,\iota)$ has associated trace form $q_{{\widetilde{h}}_0}=\langle
1\,,\,-\beta\rangle\otimes n_D$, where $n_D$ denotes the norm form of the quaternion algebra $D$. By [@KMRT p.438, Example$\;$31.44], the class of ${\widetilde{h}}_0$ has Rost invariant $$R_{{\mathbf{SU}}_{4}(L\,,\,\iota)}([{\widetilde{h}}_0])=e_3(q_{{\widetilde{h}}_0})=(\beta)\cup
(D)\;\in\;H^3(K\,,\,\mathbb{Q}/\mathbb{Z}(2))\,.$$Modifying $h_0$ by ${\widetilde{h}}_0=\pi_1(\langle 1\,,\,-\beta\rangle)$, we may further assume that the class $[h_0]\in H^1(K\,,\,{\mathbf{SU}}_{2n}(L\,,\,\iota))$ has trivial Rost invariant, i.e., $e_3(q_{h_0})=0$. Since $\mathrm{cd}_2(K)\le
3$, the Arason invariant $e_3: I^3(K)\to H^3(K\,,\,\mathbb{Z}/2)$ is injective. Hence $[q_{h_0}]=0\in W(K)$ and $[h_0]=0\in
W(L\,,\,\iota)$ by (\[para2p4NEW\]) (cf. [@Schar p.348, Thm.$\;$10.1.1]). It then follows immediately that $[h]={\widetilde{\rho}}([h_0])=0\in W(D\,,\,\sigma)$.
\[coro4p6temp\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\neq 2$ in the local henselian case. Let $(D\,,\,\sigma)$ be a quaternion division algebra with an orthogonal involution over $K$. Let $h_1,\,h_2$ be hermitian forms over $(D\,,\,\sigma)$ with the same rank and discriminant such that $$\mathscr{C}\ell(h_1\bot (-h_2))=0\,\in\;{}_2{\mathrm{Br}}(K)/(D)\,$$and $${\mathscr{R}}(h_1\bot (-h_2))=0\,\in\;
H^3(K\,,\,\mathbb{Q}/\mathbb{Z}(2))/H^1(K\,,\,\mu_2)\cup (D)\,.$$Then $h_1\cong h_2$ if and only if $(h_1)_{K_v}\cong (h_2)_{K_v}$ for every $v\in\Omega_A$.
Apply Proposition$\;$\[prop4p5temp\] to the form $h=h_1\bot
(-h_2)$ and use Witt’s cancellation theorem.
\[thm4p7temp\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\neq 2$ in the local henselian case. Let $(D,\,\sigma)$ be a quaternion division algebra with an orthogonal involution over $K$, $h$ a nonsingular hermitian form of rank $\ge
2$ over $(D\,,\,\sigma)$ and $G={\mathbf{Spin}}(h)$.
Then the natural map $$H^1(K\,,\,G){\longrightarrow}\prod_{v\in\Omega_A}H^1(K_v\,,\,G)$$has a trivial kernel.
Let $\xi\in H^1(K\,,\,{\mathbf{Spin}}(h))$ be a class which is trivial in $H^1(K_v\,,\,{\mathbf{Spin}}(h))$ for all $v\in \Omega_A$. The image of $\xi$ under the composite map $$H^1(K\,,\,G)=H^1(K\,,\,{\mathbf{Spin}}(h)){\longrightarrow}H^1(K\,,\,{\mathbf{SU}}(h)){\longrightarrow}H^1(K\,,\,{\mathbf{U}}(h))$$is the class of a hermitian form $h'$ which has the same rank and discriminant as $h$ such that $$\mathscr{C}\ell(h\bot (-h'))=0\,\in\;{}_2{\mathrm{Br}}(K)/(D)\,.$$Let $n={\mathrm{rank}}(h)$. Let ${\mathbf{Spin}}_{2n}(D\,,\,\sigma)$ and ${\mathbf{U}}_{2n}(D\,,\,\sigma)$ denote respectively the spin group and the unitary group of the hyperbolic form ${\bigl(\begin{smallmatrix} {0}& {I_n}\\ {I_n}&{0}\end{smallmatrix}\bigl)}$ over $(D\,,\,\sigma)$. Then the class $[h\bot (-h')]\in H^1(K\,,\,{\mathbf{U}}_{2n}(D\,,\,\sigma))$ lifts to an element $\xi'\in H^1(K\,,\,{\mathbf{Spin}}_{2n}(D\,,\,\sigma))$. By [@PaPr Lemma$\;$5.1], we have $$\label{eq4p7p1temp}
[R_G(\xi)]={\mathscr{R}}(h\bot(-h'))=[R_{{\mathbf{Spin}}_{2n}(D\,,\,\sigma)}(\xi')]\,\in\;\frac{H^3(K\,,\,\mathbb{Q}/\mathbb{Z}(2))}{H^1(K\,,\,\mu_2)\cup
(D)}\,.$$Since $\xi$ is locally trivial, the commutative diagram $$\begin{CD}
H^1(K\,,\,G) @>R_G>> H^3(K\,,\,\mathbb{Q}/\mathbb{Z}(2)) \\
@VVV @VVV \\
\prod_{v\in\Omega_A}H^1(K_v\,,\,G) @>R_G>>
\prod_{v\in\Omega_A}H^3(K_v\,,\,\mathbb{Q}/\mathbb{Z}(2))
\end{CD}$$shows that the Rost invariant $R_G(\xi)$ is locally trivial. By Theorem$\;$\[thm1p3temp\], noticing that the Rost invariant $R_G$ takes values in the subgroup $H^3(K\,,\,\mu_4^{\otimes 2})$, we get $R_G(\xi)=0\in H^3(K\,,\,\mathbb{Q}/\mathbb{Z}(2))$. Thus, by , $${\mathscr{R}}(h\bot
(h'))=0\,\in\;\frac{H^3(K\,,\mathbb{Q}/\mathbb{Z}(2))}{H^1(K\,,\,\mu_2)\cup (D)}\,.$$Now Corollary$\;$\[coro4p6temp\] implies that $h\cong h'$ and hence the image of $\xi\in H^1(K\,,\,G)$ in $H^1(K,\,{\mathbf{U}}(h))$ is trivial. By [@BP2 Lemma$\;$7.11], the canonical image of $\xi$ in $H^1(K,\,{\mathbf{SU}}(h))$ is also trivial.
Now consider the following commutative diagram with exact rows $$\begin{CD}
1 @>>> \frac{K^*/K^{*2}}{{\mathrm{Sn}}(h_K)} @>\varphi>> H^1(K\,,\,G) @>>> H^1(K\,,\,{\mathbf{SU}}(h))\\
&& @VVV @VVV @VVV \\
1 @>>> \prod_{v\in\Omega_A}\frac{K_v^*/K_v^{*2}}{{\mathrm{Sn}}(h_{K_v})} @>>>
\prod_{v\in\Omega_A}H^1(K_v\,,\,G) @>>> \prod_{v\in\Omega_A}
H^1(K_v\,,\,{\mathbf{SU}}(h))
\end{CD}$$which is induced by the natural exact sequence of algebraic groups $$1{\longrightarrow}\mu_2{\longrightarrow}G={\mathbf{Spin}}(h){\longrightarrow}{\mathbf{SU}}(h){\longrightarrow}1\,.$$The exactness of the first row yields $\xi=\varphi(\theta)$ for some $\theta\in \frac{K^*/K^{*2}}{{\mathrm{Sn}}(h_K)}$. The commutative diagram then shows that $\theta$ is locally trivial since $\xi$ is locally trivial. From Propositoin$\;$\[prop5p1NEW\] it follows that $\theta=1\in \frac{K^*/K^{*2}}{{\mathrm{Sn}}(h_K)}$ and hence $\xi=\varphi(\theta)$ is trivial in $H^1(K\,,\,G)$. This completes the proof.
Groups of type ${}^2A_n^*$ {#sec6}
==========================
Case of odd index
-----------------
\[prop5p1temp\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\neq 2$ in the local henselian case. Let $L/K$ be a quadratic field extension, $(D\,,\,\tau)$ a central division algebra of odd degree over $L$ with an $L/K$-involution $\tau\,($i.e., a unitary involution $\tau$ such that $L^{\tau}=K\,)$. Let $h_1,\,h_2$ be nonsingular hermitian forms over $(D\,,\,\tau)$ which have the same rank and discriminant.
If the forms $(h_1)_{K_v}\cong (h_2)_{K_v}$ over $(D_{K_v}\,,\,\tau)=(D\otimes_LL\otimes_KK_v\,,\,\tau)$ are isomorphic for all $v\in\Omega_A$, then the forms $h_1\,,\,h_2$ over $(D\,,\,\tau)$ are isomorphic.
Let $M/K$ be a field extension of odd degree such that $D_M=D\otimes_L(L\otimes_KM)$ is split over the field $LM=L\otimes_KM$. (Such an extension $M/K$ exists by [@BP1 Lemma$\;$3.3.1].) The base extension $\tau_M$ of $\tau$ is a unitary involution on the central simple $(LM)$-algebra $D_M$ such that $(LM)^{\tau_M}=M$. Let $\iota$ denote the nontrivial element of the Galois group ${\mathrm{Gal}}(L/K)$ and regard $\iota_M\in{\mathrm{Gal}}(LM/M)$ as a unitary involution on $LM$. There is a nonsingular hermitian form $(V\,,\,f)$ over $(LM\,,\,\iota_M)$ such that $(D_M\,,\,\tau_M)\cong ({\mathrm{End}}_{LM}(V)\,,\,\iota_f)$, where $\iota_f$ denotes the adjoint involution on ${\mathrm{End}}_{LM}(V)$ with respect to $f$ (cf. [@KMRT p.43, Thm.$\;$4.2 (2)]). We have a Morita equivalence between the category of hermitian forms over $(D_M\,,\,\tau_M)$ and the category of hermitian forms over $(LM\,,\,\iota_M)$ (cf. (\[para2p6NEW\])), which induces an isomorphism of Witt groups $$\phi_f\,:\;\;W(D_M\,,\,\tau_M){\xrightarrow{\sim}}W(LM\,,\,\iota_M)\,.$$
Let $h=h_1\bot (-h_2)$ and let $h_M$ be its base extension over $(D_M\,,\,\tau_M)$. Via the Morita equivalence mentioned above, $h_M$ corresponds to a hermitian form ${\widetilde{h}}_M$ over $(LM\,,\,\iota_M)$. Let $q_M:=q_{{\widetilde{h}}_M}$ be the trace form of ${\widetilde{h}}_M$ (which is a quadratic form over the field $M$). Since $h$ has even rank and trivial discriminant, the class $[q_M]\in W(M)$ of the quadratic form $q_M$ lies in $I^3(M)$. The hypothesis on the local triviality (with respect to $\Omega_A$) of $[h]=[h_1\bot
(-h_2)]$ implies that $[q_M]\in I^3(M)$ is locally trivial (with respect to the set of discrete valuations of $M$ defined in the same way as $\Omega_A$). By Lemma$\;$\[lemma2p1temp\], we have $[q_M]=0$ and hence $[{\widetilde{h}}_M]=0$ in $W(LM\,,\,\iota_M)$. Since $W(D_M\,,\,\tau_M)\cong W(LM\,,\,M)$, $[h_M]=0$ in $W(D_M\,,\,\tau_M)$. Since $M/K$ is an odd degree extension, the natural map $W(D\,,\,\tau)\to W(D_M\,,\,\tau_M)$ is injective by a theorem of Bayer-Fluckiger and Lenstra (cf. [@KMRT p.80, Coro.$\;$6.18]). So we get $[h]=0$ in $W(D\,,\,\tau)$, thus proving the proposition.
\[lemma5p2temp\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Let $L/K$ be a separable quadratic field extension and $(D\,,\,\tau)$ a central division $L$-algebra of square-free index ${\mathrm{ind}}(D)$ with a unitary involution $\tau$ such that $L^{\tau}=K$. Assume $p\nmid {\mathrm{ind}}(D)$ in the local henselian case.
Then for any nonsingular hermitian form $h$ over $(D\,,\,\tau)$, the natural map $$\frac{(R^1_{L/K}\mathbb{G}_m)(K)}{{\mathrm{Nrd}}({\mathbf{U}}(h)(K))}{\longrightarrow}\prod_{v\in\Omega_A}\frac{(R^1_{L/K}\mathbb{G}_m)(K_v)}{{\mathrm{Nrd}}({\mathbf{U}}(h)(K_v))}$$is injective.
First assume ${\mathrm{ind}}(D)=2$ so that $D$ is a quaternion division algebra over $L$. By [@KMRT p.202, Exercise$\;$III.12 (a)], we have $${\mathrm{Nrd}}({\mathbf{U}}(h)(K))={\{\,{z\tau(z)^{-1}\,|\,z\in{\mathrm{Nrd}}(D^*)}\,\}}={\mathrm{Nrd}}({\mathbf{U}}_2(D\,,\,\tau)(K))\,,$$where ${\mathbf{U}}_2(D\,,\,\tau)$ denotes the unitary group of the rank 2 hyperbolic form ${\bigl(\begin{smallmatrix} {0}& {1}\\ {1}&{0}\end{smallmatrix}\bigl)}$ over $(D\,,\,\tau)$. So we may assume that $h={\bigl(\begin{smallmatrix} {0}& {1}\\ {1}&{0}\end{smallmatrix}\bigl)}$. The exact sequence of algebraic groups $$1{\longrightarrow}{\mathbf{SU}}_2(D\,,\,\tau){\longrightarrow}{\mathbf{U}}_2(D\,,\,\tau)\overset{{\mathrm{Nrd}}}{{\longrightarrow}}(R^1_{L/K}\mathbb{G}_m){\longrightarrow}1$$gives rise to the following commutative diagram with exact rows $$\begin{CD}
1 @>>> \frac{(R^1_{L/K}\mathbb{G}_m)(K)}{{\mathrm{Nrd}}({\mathbf{U}}(h)(K))}
@>\varphi>> H^1(K\,,\,{\mathbf{SU}}_2(D\,,\,\tau))\\
&& @VVV @VVV \\
1 @>>>
\prod_{v\in\Omega_A}\frac{(R^1_{L/K}\mathbb{G}_m)(K_v)}{{\mathrm{Nrd}}({\mathbf{U}}(h)(K_v))}
@>>> \prod_{v\in\Omega_A}H^1(K_v\,,\,{\mathbf{SU}}_2(D\,,\,\tau))
\end{CD}$$We need only to show that the vertical map on the right in the above diagram is injective.
By [@KMRT p.26, Prop.$\;$2.22], there is a unique quaternion $K$-algebra $D_0$ contained in $D$ such that $D=D_0\otimes_KL$ and $\tau=\tau_0\otimes\iota$, where $\tau_0$ is the canonical involution on $D_0$ and $\iota$ is the nontrivial element in the Galois group ${\mathrm{Gal}}(L/M)$. Write $L=K(\sqrt{d})$ and let $n_{D_0}$ be the norm form of the quaternion $K$-algebra $D_0$. Then by [@KMRT p.229], we have ${\mathbf{SU}}_2(D\,,\,\tau)={\mathbf{Spin}}(q)$, where $q=\langle 1\,,\,-d\rangle\otimes n_{D_0}$. Now the result follows from Theorem$\;$\[thm3p5temp\].
Assume next ${\mathrm{ind}}(D)$ is odd (and square-free). By [@KMRT p.202, Exercise$\;$III.12 (b)], $${\mathrm{Nrd}}({\mathbf{U}}(h)(K))={\mathrm{Nrd}}(D^*)\cap (R^1_{L/K}\mathbb{G}_m)(K)\,.$$Let $\lambda\in (R^1_{L/K}\mathbb{G}_m)(K)={\{\,{z\in L^*\,|\, N_{L/K}(z)=1}\,\}}$ be such that for every $v\in\Omega_A$, $\lambda\in {\mathrm{Nrd}}({\mathbf{U}}(h)(K_v))={\mathrm{Nrd}}((D\otimes_KK_v)^*)\cap
(R^1_{L/K}\mathbb{G}_m)(K_v)$. Since ${\mathrm{ind}}(D)$ is square-free, it follows from Theorem$\;$\[thm1p4temp\] that $\lambda\in{\mathrm{Nrd}}(D^*)$. Hence $$\lambda\in{\mathrm{Nrd}}({\mathbf{U}}(h)(K))={\mathrm{Nrd}}(D^*)\cap (R^1_{L/K}\mathbb{G}_m)(K)\,.$$
Now assume $\mathrm{ind}(D)$ is even such that $\mathrm{ind}(D)/2$ is odd and square-free. In this case we have $D=H\otimes_LD'$ for some quaternion division algebra $H$ over $L$ and some central division algebra $D'$ of odd index over $L$. By [@BP1 Lemma$\;$3.3.1], there is an odd degree separable extension $K'/K$ such that $D'\otimes_KK'=D'\otimes_LLK'$ is split. By Morita theory, there is a unitary $LK'/K'$-involution $\sigma$ on $H\otimes_LLK'$ and a hermitian form $f$ over $(H\otimes_LLK'\,,\,\sigma)$ such that the involution $\tau$ on $D\otimes_LLK'$ is adjoint to $f$, and moreover, the form $h_{K'}$ over $(D\otimes_LLK'\,,\,\tau)$ corresponds to a hermitian form $h'$ over $(H\otimes_LLK'\,,\,\sigma)$. Consider the commutative diagram $$\begin{CD}
\frac{R^1_{L/K}\mathbb{G}_m(K)}{{\mathrm{Nrd}}({\mathbf{U}}(h)(K))} @>\eta>> \prod_{v\in\Omega_A}\frac{R^1_{L/K}\mathbb{G}_m(K_v)}{{\mathrm{Nrd}}({\mathbf{U}}(h)(K_v))}\\
@VVV @VVV\\
\frac{R^1_{LK'/K'}\mathbb{G}_m(K')}{{\mathrm{Nrd}}({\mathbf{U}}(h')(K'))} @>\eta'>> \prod_{v\in\Omega_A}\frac{R^1_{LK'/K'}\mathbb{G}_m(K'_v)}{{\mathrm{Nrd}}({\mathbf{U}}(h')(K'_v))}\\
\end{CD}$$The map $\eta'$ is already shown to be injective. Let $\lambda\in R^1_{L/K}\mathbb{G}_m(K)\subseteq L^*$ be an element which is a reduced norm for ${\mathbf{U}}(h)(K_v)$ for every $v$. Then, considered as an element of $R^1_{LK'/K'}(K')\subseteq (LK')^*$, $\lambda$ lies in ${\mathrm{Nrd}}({\mathbf{U}}(h')(K'))$. By [@PaPr Prop.$\;$10.2], we have $$N_{LK'/K'}({\mathrm{Nrd}}({\mathbf{U}}(h')(K')))\subseteq {\mathrm{Nrd}}({\mathbf{U}}(h)(K))\,.$$Hence, $\lambda^{2r+1}\in {\mathrm{Nrd}}({\mathbf{U}}(h)(K))$, where $2r+1=[K': K]$. It is sufficient to show that $\lambda^2\in {\mathrm{Nrd}}({\mathbf{U}}(h)(K))$. For this, we choose a quadratic extension $M/K$ such that $H\otimes_KM=H\otimes_LLM$ is split. A similar argument as above, using the result in the case of odd index this time, shows that $\lambda\in {\mathrm{Nrd}}({\mathbf{U}}(h_M)(M))$. Thus, $$\lambda^2=N_{LM/M}(\lambda)\in
N_{LM/M}({\mathrm{Nrd}}({\mathbf{U}}(h_M)(M)))\subseteq {\mathrm{Nrd}}({\mathbf{U}}(h)(K))\,.$$This completes the proof of the lemma.
\[thm5p3temp\]Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Let $L/K$ be a separable quadratic field extension and $(D\,,\,\tau)$ a central division $L$-algebra with a unitary $L/K$-involution whose index ${\mathrm{ind}}(D)$ is odd and square-free. Assume further that $p\nmid 2.{\mathrm{ind}}(D)$ in the local henselian case.
Then for any nonsingular hermitian form $h$ over $(D\,,\,\tau)$, the natural map $$H^1(K\,,\,{\mathbf{SU}}(h)){\longrightarrow}\prod_{v\in\Omega_A}H^1(K_v\,,\,{\mathbf{SU}}(h))$$has a trivial kernel.
Let $\xi\in H^1(K\,,\,{\mathbf{SU}}(h))$ be a class that is locally trivial in $H^1(K_v\,,\,{\mathbf{SU}}(h))$ for every $v\in\Omega_A$. Let $h'$ be a hermitian form whose class $[h']\in H^1(K\,,\,{\mathbf{U}}(h))$ is the image of $\xi$ under the natural map $H^1(K\,,\,{\mathbf{SU}}(h))\to
H^1(K\,,\,{\mathbf{U}}(h))$. The two forms $h'$ and $h$ have the same rank and discriminant, and they are locally isomorphic since $\xi$ is locally trivial. So by Proposition$\;$\[prop5p1temp\], $h'\cong h$ as hermitian forms over $(D\,,\,\tau)$. This means that $\xi\in
H^1(K\,,\,{\mathbf{SU}}(h))$ maps to the trivial element in $H^1(K\,,\,{\mathbf{U}}(h))$.
Consider now the following commutative diagram with exact rows $$\begin{CD}
1 @>>> \frac{(R^1_{L/K}\mathbb{G}_m)(K)}{{\mathrm{Nrd}}({\mathbf{U}}(h)(K))}
@>\varphi>> H^1(K\,,\,{\mathbf{SU}}(h)) @>>> H^1(K\,,\,{\mathbf{U}}(h)) \\
&& @V{\eta}VV @VVV @VVV \\
1 @>>>
\prod_{v\in\Omega_A}\frac{(R^1_{L/K}\mathbb{G}_m)(K_v)}{{\mathrm{Nrd}}({\mathbf{U}}(h)(K_v))}
@>>> \prod_{v\in\Omega_A}H^1(K_v\,,\,{\mathbf{SU}}(h)) @>>> \prod_{v\in\Omega_A}H^1(K_v\,,\,{\mathbf{U}}(h))
\end{CD}$$There is an element $\theta\in (R^1_{L/K}\mathbb{G}_m)(K)/{\mathrm{Nrd}}({\mathbf{U}}(h)(K))$ such that $\varphi(\theta)=\xi$. The map $\eta$ is injective by Lemma$\;$\[lemma5p2temp\]. So we have $\theta=1$ and $\xi=\varphi(\theta)$ is trivial. The theorem is thus proved.
Some observations on Suresh’s exact sequence
--------------------------------------------
\[para6p4TEMP\] Let $E$ be a field of characteristic $\neq 2$. Let $D$ be a quaternion division algebra over a quadratic field extension $L$ of $E$. Let $\tau$ be a unitary $L/E$-involution on $D$. There is a unique quaternion $E$-algebra $D_0$ contained in $D$ such that $D=D_0\otimes_EL$ and $\tau=\tau_0\otimes\iota$, where $\tau_0$ is the canonical (symplectic) involution on $D_0$ and $\iota$ is the nontrivial element of the Galois group ${\mathrm{Gal}}(L/E)$. Then we have Suresh’s exact sequence (cf. (\[para2p8v2\])) $$W(L)\overset{{\widetilde{\pi}}_1}{{\longrightarrow}}W(D_0\,,\,\tau_0)\overset{{\widetilde{\rho}}}{{\longrightarrow}}W(D\,,\,\tau)\overset{p_2}{{\longrightarrow}}W^{-1}(D_0,\,\tau_0)\,.$$The goal of this subsection is to analyze the image of the map ${\widetilde{\pi}}_1$ in this sequence.
\[para1p2TEMP\] With notation as in (\[para6p4TEMP\]), let $h_0$ be a hermitian form of rank $m$ over $(D_0,\,\tau_0)$. Let $M(h_0)\in A:={\mathrm{M}}_m(D_0)$ be a representation matrix of $h_0$. One can define the pfaffian norm $\mathrm{Pf}(h_0)$ as the pfaffian norm of $M(h_0)\in A$ with respect to the adjoint involution of $h_0$ on $A$ (cf. [@KMRT p.19]). This is a well defined element of the group $E^*/{\mathrm{Nrd}}(D_0^*)$. If $h_0=\langle {\alpha}_1,\dotsc, {\alpha}_m\rangle$ with ${\alpha}_i\in E^*$, then $\mathrm{Pf}(h_0)$ is represented by the discriminant of the quadratic form $\langle {\alpha}_1,\dotsc, {\alpha}_m\rangle$ over $E$.
\[lemma1TEMP\] With notation as in $(\ref{para6p4TEMP})$, write $L=E(\sqrt{d})$ with $d\in E^*$. Let $h_0$ be a hermitian form of even rank over $(D_0,\,\tau_0)$.
$({\mathrm{i}})$ If the class $[h_0]\in W(D_0,\,\tau_0)$ lies in the image of ${\widetilde{\pi}}_1$, then its pfaffian norm $\mathrm{Pf}(h_0)\in E^*/{\mathrm{Nrd}}(D_0^*)$ lies in the subgroup generated by $N_{L/E}(L^*)$.
$({\mathrm{ii}})$ The converse of $({\mathrm{i}})$ is true if $h_0$ is of rank $2$.
\(i) For $a+b\sqrt{d}\in L^*$ with $a\,,\,b\in E$, the form ${\widetilde{\pi}}_1(\langle a+b\sqrt{d}\rangle)$ is represented by the matrix $${\begin{pmatrix} {a}& {bd}\\ {bd}&{ad}\end{pmatrix}}\,.$$One can then verify that $${\widetilde{\pi}}_1(\langle\,a+b\sqrt{d}\,\rangle)=\begin{cases}
\langle\,a\,,\,ad(a^2-b^2d)\,\rangle\;\;&\;\text{ if }\, a\neq 0\\
\langle\,2bd\,,\,-2bd\rangle\;\;&\;\text{ if }\, a=0\neq b
\end{cases}\,.$$So it follows easily that $\mathrm{Pf}({\widetilde{\pi}}_1(\langle\,a+b\sqrt{d}\,\rangle))$ is represented by an element of $N_{L/E}(L^*)$.
\(ii) Conversely, let $h_0$ be a hermitian form of rank $2$ whose pfaffian norm $\mathrm{Pf}(h_0)$ is represented by an element of $N_{L/E}(L^*)$. We want to show $[h_0]\in\mathrm{Im}({\widetilde{\pi}}_1)$. By Suresh’s exact sequence, it suffices to show that the form ${\widetilde{\rho}}(h_0)$ is hyperbolic over $(D\,,\,\tau)$.
We may assume $h_0=\langle{\alpha}\,,\,-\gamma{\alpha}\rangle$ with ${\alpha},\,\gamma\in E^*$. The assumption on the pfaffian norm implies that $$\tau_0(u)u\gamma=
{\mathrm{Nrd}}_{D_0}(u)\gamma=a^2-b^2d\,$$ for some $u\in D_0^*$ and some $a,\,b\in E$. Since $$\langle{\alpha}\,,\,-\gamma{\alpha}\rangle\cong \langle {\alpha}\,,\,-\gamma{\alpha}\tau_0(u)u\rangle\quad\text{ over }\;\; (D_0\,,\,\tau_0)\,,$$replacing $\gamma$ by $\gamma\tau_0(u)u=\gamma.{\mathrm{Nrd}}_{D_0}(u)$ if necessary, we may assume $\gamma=a^2-b^2d$ for some $a,\,b\in E$. From the definition of the map ${\widetilde{\rho}}$, it follows easily that the form ${\widetilde{\rho}}(h_0)$ over $(D\,,\,\tau)$ is also represented by the diagonal matrix $\langle{\alpha}\,,\,-\gamma{\alpha}\rangle$. But then for $v=(a+b\sqrt{d}\,,\,1)\in D^2$, one has $${\widetilde{\rho}}(h_0)(v,\,v)=(\tau(a+b\sqrt{d})\,,\,\tau(1)){\begin{pmatrix} {{\alpha}}& {0}\\ {0}&{-\gamma{\alpha}}\end{pmatrix}}\binom{a+b\sqrt{d}}{1}={\alpha}(a^2-b^2d-\gamma)=0\,.$$This show that the rank 2 form ${\widetilde{\rho}}(h_0)$ is isotropic and hence hyperbolic.
\[lemma2TEMP\] With notation as above, assume that the field $E$ has finite $u$-invariant $u(E)=r$. Then for any hermitian form $h_0$ of rank $m>r/3$ over $(D_0,\,\tau_0)$, the form ${\widetilde{\rho}}(h_0)$ over $(D,\,\tau)$ is isotropic.
We may assume $D_0^m$ is the underlying space of the form $h_0$ and $h_0=\langle {\alpha}_1,\dotsc, {\alpha}_m\rangle$ with ${\alpha}_i\in E^*$. Then the underlying space of ${\widetilde{\rho}}(h_0)$ is $D^m=D_0^m\oplus D^m_0\sqrt{d}$. We fix a quaternion basis ${\{\,{1,\,i,\,j,\,ij}\,\}}$ for the quaternion algebra $D_0$. The subspace $\mathrm{Sym}(D,\,\tau)\subseteq D$ consisting of $\tau$-invariant elements is a 4-dimensional $E$-vector space with basis $$1,\,i\sqrt{d}\,,\,j\sqrt{d}\,,\,ij\sqrt{d}\,.$$ Let $V\subseteq \mathrm{Sym}(D,\,\tau)$ be the subspace generated by $i\sqrt{d},\,j\sqrt{d}$ and $ij\sqrt{d}$. For $w=x_1.i\sqrt{d}+x_2.j\sqrt{d}+x_3.ij\sqrt{d}$ with $x_i\in E$, a straightforward calculation yields $$w^2=di^2.x_1^2+dj^2.x_2^2+d(ij)^2.x_3^2\;\in\; E\,.$$So the map $$\phi\,:\; V^m{\longrightarrow}E\;;\quad v=(v_1,\dotsc, v_m)\longmapsto {\widetilde{\rho}}(h_0)(v,\,v)=\sum {\alpha}_iv_i^2$$defines a quadratic form of rank $3m$ over $E$. By the assumption on the $u$-invariant of $E$, the quadratic form $\phi$ is isotropic and hence the hermitian form ${\widetilde{\rho}}(h_0)$ is isotropic.
\[lemma3TEMP\] Assume that $u(E)<12$. Then for any hermitian form $h_0$ of even rank $2n$ over $(D_0,\,\tau_0)$, one has $$[h_0]\,\in \;\mathrm{Im}({\widetilde{\pi}}_1)\iff \mathrm{Pf}(h_0)\,\in\;N_{L/E}(L^*).{\mathrm{Nrd}}(D_0^*)\,.$$
In view of Lemma$\;$\[lemma1TEMP\], we need only to prove that if $\mathrm{Pf}(h_0)\in N_{L/E}(L^*){\mathrm{Nrd}}(D_0^*)$, then $[h_0]\in\mathrm{Im}({\widetilde{\pi}}_1)$.
To prove this, we use induction on $n=\mathrm{rank}(h_0)/2$, the case $n=1$ being treated in Lemma$\;$\[lemma1TEMP\]. Now we assume $\mathrm{rank}(h_0)=2n\ge 4$ and $h_0$ is anisotropic. Let $V_0$ be the underlying space of $h_0$. Then the underlying space of the form ${\widetilde{\rho}}(h_0)$ is $V=V_0\oplus V_0\sqrt{d}$. By Lemma$\;$\[lemma2TEMP\], the form ${\widetilde{\rho}}(h_0)$ is isotropic, that is, there is a nonzero vector $x_1+y_1\sqrt{d}\in V=V_0\oplus V_0\sqrt{d}$ such that $$\begin{split}
0&={\widetilde{\rho}}(h_0)(x_1+y_1\sqrt{d}\,,\,x_1+y_1\sqrt{d})\\
&=(h_0(x_1,\,x_1)-h_0(y_1,\,y_1)d)+(h_0(x_1,\,y_1)-h_0(y_1,\,x_1))\sqrt{d}\,.
\end{split}$$Thus $$\label{eq5p19p1TEMP}
h_0(x_1,\,x_1)=d.h_0(y_1,\,y_1)\quad\text{and}\;\quad h_0(x_1\,,\,y_1)=h_0(y_1,\,x_1)\,.$$Since $h_0$ is anisotropic, $h_0(x_1,\,x_1)$ and $h_0(y_1,\,y_1)$ are both nonzero and hence lie in $$E^*={\{\,{x\in D_0^*\,|\,\tau_0(x)=x}\,\}}\,.$$ In particular, $x_1\neq 0$, $y_1\neq 0$ and $$h_0(x_1,\,y_1)=h_0(y_1,\,x_1)\in E={\{\,{x\in D_0\,|\,\tau_0(x)=x}\,\}}\,.$$If $x_1=y_1\lambda$ for some $\lambda\in D_0^*$, then yields $$\tau_0(\lambda)\lambda=d\quad\text{ and }\quad \tau_0(\lambda)=\lambda$$whence $d=\lambda^2\in E^{*2}$. Since $d$ is not a square in $E$, the two vectors $x_1,\,y_1\in V_0$ generate a $D_0$-submodule $W_0:=x_1D_0+y_1D_0\subseteq V_0$ of rank 2. Put $a=h_0(y_1,\,y_1)\in E^*$ and $bd=h_0(x_1,\,y_1)=h_0(y_1,\,x_1)\in E$. Then the restriction $f_0$ of $h_0$ to $W_0$ is represented by the matrix $${\begin{pmatrix} {ad}& {bd}\\ {bd}&{a}\end{pmatrix}}={\begin{pmatrix} {0}& {1}\\ {1}&{0}\end{pmatrix}}{\begin{pmatrix} {a}& {bd}\\ {bd}&{ad}\end{pmatrix}}{\begin{pmatrix} {0}& {1}\\ {1}&{0}\end{pmatrix}}\,.$$A direct computation then gives $${\widetilde{\pi}}_1(\langle a+b\sqrt{d}\rangle)=[f_0]\,\in\;W(D_0\,,\,\tau_0)\,.$$This means that $h_0$ contains a subform $f_0$ of rank 2, which lies in the image of ${\widetilde{\pi}}_1$. Writing $h_0=f_0\bot g_0$, we get $\mathrm{Pf}(g_0)\in N_{L/E}(L^*){\mathrm{Nrd}}(D_0^*)$ since $\mathrm{Pf}(f_0)$ and $\mathrm{Pf}(h_0)$ lie in $N_{L/E}(L^*){\mathrm{Nrd}}(D_0^*)$. Now the induction hypothesis yields $[g_0]\in\mathrm{Im}({\widetilde{\pi}}_1)$, whence $[h_0]=[f_0]+[g_0]\in\mathrm{Im}({\widetilde{\pi}}_1)$.
A Hasse principle for $H^4$ of function fields of conics
--------------------------------------------------------
\[lemma2p1TEMP\] Let $F$ be a field of characteristic $\neq 2$, $\overline{F}$ a separable closure of $F$ and $C\subseteq\mathbb{P}^2_F$ a smooth projective conic over $F$. Put $\overline{C}=C\times_F\overline{F}$ and let $F(C),\, \overline{F}(C)$ denote the function fields of $C$ and $\overline{C}$ respectively.
Then the natural exact sequence $$0{\longrightarrow}\overline{F}(C)^*\otimes\mathbb{Q}_2/\mathbb{Z}_2(2){\longrightarrow}\mathrm{Div}(\overline{C})\otimes\mathbb{Q}_2/\mathbb{Z}_2(2){\longrightarrow}\mathrm{Pic}(\overline{C})\otimes\mathbb{Q}_2/\mathbb{Z}_2(2){\longrightarrow}0$$induces an injection $$H^3(F\,,\,\overline{F}(C)\otimes\mathbb{Q}_2/\mathbb{Z}_2(2)){\longrightarrow}H^3(F\,,\,\mathrm{Div}(\overline{C})\otimes\mathbb{Q}_2/\mathbb{Z}_2(2))\,.$$
Let $C^{(1)}$ be the set of closed points of $C$. For each $P\in C^{(1)}$, let $G_P$ be the absolute Galois group of the residue field $F(P)$ of $P$. This is an open subgroup of $G=\mathrm{Gal}(\overline{F}/F)$. Write $M_P=\mathrm{Hom}_{G_P}(\mathbb{Z}[G]\,,\,\mathbb{Z})$. We have an isomorphism of abelian groups $M_P\cong \bigoplus_{Q\mapsto P}\mathbb{Z}$, where the notation $Q\mapsto P$ means that $Q$ runs over the closed points of $\overline{C}$ lying over $P$. On the other hand, we have an isomorphism of $G$-modules: $$\mathrm{Div}(\overline{C})\cong\bigoplus_{P\in C^{(1)}}M_P\,.$$ Since $C$ is a smooth projective conic, $\mathrm{Pic}(\overline{C})\cong \mathbb{Z}$ as $G$-modules. The natural map $\mathrm{Div}(\overline{C})\to \mathrm{Pic}(\overline{C})$ can be identified with the summation map $$\sigma\,:\; \bigoplus_{P\in C^{(1)}}\bigoplus_{Q\mapsto P}\mathbb{Z}\longrightarrow \mathbb{Z}$$So the exact sequence in the lemma may be identified with the following $$0{\longrightarrow}\overline{F}(C)^*\otimes\mathbb{Q}_2/\mathbb{Z}_2(2){\longrightarrow}\bigoplus_{P\in C^{(1)}}M_P\otimes\mathbb{Q}_2/\mathbb{Z}_2(2)\overset{\sigma}{{\longrightarrow}}\mathbb{Q}_2/\mathbb{Z}_2(2){\longrightarrow}0\,.$$For any $i\ge 0$, $$H^i\left(F,\,M_P\otimes\mathbb{Q}_2/\mathbb{Z}_2(2)\right)=H^i(F(P)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2))\,.$$ It is thus sufficient to prove that the map $$\bigoplus_{P\in C^{(1)}} H^2(F(P)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2)){\longrightarrow}H^2(F\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2))\,$$ is surjective. In fact, we can choose a closed point $P\in C^{(1)}$ of degree 2 and consider the corresponding map $$\psi\,: \; H^2(F(P)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2)){\longrightarrow}H^2(F\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2))\,,$$which coincides with the corestriction map. We claim that this map is already surjective. To see this, consider for each $n\in\mathbb{N}$ the corestriction map $$\psi_n\,:\; H^2(F(P)\,,\,\mathbb{Z}/2^n(2)){\longrightarrow}H^2(F\,,\,\mathbb{Z}/2^n(2))\,.$$By the Merkurjev–Suslin theorem, the map $\psi_n$ may be identified with the norm map $$N_{F(P)/F}\,:\; K_2(F(P))/2^n{\longrightarrow}K_2(F)/2^n$$in Milnor’s $K$-theory. The cokernel of this norm map is killed by $2=[F(P): F]$. So taking limits yields the surjectivity of the map $\psi$. This proves the lemma.
\[thm2p2TEMP\] Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Assume $p\neq 2$ in the local henselian case. Let $C$ be a smooth projective conic in $\mathbb{P}^2_K$.
Then the natural map $$H^4(K(C)\,,\,\mathbb{Z}/2){\longrightarrow}\prod_{v\in\Omega_A} H^4(K_v(C)\,,\,\mathbb{Z}/2)$$is injective, where $v$ runs over all divisorial valuations of $K$.
By the Merkurjev–Suslin theorem, we may replace $\mathbb{Z}/2$ by $\mathbb{Q}_2/\mathbb{Z}_2(3)$. Also, we may replace the completion $K_v$ by the henselisation $K_{(v)}$ for each $v$ (cf. [@Jannsen09 Thm.$\;$2.9 and its proof]). Let $\overline{K}$ be a separable closure of $K$. Then we have a diagram of field extensions $$\xymatrix{
\overline{K}(C) & \overline{K} \ar[l]_{}\\
K_{(v)}(C) \ar[u]^{} & K_{(v)} \ar[l]_{} \ar[u]_{} \\
K(C) \ar[u]_{} & K \ar[l]_{} \ar[u]_{}
}$$which identifies the Galois groups $${\mathrm{Gal}}(\overline{K}/K)={\mathrm{Gal}}(\overline{K}(C)/K(C))\quad\text{ and }\quad {\mathrm{Gal}}(\overline{K}/K_{(v)})={\mathrm{Gal}}(\overline{K}(C)/K_{(v)}(C))\,.$$This induces Hochschild-Serre spectral sequences $$E_2^{pq}(K)=H^p(K\,,\,H^q(\overline{K}(C)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(3))) \Longrightarrow H^{p+q}(K(C)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(3))$$and $$E_2^{pq}(K_{(v)})=H^p(K_{(v)}\,,\,H^q(\overline{K}(C)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(3))) \Longrightarrow H^{p+q}(K_{(v)}(C)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(3))\,.$$Using $$\mathrm{cd}_2(\overline{K}(C))\le 1\,\quad \text{ and }\quad \mathrm{cd}_2(K_{(v)})\le \mathrm{cd}_2(K)\le 3\,,$$one finds easily that the above spectral sequences induce canonical isomorphisms $$H^4(K(C)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(3))\cong H^3(K\,,\,H^1(\overline{K}(C)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(3)))$$ and $$H^4(K_{(v)}(C)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(3))\cong H^3(K_{(v)}\,,\,H^1(\overline{K}(C)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(3)))\,.$$Since $H^1(\overline{K}(C)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(3))\cong \overline{K}(C)^*\otimes\mathbb{Q}_2/\mathbb{Z}_2(2)$, we need only prove the injectivity of the natural map $$H^3(K\,,\,\overline{K}(C)^*\otimes\mathbb{Q}_2/\mathbb{Z}_2(2)){\longrightarrow}\prod_{v\in\Omega_A}H^3(K_{(v)}\,,\,\overline{K}(C)^*\otimes\mathbb{Q}_2/\mathbb{Z}_2(2))$$is injective.
By Lemma$\;$\[lemma2p1TEMP\], we have an injection $$H^3(K\,,\,\overline{K}(C)^*\otimes\mathbb{Q}_2/\mathbb{Z}_2(2))\hookrightarrow \bigoplus_{p\in C^{(1)}}H^3(K(P)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2))\,.$$ For each $v$, let $C_{(v)}=C\times_KK_{(v)}$ be the base extension of $C$ and let $K_{(v)}(C)$ denote the function field of $C_{(v)}$. By functoriality, we may reduce to proving the injectivity of the map $$\varphi\,:\;\;\bigoplus_{P\in C^{(1)}}H^3(K(P)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2)){\longrightarrow}\prod_{v\in\Omega_A}\bigoplus_{Q\in C_{(v)}^{(1)}}H^3(K_{(v)}(Q)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2))\,.$$For fixed $v$ and $P\in C^{(1)}$, the corresponding component $\varphi_{v,\,P}$ of the map $\varphi$ is given by $$\varphi_{v,\,P}\,:\;\;H^3(K(P)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2)){\longrightarrow}\bigoplus_{Q\,|\,P}H^3(K_{(v)}(Q)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2))\,,$$where $Q$ runs over the points of the fiber $C_{(v)}\times_CP=\mathrm{Spec}(K_{(v)}\otimes_KK(P))$. An element $\alpha=(\alpha_P)\in \oplus_{P}H^3(K(P)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2))$ lies in $\ker(\varphi)$ if and only if for each $P\in C^{(1)}$, $\alpha_P$ lies in the kernel of $$\varphi_P=\prod_{v}\varphi_{v,\,P}\,:\;\;H^3(K(P)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2)){\longrightarrow}\prod_{v}H^3_{\text{\'et}}(K_{(v)}\otimes_KK(P)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2))\,.$$It suffices to prove that for every $P$, the map $\varphi_P$ is injective.
Replacing $K(P)$ by the separable closure of $K$ in $K(P)$ if necessary, we may assume that $K(P)/K$ is a finite separable extension. Then we have $$K_{(v)}\otimes_KK(P)\cong\prod_{w\,|\,v} K(P)_{(w)}\,,$$(cf. [@EGA8 IV.18.6.8]). So the map $\varphi_P$ gets identified with the natural map $$H^3(K(P)\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2)){\longrightarrow}\prod_{w}H^3(K(P)_{(w)}\,,\,\mathbb{Q}_2/\mathbb{Z}_2(2))\,,$$where $w$ runs over divisorial valuations of $K(P)$. This map is injective by Theorem$\;$\[thm1p3temp\]. The theorem is thus proved.
\[coro2p3TEMP\] Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of charachteristic $p$. Assume $p\neq 2$ in the local henselian case. Let $C$ a smooth projective conic in $\mathbb{P}^2_K$.
Then the natural map $$I^4(K(C)){\longrightarrow}\prod_{v\in\Omega_A} I^4(K_v(C))$$is injective, where $v$ runs over the set $\Omega_A$ of divisorial valuations of $K$.
For $F=K(C)$ or $K_v(C)$, we have $\mathrm{cd}_2(F)\le 4$. By the degree 4 case of the Milnor conjecture (cf. [@Voe03] and [@OVV]), we have an isomorphism $I^4(F)\cong H^4(F,\,\mathbb{Z}/2)$. (In the $p$-adic arithmetic case, we can also deduce this isomorphism from [@AEJ86 p.655, Prop.$\;$2] together with [@Le10 Thm.$\;$3.4].) The result then follows immediately from Theorem$\;$\[thm2p2TEMP\].
Case of even index
------------------
\[prop3p2TEMP\] Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Let $L/K$ be a quadratic field extension, $(D,\,\tau)$ a central division algebra over $L$ with a unitary $L/K$-involution whose index is not divisible by $4$. Let $h$ be a nonsingular hermitian form over $(D,\,\tau)$ which has even rank, trivial discriminant and trivial Rost invariant $($cf. $(\ref{para3p1TEMP}))$. Assume $p\neq 2$ if ${\mathrm{ind}}(D)$ is even. In the local henselian case, assume further that the Hasse principle with respect to divisorial valuations holds for quadratic forms of rank $6$ over $K$.
Then we have $[h]=0\in W(D,\,\tau)$ if and only if $[h\otimes_KK_v]=0\in W(D\otimes_KK_v\,,\,\tau)$ for every $v\in\Omega_A$.
If the index $\mathrm{ind}(D)$ is odd, the result is already proved in Proposition$\;$\[prop5p1temp\]. We assume next that ${\mathrm{ind}}(D)$ is even and not divisible by 4.
We first consider the case where $D$ is a quaternion algebra. As in (\[para6p4TEMP\]), we write $D=D_0\otimes_KL$ with $D_0$ a quaternion division algebra over $K$ and $L=K(\sqrt{d})$ with $d\in K^*$, and we have Suresh’s exact sequence $$\label{eq3p2p1TEMP}
W(L)\overset{{\widetilde{\pi}}_1}{{\longrightarrow}}W(D_0\,,\,\tau_0)\overset{{\widetilde{\rho}}}{{\longrightarrow}}W(D\,,\,\tau)\overset{p_2}{{\longrightarrow}}W^{-1}(D_0,\,\tau_0)$$
Let $C\subseteq \mathbb{P}^2_K$ be the smooth projective conic associated to the quaternion algebra $D_0$. Then the algebra $D\otimes_KK(C)=D_0\otimes_KL(C)$ is a split central simple algebra over $L(C)$ with a unitary $L(C)/K(C)$-involution $\tau$. By Morita theory, the hermitian form $h\otimes_KK(C)$ over $(D\otimes_KK(C)\,,\,\tau)$ corresponds to a hermitian form $h'_C$ over $(L(C)\,,\,\iota)$, where $\iota$ denotes the nontrivial element of the Galois group $\mathrm{Gal}(L(C)/K(C))$. The trace form $q_{h,\,C}$ of $h'_C$ gives a quadratic form over $K(C)$. By [@KMRT Example$\;$31.44], the quadratic form $q_{h,\,C}$ has even rank, trivial discriminant, trivial Clifford invariant and trivial Rost invariant, since $h'_C$ has even rank, trivial discriminant and trivial Rost invariant (these invariants being invariant under Morita equivalence). Hence in the Witt group $W(K(C))$ we have $[q_{h,\,C}]\in I^4(K(C))$. Since $h$ is locally hyperbolic, it follows from Corollary$\;$\[coro2p3TEMP\] that $[q_{h,\,C}]=0\in W(K(C))$, whence $[h\otimes_KK(C)]=0\in W(D\otimes_KK(C)\,,\,\tau)$. In the commutative diagram $$\begin{CD}
W(D,\,\tau) @>p_2>> W^{-1}(D_0,\,\tau_0)\\
@VVV @VVV\\
W(D\otimes_KK(C)\,,\,\tau) @>p_2>> W^{-1}(D_0\otimes_KK(C)\,,\,\tau_0)
\end{CD}$$the right vertical map is injective by [@PSS]. So we have $p_2(h)=0\in W^{-1}(D_0,\,\tau_0)$. The exactness of the sequence implies that $[h]=\tilde{\rho}([h_0])$ for some hermitian form $h_0$ over $(D_0,\,\tau_0)$ of even rank.
Let $\lambda=\mathrm{Pf}(h_0)\in K^*/{\mathrm{Nrd}}(D_0^*)$ be the pfaffian norm of $h_0$. Since $h$ is locally hyperbolic, by considering Suresh’s exact sequence locally, we see that $(h_0)_v$ lies in the image of $(\widetilde{\pi}_1)_v$ for every $v$. By Lemma$\;$\[lemma1TEMP\], this implies that $\lambda\in {\mathrm{Nrd}}((D_0)^*_v).N_{L_v/K_v}(L_v^*)$ for every $v$. In other words, the quadratic form $$\phi:=\lambda.n_{D_0}-\langle 1\,,\,-d \rangle\,,$$where $n_{D_0}$ denotes the norm form of the quaternion algebra $D_0$, is isotropic over every $K_v$. By the assumption on the Hasse principle for quadratic forms of rank 6 (and [@CTPaSu Thm.$\;$3.1] in the $p$-adic arithmetic case), $\phi$ is isotropic over $K$, which shows $\lambda\in {\mathrm{Nrd}}(D_0^*).N_{L/K}(L^*)$. As was mentioned in the proof of Corollary$\;$\[coro3p4temp\], the field $K$ has $u$-invariant 8. So by Lemma$\;$\[lemma3TEMP\], we have $[h_0]\in \mathrm{Im}(\widetilde{\pi}_1)$. Hence $[h]=\tilde{\rho}([h_0])=0\in W(D,\,\tau)$ as desired.
Consider next the general case where ${\mathrm{ind}}(D)$ is even and not divisible by 4. In this case we have $D=Q\otimes_LD'$ for some quaternion division algebra $Q$ over $L$ and some central division algebra $D'$ of odd index over $L$. By [@BP1 Lemma$\;$3.3.1], there is an odd degree separable extension $K'/K$ such that $D'\otimes_KK'=D'\otimes_LLK'$ is split. By Morita theory, there is a unitary $LK'/K'$-involution $\sigma$ on $H\otimes_LLK'$ and a hermitian form $f$ over $(H\otimes_LLK'\,,\,\sigma)$ such that the involution $\tau$ on $D\otimes_LLK'$ is adjoint to $f$, and moreover, the form $h_{K'}$ over $(D\otimes_LLK'\,,\,\tau)$ corresponds to a hermitian form $h'$ over $(H\otimes_LLK'\,,\,\sigma)$, which has even rank, trivial discriminant and trivial Rost invariant. The hypothesis that $h$ is locally hyperbolic over every $K_v$ implies that $h'$ is locally hyperbolic over every $K'_w$, where $w$ runs over the set of divisorial valuations of $K'$. By the previous case, $[h']=0\in W(H\otimes_LLK'\,,\,\sigma)$ and hence $[h]=0\in W(D\otimes_LLK'\,,\,\tau)$. Since the degree $[LK': L]=[K': K]$ is odd, the natural map $W(D,\,\tau)\to W(D\otimes_LLK'\,,\,\tau)$ is injective by a theorem of Bayer-Fluckiger and Lenstra (cf. [@KMRT p.80, Coro.$\;$6.18]). So we get $[h]=0\in W(D,\,\tau)$. This completes the proof.
\[thm3p3TEMP\] Let $K$ be the function field of a $p$-adic arithmetic surface or a local henselian surface with finite residue field of characteristic $p$. Let $L/K$ be a quadratic field extension, $(D,\,\tau)$ a central division algebra over $L$ with a unitary $L/K$-involution whose index $\mathrm{ind}(D)$ is square-free. Let $h$ be a nonsingular hermitian form over $(D,\,\tau)$.
Assume $p\neq 2$ if ${\mathrm{ind}}(D)$ is even. In the local henselian case, assume further that $p\nmid \mathrm{ind}(D)$ and that the Hasse principle with respect to divisorial valuations holds for quadratic forms of rank $6$ over $K$.
Then the natural map $$H^1(K\,,\,{\mathbf{SU}}(h)){\longrightarrow}\prod_{v\in \Omega_A}H^1(K_v\,,\,{\mathbf{SU}}(h))$$has trivial kernel.
Let $\xi\in H^1(K\,,\,{\mathbf{SU}}(h))$ be a class that is locally trivial. Let the image of $\xi$ in $H^1(K,\,{\mathbf{U}}(h))$ correspond to a hermitian form $h'$. The form $h'\bot (-h)$ has even rank, trivial discriminant and is locally hyperbolic. We claim that the Rost invariant $\mathscr{R}(h'\bot(-h))$ is trivial. Indeed, as $\xi$ is locally trivial, $R_{{\mathbf{SU}}(h)}(\xi)$ is locally trivial in $H^3(K_v,\,\mathbb{Q}/\mathbb{Z}(2))$ for every $v$. By Theorem$\;$\[thm1p3temp\], $R_{{\mathbf{SU}}(h)}(\xi)=0$. There is a group homomorphism $${\mathbf{SU}}(h){\longrightarrow}{\mathbf{SU}}(h\bot(-h))\,,\;\; f\longmapsto (f,\,\mathrm{id})$$which induces a map $$\alpha\,:\;\;H^1(K,\,{\mathbf{SU}}(h)){\longrightarrow}H^1(K\,,\,{\mathbf{SU}}(h\bot(-h)))\,.$$The image ${\alpha}(\xi)$ of $\xi$ lifts the class $[h'\bot (-h)]\in H^1(K,\,{\mathbf{U}}(h\bot(-h)))$. By general property of the (usual) Rost invariant, there is an integer $n_{{\alpha}}$ such that $$R_{{\mathbf{SU}}(h\bot(-h))}({\alpha}(\xi))=n_{{\alpha}}R_{{\mathbf{SU}}(h)}(\xi)\,.$$We have thus $\mathscr{R}(h'\bot(-h))=R_{{\mathbf{SU}}(h\bot(-h))}({\alpha}(\xi))=0$ since $\xi$ has trivial Rost invariant. Now Proposition$\;$\[prop3p2TEMP\] implies that the two forms $h', h$ over $(D,\,\tau)$ are isomorphic.
Consider the cohomology exact sequence $$\label{eq3p3p1TEMP}
1{\longrightarrow}\frac{R^1_{L/K}\mathbb{G}_m(K)}{{\mathrm{Nrd}}({\mathbf{U}}(h)(K))}\overset{\varphi}{{\longrightarrow}}H^1(K,\,{\mathbf{SU}}(h)){\longrightarrow}H^1(K,\,{\mathbf{U}}(h))$$arising from the exact sequence of algebraic groups $$1{\longrightarrow}{\mathbf{SU}}(h){\longrightarrow}{\mathbf{U}}(h)\overset{{\mathrm{Nrd}}}{{\longrightarrow}}R^1_{L/K}\mathbb{G}_m{\longrightarrow}1\,.$$The fact that $h'\cong h$ implies that $\xi$ lies in the image of the map $\varphi$ in the above cohomology exact sequence . Considering the sequence locally and using Lemma$\;$\[lemma5p2temp\], we conclude that $\xi$ is trivial in $H^1(K,\,{\mathbf{SU}}(h))$, thus proving the theorem.
*Acknowledgements.* The author thanks Prof. Jean-Louis Colliot-Thélène for helpful discussions.
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[^1]: R. Preeti [@Preeti] has proved results on the injectivity of the Rost invariant which overlap with the results in this paper. Our work was carried out independently.
[^2]: See [@Preeti] for a recent improvement of this theorem.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The unprecedented sensitivity of [*Herschel*]{} coupled with the high resolution of the HIFI spectrometer permits studies of the intensity-velocity relationship $I$($v$) in molecular outflows, over a higher excitation range than possible up to now. In the course of the CHESS Key Program, we have observed toward the bright bowshock region L1157-B1 the CO rotational transitions between $J$=5–4 and $J$=16–15 with HIFI, and the $J$=1–0, 2–1 and 3–2 with the IRAM-30m and the CSO telescopes. We find that all the line profiles $I_{\rm CO}(v)$ are well fit by a linear combination of three exponential laws $\propto \exp(-|v/v_0|)$ with $v_0= 12.5$, 4.4 and $2.5{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$. The first component dominates the CO emission at $J \geq 13$, as well as the high-excitation lines of SiO and ${\hbox{H$_{2}$O}}$. The second component dominates for $3 \le J_{\rm up} \le 10$ and the third one for $J_{\rm up} \le 2$. We show that these exponentials are the signature of quasi-isothermal shocked gas components : the impact of the jet against the L1157-B1 bowshock ($T_{\rm k}\simeq 210{\hbox{\kern 0.20em K}}$), the walls of the outflow cavity associated with B1 ($T_{\rm k}\simeq 64{\hbox{\kern 0.20em K}}$) and the older cavity L1157-B2 ($T_{\rm k}\simeq 23{\hbox{\kern 0.20em K}}$), respectively. Analysis of the CO line flux in the Large-Velocity Gradient approximation further shows that the emission arises from dense gas ($n({\hbox{H${}_2$}}) \geq 10^5-10^6{\hbox{\kern 0.20em cm$^{-3}$}}$) close to LTE up to $J$=20. We find that the CO $J$=2–1 intensity-velocity relation observed in various other molecular outflows is satisfactorily fit by similar exponential laws, which may hold an important clue to their entrainment process.'
author:
- 'B. Lefloch$^{1,2}$, S. Cabrit$^{3}$, G. Busquet$^{4}$, C. Codella$^{5,1}$, C. Ceccarelli$^1$, J. Cernicharo$^2$, J.R. Pardo$^2$, M. Benedettini$^{4}$, D.C. Lis$^6$, B. Nisini$^7$'
date: 'Received : 2012 July 03; Accepted : 2012 August 20'
title: 'The CHESS survey of the L1157-B1 shock region : CO spectral signatures of jet-driven bowshocks'
---
Introduction
============
During the earliest protostellar stages of their evolution, young stars generate fast collimated winds which impact against the parent cloud through shock fronts, generating slow “molecular outflows” of swept-up material. The intensity-velocity relationship $I_{\rm CO}$($v$) observed in low-$J$ CO lines in molecular outflows has been studied by various authors, as a possible test for discriminating between entrainment mechanisms. Downes & Cabrit (2003; hereafter DC03) showed that hydrodynamical simulations of jet-driven molecular outflows could successfully account for the observed relation $I_{\rm CO}(v)$ in CO $J$=2–1. The sensitivity and the range of excitation conditions explored were somewhat limited, however.
The heterodyne instrument, HIFI, onboard Herschel[^1] now allows studies with unprecedented sensitivity, of the dynamical evolution of gas in protostellar outflows and shocks at spectral and angular resolutions comparable to the largest ground-based single-dish telescopes (de Graauw et al. 2010). In particular, HIFI gives access to the CO ladder from $J$=5–4 up to $J$=16–15, probing a wide range of physical conditions.
As part of the CHESS Key Program dedicated to chemical surveys of star forming regions (Ceccarelli et al. 2010), the outflow shock region L1157-B1 was investigated with Herschel. The protostellar outflow driven by the Class 0 protostar L1157-mm ($d$= 250 pc; Looney et al. 2007) is the prototype of chemically rich bipolar outflows (see Bachiller et al. 2001 and references therein). Gueth et al. (1996) showed that the southern lobe of this molecular outflow consists of two cavities, likely created by the propagation of large bowshocks due to episodic events in a precessing, highly collimated jet. Located at the apex of the more recent cavity, the bright bowshock region B1 has been widely studied at millimeter and far-infrared wavelengths and has become a benchmark for magnetized shock models (see Gusdorf et al. 2008). Preliminary results (Codella et al. 2010) have confirmed the chemical richness of L1157-B1 and revealed the presence of multiple components with different excitation conditions coexisting in the B1 bowshock structure (Lefloch et al. 2010, Benedettini et al. 2012).
In this Letter, we report on high-sensitivity CO observations with HIFI of L1157-B1, from $J$=5–4 up to 16–15, and complementary observations of the $J$=1–0, 2–1 and $J$=3–2 with the IRAM 30m and the CSO telescope.
Observations and data reduction
===============================
The HIFI data
-------------
CO transitions between $J$=5–4 and $J$=16–15 were observed with HIFI at the position of L1157-B1 $\alpha_{J2000} = 20^h 39^m 10.2^s$ $\delta_{J2000} = +68^{\circ} 01\arcmin 10.5\arcsec$. The observations were carried out in double beam switching mode. The receiver was tuned in double sideband and the Wide Band Spectrometer (WBS) was used, providing a spectral resolution of 1.1 MHz, which was subsequently degraded to reach a final velocity resolution of $0.5{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$. The telescope parameters (main-beam efficiency $\eta_{mb}$, half power beamwidth HPBW) were adopted from Roelfsema et al. (2012; see Table 1).
The data were processed with the ESA-supported package HIPE 6[^2] (Herschel Interactive Processing Environment). FITS files from level 2 data were then created and transformed into GILDAS[^3] format for baseline subtraction and subsequent data analysis.
Complementary ground-based observations
---------------------------------------
The CO $J$=3–2 line emission was mapped at the Nyquist [**spatial**]{} frequency across a region of $72\arcsec \times 120\arcsec$ in the southern lobe of the L1157 outflow in June 2009 using the facility receivers and spectrometers of the Caltech Submillimeter Observatory (CSO) on Mauna Kea, Hawaii. Observations were carried out in position switching mode using a reference position 10$^{\prime}$ East from the nominal position of B1. Small contamination from the cloud was observed, resulting in a narrow dip at the cloud velocity $v_{lsr}= +2.6{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$ (Bachiller & Perez-Gutierrez, 1997). The data were taken under good to average weather conditions, with system temperatures $\rm T_{sys}$ in the range $750-850{\hbox{\kern 0.20em K}}$. An FFTS was used as a spectrometer, which provided a nominal resolution of $0.1{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$. The final resolution was degraded to $0.25{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$. The final rms of the map is $\approx 0.25{\hbox{\kern 0.20em K}}$ per $0.25{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$ velocity interval.
Deep integrations were performed at the frequency of the CO $J$=1–0 and $J$=2–1 transitions in June and August 2011 as part of an unbiased spectral survey of L1157-B1 at the IRAM 30m telescope (Lefloch et al. 2012, in prep). The EMIR receivers were connected to the 200 kHz resolution (FTS) spectrometers. Observations were carried out using a nutating secondary with a throw of $3\arcmin$, resulting in a narrow absorption feature at the cloud velocity.
The observations (frequency, $\rm Obs\_Id$) and the telescope parameters are summarized in Table 1. Line intensities are expressed in units of antenna temperature corrected for atmospheric attenuation (for ground-based observations) $T_A{}^{*}$.
Results
=======
{width="\columnwidth"}
{width="1.3\columnwidth"}
CO Spectral signatures
----------------------
The CO line profiles are displayed in Fig. 1 ($J_{\rm up} \ge 5$), Fig. 2a ($J=$1–0, 2–1) and Fig. 3 ($J$=3–2) on a log-linear scale. This permits identification of three underlying components in the line profiles, denoted hereafter $g_1$, $g_2$ and $g_3$. Each component is well described by an exponential law $I_{\rm CO}(v)= I_{\rm CO}(0)\exp(-|v/v_0|)$ showing the same slope at all $J$, but differing relative intensities.
The high-excitation CO transitions $J$=13–12 and $J$=16–15 are well fit by the component $g_1$ [*alone*]{}, with $v_0= 12.5{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$ (Fig. 1, top). In the lower $J$ transitions, $g_1$ still dominates the emission at high velocity $v \le -20 {\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$ (Figs. 1–2) and a simple scaling to the $J$=16–15 line profile in this velocity range allows us to determine the total $g_1$ contribution to the integrated intensity of each CO line (Table 1). After removing the contribution of $g_1$, the $J$=10–9 and $J$=9–8 line profiles appear to be well reproduced by the $g_2$ component alone, with $v_0= 4.4{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$ (Fig. 1). Hence, the same procedure as above was applied to estimate the total contribution of $g_2$ across the CO ladder (Table 1). An emission excess with respect to the $g_2$ and $g_1$ contributions is observed at velocities close to the cloud velocity at $J \le 7$ and actually dominates the emission in the low excitation transitions $J \leq 2$ (Table 1). This “residual emission” is well fit by the third exponential function $g_3 \propto \exp(-|v/2.5|)$, except for the transitions $\rm J\leq 2$ for which $v_0$ has a slightly steeper value $\approx 2.0$.
This profile decomposition is justified by the fact the the $^{12}$CO line emission is optically thin, as shown by comparison with spectra, except at velocities very close to that of the ambient cloud for the low-$J$ transitions.
The fact that the slopes of $g_1$, $g_2$, and $g_3$ are [*independent*]{} of the CO transition considered is quite remarkable, and somewhat unexpected as one would naively assume the temperature gradients in shocked, accelerated gas to alter the shape of $I_{\rm CO}$($v$) depending on the CO rotational level. Instead, it seems that the upper energy of the level only changes the relative importance of each exponential component in the resulting profile. In the following section, we show that this behavior is due to each component probing a distinct spatial region with almost uniform excitation conditions.
Shock origin and physical conditions
------------------------------------
Figures 2a and 1 show that first $g_3$, then $g_2$ and finally $g_1$ dominate at progressively higher $J_{up}$, which implies that the three components trace gas with progressively higher excitation conditions. We note that all three components emit over a wide range of velocities and all display an emission peak at velocities close to the cloud velocity. Therefore, their excitation conditions cannot be determined from a simple analysis (e.g. line ratios) in different velocity intervals. Instead, we use our CO profile decomposition, which yields the total flux of each component as a function of $J_{\rm up}$ (Table 1). The excitation conditions in each component are first obtained from a simple rotational diagram analysis of the HIFI and IRAM CO fluxes, after convolving to a common angular resolution of $20\arcsec$ (Fig. 2c). Further constraints on the kinetic temperature and density of the CO gas are then obtained using a radiative transfer code in the Large Velocity Gradient (LVG) approximation assuming a plane parallel geometry. We used the collisional rate coefficients of Yang et al. (2010) and built a grid of models with density between $10^4 {\hbox{\kern 0.20em cm$^{-3}$}}$ and $10^7 {\hbox{\kern 0.20em cm$^{-3}$}}$ and temperature between $10{\hbox{\kern 0.20em K}}$ and $1000{\hbox{\kern 0.20em K}}$ ($250{\hbox{\kern 0.20em K}}$) to determine the region of minimum $\chi^2$ as a function of density and temperature for $g_1$ ($g_2$ and $g_3$). We adopted a typical line width $\Delta v$ of $10{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$ ($g_1$), and $5{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$ for ($g_2$ and $g_3$).
### The $ g_1$ component
PACS observations of L1157-B1 have shown that the CO $J$=16–15 emission arises from a small ($\approx 7-10\arcsec$) region, which peaks at $\approx 5\arcsec$ North with respect to the nominal position of B1, and is associated with a partly-dissociative J-type shock in the region where the protostellar jet impacts the cavity (Benedettini et al. 2012). We conclude that $g_1$ is the spectral signature of this shock and refer to this gas component as the “ $g_1$ shock” in the subsequent discussion.
As shown in Fig. 2b, the $g_1$ component alone dominates the profiles of the ${\hbox{H$_{2}$O}}$ $3_{12}$–$3_{03}$ ($E_{up}=215{\hbox{\kern 0.20em K}}$) and SiO $J$=8–7 ($E_{up}=75{\hbox{\kern 0.20em K}}$) lines, observed respectively with HIFI (Busquet et al., in prep) and the IRAM 30m telescope (Codella et al. in prep). In spite of large differences in their upper level energies, an excellent match is observed at all velocities between those tracers and the CO $J$=16–15 ($E_{up}=751.8{\hbox{\kern 0.20em K}}$), except in the low-velocity range of the SiO $J$=8–7 transition, where a slight excess is observed. These three transitions are therefore probing the same physical region and the similarity of their line profiles justifies our use of the CO $J$=16–15 as a template for determining the contribution of $g_1$ to each CO transition.
Given the wide range of CO transitions and frequencies considered, the variations of the coupling of the telescope beam with the (off-centered) compact $g_1$ shock must be taken into account. This was done by convolving a fully sampled map of the SiO J=8-7 emission obtained at the IRAM 30m telescope (Codella et al., in prep) to the resolution of the HIFI beams of the CO transitions $J$=5–4 up to $J$=8–7. The $J$=16–15 line flux in a $20\arcsec$ beam was directly obtained by convolving the PACS image of Benedettini et al. (2012) and the $J$=13–12 line flux was estimated under the assumption that the ratio of $J$=13–12/$J$=16–15 is preserved when degrading the resolution from $14\arcsec$ ($10.7\arcsec$) to $20\arcsec$. This is supported by the fact that both lines show similar profiles. Since the IRAM $J$=2–1 and HIFI $J$=16–15 observations have very similar angular resolution (Table 1), the $J$=2–1 line flux was estimated under the same assumption as for $J$=13–12.
The rotational diagram of $g_1$ is shown in Fig. 2c. The level populations in the HIFI range are well fit by a single rotational temperature $T_{rot}= 206{\hbox{\kern 0.20em K}}$ and a beam-averaged column density $N({\rm CO})= 2.3\times 10^{15} {\hbox{\kern 0.20em cm$^{-2}$}}$. The populations of the levels $J_{\rm up}$=1 and $J_{\rm up}$=2 lie a factor 3–7 above this trend, and could be the signature of a lower excitation component. LVG calculations were then carried out taking into account the CO line fluxes from $J$=5–4 to $J$=20–19, using the HIFI and PACS data (see Benedettini et al. 2012). The results are shown in Fig. 2d in the form of $\chi^2$ contours. The best-fit solution ($\chi^2= 0.35$) is obtained for $T_{\rm k}= 210{\hbox{\kern 0.20em K}}$ and $\rm n({\hbox{H${}_2$}})\ge $ a few $10^6{\hbox{\kern 0.20em cm$^{-3}$}}$, and a source-averaged column density $N({\rm CO})= 0.9\times 10^{16}{\hbox{\kern 0.20em cm$^{-2}$}}$ for a source size of $\approx 10\arcsec$. While our results are consistent with our previous PACS analysis (Benedettini et al. 2012), they favor dense solutions close to LTE with $n({\hbox{H${}_2$}}) \geq 10^6{\hbox{\kern 0.20em cm$^{-3}$}}$ and kinetic temperatures in the range $200-300{\hbox{\kern 0.20em K}}$ (Fig. 2d).
### The $g_2$ component
Our CSO map shows that the CO $J$=3–2 outflow emission is dominated by the $g_2$ component all over the B1 cavity,from the driving protostar L1157-mm down to the bowshock L1157-B1, at the cavity apex (Fig. 3). The exponent of $g_2$ remains constant at all the positions observed. This suggests that the $g_2$ component arises from the shocked gas in the walls of the B1 cavity.
The excitation conditions in the $g_2$ component were derived by scaling the CO fluxes at each $J$ to a common angular resolution of $20\arcsec$, following the same procedure as described above. The coupling between the telescope beam and the source was estimated from the convolution of our CO $J$=3–2 map to the resolution of the different HIFI beams. The data are well fit by a single rotational temperature $T_{rot}\simeq 64{\hbox{\kern 0.20em K}}$ and a beam-averaged column density $N({\rm CO})= 4.0\times 10^{16}{\hbox{\kern 0.20em cm$^{-2}$}}$. Our LVG calculations again favor an LTE solution with density above $10^5{\hbox{\kern 0.20em cm$^{-3}$}}$ and kinetic temperature in the range $60-80{\hbox{\kern 0.20em K}}$ (see Fig. 2d). The best-fit solution ($\chi^2= 0.27$) is obtained for $n({\hbox{H${}_2$}})\simeq 1.0\times 10^7{\hbox{\kern 0.20em cm$^{-3}$}}$, $T_{\rm k}= 64{\hbox{\kern 0.20em K}}$, and a source-averaged column density $N({\rm CO})= 0.9\times 10^{17}{\hbox{\kern 0.20em cm$^{-2}$}}$ for a typical source size of $20\arcsec$. Such a value of $T_{\rm k}$ is in good agreement with the value derived by Tafalla & Bachiller (1995) from multi-transition NH$_3$ observations using the VLA. The bulk of $\rm NH_3$ emission arises from the cavity walls and peaks at the apex of B1 (Tafalla & Bachiller, 1995), supporting our interpretation of the $g_2$ component as tracing this cavity.
{width="\columnwidth"}
### The $g_3$ component
The $g_3$ component was identified as the residual emission in the CO line profile, in addition to the contributions of $g_1$ and $g_2$. The CSO data bring some insight into the spatial origin of this component. The CO $J$=3–2 emission from the older B2 cavity, south of B1, is seen to follow the same intensity-velocity distribution $I_{\rm CO}$($v$) $\propto \exp(-|v/2.0|)$ as the $g_3$ component observed toward B1 (see bottom panel in Fig. 3). We thus speculate that $g_3$ is actually tracing shocked gas from the previous ejection, which led to the formation of the B2 outflow cavity.
The excitation conditions in the $g_3$ component were derived by scaling the CO fluxes at each $J$ to a common angular resolution of $20\arcsec$, following the same procedure as described above. A rotational diagram analysis of the $g_3$ emission yields $T_{rot}= 26{\hbox{\kern 0.20em K}}$ and a beam-averaged gas column density $N({\rm CO})= 2.6\times 10^{16}{\hbox{\kern 0.20em cm$^{-2}$}}$. Our LVG calculations again favor an LTE solution with density $\ge 10^5{\hbox{\kern 0.20em cm$^{-3}$}}$ and $T_{\rm k}\simeq 23{\hbox{\kern 0.20em K}}$ (Fig. 2d). The best-fit solution was obtained for a source size of $25\arcsec$ and a source-averaged column density $N({\rm CO})\simeq 1.0\times 10^{17}{\hbox{\kern 0.20em cm$^{-2}$}}$. The lower temperature of $g_3$ compared to $g_2$ is consistent with the B2 cavity being older than B1, thus having experienced more post-shock cooling (Gueth et al. (1996) estimated an age of $3000{\hbox{\kern 0.20em yr}}$ and $2000{\hbox{\kern 0.20em yr}}$ for the outflow cavities associated with bowshocks B2 and B1, respectively).
The relation $I_{\rm CO}$($v$) revisited
----------------------------------------
{width="7.cm"}
Previous work described the relation $I_{\rm CO}$($v$) in outflows by a broken power law, $I_{\rm CO}$($v$)$\propto v^{-\gamma}$ with $\gamma \simeq 1.8$ up to line-of-sight velocities $v_{break} \approx 10-30 {\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$ and a steeper slope $\gamma \simeq$ 3–7 at higher velocities. This behavior is successfully reproduced by jet-driven flows, as a result of CO dissociation above shock speeds of $20{\hbox{\kern 0.20em km\kern 0.20em s$^{-1}$}}$ and of the temperature dependence of the line emissivity (see DC03). However, the underlying bowshock model predicts a power-law at low velocities on long time scales, rather than an exponential law, as observed in L1157-B1. It also predict a continuous range of temperatures in the swept-up gas, at odds with our finding that the L1157-B1 line profiles seem to be composed of three quasi-isothermal spectral components.
We show in Fig. 4 the CO $J$=2–1 observations of five outflows previously studied by Bachiller & Tafalla (1999) and modelled by DC03 : L1448, Orion A, NGC2071, L1551, Mon R2. We display in dashed the best fit to the data with a single exponential $I_{\rm CO}$($v$) $\propto \exp(-|v/v_0|)$. A very good agreement is observed in all cases (Fig. 4), with values of $v_0$ well in the range of those determined in L1157-B1. We conclude that an exponential relation $I_{\rm CO}$($v$) $\propto \exp(-|v/v_0|)$ is a good approximation to the observed intensity-relation not only in L1157-B1 but in molecular outflows in general, with a reduced number of free parameters compared to a broken power law.
Our second main finding, that the exponential components in L1157-B1 appear quasi-isothermal and close to LTE, also has important implications. First, it shows that HIFI data are crucial to resolve ambiguities between sub-LTE vs LTE fits to CO excitation diagrams based on PACS data (Benedettini et al. 2012, Neufeld 2012). Second, it shows that the CO flux up to $J_{\rm up}= 16$ is dominated in each component by the densest, coolest postshock gas. Since such dense gas already reached a final constant speed, the broad velocity range of each exponential may require a broad range of view angles and/or shock speeds within the telescope beam. The exact origin of this spectral shape remains to be explained and may hold an important clue to entrainment and shock dynamics in molecular outflows.
We thank R. Bachiller and T. Downes for providing us with the observational data presented in Fig. 3. HIFI has been designed and built by a consortium of institutes and university departments from across Europe, Canada and the United States under the leadership of SRON Netherlands Institute for Space Research, Groningen, The Netherlands and with major contributions from Germany, France and the US. Consortium members are: Canada: CSA, U.Waterloo; France: CESR, LAB, LERMA, IRAM; Germany: KOSMA, MPIfR, MPS; Ireland, NUI Maynooth; Italy: ASI, IFSI-INAF, Osservatorio Astrofisico di Arcetri-INAF; Netherlands: SRON, TUD; Poland: CAMK, CBK; Spain: Observatorio Astronómico Nacional (IGN), Centro de Astrobiología (CSIC-INTA). Sweden: Chalmers University of Technology - MC2, RSS & GARD; Onsala Space Observatory; Swedish National Space Board, Stockholm University - Stockholm Observatory; Switzerland: ETH Zurich, FHNW; USA: Caltech, JPL, NHSC. C. Codella and C. Ceccarelli acknowledge the financial support from the COST Action CM0805 “The Chemical Cosmos”. S. Cabrit and C. Ceccarelli acknowledge the financial support from the french spatial agency CNES. G.Busquet is supported by an Italian Space Agency (ASI) fellowship under contract number I/005/007. B. Lefloch thanks the Spanish MEC for funding support through grant SAB2009-0011. J. Cernicharo thanks the Spanish MICINN for funding support through grants AYA2009-07304, and CSD2009-00038. Support for this work was provided by NASA through an award issued by JPL/Caltech. The CSO is supported by the National Science Foundation under the contract AST-08388361.
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[crrcrrccccl]{} 1–0 & 115.27120 & 5.5 & - & 0.78 & 21.4 & 2.0 & 1.82 & 13.8 & 27.8 & IRAM\
2–1 & 230.53800 & 16.6 & - & 0.59 & 10.7 & 2.7 & 2.39 & 29.7 & 52.1 & IRAM\
3–2 & 345.79599 & 33.2 & - & 0.65 & 22.0 & 130 & - & 42.9 & 17.5 & CSO\
5–4 & 576.26793 & 83.0 & 1342181160 & 0.75 & 37.4 & 8.0 & 2.58 & 39.8 & 8.4 & HIFI\
6–5 & 691.47308 & 116.2& 1342207606 & 0.75 & 30.7 & 5.0 & 3.03 & 45.3 & 7.5& HIFI\
7–6 & 806.65180 & 154.9& 1342201707 & 0.75 & 26.3 & 7.1 & 3.03 & 36.9 & 3.0& HIFI\
& & & 1342207624 & 0.75 & 26.3 & & & & & HIFI\
8–7 & 921.79970 & 199.1& 1342201554 & 0.74 & 23.0 & 10.0& 4.55 & 27.4 & - & HIFI\
& & & 1342207323 & 0.74 & 23.0 & & & & & HIFI\
9–8 & 1036.91239& 248.9& 1342200962 & 0.74 & 20.5 & 7.7 & 4.09 & 23.8 & - & HIFI\
& & & 1342207641 & 0.74 & 20.5 & & & & & HIFI\
10–9&1151.98544 & 304.2& 1342207691 & 0.64 & 18.4 & 36 & 3.79 & 9.61 & - & HIFI\
& & & 1342196511 & 0.64 & 18.4 & & & & & HIFI\
13-12&1496.92291 & 503.2& 1342214390 & 0.72 & 14.1 & 46 & 4.55 & - & - & HIFI\
16-15&1841.34551 & 751.8& 1342196586 & 0.70 & 11.5 & 26 & 2.27 & - & - & HIFI\
[^1]: Herschel is an ESA space observatory with science instruments provided by European-led principal Investigator consortia and with important participation from NASA.
[^2]: HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia.
[^3]: http://www.iram.fr/IRAMFR/GILDAS
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Kevin Whyte
title: 'The Large Scale Geometry of the Higher Baumslag-Solitar Groups'
---
Introduction {#introduction .unnumbered}
============
The Baumslag-Solitar groups:
$$BS(m,n)=<x,y| x y^{m} x^{-1} = y^{n}>$$
are some of the simplest interesting infinite groups which are not lattices in Lie groups. They have been studied in depth from the point of view of combinatorial group theory. It is natural to ask if the geometric approach to the theory of infinite groups, which has been so successful in the study of lattices, can yield any insights in this nonlinear case.
The first step towards a geometric understanding of the Baumslag-Solitar groups is to decide which among the $BS(m,n)$ are quasi-isometric. The groups $BS(1,n)$ are solvable, hence amenable, and so are not quasi-isometric to any of the $BS(m,n)$ with $1 < m \leq n$ which contain free subgroups and hence are are nonamenable.
The solvable groups $BS(1,n)$ are in many respects the most lattice-like of the Baumslag-Solitar groups. They are discrete subgroups in products of real and $p$-adic Lie groups. The groups $BS(1,n)$ are classified up to quasi-isometry by Farb and Mosher in [@FM1]. They prove that $BS(1,n)$ and $BS(1,m)$ are quasi-isometric only if $n$ and $m$ have common powers. When $n$ and $m$ have common powers $BS(1,n)$ and $BS(1,m)$ are not only quasi-isometric, but are commensurable (have isomorphic subgroups of finite index). This is the same rigidity phenomenon as occurs for nonuniform lattices in higher rank. Despite this rigidity, their full group of self quasi-isometries is quite large, and in this they more closely resemble uniform lattices.
In this paper we classify all the Baumslag-Solitar groups up to quasi-isometry. The higher Baumslag-Solitar groups, namely those with $1<m<n$, are unlike the groups $BS(1,n)$ in many ways. They are nonlinear, not residually finite, and usually not Hopfian. Indeed, this “bad” behavior was the motivation for their discovery. Our results show that the higher Baumslag-Solitar groups exhibit a surprising lack of rigidity; all the higher Baumslag-Solitar groups, aside from the degenerate case of $BS(n,n)$, are quasi-isometric to each other. The quasi-isometries we construct do not reflect any clear algebraic relationship between the groups. In particular, many of the groups we prove to be quasi-isometric are not commensurable. Our method of constructing quasi-isometries seems to be fundamentally different from earlier constructions.
Our main results concern a class of groups somewhat larger than the class of Baumslag-Solitar groups. We define a [*graph of ${\mathbb Z}$s*]{} as a finite graph of groups, in the sense of Serre ([@Se]), with all vertex and edge groups infinite cyclic. This class includes the Baumslag-Solitar groups, which are precisely the HNN extensions of ${{\mathbb Z}}$.
\[gen\] If $G$ is a graph of ${\mathbb Z}$s and $\Gamma=\pi_{1} G$ then exactly one of the following is true:
1. $\Gamma$ contains a subgroup of finite index of the form $F_{n} \times {\mathbb Z}$.
2. $\Gamma = BS(1,n)$ for some $n>1$.
3. $\Gamma$ is quasi-isometric to $BS(2,3)$.
Here $F_{n}$ is the free group on $n$ generators.
\[bs\] All the groups $BS(m,n)$ with $1<m<n$ are quasi-isometric to each other.
Outline
-------
Let $G$ be the fundamental groups of a graph of ${{\mathbb Z}}$s. We begin by constructing a geometric model space for the geometry of $G$. This model is a contractible $2$-complex, $X_{G}$ on which $G$ acts cocompactly, freely and properly discontinuously by isometries. The algebraic fact that $G$ is a graph of ${{\mathbb Z}}$s translates into the geometric fact that $X_{G}$ is a warped product of a tree with ${{\mathbb R}}$. In other words, $X_{G}$ is topologically $T \times {{\mathbb R}}$, with a metric which differs from the product metric in that the metric on $v \times {{\mathbb R}}$ is scaled by a warping function $T \to {{\mathbb R}}^{+}$.
The tree $T$ is the Bass-Serre tree of the graph of groups, and the warping function is induced by a $G$ invariant orientation on $T$. If two graphs of ${{\mathbb Z}}$s are quasi-isometric, we show that there is a quasi-isometry between their Bass-Serre trees which coarse respects, in an appropriate sense, the orientations. Conversely, any coarsely orientation preserving quasi-isometry between Bass-Serre trees induces a quasi-isometry between the groups. Thus the classification of Baumslag-Solitar groups, and graphs of ${{\mathbb Z}}$s in general, reduces to classifying coarsely oriented trees.
The heart of our construction is the construction of coarsely orientation preserving quasi-isometries between trees. We first decompose the trees into lines of constant “slope” (see § \[laminate\]) with respect to the orientation. The quasi-isometries are built line by line. This also requires a quasi-isometry between the spaces of lines, with nice properties with respect to the orientation. Building this matching of lines uses Hall’s Selection theorem, and the axiom of choice.
This construction is sufficient to allow us to completely classify graphs of ${{\mathbb Z}}$s up to quasi-isometry. We also explore the issue of commensurabilities among graphs of ${{\mathbb Z}}$s sufficiently to show that although all the groups $BS(m,n)$ for $1<m<n$ are quasi-isometric, they are in general not commensurable. Thus we have many new explicit examples of groups which are quasi-isometric but not commensurable.
We next turn to describing the quasi-isometry groups of these graphs of ${{\mathbb Z}}$s. For the quasi-isometry groups of the solvable Baumslag-Solitar groups, $BS(1,n)$, there is a nice description in [@FM1]. The situation for the higher Baumslag-Solitar groups is substantially more complex. We discuss the complications and give several descriptions, none entirely satisfactory, of these quasi-isometry groups.
We discuss some generalizations. It is natural to ask what an arbitrary finitely generated groups quasi-isometric to a graph of ${{\mathbb Z}}$s can be. It follows from [@MSW] that any such group is a finite graph of virtual ${{\mathbb Z}}$s. As our classification extends without change to that larger class of groups, we get a complete description of the class of groups quasi-isometric to the higher Baumslag-Solitar groups. We also discuss some further classes of graphs of groups to which our classification of coarsely oriented trees is relevant.
The geometric models {#models}
====================
One of the basic principles of geometric group theory is the Milnor-Svarc theorem, which says that if $G$ is a finitely generated group, and $G$ acts properly discontinuously and cocompactly by isometries on a proper geodesic metric space $X$, then $X$ is quasi-isometric to $G$. Thus, for questions about the quasi-isometric geometry of $G$, one can work instead with $X$.
The $2$-complexes
-----------------
Let $\Gamma$ be a graph of ${\mathbb Z}$s, $G=\pi_{1}\Gamma$, and $T$ the Bass-Serre tree of $G$.
We first describe a 2-complex $X_{G}$ on which $G$ acts properly discontinuously and cocompactly by isometries. Build a compact complex with $\pi_{1}=G$ out of the graph $\Gamma$ as follows: start with a disjoint collection of circles, one for each vertex of $\Gamma$. For each edge of $\Gamma$ glue in an $S^{1}\times [0,1]$ where the attaching maps at each end are covering maps inducing the same map on fundamental groups as the inclusions of the corresponding edge groups. The universal cover of the complex is the desired $X_{G}$.
Following [@FM1], we give another description of $X_{G}$. Topologically, $X_{G}$ is $T \times {{\mathbb R}}$. Let $e$ be an edge of $\Gamma$, and let the index of the inclusion into its vertex groups be $n \geq m$. The action of the edge group of $e$ on the strip $e
\times {{\mathbb R}}$ is translation by $n$ over one endpoint and by $m$ over the other. This becomes isometric if we metrize the strip as a warped product $dt^{2} + (\frac{n}{m})^{2t}ds^{2}$, where $t$ is the parameter along $e$, and $x$ along ${{\mathbb R}}$. This makes $e
\times {{\mathbb R}}$ isometric to a horostrip (the region between two concentric horoballs) of width 1 in a space of constant curvature $-
\ln \frac{n}{m}$.
For any vertex $v$ in $T$, we call the subspace $v \times {{\mathbb R}}$ of $X_{G}$ the [*vertex space*]{} over $v$. Likewise, for any edge $e$, $e \times {{\mathbb R}}$ is the [*edge space*]{} over $e$.
Given any two vertices of $T$, $v_{1}$ and $v_{2}$, let $G_1$ and $G_{2}$ be their stabilizers. Let $G_{12}$ be their intersection, which is the stabilizer of the path between them. $G_{12}$ has finite intersection in both $G_{1}$ and $G_{2}$. We call the ratio $\frac{[G_{12}:G_{1}]}{[G_{12}:G_{2}]}$ the [*contraction factor*]{} between $v_{1}$ and $v_{2}$. The terminology is justified by the geometric interpretation as the contraction factor of the closest point projection map from the vertex space over $v_{1}$ to the vertex space over $v_{2}$.
It is more convenient to work with an additive rather than multiplicative invariant. We define the [*height change*]{} between two vertices as the logarithm of the contraction factor. By choosing a base point of $T$, we can define the [*height*]{} of a vertex $v$ as the height change between the base point and $v$. We extend the height function, $h$, to all of $T$ by linear interpolation along edges.
The metric on $X_{G}$ can be described in terms of the height function as a warped product $T {\propto}{{\mathbb R}}$ with warping function $e^{-h}$. Thus $T$, together with the height function, determines the complex $X_{G}$ up to isometry.
The coarsely oriented Bass-Seree tree
-------------------------------------
We view the height change along edges as giving a quantitative analogue of an orientation. If $S$ is an oriented tree, then we can define a height change function by declaring the height change across an edge to be $1$ or $-1$ depending on the orientation. With this height change function, an isometry of $S$ preserves the orientation if and only if it preserves the height change function. Just as we view a quasi-isometry as a “large scale isometry”, we view a quasi-isometry which preserves the height change function, on a large scale, as coarsely orientation preserving.
[**Definition**]{} A quasi-isometry $f:T_{1} \to T_{2}$ between trees with height functions $h_{1}$ and $h_{2}$, is [*coarsely orientation preserving*]{} is there is $C>0$ so that for all $v_{1}$ and $v_{2}$ in $T_{1}$: $$|h_{1}(v_{1},v_{2}) - h_{2}(f(v_{1}),f(v_{2}))| \leq C$$
Notice that only the height change between two points is involved in this definition, so the notion of coarsely orientation preserving is independent of the choice of base points.
\[extend\] For $i=1,2$, let $G_{i}$ be a graph of ${\mathbb Z}$s, with Bass-Serre tree $T_{i}$. If $f$ is a quasi-isometry from $T_{1}$ to $T_{2}$ which is coarsely orientation preserving then $f \times Id : X_{G_{1}} \to X_{G_{2}}$ is a quasi-isometry.
The cases of bounded and unbounded height functions are fundamentally different. In the former, the vertex spaces are isometrically embedded, while in the latter they are exponentially distorted. Clearly, boundedness of height function is a coarse orientation preserving quasi-isometry invariant. The theorem holds in this case as, when the height function is bounded, the complex $X$ is bilipschitz equivalent to the product $T \times {{\mathbb R}}$. Thus we assume the height functions are both unbounded.
If the height function on $T$ is unbounded then the distance in $X_{G}$ is quasi-isometric to:
$$d_{T}(t_{1},t_{2}) + max(0, -h(t_{1},t_{2}) + \ln |x_{1}-x_{2}|)$$
where $h(t_{1},t_{2})$ is the maximum height along the geodesic $t_{1}t_{2}$.
As the set of vertex spaces is coarsely dense in $X$, we may assume that $t_{1}$ and $t_{2}$ are vertices of $T$. Given any path $p$ from $(t_{1},x_{1})$ to $(t_{2},x_{2})$, we can replace $p$ by a path which is piecewise horizontal (constant ${{\mathbb R}}$ coordinate) or vertical (constant $T$ coordinate) without multiplying the height by more that a constant factor.
The total length of such a path is the length of its projection to $T$ (= the length of the horizontal segments) plus the length of the lengths of the vertical segments. The length of a vertical segment in the vertex space over $t$ is $e^{-h(t)}$ times the change in the ${{\mathbb R}}$ coordinate. Thus any path can be shortened by moving all the vertical changes to occur in the vertex space over the point of maximal height on the projection of the path to $T$. Thus the distance between the points $(t_{1},x_{1})$ and $(t_{2},x_{2})$ is bounded below by a multiple of the minimal length of a path which is a horizontal path from $t_{1}$ to a point $t$, followed by a vertical path from $(t,x_{1})$ to $(t,x_{2})$ an then a horizontal path from $t$ to $t_{2}$. The length of such a path is $d_{T}(t_{1},t) + d_{T}(t_{2},t) + e^{-h(t)}|x_{1}-x_{2}|$.
This length is equal to $d_{T}(t_{1},t_{2}) + 2d_{T}(t,t_{1}t_{2}) +
e^{-h(t)}|x_{1}-x_{2}|$. Replacing $t$ by the closest point at the same height as $t$ to $t_{1}t_{2}$ shortens the path, so we may assume that $t$ is this closet point.
Since there is a cocompact symmetry group, it is easy to see that there are $\beta > 0$ and $C >0$ so that the distance of any point $t$ of $T$ to the set of points of height at least $h$ in $T$ is within $C$ of $\beta
max(0,h-h(t))$. Thus the minimal length of a path from $(t_{1},x_{2})$ to $(t_{2},x_{2})$ is, to within $C$, the minimum over $h$ of:
$$d_{T}(t_{1},t_{2}) + 2max(0,\beta|h-h(t_{1},t_{2})|) + e^{-h}|x_{1}-x_{2}|$$
The minimum of this over all $h$ occurs at $h=h(t_{1},t_{2})$ if $|x_{1} - x_{2}| \leq \frac{2e^{h(t_{1},t_{2})}}{\beta}$, and at $h=\ln \frac{\beta |x_{1}-x_{2}|}{2}$ otherwise. Substituting this value for $h$ finishes the proof of the lemma.
Using this lemma, we complete the proof of the theorem. By choosing basepoints so that $f$ is basepoint preserving, we may assume the difference $|h_{1}(t)-h_{2}(f(t))|$ is bounded. Since $f$ is a quasi-isometry, the image of the geodesic from $t$ to $t'$ is within a uniformly bounded distance of the geodesic from $f(t)$ to $f(t')$. Combining these facts, we see that the difference between $h_{1}(t,t')$ and $h_{2}(f(t),f(t'))$ is uniformly bounded. The approximation to the distance in $X$ in the lemma is thus quasi-preserved by $f \times Id$, and thus $f \times Id$ is a quasi-isometry.
Thus, if we can construct a coarse orientation preserving quasi-isometry between the Bass-Serre trees of two graphs of ${{\mathbb Z}}$s, this gives a quasi-isometry between the groups. In fact, all quasi-isometries among graphs of ${{\mathbb Z}}$s arise this way, see § \[qigroups\].
Constructing quasi-isometries {#trees}
=============================
[**Definition**]{} A coarsely oriented tree is [*homogeneous*]{} if the multiset of height changes of edges incident to a vertex $v$ is the same for all $v$. This is equivalent to the transitivity of height change preserving isometries.
The Bass-Serre trees of the Baumslag-Solitar groups are homogeneous. We show in section §\[classify\] that any coarsely oriented tree with cocompact symmetry group is coarsely orientation preserving quasi-isometric to a homogeneous tree.
In this section we classify homogeneous coarsely oriented tree up to coarsely orientation preserving quasi-isometry. Recall that is $T$ is an oriented tree, there is an induced coarse orientation in which the height change across an edge is either $1$ or $-1$ depending on whether is edge is crossed with or against the orientation. Homogeneous oriented trees are determined by their [*type*]{}, which is the ordered pair $(n,m)$ of the number of edges oriented away from and the number of edges oriented towards any vertex.
\[homo\] Let $T$ be a homogeneous coarsely oriented tree with height function $h$. Precisely one of the following holds:
- $h$ is constant.
- At every vertex of $T$ there is one edge which strictly increases (resp. decreases) height, and all the other edges at the vertex strictly decrease (resp. increase) height.
- $T$ is coarsely orientation preserving quasi-isometric to the oriented tree of type $(2,2)$.
The bulk of the proof of this theorem is constructing coarsely orientation preserving quasi-isometries to show
\[main\] If $T$ is a homogeneous coarsely oriented tree for which, at every vertex, there are at least two edges which strictly increase height, and two edges which strictly decrease height, then $T$ is coarsely orientation preserving quasi-isometric to the homogeneous oriented tree of type $(2,2)$.
Assuming the lemma we complete the proof of the theorem.
If there are no edges which change height, then $h$ is constant. Otherwise there are, at every vertex, both an edge which strictly increases height and an edge which strictly decreases height.
Suppose there are edges which do not change height. Let $F$ be the forest of such edges. The components of $F$ are either edges, with one at every vertex, or infinite trees without valence one vertices. In the former case, collapsing $F$ is a coarse orientation preserving quasi-isometry to a tree which satisfies the hypothesis of lemma \[main\]. In the latter case, the following lemma produces a subset $F$, the collapsing of which has the same result.
Let $S$ be an infinite tree without valence one vertices. There is a subset of the edges of $S$ which contains exactly one edge at every vertex.
Let $S'$ be a maximal subtree of $S$ for which there is such a subset of edges. If $S' \neq S$ then there is a vertex $v$ of $S-S'$ and an edge $e$ with one endpoint $v$ and the other endpoint, $u$, in $S'$. If $u$ is not in an edge of the subset, then one can extend $S'$ to $S' \cup e$ and add $e$ to the subset of edges. If $u$ is in one of the subset of edges of $S'$ then let $e'$ be any edge at $v$ other than $e$, and extend $S'$ to $S' \cup e \cup e'$ adding $e'$ to the subset of edges. In either case this contradicts maximality.
Finally, if there are no edges which do not change height, then one is clearly either in the second case of the theorem or satisfy the hypotheses of lemma \[main\] and hence in the third case. This completes the proof of the theorem, assuming lemma \[main\].
We now turn to the proof of lemma \[main\]. There are two steps in this proof. The first step is to decompose the tree into lines along which the height function changes at essentially a constant rate with respect to length. The second step is to find a matching of the lines in one tree with the lines in the other so that we can assemble a coarsely orientation preserving quasi-isometry line by line.
Constant slope laminations {#laminate}
--------------------------
[bf Definition]{} Let $\beta$ and $C$ in ${\mathbb R}$ be given. We call a bi-infinite geodesic $\gamma$ in $T$ a [*line of slope $(\beta,C)$*]{} if and only if for all $n$ and $m$ in ${\mathbb Z}$
$$|h(\gamma(n))-h(\gamma(m)) - \beta (n-m)| \leq C$$
\[cover\] If $T$ is a homogeneous tree with height function which has at each vertex at least two edges along which the height increases, and two along which it decreases, then there is $\beta_{0} >0$ so that for any $0 \leq \beta \leq \beta_{0}$ there is a $C$ and a family of lines of slope $(\beta, C)$ exactly one of which passes through each vertex of $T$.
We will call such a collection a [*lamination by lines of slope $\beta$*]{}.
Take $\beta_{0}$ to be such that there are two or more edges, at each vertex, which increase height by at least $\beta_{0}$ and two or more which decrease it by at least $\beta_{0}$. Fix $0 \leq \beta \leq
\beta_{0}$. Let $M$ be the maximal amount height changes along any edge, and take $C=2M$.
Given any vertex $v$ and edge $e$ at $v$ which increases height by at least $\beta_{0}$ we can find a ray of slope $(\beta,M)$ starting at $v$ and beginning with $e$. We build this ray inductively. If a ray $r$ of length $n$ has been constructed, extend it to length $n+1$ by choosing an edge which increases height by at least $\beta$ if $\beta (n+1)
\geq h(r(n))-h(v)$ and choosing one which decreases height by at least $\beta$ otherwise. It is easy to see that $r$ has the desired properties. It is likewise possible to build a ray of slope $(-\beta,M)$ through any edge $e'$ at $v$ along which height decreases by at least $\beta$. By gluing the two we get a line of slope $(\beta,C)$.
Now suppose we have $T'$ a subtree of $T$ which has been given a covering by lines of slope $(\beta,C)$. If $T' \neq T$ then there is a $v$, a vertex of $T$, which is adjacent to $T'$. Since only one edge connects $v$ to $T'$ we can build a line of slope $(\beta,C)$ through $v$ disjoint from $T'$. Then we can enlarge $T'$ to include $v$, the edge connecting $v$ to $T'$, and the new line. Continuing in this way we cover all of $T$.
Matching the lines
------------------
Given two trees, $T_{1}$ and $T_{2}$, covered by lines of slope $\beta_{1}$ and $\beta_{2}$ we try to find an coarsely orientation preserving quasi-isometry from $T_{1}$ to $T_{2}$ one line at a time. Given two lines there is an coarsely orientation preserving quasi-isometry between the lines, which is unique up to bounded distance. Given a bijection between the sets of lines covering $T_{1}$ and those covering $T_{2}$ we get almost orientation preserving maps $T_{1} \to T_{2}$ and $T_{2} \to T_{1}$ with compositions at bounded distance from the identity maps of $T_{1}$ and $T_{2}$. We now discuss the precise conditions which make this map a quasi-isometry of the trees.
Let $T_{1}'$ and $T_{2}'$ be the trees obtained from $T_{1}$ and $T_{2}$ by collapsing the lines of the laminations to points. Suppose we have a tree isomorphism, $f$, between these quotients. This gives, as above, $\hat{f}:T_{1} \to T_{2}$. This $\hat{f}$ has bounded stretch along the lines of the laminations and is coarsely orientation preserving. If we have an edge $e$ at height $h$ in $T_{1}$ which connects two lines, $a$ and $b$, it maps to an edge of $T_{1}'$ and so its image under $\hat{f}$ maps to an edge of $T_{2}'$. There is a unique edge $e'$ of $T_{2}$ which maps to edge of $T'_{2}$. The edge $e'$ connects two lines, $a'$ and $b'$, in $T_{2}$.
Since $\hat{f}$ is coarsely orientation preserving, the end points of $e'$ are near the points of height $h$ on $a'$ and $b'$. If $e'$ is at height $h'$ then these points are at distance $2|h'-h| + 1$ in $T_{2}$. Thus $\hat{f}$ is a quasi-isometry of $T_{1}$ and $T_{2}$ if, for every edge $e$ of $T'_{1}$, the heights of the edge in $T_{1}$ mapping to $e$ and the edge of $T_{2}$ mapping to $f(e)$ differ by a uniformly bounded amount.
For $i=1,2$ let $T_{i}$ be a homogeneous tree of valence $n_{i}$ and be covered by lines of slopes $\beta_{i}$. If $\frac{\beta_{1}}{\beta_{2}} = \frac{n_{1}-2}{n_{2}-2}$ then there are a $K>0$ and a tree isomorphism between $T_{1}'$ and $T_{2}'$ so that corresponding edges, when lifted to $T_{1}$ and $T_{2}$, differ in height by at most $K$.
Pick base points $v_{1}$ in $T_{1}'$ and in $T_{2}'$, let $f(v_{1})=v_{2}$. Assume we can biject the edges at $v_{1}$ and $v_{2}$ in such a way as to change heights by at most $K$. This gives $f$ on the balls of radius $1$ around the basepoints. Suppose we have the map $f$ defined between the balls of radius $n$. If, for each $v$ in the sphere of radius $n$, we can biject the edges at $v$ which connect to the sphere of radius $n+1$ with those at $f(v)$ which connect to the sphere or radius $n+1$, in such a way as to change height by at most $K$, then we can extend $f$ to the balls of radius $n+1$. Then, by induction, we would have the desired $f$.
To construct the edge bijections needed in this construction we use Hall’s selection theorem. In this context this says that bijections will exist between the edges at $w_{1}$ and $w_{2}$ if and only if for every interval $[a,b]$ in ${\mathbb R}$ the number of edges at $w_{1}$ with heights in $[a,b]$ is no more than the number at $w_{2}$ with heights in $[a-K,b+K]$, and vice versa. This holds because the number of vertices on a line $l$ of slope $\beta$ with heights in the range $[a,b]$ is, to within a uniform additive error, $\frac{b-a}{\beta}$. Thus, by the condition on the slopes, Hall’s theorem applies for $K$ large enough.
Lemma \[cover\] gives laminations of $T_{1}$ and $T_{2}$ by lines of constant slope, with slopes of arbitrary ratio. This completes the proof of lemma \[main\].
The classification of graphs of ${{\mathbb Z}}$s
================================================
Using theorem \[extend\] and the previous construction, we can construct quasi-isometries between many graphs of ${{\mathbb Z}}$s. For the solvable Baumslag-Solitar groups, the classification in [@FM1] proves that quasi-isometry implies abstract commensurability. The quasi-isometries we construct are very different in nature, relying on the axiom of choice. We investigate when these graphs of ${{\mathbb Z}}$s are commensurable in sufficient detail to see that many of the groups we prove are quasi-isometric are not commensurable. In particular, while all of the higher Baumslag-Solitar groups $BS(m,n)$ for $1<m<n$ are quasi-isometric, they are, in general, not commensurable.
The quasi-isometric classification {#classify}
----------------------------------
Theorem \[homo\] allows us to construct the quasi-isometries we need to prove Theorem \[gen\]. We first show that if $G$ is any graph of ${{\mathbb Z}}$s then its Bass-Serre tree is coarsely orientation preserving quasi-isometric to a homogeneous tree.
We can assume that there are no edges in the graph of groups, $\Gamma$, which have distinct endpoints and for which the edge group includes isomorphically to either of its vertex group. If there were any such edges, they could be collapsed to give a graph of groups with the same fundamental group and fewer edges.
Let $F$ be a maximal tree in $\Gamma$. There is a family of lifts of $F$ to $T$ so that every vertex of $T$ is contained in exactly one of the lifts in the family.
This is done exactly as in Theorem \[cover\]. Since every edge group of $F$ includes as a subgroup of index at least two in both of its vertex groups, for any lift of an endpoint to $T$ there are at least two lifts of the edge at that vertex. If we have lifts which cover a subtree $T'$ of $T$ then there is a $v$ in $T$ adjacent to $T'$. As each edge of $F$ has more than one lift at each lift of its endpoints there is a lift of $F$ through $v$ disjoint from $T'$.
Pick a base point in $\Gamma$ and define the height of a lift of $F$ as the height of the lift of the base point it contains. Then the tree $\hat{T}$ of these lifts, or equivalently the tree obtained by collapsing each lift, is a homogeneous tree coarsely orientation preserving quasi-isometric to $T$.
If $T$ has bounded height function then it easy to see that $G$ has a subgroup of finite index which is $F_{n} \times {{{\mathbb Z}}}$.
If the height function on $T$ is unbounded then the height function on $\hat{T}$ is also unbounded. Each vertex then must have at least one edge increasing height and one decreasing height. If $F$ contains any edges then there is at least one edge which does not change height at each vertex of the collapsed tree. As in §\[trees\] this implies that $T$ is coarsely orientation preserving quasi-isometric to the oriented tree of type $(2,2)$.
If $F$ contains no edges, then $\Gamma$ has only one vertex. If there is a loop in $\Gamma$ which does not change height then again $T$ is coarsely orientation preserving quasi-isometric to the oriented tree of type $(2,2)$. The same holds, by lemma \[main\], if there are two or more loops that do change height, or a single loop which changes height which has more than one lift at both of its endpoints. Thus the only graph of ${{\mathbb Z}}$s with unbounded height function not coarsely orientation preserving quasi-isometric to the oriented tree of type $(2,2)$ is a graph of ${{\mathbb Z}}$s with a single vertex and a single edge which includes isomorphically at one end. These are precisely the solvable Baumslag-Solitar group, which are classified up to quasi-isometry in [@FM1].
This completes the proof of Theorem \[gen\].
Noncommensurability
-------------------
We investigate when graphs of ${{\mathbb Z}}$s are commensurable. While we do not get a complete classification, we show that many of the groups we have shown to be quasi-isometric are not commensurable.
Suppose $(a,b)=(c,d)=1$ and $\frac{a}{b}\neq
\frac{c}{d}$, then the groups $BS(a,b)$ and $BS(c,d)$ are not commensurable.
Let $\Gamma$ be any graph of ${\mathbb Z}$s not quasi-isometric to $F_{n} \times {\mathbb Z}$ or to a solvable Baumslag-Solitar group.
An element $\gamma$ is of [*vertex type*]{} if and only if for any $\sigma \in \Gamma$ there are $n$ and $m$ nonzero for which $\gamma^{n}=\sigma \gamma^{m} {\sigma}^{{-1}}$.
$\gamma$ is of vertex type if and only if $\gamma$ stabilizes a vertex.
Since the tree has bounded valence any two vertex stabilizers are commensurable, so certainly any element which fixes a vertex is of vertex type. Conversely, if $\gamma$ does not stabilize a vertex then it is a hyperbolic tree automorphism, so any element which conjugates one power of $\gamma$ to another must preserve its axis. This can only be the entire group if $T$ is quasi-isometric to ${\mathbb Z}$ which does not happen for graphs of ${\mathbb Z}$s in this quasi-isometry class.
If $\Gamma$ is represented by a graph without any edge groups which include isomorphically into either vertex group, for example $BS(m,n)$ for $m$ and $n$ both greater than one, then no vertex stabilizer is contained in another. In that case, the maximal cyclic subgroups of vertex type are precisely the vertex stabilizers, so the vertex set of $T$ is determined as a $\Gamma$ set by $\Gamma$. The height function is also determined, as it is defined in terms of the modular homomorphism which is the ratio of indices of the intersections of two vertex stabilizers in each one. It is not difficult to modify this to cope with loops which include isomorphically into one end. There is some ambiguity in identifying the edges do to the possibility of sliding.
For the special case of the groups $BS(m,n)$ with $m$ and $n$ relatively prime and larger than one, the only finite index subgroups are graphs of ${\mathbb Z}$s with underlying graph a circle and all edge groups including as subgroups of index $m$ and $n$ in its vertex groups. As discussed above, we can therefore recover the number $\frac{n}{m}$ just from the isomorphism type of such a group. Thus we see that $\frac{m}{n}$, and therefore $m$ and $n$, are commensurability invariants. In other words, no two of these Baumslag-Solitar groups are commensurable.
The group quasi-isometries {#qigroups}
==========================
In this section we calculate the quasi-isometry group of the groups $BS(m,n)$, for $1<m<n$. As all these groups, and most graphs of ${\mathbb
Z}$s, are quasi-isometric they all have the same quasi-isometry group. The quasi-isometry groups of the solvable Baumslag-Solitar $BS(1,n)$ is the product $Bilip({{\mathbb R}}) \times Bilip({{\mathbb Q}}_{n})$ [@FM1]. We give a similar description of the quasi-isometry group of the higher Baumslag-Solitar groups, although the final form is substantially more complicated.
We start by proving that the special form of the quasi-isometries we construct in §\[trees\] are, in fact, the general case. According to [@FM3], if $F: X \to X$ is a quasi-isometry, there is a quasi-isometry $f: T \to T$ so that $\pi(F(x))=f(\pi(x))$ for $\pi$ the projection of $X$ to $T$.
If $F: X \to X$ is a quasi-isometry covering $f: T
\to T$ then $f$ is coarsely orientation preserving.
For any $t$ in $T$, we define the fiber distance on $\{t\}\times {\mathbb R}$ as the induced path metric. Since any quasi-isometry quasi-preserves the vertex spaces, it quasi-preserves the fiber distance. In terms of the ${\mathbb R}$ coordinate this distance is just $e^{-h(t)}|x_{1}-x_{2}|$.
For any two $t$ and $t'$ in $T$, let $p:\{t\}\times{\mathbb R} \to
\{t'\}\times {\mathbb R}$ be closest point projection. We define the fiber distortion of $p$ as:
$$\frac{d_{F}(p(x),p(y))}{d_{F}(x,y)}$$
where $x$ and $y$ are any points on the vertex space over $t$, and $d_{F}$ is the fiber distance.
This distortion is $e^{h(t)-h(t')}$. Closest point projection between the vertex spaces is preserved by the quasi-isometry, to within a distance determined by $d(t,t')$. As we let $|x-y|$ go to infinity this additive constant has less and less effect on the distortion. Thus the limit of distortion of points farther and farther apart is bounded above and below by multiples, depending only on the quasi-isometry constants of $F$, of $e^{h(t)-h(t')}$. This shows that the height change $h(t)-h(t')$ differs from $h(f(t))-h(f(t'))$ by at most some uniform additive error. This is precisely the definition of coarsely orientation preserving.
Thus any quasi-isometry $F$ of $X$ covers an almost orientation preserving quasi-isometry $f$ of $T$. According to the results of §1, $f \times Id$ is a quasi-isometry of $X$. The quasi-isometry constants of $f \times Id$ may be much larger than those of $F$. Even if $F$ was an isometry, $f \times Id$ need not be. There is an extension which is better. If $f$ is any coarsely orientation preserving map then define the height change of $f$, $h(f)$, as the height change between $t$ and $f(t)$ for some $t$ in $T$. This change is defined up to an error determined by the $C$ in the definition of coarsely orientation preserving. The map $f \times
e^{-h(f)}$ is a quasi-isometry with constants that depend only on the quasi-isometry and coarsely orientation preserving constants of $f$.
The lemma shows that the group of quasi-isometries of $X$ splits as a semi-direct product of the group of coarsely orientation preserving quasi-isometries of $T$ and those quasi-isometries of $X$ which lie over the identity on $T$. In the case of $BS(1,n)$ we can identify the coarsely orientation preserving quasi-isometries as $Bilip({\mathbb
Q}_{n})$ and the quasi-isometries covering the identity as $Bilip({\mathbb R}$. In this case the full quasi-isometry group is the product of the two. The situation is more complicated in the case of $BS(m,n)$.
A quasi-isometry, $F$, covering the identity takes each vertex space to itself. For each $t$ in $T$, let $f_{t}: {\mathbb R} \to {\mathbb
R}$ be the restriction of the quasi-isometry to the vertex space $\{t\} \times {\mathbb R}$. If $F$ is an $(A,B)$ quasi-isometry then there are some $(A',B')$ for which $F$ restricted to each vertex space is an $(A',B')$ quasi-isometry with respect to fiber distance. The fiber distance between $(t,x)$ and $(t,y)$ is $e^{-h(t)}|x-y|$, so $f_{t}$ is an $(A',B'e^{h(t)}$ quasi-isometry.
Let $e$ be an edge of $T$ with endpoints $t$ and $t'$, where $h(t)
\geq h(t')$. The distance between $(t,x)$ and $(t',y)$ is $1+e^{-h}|x-y|$, and the distance between $(t,f_{t}(x))$ and $(t',f_{t'}(y))$ is $1+e^{-h}|f_{t}(x)-f_{t'}(y)|$. Given that $f_{t}$ and $f_{t'}$ are quasi-isometries with constants as above, $F$ will be an $(A',B')$ quasi-isometry on the strip $e \times
{\mathbb R}$ if and only if $|f_{t}(x)-f_{t'}(x)| \leq e^{h}(A'+B'-1)$.
In summary, $F$ is a quasi-isometry covering the identity if and only if, for some $A,B,$ and $C$, for each $t$ in $T$, $f_{t}$ is an $(A,Be^{h(t)})$ quasi-isometry and, for each edge in $e$ in $T$, we have $d(f_{t},f_{t'})<Ce^{h}$.
Consider the metric $d^{l}$ on $T$ where each edge has length $e^{h}$, where $h$ is the height of the higher endpoint. We define the [*lower boundary*]{}, $\partial^{l}T$ as the ideal points of the metric completion of $T$ with respect to this metric. The previous paragraph shows that a quasi-isometry covering the identity is a Lipschitz map from $(T,d^{l})$ to $QI({\mathbb R})$ so that the map $f_{t}$ is an $(A,Be^{h})$ quasi-isometry. This gives a Lipschitz map from $\partial^{l}$ to $Bilip({\mathbb R})$ so that the image has uniformly bounded Lipschitz constants.
Let $Bilip_{L}({\mathbb R})$ be the space of bilipschitz maps with bilipschitz constant at most $L$, equipped with the metric $d(f,g)=|f-g|_{\infty} + |f^{-1}-g^{-1}|_{\infty}$. The quasi-isometries of $X$ covering the identity on $T$ are the bilipschitz maps $\partial^{l} T \to Bilip({\mathbb R})$ which are contained in $Bilip_{L}$ for some $L$ for some $L$.
We saw above that any quasi-isometry of $X$ induces such a map from $\partial^{l}T$ to $Bilip({\mathbb R})$. So we need to see that any such map extends to a quasi-isometry of $X$, and that this extension is unique up to bounded distance.
For any $t$ in $T$, the distance from $t$ to $\partial^{l}T$ in the metric $d^{l}$ is bounded above and below by multiples of $e^{h}$. Let $F$ and $F'$ induce the same maps on $\partial^{l}T$. For any $t$ in $T$, pick $a$ in $\partial^{l}T$ at minimal distance. We must have a constant $K$ so that $d(F_{t},F_{a})\leq Ke^{h(t)}$ and the same for $F'_{t}$. So $d(F_{t},F'_{t})\leq 2Ke^{h(t)}$. As the distance along the vertex space over $t$ is scaled by $e^{-h(t)}$, this shows $F$ and $F'$ are at bounded distance.
Essentially the same argument allows us to construct an extension. Given a map on $\partial^{l} T$, and any $t$ in $T$, we pick any $a$ in $\partial^{l}T$ at minimal distance from $t$ and define $f_{t}$ to be equal to $f_{a}$. For $t$ and $t'$ in $T$, and any $a$ and $a'$ in $\partial^{l}T$ at minimal distance from them, we know that $d^{l}(a,a')\leq Ke^{h(t)}+Ke^{h(t')}+d^{l}(t,t')$. Since the map on the lower boundary is $M$ Lipschitz
$$d(f_{a},f_{a'}) \leq Md^{l}(a,a') \leq MK(e^{h(t)}+e^{h(t')})+Md^{l}(t,t')$$
So long as $t$ and $t'$ are not equal, $d(t,t')\geq e^{max(h(t),h(t'))}$ so we have:
$$d(f_{a},f_{a'}) \leq M(2K+1)d^{l}(t,t')$$
Thus the extension is a quasi-isometry $T$ to $T$.
We can express this bilipschitz map from $\partial^{l}T$ to $Bilip({\mathbb R})$ differently: it can all be assembled into a single bilipschitz map $\partial^{l}T \times {\mathbb R}$ to itself which covers the identity map of $\partial^{l}T$.
Any coarsely orientation preserving quasi-isometry of $T$ induces a bilipschitz map of $\partial^{l}T$. If $T$ has at least two edges decreasing height at each vertex, $\partial^{l}T$ is dense in the boundary of $T$, so the map on $\partial^{l}T$ determines the quasi-isometry up to bounded distance.
We have proven:
Let $T$ be the Bass-Serre tree of $BS(m,n)$ for $1<m<n$. The group of coarsely orientation preserving of $T$ is a subgroup, $G$, of $Bilip(\partial^{l}T)$, and the group of quasi-isometries of $BS(m,n)$ is the group of bilipschitz bundle maps of $\partial^{l}T \times {\mathbb R}$ covering $G$.
It would be nice to understand which bilipschitz maps of the lower boundary come from coarsely orientation preserving quasi-isometries. It seems likely that some sort of conformal structure should do the trick.
There is also an upper boundary, defined as the limit points of $T$ with edges scaled by $e^{-h}$. An coarsely orientation preserving quasi-isometry of $T$ also induces a bilipschitz map of this upper boundary. As with the lower boundary, this boundary is typically dense in the full boundary and so a quasi-isometry is determined by its action on the upper boundary. In the case of $BS(1,n)$ this boundary is ${\mathbb Q}_{n}$ and the bilipschitz group of the upper boundary is exactly the group of coarsely orientation preserving quasi-isometries.
Other Applications
==================
The results of [@MSW] show that any group quasi-isometric to a graph of ${{\mathbb Z}}$s is a graph of virtual ${{\mathbb Z}}$s. Graphs of virtual ${{\mathbb Z}}$s have models like those of §\[models\], except that the vertex spaces are only quasi-isometric to ${{\mathbb Z}}$ rather than isomorphic to ${{\mathbb Z}}$. This is all that we use about the vertex spaces, so our results apply in this slightly greater generality.
Let $\Gamma$ be a finitely generated group. $\Gamma$ is quasi-isometric to $BS(2,3)$ iff $\Gamma$ is a graph of virtual ${{\mathbb Z}}$s which is neither commensurable to $F_n \times {{\mathbb Z}}$ nor virtually solvable.
More generally, the techniques of this paper can be used to study more general graphs of groups. One certainly needs to assume that the Bass-Serre tress has bounded valence, which means that all of the edge-to-vertex inclusions have finite index image. In this case, all of the edge and vertex groups are commensurable. We call such a graph of groups [*homogeneous*]{}.
Very little can be said in general, as one needs to understand the large scale dynamics of isomorphisms among finite index subgroups of the vertex groups. One case where this is possible is graphs of groups in which every vertex and edge groups is ${{\mathbb Z}}^n$ for some fixed $n$. The isomorphisms among the finite index subgroups can be represented as elements of $SL_n({{\mathbb Q}})$.
In order for the geometry to reduce to coarsely oriented trees, one needs all of these isomorphisms to lie on a single one parameter subgroup of $GL_n({{\mathbb R}})$. The natural examples of this type are HNN extensions of $Z^n$ along finite index subgroups. Let $G$ be ${{\mathbb Z}}^n$, $G'$ and $G''$ finite index subgroups, and $T:G' \to G''$ an isomorphism. Abstractly, these HNN extensions are the groups: $$\Gamma_T=<G,t| t^{-1} g t = Tg \hbox{, for } g \in G'>$$
We assume that at least one of the groups $G'$ or $G''$ is a proper subgroup of $G$. Groups of this type are studied in [@FM3]. Recall that the [*Absolute Jordan form*]{} of $T$ is the matrix which is the Jordan form of $T$ except that the values on the diagonal are the norms of the eigenvalues rather than the eigenvalues themselves.
[@FM3] Let $\Gamma_T$ and $\Gamma_{T'}$ be as above.
- If $\Gamma_T$ and $\Gamma_{T'}$ are quasi-isometric, then for some $\alpha \in {{\mathbb R}}^+$ the absolute Jordan forms of $T^\alpha$ and $T'$ are equal.
- If $G_T$ and $G_{T'}$ are solvable (which is equivalent to one of the subgroups nonproper) then $G_T$ and $G_{T'}$ are quasi-isometric iff there is an $\alpha \in {{\mathbb Q}}^+$ for which the absolute Jordan forms of $T^\alpha$ and $T'$ are equal.
The results of this paper allow us to complete the classification.
Let $\Gamma_T$ and $\Gamma_{T'}$ be as above. If neither is solvable, and the there is an $\alpha \in {{\mathbb R}}^+$ so that the absolute Jordan forms of $T^\alpha$ and $T'$ are the same, then $\Gamma_T$ and $\Gamma_{T'}$ are quasi-isometric.
It is interesting that for the nonsolvable cases one has a complete invariant of the quasi-isometry type, and a continuous family of quasi-isometry types, while in the solvable cases one has a discrete refinement of the invariant.
We hope to explore more general homogeneous graphs of groups, and the nature of their invariants, in future work.
B. Farb and L. Mosher (appendix by D. Cooper), A rigidity theorem for the solvable Baumslag-Solitar groups, [*Inventiones*]{}, Vol. 131, No. 2 (1998), pp. 419-451.
B. Farb and L. Mosher, Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II, to appear in [*Inventiones*]{}.
B. Farb and L. Mosher, On the asymptotic geometry of abelian-by-cyclic groups, I, preprint.
L. Mosher, M. Sageev, and K. Whyte, Quasi-actions on trees I: Bounded valence, preprint.
J.P. Serre, [*Trees*]{}, translated by J. Stillwell, Springer-Verlag, 1980.
Kevin Whyte\
Dept. of Mathematics\
University of Chicago\
Chicago, Il\
E-mail: [email protected]\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Primordial Inflation Explorer (PIXIE) is an Explorer-class mission concept to measure the gravitational-wave signature of primordial inflation through its distinctive imprint on the linear polarization of the cosmic microwave background (CMB). Its optical system couples a polarizing Fourier transform spectrometer to the sky to measure the differential signal between orthogonal linear polarization states from two co-pointed beams on the sky. The double differential nature of the four-port measurement mitigates beam-related systematic errors common to the two-port systems used in most CMB measurements. Systematic errors coupling unpolarized temperature gradients to a false polarized signal cancel to first order for any individual detector. This common-mode cancellation is performed optically, prior to detection, and does not depend on the instrument calibration. Systematic errors coupling temperature to polarization cancel to second order when comparing signals from independent detectors. We describe the polarized beam patterns for PIXIE and assess the systematic error for measurements of CMB polarization.'
author:
- 'A. Kogut'
- 'D. J. Fixsen'
title: 'Systematic error cancellation for a four-port interferometric polarimeter'
---
[**\*** Address all correspondence to: Alan Kogut, ]{}
[1]{}
Introduction
============
Polarization of the cosmic microwave background (CMB) provides a powerful test of the physics of the early universe. An arbitrary pattern of linear polarization mapped over the sky may be decomposed into a spatially symmetric component (even parity E-modes) and an anti-symmetric component (odd parity B-modes). Scalar sources such as temperature or density perturbations can only generate even-parity E-modes, while gravitational waves created during an inflationary epoch in the early universe can generate either parity. Detection of the B-mode signal in the CMB polarization field is thus recognized as a “smoking gun” signature of inflation, testing physics at energies inaccessible through any other means [@rubakov/etal:1982; @fabbri/pollock:1983; @abbott/wise:1984; @polnarev:1985; @davis/etal:1992; @grishchuk:1993; @kamionkowski/etal:1997; @seljak/zaldarriaga:1997].
![A two-port system (left) couples a single linear polarization from the sky to a single detector. PIXIE’s polarizing Fourier Transform Spectrometer operates as a four-port device (right) with two input ports open to the sky and two output ports terminated by polarization-sensitive detectors. Interfering the two beams cancels the effects of common mode beam ellipticity, as each detector then couples to both linear polarizations from the sky. \[4\_port\_fig\]](pixie_4_port_syserr_jatis_fig_1.pdf){height="2.5in"}
The amplitude of the gravitational wave signal depends on the energy scale of inflation as $$E = 1.06 \times 10^{16}
\left(
\frac{r}{0.01}
\right)^{1/4}
~{\rm GeV}
\label{potential_eq}
\vspace{-2mm}$$ where $r$ is the power ratio of gravitational waves to density fluctuations [@lyth/riotto:1999]. In most large-field models, $r$ is predicted to be of order 0.01, corresponding to polarized amplitude 30 nK or energy near the Grand Unified Theory scale, $10^{16}$ GeV. Signals at this amplitude could be detected by a dedicated polarimeter, providing a critical test of a central component of modern cosmology. Detection of a gravitational-wave component in the CMB polarization would have profound implications for both cosmology and high-energy physics. It would provide strong evidence for inflation, provide a direct, model-independent determination of the relevant energy scale, and test physics at energies a trillion times beyond those accessible to particle accelerators. Generation of gravitational waves during inflation is purely a quantum-mechanical process: a detection of the B-mode signal provides direct observational evidence that gravity obeys quantum mechanics.
Characterizing the CMB to measure polarization at the parts-per-billion level requires careful control of systematic errors. A particular concern are systematic errors related to the instrument optics, which can couple the much brighter unpolarized temperature fluctuations into a false polarization signal. All CMB instruments must couple the detectors to the sky, and must therefore account for potential beam-related systematic errors. An extensive literature discusses common effects and mitigation strategies [@hu/etal:2003; @odea/etal:2007; @rosset/etal:2007; @shimon/etal:2008].
The Primordial Inflation Explorer (PIXIE) is an Explorer-class mission designed to measure the inflationary signature in polarization as well as distortions from the 2.725 K blackbody spectrum induced by energy-releasing processes at more recent cosmological epochs [@kogut/etal:2011]. Its projected sensitivity of a few nK on degree angular scales or larger corresponds to a limit $r < 10^{-3}$ at 5 standard deviations. PIXIE differs from most CMB polarimeters in its use of a polarizing Fourier transform spectrometer coupled to the sky through a multi-moded optical system. The double differential nature of the resulting four-port measurement minimizes beam-related systematic errors common to the two-port systems used in most CMB measurements. We describe the polarized beam patterns for PIXIE and assess the systematic error for measurements of CMB polarization.
PIXIE Optical System
====================
A common implementation for CMB polarimetry images the sky onto a set of polarization-sensitive detectors. Since each detector is sensitive to a single linear polarization from the sky (although two or more detectors may share a physical pixel), the resulting system may be described as a two-port device with any polarization comparison between detectors occurring post-detection. In contrast, the PIXIE optical system forms a four-port device (Fig \[4\_port\_fig\]). Reflective optics couple a polarizing Fourier Transform Spectrometer (FTS) to the sky. The FTS introduces an optical phase delay between the two input beams, and routes recombined beams to non-imaging concentrators at each of two output ports. Within each concentrator, a pair of polarization-sensitive detectors measure the power as a function of optical phase delay. Let $\vec{E} = E_x \hat{x} + E_y \hat{y}$ represent the electric field incident from the sky. The power $P$ at the detectors as a function of the phase delay $z$ may be written $$\begin{aligned}
P_{Lx} &=& 1/2 ~\smallint \{ ~(E_{Ax}^2+E_{By}^2)+(E_{Ax}^2-E_{By}^2) \cos(4z\omega /c) ~\}d\omega \nonumber \\
P_{Ly} &=& 1/2 ~\smallint \{ ~(E_{Ay}^2+E_{Bx}^2)+(E_{Ay}^2-E_{Bx}^2) \cos(4z\omega /c) ~\}d\omega \nonumber \\
P_{Rx} &=& 1/2 ~\smallint \{ ~(E_{Ay}^2+E_{Bx}^2)+(E_{Bx}^2-E_{Ay}^2) \cos(4z\omega /c) ~\}d\omega \nonumber \\
P_{Ry} &=& 1/2 ~\smallint \{ ~(E_{Ax}^2+E_{By}^2)+(E_{By}^2-E_{Ax}^2) \cos(4z\omega /c) ~\}d\omega~,
\label{full_p_eq}\end{aligned}$$ where $\hat{x}$ and $\hat{y}$ refer to orthogonal linear polarizations, L and R refer to the detectors in the left and right concentrators, A and B refer to the two input beams, $\omega$ is the angular frequency of incident radiation, and the factor of 4 reflects the symmetric folding of the optical path. When both input ports are open to the sky, the power at each detector consists of a dc term proportional to the intensity $E_x^2 + E_y^2$ (Stokes $I$) plus a term modulated by the phase delay $z$, proportional to the linear polarization $E_x^2 - E_y^2$ (Stokes $Q$) in instrument-fixed coordinates. Each detector is thus sensitive to the difference between orthogonal linear polarizations from the two input ports, with the difference now occurring pre-detection. Rotation of the instrument about the beam axis rotates the instrument coordinate system relative to the sky to allow separation of Stokes $Q$ and $U$ parameters on the sky.
Rotation of the instrument relative to the sky can produce systematic errors in the recovered polarization if the instrument beams are not azimuthally symmetric. This effect has been well studied for two-port devices which couple the sky directly to a single polarization-sensitive detector. The dominant systematic error for such a device is temperature–polarization coupling as the beam ellipticity interacts with local gradients in the unpolarized sky intensity, producing a spin-dependent signal degenerate with true polarization. Temperature–polarization coupling can be mitigated in hardware using such techniques as polarization modulation (rapidly switching a single detetor between orthogonal polarization states) or in analysis using a well-measured beam profile.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![PIXIE optical signal path. The left panel shows the optical elements within the Fourier Transform Spectrometer while the right panel shows the physical layout. \[pixie\_optics\_layout\] ](pixie_4_port_syserr_jatis_fig_2.pdf "fig:"){height="3.5in"}
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The beam response of a four-port system is considerably more complicated than the two-port systems commonly used for CMB measurements. Although a four-port system may still employ mitigation strategies such as rotation or beam mapping, the double differential nature of the four-port measurement provides additional mitigation against common spin-dependent polarization errors while simultaneously providing a means to identify and correct residual effects. Figure \[pixie\_optics\_layout\] shows the PIXIE optical path. Consider (in a time-reversed sense) the path through the optics taken by photons leaving the $\hat{x}$ detector in the left-side concentrator. Since this detector is sensitive to a single linear polarization, the photons exiting the left-side concentrator are entirely in the $\hat{x}$ polarization. A series of polarizing wire grids within the FTS splits the beam and rotates the polarization so that half the initial power exits through port A in the $\hat{y}$ polarization while the other half exits through port B in the $\hat{x}$ polarization (see, e.g., Appendix A of reference ). A set of reflective mirrors then couples ports A and B to the sky while preserving the polarization state. Stops at transfer mirror 5 and at the entrance to the concentrator circularize the beam. The entire instrument, including all baffling, remains isothermal at 2.725 K so that rays scattered out of the beam terminate on an isothermal black surface. Such rays contribute to the photon noise budget but do not introduce artifacts in the beams.
Let us define the beam pattern of the concentrator as $H_x(\theta, \phi)$ for the $\hat{x}$ polarization and $H_y(\theta, \phi)$ for the $\hat{y}$ polarization, where the angular coordinates $\theta$ and $\phi$ are referred to the sky. Similarly, we define the beam pattern for the fore-optics (defined as all elements in the optical chain skyward of the concentrator feed) as $F_x(\theta, \phi)$ and $F_y(\theta, \phi)$. Using subscripts $L$ and $R$ to distinguish the two concentrator ports and $A$ and $B$ for the two fore-optic ports, we may re-write Equation \[full\_p\_eq\] as $$\begin{aligned}
P_{Lx} &\propto& H_{Lx} \left[ F_{Ax} E_{x}^2 - F_{By} E_{y}^2 \right] \nonumber \\
P_{Ly} &\propto& H_{Ly} \left[ F_{Ay} E_{y}^2 - F_{Bx} E_{x}^2 \right] \nonumber \\
P_{Rx} &\propto& H_{Rx} \left[ F_{Bx} E_{x}^2 - F_{Ay} E_{y}^2 \right] \nonumber \\
P_{Ry} &\propto& H_{Ry} \left[ F_{By} E_{y}^2 - F_{Ax} E_{x}^2 \right] ,
\label{full_beam_eq}\end{aligned}$$ where for clarity we suppress the dependence on angular coordinates $(\theta, \phi)$ as well as the phase delay integral over frequency. Two points are apparent. First, the signal at any single detector depends on the convolution of the concentrator beam profile with the [*differential*]{} beam profile generated by the A- and B-side fore-optics. To the extent that the A- and B-side optics have identical beam patterns, the detectors produce no response from an unpolarized sky, regardless of the intensity gradient on the sky or the ellipticity of the fore-optics. This common mode cancellation is performed optically, prior to detection, and does not depend on the instrument calibration. Second, the beam pattern for the concentrator horn appears only as a common-mode multiplicative factor. Systematic errors coupling temperature anisotropy to polarization thus cancel to second order when comparing signals from independent detectors.
Figure \[common\_mode\_fig\] illustrates the multiple levels of common-mode subtraction. An ideal azimuthally symmetric beam would introduce no temperature-polarization coupling. Real beams, however, will have some ellipticity (left column). Rotation of an elliptical beam couples to unpolarized gradients in the sky to produce a time-dependent signal degenerate with a true polarization signal. If, however, two beams sensitive to opposite polarization states but with the same ellipticity are compared, the common-mode ellipticity cancels for unpolarized emission, leaving no net temperature-polarization coupling (second column). Only the *differential* ellipticity produces a net temperature-polarization coupling, which appears at second order in the beam difference. The PIXIE optics employ such beam cancellation in the $A-B$ comparison for a single detector (third column, with the differential ellipticity greatly exaggerated). Comparisons between different detectors provide an additional level of cancellation. Each concentrator contains two detectors sensitive to orthogonal polarization states (Eq. \[full\_p\_eq\]), which view the same sky through the same fore-optics. If the differential ellipticity between the $A$ and $B$ sky beams is the same for the $\hat{x}$ polarization as for the $\hat{y}$ polarization, the net temperature-polarization coupling for the single-detector output will cancel to *second* order in the detector-pair difference (right-most column). Alternatively, an orthogonal linear combination of detector pairs can be chosen to cancel the sky signal, thereby isolating any beam effects. Such measurements can be used both as confirmation of the expected amplitude of the beam differences and to correct residual beam effects in the sky data.
![Cartoon illustrating signal cancellation from differential beam profiles. Colored regions indicate the beam shape, while the white lines indicate the polarization state accepted by each beam. \[common\_mode\_fig\] ](pixie_4_port_syserr_jatis_fig_3.pdf){width="6.0in"}
The double-differential beam cancellation of PIXIE’s four-port optical system reduces the sensitivity to unpolarized gradients on the sky. The following sections use Monte Carlo ray-trace code to evaluate the common-mode and differential beam patterns. We quantify the expected systematic error response for ideal optics and show the minimal degradation in performance after accounting for machining and assembly tolerances.
![Linear combinations of the PIXIE fore-optics showing the common-mode and differential beam patterns. The spatial ($\Delta$) and polarization ($\delta$) asymmetries are small compared to the mean beam pattern $F$. Contours for the common-mode response $F$ are shown at amplitude 0.3, 0.7, and 0.9 to highlight the circular tophat beam structure. Note the change in scale for the three differential beam patterns. \[fore\_optics\_beams\] ](pixie_4_port_syserr_jatis_fig_4.pdf){width="4.0in"}
Single-Detector Response
========================
Systematic errors in the PIXIE four-port optical system depend on successive differences in the beam patterns. We may write the individual fore-optics beam patterns in terms of the linear combinations $$\begin{aligned}
F &=& \left( F_{Ax} + F_{Ay} + F_{Bx} + F_{By} \right) / 4 \nonumber \\
\Delta &=& \left( F_{Ax} + F_{Ay} - F_{Bx} - F_{By} \right) / 4 \nonumber \\
\delta &=& \left( F_{Ax} - F_{Ay} + F_{Bx} - F_{By} \right) / 4 \nonumber \\
\epsilon &=& \left( F_{Ax} - F_{Ay} - F_{Bx} + F_{By} \right) / 4
\label{F_def}\end{aligned}$$ to distinguish the common-mode beam pattern $F = F(\theta, \phi)$ from the differential beam patterns $\Delta$ (A–B spatial asymmetry), $\delta$ ($\hat{x} - \hat{y}$ polarization asymmetry), and $\epsilon$ (spatial/polarization cross term). With these definitions, the individual beam patterns become $$\begin{aligned}
F_{Ax} &=& F + \delta + \Delta + \epsilon \nonumber \\
F_{Ay} &=& F - \delta + \Delta - \epsilon \nonumber \\
F_{Bx} &=& F + \delta - \Delta - \epsilon \nonumber \\
F_{By} &=& F - \delta - \Delta + \epsilon ~.
\label{FAX_def}\end{aligned}$$ Note that these four linear combinations represent a complete set, carrying all information for 2 ports in 2 linear polarizations.
Figure \[fore\_optics\_beams\] shows the common-mode and differential beam patterns, using a Monte Carlo ray-trace code to propagate $10^{11}$ rays through the PIXIE fore-optics. As expected, the beams are dominated by the common-mode illumination $F$. Since $F$ by definition is the average of the beams, it can not generate any *differential* ellipticity and thus can not generate temperature-polarization coupling regardless of its azimuthal structure.
Differences between the left and right beams are are measured by the $A-B$ spatial asymmetry $\Delta(\theta, \phi)$. Out-of-plane reflections at the secondary mirror and folding flat generate a dipolar modulation in $\Delta$ with rms amplitude 0.015 of the common-mode beam pattern. This is the largest differential mismatch between the beams. The instrument is symmetric about the left-right midplane so that, by design, the $A$ and $B$ beams are mirror images of each other (Figure \[pol\_symmetry\_cartoon\]). Structure within one of the beams will thus be reflected left-to-right in the other beam, maximizing the net effect along the left-right direction.
Differences between polarization states are measured by the polarization asymmetry $\delta(\theta, \phi)$. Since both the $\hat{x}$ and $\hat{y}$ polarizations from the detectors are launched at 45 relative to the symmetry plane of the instrument ($\S$4), this term is small (of order $10^{-4}$ of the common-mode pattern). For completeness, there is also a spatial/polarization cross term $\epsilon(\theta, \phi)$. This term is also small (of order $10^{-4}$). As shown below, it does not couple to temperature-polarization mixing, but appears as a small perturbation on the amplitude of the measured polarization signal.
![ Schematic of the PIXIE optical system showing the symmetric polarization response at the beam apertures. The Fourier transform spectrometer interferes a single linear polarization from one side of the instrument with the orthogonal polarization from the other side. By construction, the $\hat{x}$ polarization on the A side is simply the mirror reflection of the $\hat{y}$ polarization on the B side. \[pol\_symmetry\_cartoon\] ](pixie_4_port_syserr_jatis_fig_5.pdf){width="3.0in"}
Using these definitions, it is straightforward (if somewhat tedious) to show that $$\begin{aligned}
P_{Lx} &=& H_{Lx} [\, \, ~ ~~~{\boldsymbol Q} \, F
~ + ~{\boldsymbol Q} \epsilon
~ + ~{\boldsymbol I} ( \, \, ~~\delta + \Delta ) ~~] \nonumber \\
P_{Ly} &=& H_{Ly} [~ -{\boldsymbol Q} \, F
~ + ~{\boldsymbol Q} \epsilon
~ + ~{\boldsymbol I} ( -\delta + \Delta ) ~~] \nonumber \\
P_{Rx} &=& H_{Rx} [\, \, ~ ~~~{\boldsymbol Q} \, F
~ - ~{\boldsymbol Q} \epsilon
~ + ~{\boldsymbol I} (\, \, ~~\delta - \Delta ) ~~] \nonumber \\
P_{Ry} &=& H_{Ry} [~ -{\boldsymbol Q} \, F
~ - ~{\boldsymbol Q} \epsilon
~ + ~{\boldsymbol I} ( -\delta - \Delta ) ~~] ~.
\label{4_det_with_delta}\end{aligned}$$
The first term in brackets represents the desired polarized sky signal ${\boldsymbol Q(\theta, \phi)}$, convolved with the mean fore-optics beam pattern. The second term, ${\boldsymbol Q} \, \epsilon$, convolves the true sky polarization with the cross beam pattern $\epsilon(\theta, \phi)$. This term is small. The cross beam pattern may be written as the double difference $$\epsilon = (F_{Ax} - F_{Ay}) - (F_{Bx} - F_{By})
\label{eps_def}$$ and is thus second order in the beam difference. Furthermore, since this term does not mix the Stokes parameter ${\boldsymbol Q}$ with either ${\boldsymbol U}$ or ${\boldsymbol I}$, it only appears as a scale error in the amplitude of the true sky polarization and may be absorbed by the calibration. The final term represents systematic temperature–polarization coupling.
The left–right symmetry of the PIXIE optics minimizes temperature-polarization coupling. PIXIE’s optical design interferes the $\hat{x}$ polarization from one beam with the $\hat{y}$ polarization from the other beam (Eq. \[full\_beam\_eq\]). The optical system is symmetric about the central plane, so that the $\hat{x}$ polarization from one beam is the mirror reflection of the $\hat{y}$ polarization from the other beam (Fig \[pol\_symmetry\_cartoon\]). This enforces a reflection symmetry such that $$\begin{aligned}
F_{Ax}(\theta, \phi) &=& F_{By}(\theta, -\phi) \nonumber \\
F_{Ay}(\theta, \phi) &=& F_{Bx}(\theta, -\phi)
\label{F_symmetry_eq}\end{aligned}$$ where the azimuthal angle $\phi$ is defined from the midline. Note that this left–right symmetry is not equivalent to an $\hat{x}$–$\hat{y}$ symmetry since the $\hat{x}$–$\hat{y}$ coordinate system is rotated by 45 with respect to the optical midline. Temperature-polarization mixing thus depends on the linear combinations $$\begin{aligned}
\delta + \Delta &=& F_{Ax} - F_{By} \nonumber \\
\delta - \Delta &=& F_{Bx} - F_{Ay}
\label{delta_diff}\end{aligned}$$ proportional to the anti-symmetric component of the [*difference*]{} between the beams.
The spacecraft spin combines with the mirror symmetry of the instrument optics to further minimize temperature-polarization coupling. Each detector is sensitive to a single linear polarization (Stokes $Q$ in a coordinate system fixed with respect to the instrument). The entire spacecraft rotates about the instrument boresight to interchange the roles of $\hat{x}$ and $\hat{y}$ polarization at the detectors, allowing full characterization of the Stokes $Q$ and $U$ parameters on the sky. True sky polarization is modulated at twice the spacecraft spin frequency, $$Q_{\rm inst} = Q_{\rm sky} \cos(2 \gamma) + U_{\rm sky} \sin(2 \gamma)
\label{qu_sky}$$ where $\gamma$ is the spin angle of the instrument with respect to the sky. Temperature–polarization mixing is dominated by the anti-symmetric component of the differential beam pattern from the instrument fore-optics. Anti-symmetric signals can only appear at odd harmonics of the spacecraft spin, and may readily be distinguished from true sky polarization.
We quantify the suppression of temperature–polarization systematic errors using the spin-dependent moments of the differential beam patterns. The instantaneous power at each detector depends on the convolution of the beam pattern (in instrument-fixed coordinates) with the sky signal (rotated from sky to instrument coordinates). Azimuthal asymmetry in the beam patterns causes the measured power to vary with the spacecraft spin angle. We thus compute the coefficients $$\begin{aligned}
a_m &=& \int B(\Omega) \cos(m \phi) d\Omega \nonumber \\
b_m &=& \int B(\Omega) \sin(m \phi) d\Omega
\label{am_def}\end{aligned}$$ where $B$ represents one of the linear combinations of beam patterns (Eq. \[F\_def\]) and $d \Omega = \sin(\theta) d\theta d\phi$ is computed in instrument coordinates centered on the boresight.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Decomposition of the PIXIE differential beam patterns by spin angle. The common-mode beam $F$ is sensitive only to polarized emission and does not contribute to temperature–polarization systematic errors. The mirror symmetry of the PIXIE optics suppresses temperature–polarization mixing from the A–B spatial asymmetry (beam $\Delta$) by a factor of $10^{-6}$ (see text). \[fore\_optics\_vs\_m\] ](pixie_4_port_syserr_jatis_fig_6.pdf "fig:"){height="2.6in"}
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Figure \[fore\_optics\_vs\_m\] shows the power $P_m = a_m^2 + b_m^2$ as a function of spin moment $m$. The odd-even asymmetry in spin moment $m$ is superposed atop an overall decrease in power with increasing $m$. The noise floor at $P \approx 10^{-12}$ reflects shot noise from the discrete ray-trace simulation. Recall that the common-mode beam pattern $F$ is sensitive only to polarized emission on the sky (Eq. \[4\_det\_with\_delta\]) and does not create temperature–polarization errors even for the $m=2$ case. Systematic errors from temperature–polarization coupling are dominated by the $m=2$ mode of the A–B spatial asymmetry $\Delta$, and are suppressed by a factor $10^{-6}$ relative to the polarization response in the common-mode beam.
Additional Symmetries
=====================
The mirror symmetry of PIXIE’s differential 4-port interferometer suppresses systematic errors from temperature–polarization coupling by 6 orders of magnitude for the single-detector response. Additional symmetries between different detectors allow further suppression and identification of beam-related systematic errors. The left and right concentrators are identical, resulting in left–right symmetry $$\begin{aligned}
H_{Lx}(\theta, \phi) &=& H_{Rx}(\theta, -\phi) \nonumber \\
H_{Ly}(\theta, \phi) &=& H_{Ry}(\theta, -\phi)
\label{H_symmetry_eq}\end{aligned}$$ for identical polarization states. This is similar to the left–right symmetry in Eq. \[F\_symmetry\_eq\] except that the symmetry is now between identical polarization states on opposite sides of the instrument.
![ Schematic showing the orientation of the PIXIE concentrator. The square aperture is rotated 45to minimize any differences between the $\hat{x}$ and $\hat{y}$ polarization. \[horn\_cartoon\] ](pixie_4_port_syserr_jatis_fig_7.pdf){width="2.5in"}
Differences between the two polarizations $\hat{x}$ and $\hat{y}$ within a single concentrator can occur, corresponding to the difference between the E-plane and H-plane beam patterns for a single-moded feed. PIXIE’s multi-moded operation reduces this effect, which vanishes in the geometric optics limit. We further reduce the effect by rotating the concentrator so that the symmetry axes of the square aperture lie at $\pm 45\deg$ relative to the $\hat{x}$ and $\hat{y}$ polarization vectors (Fig \[horn\_cartoon\]). The resulting beams in $\hat{x}$ and $\hat{y}$ are equivalent linear combinations of the E-plane and H-plane beam patterns, so that $$\begin{aligned}
H_{Lx} &\approx& H_{Ly} \nonumber \\
H_{Rx} &\approx& H_{Ry}
\label{horn_pol_eq}\end{aligned}$$ with residuals resulting from small displacements in the rotation angle [@kogut/fixsen:2018]. Without loss of generality, we may follow Eq. \[FAX\_def\] to decompose the beam pattern from each horn into a component common to all four detectors plus a set of differential beam patterns, $$\begin{aligned}
H &=& \left( H_{Lx} + H_{Ly} + H_{Rx} + H_{Ry} \right) / 4 \nonumber \\
\rho &=& \left( H_{Lx} - H_{Ly} + H_{Rx} - H_{Ry} \right) / 4 \nonumber \\
\tau &=& \left( H_{Lx} + H_{Ly} - H_{Rx} - H_{Ry} \right) / 4 \nonumber \\
\kappa &=& \left( H_{Lx} - H_{Ly} - H_{Rx} + H_{Ry} \right) / 4
\label{H_def}\end{aligned}$$
so that $$\begin{aligned}
H_{Lx} &=& H + \rho + \kappa + \tau \nonumber \\
H_{Ly} &=& H - \rho + \kappa - \tau \nonumber \\
H_{Rx} &=& H + \rho - \kappa - \tau \nonumber \\
H_{Ry} &=& H - \rho - \kappa + \tau
\label{horn_def}\end{aligned}$$ where the horn parameters are defined analogously to the fore-optics in Eq. \[F\_def\]. Figure \[horn\_beam\_fig\] shows the common-mode and differential beam patterns from the concentrator horn. Asymmetries from the off-axis orientation appear at the few-percent level. As with the fore-optics, the differential beams are dominated by an anti-symmetric (dipolar) component.
Appendix A shows the full response for each detector. Retaining only terms to first order in the beam differences, the signals at each detector become $$\begin{aligned}
P_{Lx} = ~~{\boldsymbol Q} \, HF
&+& {\boldsymbol Q} \, ( ~H \epsilon + F\rho + F\kappa + F\tau) \nonumber \\
&+& {\boldsymbol I} \, H (\delta + \Delta) \nonumber \\
P_{Ly} = -{\boldsymbol Q} \, HF
&+& {\boldsymbol Q} \, ( ~H \epsilon + F\rho - F\kappa + F\tau ) \nonumber \\
&-& {\boldsymbol I} \, H (\delta - \Delta) \nonumber \\
P_{Rx} = ~~{\boldsymbol Q} \, HF
&+& {\boldsymbol Q} \, ( -H \epsilon + F\rho - F\kappa + F\tau ) \nonumber \\
&+& {\boldsymbol I} \, H (\delta - \Delta) \nonumber \\
P_{Ry} = -{\boldsymbol Q} \, HF
&+& {\boldsymbol Q} \, ( -H \epsilon + F\rho + F\kappa - F\tau ) \nonumber \\
&-& {\boldsymbol I} \, H (\delta + \Delta) ~~.
\label{big_4_det_first}\end{aligned}$$ Systematic errors from the concentrator beam pattern appear in the second (polarization amplitude) term. Although larger in amplitude than the spatial-polarization error ${\boldsymbol Q} \epsilon$, these terms do not couple temperature to polarization and so may be absorbed into the calibration. The final term representing temperature–polarization mixing is dominated at lowest order by the differential error from the fore-optics.
![Common-mode and differential beam patterns for the PIXIE feed horn concentrators. The feed horn beam pattern does not directly source $T \rightarrow B$ systematic errors, but only modulates the effect from the differential fore-optics. The off-axis design creates dipolar modulation in the differential beam patterns $\rho$ and $\tau$, while the square shape is reflected in the quadrupolar modulation for $\kappa$. Contours for the common-mode response $H$ are shown at amplitude 0.3, 0.7, and 0.9. \[horn\_beam\_fig\] ](pixie_4_port_syserr_jatis_fig_8.pdf){width="4.0in"}
Combined Detector Response
==========================
PIXIE’s four detectors share different portions of the optical system (left or right concentrator, $\hat{x}$ or $\hat{y}$ polarization). Linear combinations of the post-detection signals can either eliminate or isolate specific systematic error signals, providing additional safeguards against temperature–polarization mixing. For example, we may combine all 4 detectors to yield the sum signal $$\begin{aligned}
\left[ P_{Lx} - P_{Ly} + P_{Rx} - P_{Ry} \right] / 4 &=&
{\boldsymbol Q} \, HF \nonumber \\
&+& {\boldsymbol Q} \, \epsilon \tau \nonumber \\
&+& {\boldsymbol I} \, H \delta \nonumber \\
&+& {\boldsymbol I} \, \Delta \tau ~~,
\label{4_det_coadd}\end{aligned}$$ where we now retain terms to second order in the differential beam patterns. As before, the first term is the true sky polarization, convolved with the combined common-mode beam pattern from the feed horn and fore-optics. The second term affects only the amplitude of the true sky polarization and may be absorbed into the calibration. The final two terms represent systematic errors coupling temperature anisotropy to polarization.
We use Monte Carlo ray-trace simulations to quantify the expected amplitude of these terms. Table \[beam\_summary\_table\] summarizes the common-mode and differential beam patterns for the PIXIE optical system. The differential beam patterns are small compared to the common-mode response. We compare the weighted beam area of the differential beams to the weighted area of the common-mode beam pattern, $$f = \frac{ \int | \Delta(\theta,\phi)| d\Omega }
{ \int | F(\theta,\phi)| d\Omega } ~,
\label{fractional_area_eq}$$ computed similarly for each of the 6 differential beam patterns. The differential beams have fractional area of a few percent for the concentrator, and $10^{-2}$ to $10^{-5}$ for the more symmetric fore-optics. The differential beams are dominated by a dipolar modulation ($m=1$) which does not lead to temperature–polarization mixing. The systematic error response to spin modulation at $m=2$ is typically of order $10^{-6}$ or smaller.
------------------------ -------------------- --------------------- --------------------- ---------------------
Parameter $F$ $\Delta$ $\delta$ $\epsilon$
Peak Amplitude 1 $3 \times 10^{-2}$ $2 \times 10^{-4}$ $1 \times 10^{-4}$
Relative Beam Area $f$ 1 $2 \times 10^{-2}$ $8 \times 10^{-5}$ $4 \times 10^{-5}$
Power (m=1) $7 \times 10^{-5}$ $2 \times 10^{-5}$ $9 \times 10^{-11}$ $2 \times 10^{-10}$
Power (m=2) $3 \times 10^{-4}$ $1 \times 10^{-6}$ $4 \times 10^{-9}$ $7 \times 10^{-11}$
Parameter $H$ $\rho$ $\tau$ $\kappa$
Peak Amplitude 1 $6 \times 10^{-2}$ $8 \times 10^{-2}$ $1 \times 10^{-2}$
Relative Beam Area $f$ 1 $3 \times 10^{-2}$ $5 \times 10^{-2}$ $7 \times 10^{-3}$
Power (m=1) $5 \times 10^{-8}$ $2 \times 10^{-4}$ $9 \times 10^{-4}$ $4 \times 10^{-8}$
Power (m=2) $6 \times 10^{-4}$ $8 \times 10^{-10}$ $4 \times 10^{-6}$ $3 \times 10^{-5}$
------------------------ -------------------- --------------------- --------------------- ---------------------
: Spin Modulation of the Common-Mode and Differential Beam Patterns[]{data-label="beam_summary_table"}
We may now quantify the systematic error terms in the post-detection linear combination. The third term ${\boldsymbol I} \, H \delta$ in Eq. \[4\_det\_coadd\] is similar to the temperature–polarization mixing ${\boldsymbol I} \, H \Delta$ from a single detector (Eq. \[delta\_diff\]), but reduced in amplitude by a factor of 200 due to replacing the A–B differential beam pattern $\Delta$ with the smaller $\hat{x} - \hat{y}$ differential beam pattern $\delta$. The lower response to $m=2$ modulation from the $\delta$ differential beam (compared to the $\Delta$ beam) produces additional systematic error suppression. The final term ${\boldsymbol I} \, \Delta \tau$ also represents temperature–polarization mixing, but now appears at second order in small beam differences and is reduced by a factor $20$ in amplitude from the single-detector error. The $m=2$ spin modulation of the $\tau$ differential beam yields additional suppression.
We may also choose linear combinations of detectors to cancel the polarized sky signal ${\boldsymbol Q} \, HF$, thereby isolating specific systematic error signals. Such measurements of the systematic error signals can be used both to correct the sky measurements and as confirmation of the expected effect from beam pattern differences. For example, the orthogonal combination of four detectors becomes $$\begin{aligned}
\left[ P_{Lx} - P_{Ly} - P_{Rx} + P_{Ry} \right] / 4 &=&
{\boldsymbol Q} \, F \kappa \nonumber \\
&+& {\boldsymbol Q} \, \epsilon \tau \nonumber \\
&+& {\boldsymbol I} \, \Delta \rho \nonumber \\
&+& {\boldsymbol I} \, \delta \kappa ~~.
\label{4_det_diff}\end{aligned}$$ We may again use Table \[beam\_summary\_table\] to estimate the amplitude of each term. Unpolarized CMB signals ${\boldsymbol I}$ have amplitude of order 100 $\mu$K, while the E-mode polarization ${\boldsymbol Q}$ is of order 1 $\mu$K. Multiplying each CMB term by the relative beam area of each beam pattern yields an estimate of the relative amplitude of each term (prior to spin modulation). The difference signal is dominated by the term ${\boldsymbol I} \, \Delta \rho$, representing the convolution of the unpolarized CMB anisotropy with the double beam difference $\Delta \rho$. We may instead choose to compare signals from the two detectors sharing a common concentrator. A similar analysis shows that the detector-pair combination $( P_{Lx} + P_{Ly} )/2$ is dominated by the term ${\boldsymbol I} \, H \Delta$ which isolates a single differential beam for measurement and correction. Similar linear combinations can isolate other terms.
Tolerance
=========
Mirror symmetries within PIXIE’s differential optics suppress systematic errors coupling unpolarized structure in the sky to a false polarized signal. Positioning errors in the optical components during assembly can distort the beams from the ideal beam patterns. We quantity the resulting degradation in optical performance using 30 Monte Carlo realizations of the PIXIE optical system. For each realization, we adjust the position of each optical element allowing both translation and rigid-body rotation about its nominal orientation assuming assembly and machining tolerances of $\pm$0.05 mm drawn from a random Gaussian distribution. After adjusting all optical elements, we follow the paths of $10^9$ rays through the adjusted optical system to define the distorted beam patterns.
![Differences between the nominal beam patterns from Figure \[fore\_optics\_beams\] and the distorted beam patterns after allowing for machining and assembly tolerances. Beam patterns are shown from a single Monte Carlo realization in which the position and orientation of each optical element are perturbed about the nominal configuration. \[fore\_optics\_diff\_beams\] ](pixie_4_port_syserr_jatis_fig_9.pdf){width="4.0in"}
The PIXIE optical system is robust to typical machining and assembly tolerances. The FTS left and right transfer mirror sets and the mid-plane septum containing the polarizing grids are each machined from a single block of aluminum. The relative position and orientation of the mirrors or grids within each set have the $\pm$0.02 mm tolerance of computer-aided milling machines. This minimizes relative displacement of these components during assembly (although each set can still be displaced as a rigid body). All optical elements as well as the supporting structure are fabricated from the same material (aluminum) so that self-similar thermal contraction retains optical alignment. Alignment of the primary, secondary, and folding flat mirrors relative to the FTS assumes somewhat looser $\pm$0.05 mm tolerance typical of pinned construction. Since these components are machined individually, tolerancing errors should be uncorrelated. Figure \[fore\_optics\_diff\_beams\] shows the difference between the nominal beam patterns and the distorted patterns for a single Monte Carlo realization of the distorted optical system. The dominant effect is an angular displacement of order 3$\amin$ between the $A$ and $B$ beams on the sky, caused by displacements of the primary and secondary mirrors. This in turn creates an anti-symmetric (dipolar) pattern in both the mean beam ($F$) and the A–B spatial asymmetry ($\Delta$). Angular displacement of the beam centroid couples to spin moment $m=1$ and does not induce additional temperature–polarization mixing.
Figure \[distorted\_beams\_vs\_m\] compares the spin dependence of the nominal beam patterns to the distorted beams generated from a single Monte Carlo realization of the full optical system. It is similar to the ideal beam patterns shown in Figure \[fore\_optics\_vs\_m\], except that the position and pointing of each optical element has been perturbed by an amount randomly chosen from a Gaussian normal distribution of width 0.05 mm. We now also include the illumination of the (perturbed) fore-optics by the (perturbed) concentrator. For clarity, we compare the spin decomposition for the nominal and distorted configurations for a single choice of differential beam. Temperature-polarization coupling for a single detector is dominated by the A–B differential beam $\Delta(\theta,\phi)$ (Eq. \[big\_4\_det\_first\]). PIXIE has 4 detectors; we show the distorted beam decomposition for detectors observing the same ($\hat{x}$) sky polarization from either the left or right concentrator. Compared to the ideal system, the distorted optical system has a larger response to systematic error coupling at $m=2$, but the response is still suppressed by a factor of $10^5$ compared to the true sky polarization.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Effects of machining and assembly tolerances on the differential beam pattern $\Delta(\theta, \phi$). We compare the differential beam pattern for the nominal optical configuration to a Monte Carlo realization with all optical elements perturbed from their nominal positions. The distorted patterns are shown as a function of spin moment $m$ for the detectors sensitive to $\hat{x}$ sky polarization in both the left-side and right-side concentrators. The distorted optical system still shows suppression of order $10^{-5}$ for temperature-polarization coupling at $m=2$. \[distorted\_beams\_vs\_m\] ](pixie_4_port_syserr_jatis_fig_10.pdf "fig:"){height="2.6in"}
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Discussion
==========
Systematic errors coupling unpolarized anisotropy to a false polarized signal are a common concern to CMB polarimeters. PIXIE’s optical design provides several layers of mitigation compared to instruments imaging the CMB across a large (kilo-pixel) focal plane. Missions employing kilo-pixel arrays across large fields of view must account for the systematic degradation in beam shape from coma and shear for detectors farther from the center of the focal plane. All four PIXIE detectors, in contrast, lie at the center of the focal plane, allowing eam deformation to be minimized (Fig \[fore\_optics\_beams\]).
More importantly, PIXIE’s four-port optical system provides three distinct levels of differential measurement. The Fourier transform spectrometer produces a signal that depends on the difference between two nearly-identical beams on the sky. This differential measurement is performed optically, prior to detection, and is independent of detector calibration. We use ray-trace simulations to evaluate the differential beam patterns after removing the common-mode response. The differential beams can be described in terms of the spatial asymmetry between the A- and B-sides of the instrument, the polarization asymmetry between the $\hat{x}$ and $\hat{y}$ response, plus a cross term for the mixed spatial-polarization difference. All of the differential beams are small compared to the common-mode response. The largest effect is the spatial (A–B) asymmetry, which has only 1.5% of the common-mode response. The other differential beams have response below 0.01%.
PIXIE’s symmetric design further reduces systematic error response from the differential beam cancellation. The FTS interferes the $\hat{x}$ polarization from the A-side beam with the $\hat{y}$ polarization from the B-side beam (Fig \[pol\_symmetry\_cartoon\]). The optical system is symmetric about the mid-plane between the two sides, which forces the $\hat{x}$ polarization from one beam to be the mirror reflection of the $\hat{x}$ polarization from the other beam. The A–B mirror reflection combines with the A–B beam subtraction to produce an anti-symmetric (dipole) response in the differential beam patterns. The anti-symmetric part of the differential beam pattern does not contribute to the systematic error from temperature-polarization coupling. Each detector samples a single polarization state; the entire instrument spins about the boresight to allow full sampling of the sky polarization. True polarized signals appear at twice the spin frequency, while anti-symmetric signals can only appear at odd harmonics of the spin. Systematic errors from temperature–polarization coupling thus depend only on the $m=2$ component of the differential beam patterns, which are dominated by a dipole ($m=1$) response. Ray-trace models of the PIXIE beams show that the response at $m=2$ is reduced by an additional factor of $10^6$ or more. In principle, the optical system could further be optimized to suppress the $m=2$ differential beam response, moving power to other $m$ values that do not participate in temperature–polarization mixing. This has not yet been done but is planned for future development.
Finally, we may follow the common practice for CMB measurements and combine the post-detection signals from individual detectors. The four detectors are mounted in identical concentrators and view the same sky direction through the same optical path. Combining all four detectors cancels the leading effects from differential beams in the single-detector signal, reducing the systematic error response by a factor of 1000 or more compared to the individual detectors. Conversely, orthogonal linear combinations of 2 or 4 detectors can cancel the polarized sky signal to isolate, identify, and model specific systematic effects from the individual differential beam patterns.
Systematic error suppression in the differential PIXIE optics is robust against typical machining and assembly tolerances. We combine ray-trace optical simulations with Monte Carlo realizations of distorted PIXIE optics to evaluate both the individual beam patterns and the resulting systematic error response. Each Monte Carlo realization of then PIXIE optics perturbs each optical element (mirrors, folding flats, polarizing grids, etc) in both position and orientation by an amount drawn from a Gaussian distribution whose width is set by typical machining/assembly tolerances of 0.05 mm. The dominant effect of such tolerance errors is an angular displacement of the A-side beams relative to the B-side beams. The two beams are normally co-pointed on the sky; after accounting for tolerances the beams are typically mis-aligned by 3$\amin$. This is small compared to the 2.6 width of the common-mode beams; the resulting dipolar beam asymmetries predominantly effect the $m=1$ spin moment and do not couple efficiently to polarization. The distorted optical system still provides suppression of the $m=2$ temperature–polarization systematic error by factor of order $10^5$.
Full Single-Detector Systematic Error
=====================================
Expanding Eqs. \[4\_det\_with\_delta\] and \[horn\_def\] yields individual detector signals $$\begin{aligned}
P_{Lx} = ~~{\boldsymbol Q} \, HF
& & ~~~~~~~~
~~+~~ {\boldsymbol Q} \, H \epsilon
~+~~ {\boldsymbol I} \, H \delta
~+~~ {\boldsymbol I} \, H \Delta \nonumber \\
&+& {\boldsymbol Q} \, F \rho
~~+~~ {\boldsymbol Q} \, \rho \epsilon
~~+~~ {\boldsymbol I} \, \rho \delta
~~+~~ {\boldsymbol I} \, \rho \Delta \nonumber \\
&+& {\boldsymbol Q} \, F \kappa
~~+~~ {\boldsymbol Q} \, \kappa \epsilon
~~+~~ {\boldsymbol I} \, \kappa \delta
~~+~~ {\boldsymbol I} \, \kappa \Delta \nonumber \\
&+& {\boldsymbol Q} \, F \tau
~~+~~ {\boldsymbol Q} \, \tau \epsilon
~~+~~ {\boldsymbol I} \, \tau \delta
~~+~~ {\boldsymbol I} \, \tau \Delta \\
& & \nonumber \\
P_{Ly} = -{\boldsymbol Q} \, HF
& & ~~~~~~~~
~~+~~ {\boldsymbol Q} \, H \epsilon
~-~~ {\boldsymbol I} \, H \delta
~+~~ {\boldsymbol I} \, H \Delta \nonumber \\
&+& {\boldsymbol Q} \, F \rho
~~-~~ {\boldsymbol Q} \, \rho \epsilon
~~+~~ {\boldsymbol I} \, \rho \delta
~~-~~ {\boldsymbol I} \, \rho \Delta \nonumber \\
&-& {\boldsymbol Q} \, F \kappa
~~+~~ {\boldsymbol Q} \, \kappa \epsilon
~~-~~ {\boldsymbol I} \, \kappa \delta
~~+~~ {\boldsymbol I} \, \kappa \Delta \nonumber \\
&+& {\boldsymbol Q} \, F \tau
~~-~~ {\boldsymbol Q} \, \tau \epsilon
~~+~~ {\boldsymbol I} \, \tau \delta
~~-~~ {\boldsymbol I} \, \tau \Delta \\
& & \nonumber \\
P_{Rx} = ~~{\boldsymbol Q} \, HF
& & ~~~~~~~
~~-~~ {\boldsymbol Q} \, H \epsilon
~+~~ {\boldsymbol I} \, H \delta
~-~~ {\boldsymbol I} \, H \Delta \nonumber \\
&+& {\boldsymbol Q} \, F \rho
~~-~~ {\boldsymbol Q} \, \rho \epsilon
~~+~~ {\boldsymbol I} \, \rho \delta
~~-~~ {\boldsymbol I} \, \rho \Delta \nonumber \\
&-& {\boldsymbol Q} \, F \kappa
~~+~~ {\boldsymbol Q} \, \kappa \epsilon
~~-~~ {\boldsymbol I} \, \kappa \delta
~~+~~ {\boldsymbol I} \, \kappa \Delta \nonumber \\
&-& {\boldsymbol Q} \, F \tau
~~+~~ {\boldsymbol Q} \, \tau \epsilon
~~-~~ {\boldsymbol I} \, \tau \delta
~~+~~ {\boldsymbol I} \, \tau \Delta \\
& & \nonumber \\
P_{Ry} = -{\boldsymbol Q} \, HF
& & ~~~~~~~
~~-~~ {\boldsymbol Q} \, H \epsilon
~-~~ {\boldsymbol I} \, H \delta
~-~~ {\boldsymbol I} \, H \Delta \nonumber \\
&+& {\boldsymbol Q} \, F \rho
~~+~~ {\boldsymbol Q} \, \rho \epsilon
~~+~~ {\boldsymbol I} \, \rho \delta
~~+~~ {\boldsymbol I} \, \rho \Delta \nonumber \\
&+& {\boldsymbol Q} \, F \kappa
~~+~~ {\boldsymbol Q} \, \kappa \epsilon
~~+~~ {\boldsymbol I} \, \kappa \delta
~~+~~ {\boldsymbol I} \, \kappa \Delta \nonumber \\
&-& {\boldsymbol Q} \, F \tau
~~-~~ {\boldsymbol Q} \, \tau \epsilon
~~-~~ {\boldsymbol I} \, \tau \delta
~~-~~ {\boldsymbol I} \, \tau \Delta
\label{big_4_det_2nd}\end{aligned}$$ where now we retain all terms to second order.
**Alan Kogut** is an astrophysicist at NASA’s Goddard Space Flight Center. He received his A.B. from Princeton University and his PhD from the University of California at Berkeley. His research focuses on observations of the frequency spectrum and linear polarization of the cosmic microwave background and diffuse astrophysical foregrounds at millimeter and sub-mm wavelengths.
**Dale Fixsen** is an astrophysicist at the University of Maryland. He received his B.S. from Pacific Lutheran University, Tacoma Washington and his PhD from Princeton University. His research focuses on the spectrum, temperature, and polarization of the cosmic microwave background and the radio and infrared cosmic backgrounds.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, the authors deal with the $q$-Genocchi numbers and polynomials with weight zero. They discover some interesting relations via the $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and familiar basis Bernstein polynomials. Finally, the authors show that the $p$-adic $\log$ gamma functions are associated with the $q$-Genocchi numbers and polynomials with weight zero.'
address:
- 'Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, TURKEY'
- 'Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, TURKEY'
- 'School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, 300387, China'
author:
- Serkan Araci
- Mehmet Açikgöz
- Feng Qi
title: 'On the $\boldsymbol{q}$-Genocchi numbers and polynomials with weight zero and their applications'
---
Preliminaries
=============
Let $p$ be an odd prime number. Denote the ring of the $p$-adic rational integers by $\mathbb{Z}_{p}$, the field of rational numbers by $\mathbb{Q}$, the field of the $p$-adic rational numbers by $\mathbb{Q}_{p}$, and the completion of algebraic closure of $\mathbb{Q}_{p}$ by $\mathbb{C}_{p}$, respectively. Let $\mathbb{N}$ be the set of positive integers and $\mathbb{N}^{\ast}=\mathbb{N}\cup\{0\}$ the set of all non-negative integers. The $p$-adic absolute value is defined by $$|p|_{p}=\frac1p.$$ Assume $|q-1|_{p}<1$ is an indeterminate number in the sense that either $q\in\mathbb{C}$ or $q\in\mathbb{C}_p$. A $q$-analogue of $x$ may be defined by $$[x]_{q}=\frac{1-q^{x}}{1-q}$$ satisfying $\lim_{q\to 1}[x]_{q}=x$.
A function $f$ is said to be uniformly differentiable at a point $a\in\mathbb{Z}_{p}$ if the divided difference $$F_{f}(x,y)=\frac{f(x)-f(y)}{x-y}$$ converges to $f'(a)$ as $(x,y)\to (a,a)$. The class of all the uniformly differentiable functions is denoted by $UD(\mathbb{Z}_{p})$.
For $f\in UD(\mathbb{Z}_{p})$, the $p$-adic $q$-analogue of Riemann sum for $f$ was defined by $$\label{Reieman-sum-q-qnqlogue}
\frac{1}{[p^{n}]_q}\sum_{0\le\xi <p^{n}}f(\xi)q^{\xi}
=\sum_{0\le\xi <p^{n}}f(\xi)\mu_{q}\bigl(\xi+p^{n}\mathbb{Z}_{p}\bigr)$$ in [@Araci-Genocchi-Kim0; @Araci-Genocchi-kim7], where $n\in\mathbb{N}$. The integral of $f$ on $\mathbb{Z}_{p}$ is defined as the limit of as $n$ tends to $\infty$, if it exists, and represented by $$I_{q}(f)=\int_{\mathbb{Z}_{p}}f(\xi) \operatorname{d\mspace{-2mu}}\mu_{q}(\xi). \label{Genocchi-eq1}$$ The bosonic integral and the fermionic $p$-adic integral on $\mathbb{Z}_{p}$ are defined respectively by $$I_{1}(f)=\lim_{q\to 1}I_{q}(f)$$ and $$I_{-q}(f)=\lim_{q\to -q}I_{q}(f). \label{Genocchi-eq2}$$ For a prime $p$ and a positive integer $d$ with $(p,d)=1$, set $$\begin{gathered}
X=X_{d}=\lim_{\overleftarrow{n}}\mathbb{Z}/dp^{n}\mathbb{Z}, \quad X_{1}=\mathbb{Z}_{p}, \\
X^{\ast}=\bigcup_{\substack{(a,p)=1\\0<a<dp}}a+dp\mathbb{Z}_{p},\end{gathered}$$ and $$a+dp^{n}\mathbb{Z}_{p}=\bigl\{ x\in X\mid x\equiv a\mod dp^{n}\bigr\},$$ where $a\in\mathbb{Z}$ satisfies $0\le a<dp^{n}$ and $n\in\mathbb{N}$.
Main results
============
In [@Araci-Genocchi-1; @Araci-Genocchi-2], Arací, Açikgöz, and Seo considered the $q$-Genocchi polynomials with weight $\alpha$ in the form $$\frac{\widetilde{G}_{n+1,q}^{(\alpha)}(x)}{n+1}
=\int_{\mathbb{Z}_{p}}[ x+\xi]_{q^{\alpha}}^{n}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi),
\label{Genocchi-eq3}$$ where $\widetilde{G}_{n+1,q}^{(\alpha)}=\widetilde{G}_{n+1,q}^{(\alpha)}(0)$ is called the $q$-Genocchi numbers with weight $\alpha$. Taking $\alpha=0$ in , we easily see that $$\frac{\widetilde{G}_{n+1,q}}{n+1}\triangleq\frac{\widetilde{G}_{n+1,q}^{(0)}}{n+1}=\int_{\mathbb{Z}_{p}}\xi^{n}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi), \label{Genocchi-eq4}$$ where $\widetilde{G}_{n,q}$ are called the $q$-Genocchi numbers and polynomials with weight $0$. From , it is simple to see $$\sum_{n=0}^{\infty}\widetilde{G}_{n,q}\frac{t^{n}}{n!}=t\int_{\mathbb{Z}_{p}}e^{\xi t}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi) . \label{Genocchi-eq5}$$ By , we have $$q^{n}I_{-q}(f_{n})+(-1)^{n-1}I_{-q}(f)
=[2]_{q}\sum_{0\le\ell<n}q^{\ell}(-1)^{n-1-\ell}f(\ell), \label{Genocchi-eq6}$$ where $f_{n}(x)=f(x+n)$ and $n\in\mathbb{N}$. See [@Araci-Genocchi-kim16; @Araci-Genocchi-kim15; @Araci-Genocchi-kim17]. Taking $n=1$ in leads to the well-known equality $$qI_{-q}(f_{1})+I_{-q}(f)=[2]_{q}f(0), \label{Genocchi-eq7}$$ When setting $f(x)=e^{xt}$ in , we find $$\sum_{n=0}^{\infty}\widetilde{G}_{n,q}\frac{t^{n}}{n!}=\frac{[2]_{q}t}{qe^{t}+1}. \label{Genocchi-eq8}$$ By , we obtain the $q$-Genocchi polynomials with weight $0$ as follows $$\label{Genocchi-eq9}
\sum_{n=0}^{\infty}\widetilde{G}_{n,q}(x)\frac{t^{n}}{n!}=
\frac{[2]_{q}t}{qe^{t}+1}e^{xt}.$$ By , we see that $$\sum_{n\geq 0}\widetilde{G}_{n,q}(x)\frac{t^{n}}{n!}=t\frac{
1-\bigl(-q^{-1}\bigr)}{e^{t}-\bigl(-q^{-1}\bigr)}e^{xt}=t\sum_{n\geq
0}H_{n}\bigl(-q^{-1},x\bigr) \frac{t^{n}}{n!}.$$ By equating coefficients of $t^{n}$ on both sides of the above equality, we derive the following theorem.
For $n\in\mathbb{N}$, we have $$\frac{\widetilde{G}_{n+1,q}(x)}{n+1}=H_{n}\bigl(-q^{-1},x\bigr),$$ where $H_{n}\bigl(-q^{-1},x\bigr)$ are the $n$-th Frobenius-Euler polynomials.
By , we discover that $$\begin{aligned}
[2]_{q}\sum_{n=0}^{\infty}x^{n}\frac{t^{n}}{n!} &=q\int_{\mathbb{Z}_{p}}e^{(x+\xi+1) t}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)+\int_{\mathbb{Z}_{p}}e^{(x+\xi) t}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi) \\
&=\sum_{n=0}^{\infty}\biggl[q\int_{\mathbb{Z}_{p}}(x+\xi+1)^{n}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)
+\int_{\mathbb{Z}_{p}}(x+\xi)^{n}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)\biggr]\frac{t^{n}}{n!} \\
&=\sum_{n=0}^{\infty}\bigl[qH_{n}\bigl(-q^{-1},x+1\bigr)+H_{n}\bigl(-q^{-1},x\bigr)\bigr]\frac{t^{n}}{n!}.\end{aligned}$$ Equating coefficients of $\frac{t^{n}}{n!}$ on both sides above equation, we deduce the following theorem.
\[Genocchi-eq10-thm\] For $n\in\mathbb{N}$, the identity $$qH_{n}\bigl(-q^{-1},x+1\bigr)+H_{n}\bigl(-q^{-1},x\bigr)=[ 2
]_{q}x^{n} \label{Genocchi-eq10}$$ is valid.
In particular, when letting $q=1$, the identity becomes $$G_{n}(x+1)+G_{n}(x)=2nx^{n-1},$$ where $G_{n}(x)$ are called the Genocchi polynomials.
If we substitute $x=0$ into , then Theorem \[Genocchi-eq10-thm\] can be rewritten as Theorem \[Genocchi-eq13-thm\] below.
\[Genocchi-eq13-thm\] The identity $$q{\widetilde{G}_{n,q}(1)}+{\widetilde{G}_{n,q}}=\begin{cases}[2]_{q},& n=1\\
0, & n\ne 1
\end{cases} \label{Genocchi-eq13}$$ is true, where $\widetilde{G}_{n,q}$ are called the Genocchi numbers and polynomials with weight $0$.
When we substitute $x$ by $1-x$ and $q$ by $q^{-1}$ in , it follows that $$\begin{gathered}
\sum_{n=0}^{\infty}\widetilde{G}_{n,q^{-1}}(1-x)\frac{t^{n}}{n!
}=t\frac{1+q^{-1}}{q^{-1}e^{t}+1}e^{(1-x) t}
=\frac{1+q}{e^{t}+q}e^{t}e^{xt} \\*
=-\frac{[2]_{q}(-t)}{qe^{-t}+1}e^{(-t) x}
=\sum_{n=0}^{\infty}(-1)^{n+1}\widetilde{G}_{n,q}(x)\frac{t^{n}}{n!}.\end{gathered}$$ From this, we procure symmetric properties of this type polynomials.
The following identity holds $$\widetilde{G}_{n,q^{-1}}(1-x)=(-1)^{n+1}\widetilde{
G}_{n,q}(x). \label{Genocchi-eq11}$$
By using for $\alpha=0$ and the binomial theorem, we readily obtain that $$\begin{aligned}
\frac{\widetilde{G}_{n+1,q}(x)}{n+1}&=\int_{\mathbb{Z}_{p}}(x+\xi)^{n}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi) \\
&=\sum_{k=0}^{n}\binom{n}{k}\biggl[\int_{\mathbb{Z}_{p}}\xi^{k}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)\biggr] x^{n-k} \\
&=\sum_{k=0}^{n}\binom{n}{k}\frac{\widetilde{G}_{k+1,q}}{k+1}x^{n-k}.\end{aligned}$$ Further using $$\frac{n+1}{k+1}\binom{n}{k}=\binom{n+1}{k+1},$$ we obtain $$\widetilde{G}_{n+1,q}(x)=\sum_{k=0}^n\binom{n+1}{k+1}\widetilde{G}_{k+1,q}x^{n-k} =\sum_{k=1}^{n+1}\binom{n+1}k\widetilde{G}_{k,q}x^{n+1-k}.$$ Thus, we get the following conclusion.
The identity $$\widetilde{G}_{n,q}(x)=\sum_{k=0}^{n}\binom{n}{k}\widetilde{G}
_{k,q}x^{n-k} \label{Genocchi-eq12}$$ is true, where the usual convention of replacing $\bigl(\widetilde{G}_{q}\bigr)^{n}$ by $\widetilde{G}_{n,q}$ is used.
Combining with leads to the following proposition.
The identity $$\widetilde{G}_{0,q}=0\quad \text{and}\quad {q\bigl(\widetilde{G}_{q}+1\bigr)^{n}} +{\widetilde{G}_{n,q}}=\begin{cases}[2]_{q},& n=1\\
0, & n\ne 1
\end{cases} \label{Genocchi-eq14}$$ is true, where the usual convention of replacing $\bigl(\widetilde{G}_{q}\bigr)^{n}$ by $\widetilde{G}_{n,q}$ is used.
From , it follows that $$\begin{aligned}
q^{2}\widetilde{G}_{n+1,q}(2) &=q^{2}\bigl(\widetilde{G}_{q}+1+1\bigr)^{n+1}\\
&=q^{2}\sum_{k=0}^{n+1}\binom{n+1}{k}\bigl(\widetilde{G}_{q}+1\bigr)^{k} \\
&=(n+1) q^{2}\bigl(\widetilde{G}_{q}+1\bigr)
^{1}+q\sum_{k=2}^{n+1}\binom{n+1}{k}q\bigl(\widetilde{G}_{q}+1\bigr)^{k}\\
&=(n+1) q\bigl([2]_{q}-\widetilde{G}_{1,q}\bigr)
-q\sum_{k=2}^{n+1}\binom{n+1}{k}\widetilde{G}_{k,q} \\
&=(n+1) q[2]_{q}-\Biggl[q\sum_{k=2}^{n+1}\binom{n+1}{k}\widetilde{G}_{k,q} +(n+1)q\widetilde{G}_{1,q}\Biggr]\\
&=(n+1) q[2]_{q}-q\sum_{k=0}^{n+1}\binom{n+1}{k}\widetilde{G}_{k,q}\\
&=(n+1) q[2]_{q}-q\bigl(\widetilde{G}_{q}+1\bigr)^{n+1} \\
&=(n+1) q[2]_{q}+\widetilde{G}_{n+1,q}\end{aligned}$$ for $n>1$. Therefore, we deduce the following proposition.
For $n>1$, $$\widetilde{G}_{n+1,q}(2)=\frac{(n+1)}{q}[2]_{q}+\frac{1}{q^{2}}\widetilde{G}_{n+1,q}.
\label{Genocchi-eq15}$$
By virtue of , , and , we find $$\begin{gathered}
(n+1)\int_{\mathbb{Z}_{p}}(1-\xi)^{n}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)=(n+1) (-1)^{n}\int_{\mathbb{Z}_{p}}(\xi -1)^{n}\operatorname{d\mspace{-2mu}}\mu
_{-q}(\xi) \\
=(-1)^{n}\widetilde{G}_{n+1,q}(-1)=\widetilde{G}_{n+1,q^{-1}}(2)
=(n+1) [2]_{q}+q^{2}\widetilde{G}_{n+1,q^{-1}}.\end{gathered}$$ As a result, we may concluded Theorem \[Genocchi-eq16-thm\] below.
\[Genocchi-eq16-thm\] The identity $$\int_{\mathbb{Z}_{p}}(1-\xi)^{n}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)=[2]
_{q}+q^{2}\frac{\widetilde{G}_{n+1,q^{-1}}}{n+1} \label{Genocchi-eq16}$$ is valid.
Let $UD(\mathbb{Z}_{p})$ be the space of continuous functions on $\mathbb{Z}_{p}$. For $f\in UD(\mathbb{Z}_{p})$, the $p$-adic analogue of Bernstein operator for $f$ is defined by $$\boldsymbol{B}_{n}(f,x)=\sum_{k=0}^{n}f\biggl(\frac{k}{n}\biggr) B_{k,n}(x)
=\sum_{k=0}^{n}f\biggl(\frac{k}{n}\biggr)\binom{n}{k}x^{k}(1-x)^{n-k},$$ where $n,k\in\mathbb{N}^{\ast}$ and the $p$-adic Bernstein polynomials of degree $n$ is defined by $$B_{k,n}(x)=\binom{n}{k}x^{k}(1-x)^{n-k},\quad
x\in\mathbb{Z}_{p}.\label{Genocchi-eq19}$$ See [@Araci-Genocchi-3; @Araci-Genocchi-Kim-3; @Araci-Genocchi-k-2; @Araci-Genocchi-Kim-4]. Via the $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and Bernstein polynomials in , we can obtain that $$\begin{aligned}
I_{1} &=\int_{\mathbb{Z}_{p}}B_{k,n}(\xi) \operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)\\
&=\binom{n}{k}\int_{\mathbb{Z}_{p}}\xi^{k}(1-\xi)^{n-k}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi) \\
&=\binom{n}{k}\sum_{\ell=0}^{n-k}\binom{n-k}{\ell}(-1)^{\ell} \biggl[\int_{\mathbb{Z}_{p}}\xi^{\ell+k}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)\biggr] \\
&=\binom{n}{k}\sum_{\ell=0}^{n-k}\binom{n-k}{\ell}(-1)^{\ell}\frac{
\widetilde{G}_{\ell+k+1,q}}{\ell+k+1}.\end{aligned}$$ On the other hand, by symmetric properties of Bernstein polynomials, we have $$\begin{aligned}
I_{2} &=\int_{\mathbb{Z}_{p}}B_{n-k,n}(1-\xi) \operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)\\
&=\binom{n}{k}\sum_{s=0}^{k}\binom{k}{s}(-1)^{k+s}\int_{\mathbb{Z}_{p}}(1-\xi)^{n+s}\operatorname{d\mspace{-2mu}}\mu_{-q}(x)\\
&=\binom{n}{k}\sum_{s=0}^{k}\binom{k}{s}(-1)^{k+s} \biggl([2]_{q}+q^{2}\frac{\widetilde{G}_{n+s+1,q^{-1}}}{n+s+1}\biggr) \\
&=
\begin{cases}\displaystyle
[2]_{q}+q^{2}\frac{\widetilde{G}_{n+s+1,q^{-1}}}{n+s+1},& k=0\\\displaystyle
\binom{n}{k}\sum_{s=0}^{k}\binom{k}{s}(-1)^{k+s}\biggl([2]_{q}+q^{2}\frac{\widetilde{G}
_{n+s+1,q^{-1}}}{n+s+1}\biggr), & k\ne 0.
\end{cases}\end{aligned}$$ Equating $I_{1}$ and $I_{2}$ yields Theorem \[thm7-araci-qi\] below.
\[thm7-araci-qi\] The following identity holds: $$\sum_{\ell=0}^{n-k}\binom{n-k}{\ell}(-1)^{\ell}\frac{\widetilde{G}_{\ell+k+1,q}}{\ell+k+1}=
\begin{cases}\displaystyle
[2]_{q}+q^{2}\frac{\widetilde{G}_{n+s+1,q^{-1}}}{n+s+1},& k=0;\\\displaystyle
\sum_{s=0}^{k}\binom{k}{s}(-1)^{k+s}\biggl([2]_{q}+q^{2} \frac{\widetilde{G}_{n+s+1,q^{-1}}}{n+s+1}\biggr), & k\ne 0.
\end{cases}$$
The $p$-adic $q$-integral on $\mathbb{Z}_{p}$ of the product of several Bernstein polynomials can be calculated as $$\begin{aligned}
I_{3} &=\int_{\mathbb{Z}_{p}}\prod_{s=1}^{m}B_{k,n_{s}}(\xi) \operatorname{d\mspace{-2mu}}\mu_{-q}(
\xi) \\
&=\prod_{s=1}^{m}\binom{n_{s}}{k}\int_{\mathbb{Z}_{p}} \xi^{mk}(1-\xi)^{n_{1}+\dotsm+n_{m}-mk}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)\\
&=\prod_{s=1}^{m}\binom{n_{s}}{k}\sum_{\ell=0}^{n_{1}+\dotsm+n_{m}-mk}
\binom{n_{1}+\dotsm+n_{m}-mk}{\ell}(-1)^{\ell} \biggl[\int_{\mathbb{Z}_{p}}\xi^{\ell+mk}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi)\biggr]\\
&=\prod_{s=1}^{m}\binom{n_{s}}{k}\sum_{\ell=0}^{n_{1}+\dotsm+n_{m}-mk}
\binom{n_{1}+\dotsm+n_{m}-mk}{\ell}(-1)^{\ell}\frac{\widetilde{G}
_{\ell+mk+1,q^{-1}}}{\ell+mk+1}.\end{aligned}$$ On the other hand, by symmetric properties of Bernstein polynomials and , we have $$\begin{aligned}
I_{4} &=\int_{\mathbb{Z}_{p}}\prod_{s=1}^{m}B_{n_{s}-k,n_{s}}(1-\xi) \operatorname{d\mspace{-2mu}}\mu
_{-q}(\xi) \\
&=\binom{n}{k}\sum_{\ell=0}^{mk}\binom{mk}{\ell}(
-1)^{mk+\ell}\int_{\mathbb{Z}_{p}}(1-\xi)^{n_{1}+\dotsm+n_{m}+\ell}\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi) \\
&=\prod_{s=1}^{m}\binom{n_{s}}{k}\sum_{\ell=0}^{mk}\binom{mk}{\ell}(
-1)^{mk+\ell}\biggl([2]_{q}+q^{2}\frac{\widetilde{G}
_{n_{1}+\dotsm+n_{m}+\ell+1,q^{-1}}}{n_{1}+\dotsm+n_{m}+\ell+1}\biggr) \\
&=\begin{cases}\displaystyle
[2]_{q}+q^{2}\frac{\widetilde{G}
_{n_{1}+\dotsm+n_{m}+1,q^{-1}}}{n_{1}+\dotsm+n_{m}+1}, & k=0\\\displaystyle
\prod_{s=1}^{m}\binom{n_{s}}{k}\sum_{\ell=0}^{mk}\binom{mk}{\ell}(
-1)^{mk+\ell}\biggl([2]_{q}+q^{2}\frac{\widetilde{G}
_{n_{1}+\dotsm+n_{m}+\ell+1,q^{-1}}}{n_{1}+\dotsm+n_{m}+\ell+1}\biggr), & k\ne 0.
\end{cases}\end{aligned}$$ Equating $I_{3}$ and $I_{4}$ results in an interesting identity for $q$-analogue of Genocchi polynomials with weight $0$.
The identity $$\begin{gathered}
\sum_{\ell=0}^{n_{1}+\dotsm+n_{m}-mk}\binom{n_{1}+\dotsm+n_{m}-mk}{\ell}(-1)
^{\ell}\frac{\widetilde{G}_{\ell+mk+1,q^{-1}}}{\ell+mk+1}\\
=\begin{cases}\displaystyle
[2]_{q}+q^{2}\frac{\widetilde{G}_{n_{1}+\dotsm+n_{m}+1,q^{-1}}}{n_{1}+\dotsm+n_{m}+1}, & k=0\\\displaystyle
\sum_{\ell=0}^{mk}\binom{mk}{\ell}(-1)^{mk+\ell} \biggl([2]_{q}+q^{2}\frac{\widetilde{G}_{n_{1}+\dotsm+n_{m}+\ell+1,q^{-1}}} {n_{1}+\dotsm+n_{m}+\ell+1}\biggr), & k\ne 0
\end{cases}\end{gathered}$$ is true.
Other identities
================
In this section, we consider Kim’s $p$-adic $q$-$\log$ gamma functions related to the $q$-analogue of Genocchi polynomials.
For $x\in\mathbb{C}_{p}\setminus\mathbb{Z}_{p}$, $$(1+x)\log(1+x)=x+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n(n+1)}x^{n+1}.$$
Kim’s $p$-adic locally analytic function on $x\in\mathbb{C}_{p}\setminus\mathbb{Z}_{p}$ can be defined as follows.
For $x\in\mathbb{C}_{p}\setminus\mathbb{Z}_{p}$, $$G_{p,q}(x)=\int_{\mathbb{Z}_{p}}[ x+\xi]_{q}(\log[ x+\xi]_{q}-1)
\operatorname{d\mspace{-2mu}}\mu_{-q}(\xi).$$
If $q\to 1$, then $$\label{Genocchi-eq17}
G_{p,1}(x)\triangleq G_{p}(x)=\int_{\mathbb{Z}_{p}}(x+\xi)[\log(x+\xi)-1] \operatorname{d\mspace{-2mu}}\mu_{-q}(\xi).$$ Replacing $x$ by $\frac{\xi}{x}$ in leads to $$(x+\xi)[\log(x+\xi)-1]=(x+\xi)\log x+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{
n(n+1)}\frac{\xi^{n+1}}{x^{n}}-x. \label{Genocchi-eq18}$$ From and , we can establish an interesting formula which is useful for studying in the theory of the $p$-adic analysis and the analytic number.
For $x\in\mathbb{C}_{p}\setminus\mathbb{Z}_{p}$, $$\label{araci-qi-final-eq}
G_{p}(x)=\biggl(x+\frac{\widetilde{G}_{2,q}}{2}\biggr)\log x+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n(n+1)
(n+2)}\frac{\widetilde{G}_{n+2,q}}{x^{n}}-x.$$
[99]{}
S. Araci, M. Açikgöz, and J. J. Seo, *A note on the weighted $q$-Genocchi numbers and polynomials with their interpolation function*, Honam Math. J. **28** (2012), in press.
S. Araci, D. Erdal, and J. J. Seo, *A study on the Fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials*, Abstr. Appl. Anal. **2011** (2011), Article ID 649248, 10 pages; Available online at <http://dx.doi.org/10.1155/2011/649248>.
S. Araci, J. J. Seo, and D. Erdal, *New construction weighted $(h,q)$-Genocchi numbers and polynomials related to Zeta type function*, Discrete Dyn. Nat. Soc. **2011** (2011), Article ID 487490, 7 pages; Available online at <http://dx.doi.org/10.1155/2011/487490>.
T. Kim, *A note on the $q$-analogue of $p$-adic log gamma function*, Available online at <http://arxiv.org/abs/0710.4981>.
T. Kim, *New approach to $q$-Euler polynomials of higher order*, Russ. J. Math. Phys. **17** (2010), no. 2, 218–225; Available online at <http://dx.doi.org/10.1134/S1061920810020068>.
T. Kim, *On a $q$-analogue of the $p$-adic log gamma functions and related integrals*, J. Number Theory **76** (1999), no. 2, 320–329; Available online at <http://dx.doi.org/10.1006/jnth.1999.2373>.
T. Kim, *$q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals*, J. Nonlinear Math. Phys. **14** (2007), no. 1, 15–27.
T. Kim, *$q$-Volkenborn integration*, Russ. J. Math. phys. **9** (2002), no. 3, 288–299.
T. Kim, *Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the Fermionic $p$-adic integral on $\mathbb{Z}_{p}$*, Russ. J. Math. Phys. **16** (2009), no. 4,484–491; Available online at <http://dx.doi.org/10.1134/S1061920809040037>.
T. Kim and J. Choi, *On the $q$-Euler numbers and polynomials with weight $0$*, Abstr. Appl. Anal. **2012** (2012), in press.
T. Kim, J. Choi, and Y.-H. Kim, *Some identities on the $q$-Bernoulli numbers and polynomials with weight $0$*, Abstr. Appl. Anal. **2011** (2011), Article ID 361484, 8 pages; Available online at <http://dx.doi.org/10.1155/2011/361484>.
D. S. Kim, T. Kim, S.-H. Lee, D.-V. Dolgy, and S.-H. Rim, *Some new identities on the Bernoulli numbers and polynomials*, Discrete Dyn. Nat. Soc. **2011** (2011) Article ID 856132, 11 pages; Available online at <http://dx.doi.org/10.1155/2011/856132>.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Arkadiusz Madej^1^, Nan Wang^2^, Nikolaos Athanasopoulos^1^, Rajiv Ranjan^3^, and Blesson Varghese^1^'
bibliography:
- 'references.bib'
title: 'Priority-based Fair Scheduling in Edge Computing'
---
Introduction {#sec:introduction}
============
Fair Scheduling on the Edge {#sec:scheduler}
===========================
Job Management in the Scheduler {#sec:jobmanagement}
===============================
Experimental Studies {#sec:experimentalstudies}
====================
Related Work {#sec:relatedwork}
============
Conclusions {#sec:conclusions}
===========
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Prateek Jain[^1]'
- 'Om Thakkar[^2]'
- 'Abhradeep Thakurta[^3]'
bibliography:
- 'reference.bib'
title: Differentially Private Matrix Completion Revisited
---
=1
[^1]: Microsoft Research. Email: `[email protected]`.
[^2]: Department of Computer Science, Boston University. Email: `[email protected]`.
[^3]: Computer Science Department, University of California Santa Cruz. Email: `[email protected]`.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In a recent paper in Journal of Convex Analysis the authors studied, in non-reflexive Banach spaces, a class of maximal monotone operators, characterized by the existence of a function in Fitzpatrick’s family of the operator which conjugate is above the duality product. This property was used to prove that such operators satisfies a restricted version of Brøndsted-Rockafellar property.
In this work we will prove that if a single Fitzpatrick function of a maximal monotone operator has a conjugate above the duality product, then all Fitzpatrick function of the operator have a conjugate above the duality product. As a consequence, the family of maximal monotone operators with this property is just the class NI, previously defined and studied by Simons.
We will also prove that an auxiliary condition used by the authors to prove the restricted Brøndsted-Rockafellar property is equivalent to the assumption of the conjugate of the Fitzpatrick function to majorize the duality product.\
\
2000 Mathematics Subject Classification: 47H05, 49J52, 47N10.\
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Key words: Maximal monotone operators, Brøndsted-Rockafellar property, non-reflexive Banach spaces, Fitzpatrick functions.\
author:
- 'M. Marques Alves[^1][^2]'
- 'B. F. Svaiter[^3] [^4]'
title: A new old class of maximal monotone operators
---
Introduction
============
Let $X$ be a real Banach space. We use the notation $X^*$ for the topological dual of $X$ and ${\langle{\cdot},{\cdot}\rangle}$ for the duality product in $X\times X^*$: $${\langle{x},{x^*}\rangle}=x^*(x).$$ Whenever necessary, we will identify $X$ with its image under the canonical injection of $X$ into $X^{**}$. To simplify the notation, from now on $\pi$ and $\pi_*$ stands for the duality product in $X\times X^*$ and $X^*\times
X^{**}$ respectively: $$\begin{aligned}
\nonumber
&\pi:X\times X^*\to{\mathbb{R}}, &&\pi_*:X^*\times X^{**}\to{\mathbb{R}}\\
\label{eq:df.pps}
&\pi(x,x^*)={\langle{x},{x^*}\rangle},& &\pi_*(x^*,x^{**})={\langle{x^*},{x^{**}}\rangle}.\end{aligned}$$ The *indicator function* of $A\subset X$ is $\delta_A:X\to{\bar{\mathbb{R}}}$, $$\delta_A(x):=
\begin{cases}
0,& x\in A\\
\infty,& \mbox{ otherwise.}
\end{cases}$$ For $f:X\to{\bar{\mathbb{R}}}$, the lower semicontinuous convex closure of $f$ is $\operatorname{cl\,conv}f:X\to{\bar{\mathbb{R}}}$, the largest lower semicontinuous convex function majorized by $f$. The conjugate of $f$ is $f^*:X^*\to {\bar{\mathbb{R}}}$, $$f^*(x^*)=\sup_{x\in X} {\langle{x},{x^*}\rangle}-f(x).$$ It is trivial to check that $ f^*=(\operatorname{cl\,conv}f)^*$.
A point point-to-set operator $T:X{\rightrightarrows}X^*$ is a relation on $X\times X^*$: $$T\subset X\times X^*$$ and $x^*\in T(x)$ means $(x,x^*)\in T$. An operator $T:X{\rightrightarrows}X^*$ is [*monotone*]{} if $${\langle{x-y},{x^*-y^*}\rangle}\geq 0,\forall (x,x^*),(y,y^*)\in T.$$ and it is [*maximal monotone*]{} if it is monotone and maximal (with respect to the inclusion) in the family of monotone operators of $X$ in $X^*$. Maximal monotone operators in Banach spaces arises, for example, in the study of PDE’s, equilibrium problems and calculus of variations.
Given a maximal monotone operator $T:X{\rightrightarrows}X^*$, Fitzpatrick defined [@Fitz88] the family $\mathcal{F}_T$ as those convex, lower semicontinuous functions in $X\times X^*$ which are bounded bellow by the duality product and coincides with it at $T$: $$\label{eq:def.ft}
{\mathcal{F}}_T=\left\{ h\in {\bar{\mathbb{R}}}^{X\times X^*}
\left|
\begin{array}{ll}
h\mbox{ is convex and lower semicontinuous}\\
{\langle{x},{x^*}\rangle}\leq h(x,x^*),\quad \forall (x,x^*)\in X\times X^*\\
(x,x^*)\in T
\Rightarrow
h(x,x^*) = {\langle{x},{x^*}\rangle}
\end{array}
\right.
\right\}.$$ Fitzpatrick found an explicit formula for the minimal element of $\mathcal{F}_T$, from now on *Fitzpatrick function* of $T$, $\varphi_T:X\times X^*\to{\bar{\mathbb{R}}}$ $$\label{eq:def.f.fitz}
\varphi_T(x,x^*)=\sup_{(y,y^*)\in T} {\langle{x},{y^*}\rangle}+{\langle{y},{x^*}\rangle}-
{\langle{y^*},{y}\rangle}.$$ Moreover, he also proved that if $h\in {\mathcal{F}}_T$ then $h$ represents $T$ in the following sense: $$(x,x^*)\in T\iff h(x,x^*)={\langle{x},{x^*}\rangle}.$$ Note that $$\varphi_T(x,x^*)=(\pi+\delta_T)^*(x^*,x).$$ The supremum of Fitzpatrick family is the $\mathcal{S}$-function, defined and studied by Burachik and Svaiter in [@BuSvSet02], $\mathcal{S}_T:X\times X^*\to{\bar{\mathbb{R}}}$ $$\mathcal{S}_T(x,x^*)=\sup \left\{ h(x,x^*)\;\left|\;
\begin{array}{l}
h: X\times X^*\to {\bar{\mathbb{R}}}\mbox{ convex lower semicontinuous}\\
h(x,x^*)\leq {\langle{x},{x^*}\rangle}, \quad\forall (x,x^*)\in T
\end{array}
\right\}\right.$$ or, equivalently (see [@BuSvSet02 Eq.(35)], [@BuSvIMPA01 Eq. 29]) $$\label{eq:def.sf}
\mathcal{S}_T=\operatorname{cl\,conv}(\pi+\delta_T).$$ Some authors [@BorJCA06; @VosSet06; @BorProc07] attribute the $\mathcal{S}$-function to [@PenRelv04] although [@PenRelv04] was submitted after the publication of [@BuSvSet02]. Moreover, the content of [@BuSvSet02], and specifically the $\mathcal{S}$ function, was presented on Erice workshop on July 2001, by R. S. Burachik [@BuErice01b]. A list of the talks of this congress, which includes [@PenErice01], is available on the www[^5]. It shall also be noted that [@BuSvIMPA01], the preprint of [@BuSvSet02], was published ( and available on www) at IMPA preprint server in August 2001.
Burachik and Svaiter defined [@BuSvSet02], for $h:X\times X^*\to {\bar{\mathbb{R}}}$, $$\mathcal{J}h:X\times X^*\to{\bar{\mathbb{R}}},
\qquad \mathcal{J}h(x,x^*)=h^*(x^*,x)$$ and proved that if $T$ is maximal monotone, then $\mathcal{J}$ maps ${\mathcal{F}}_T$ into itself and $\mathcal{J}\;\mathcal{S}_T=\varphi_T$: $$\mathcal{S}_T^*(x^*,x)=\varphi_T(x,x^*).$$ Note that any $h\in {\mathcal{F}}_T$ satisfies the condition bellow: $$\label{eq:ca.r}
\begin{array}{l}
h(x,x^*)\geq {\langle{x},{x^*}\rangle}\\
h^*(x^*,x)\geq {\langle{x},{x^*}\rangle}
\end{array}\qquad
\forall\, (x,x^*)\in X\times X^*.$$ What about the converse? Burachik and Svaiter proved in [@BuSvProc03 Theorem 3.1] that if a closed convex function $h$ satisfies in a *reflexive* Banach space, then $h$ represents a maximal monotone operator and $h$ belongs to the Fitzpatrick function of this operator. This result has been used for ensuring maximal monotonicity in reflexive Banach spaces [@SimZaProc04; @PenRelv04; @PenZaProc06; @BorProc06; @BoGraWanSIAM06; @BarBauBorNA07; @BauBorWanSIAM07; @BauWanSet07; @BorProc07; @SimJAC07].
For the case of a non-reflexive Banach space, Marques-Alves and Svaiter proved [@MASvJCA08] that if $h$ is a convex lower semicontinuous function in $X\times X^*$ and $$\label{eq:ca}
\begin{array}{l}
h(x,x^*)\geq {\langle{x,},{x^*}\rangle},\qquad
\forall\, (x,x^*)\in X\times X^* \\
h^*(x^*,x^{**})\geq {\langle{x^*},{x^{**}}\rangle},\qquad
\forall\, (x^*,x^{**})\in X^*\times X^{**}
\end{array}\qquad$$ then again $h$ and $\mathcal{J}h$ represent a maximal monotone operator and belong to Fitzpatrick family of this operator. Moreover, the operator $T$ satisfies a restricted version of the Brøndsted-Rockafellar property. In particular, Marques-Alves and Svaiter proved that if $T$ is maximal monotone and *one* Fitzpatrick function of $T$ satisfies , then $T$ satisfies the restricted Brøndsted-Rockafellar property. The case of $h$ convex (but not lower semicontinuous) and satisfying was also examined in [@MASvJCA08].
Martínez-Legaz and Svaiter [@LegSvSet05] defined (with a different notation), for $h:X\times X^*\to{\bar{\mathbb{R}}}$ and $(x_0,x_0^*)\in X\times X^*$ $$\label{eq:def.hx}
\begin{array}{l}
h_{(x_0,x_0^*)}:X\times X^*\to{\bar{\mathbb{R}}},\\[.4em]
h_{(x_0,x_0^*)}(x,x^*):=h(x+x_0,x^*+x_0^*)-[{\langle{x},{x_0^*}\rangle}+{\langle{x_0},{x^*}\rangle}
+{\langle{x_0},{x_0^*}\rangle}].
\end{array}$$ The operation $h\mapsto h_{(x_0,x_0^*)}$ preserves many properties of $h$, as convexity, lower semicontinuity and can be seen as the action of the group $(X\times X^*,+)$ on ${\bar{\mathbb{R}}}^{X\times X^*}$, because $$\left(h_{(x_0,x_0^*)}\right)_{(x_1,x_1^*)}=h_{(x_0+x_1,x_0^*+x_1^*)}.$$ Moreover $$\left(h_{(x_0,x_0^*)}\right)^*=\left(h^*\right)_{(x_0^*,x_0)},$$ where the rightmost $x_0$ is identified with its image under the canonical injection of $X$ into $X^{**}$. Therefore,
1. $h\geq \pi\iff h_{(x_0,x_0)}\geq \pi$,
2. $
\left(h_{(x_0,x_0^*)}\right)^*\geq \pi_*\iff
\left(h^*\right)_{(x_0^*,x_0)}
\geq \pi_*$,
and finally, $$h\in{\mathcal{F}}_T\iff h_{(x_0,x_0^*)}\in {\mathcal{F}}_{T-\{(x_0,x_0^*)\}}.$$ Marques-Alves and Svaiter work [@MASvJCA08] was heavily based on these nice properties of the map $h\mapsto
h_{(x_0,x_0^*)}$. These authors also used the fact that if $h$ satisfies condition , then it also satisfies the following *auxiliary condition*: $$\label{eq:cb}
\inf_{(x,x^*)\in X\times X^*} h_{(x_0,x_0^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+
\frac{1}{2}
{ {\|x^*\|}^2 }=0,\qquad \forall (x_0,x_0^*)\in X\times X^*.$$ A possible generalization of [@MASvJCA08] would be to require only the auxiliary condition for one Fitzpatrick function of a maximal monotone operator $T$ and then conclude that this operator satisfies the restricted Brøndsted-Rockafellar property. Unfortunately, condition is not more general than condition , as we will prove.
The class of operators studied in [@MASvJCA08] is the class of maximal [**M**]{}onotone operators for which there exists a function in Fitzpatrick family with a conjugate [**A**]{}bove the duality product. So, for the time being, we will call these operators type MA. We will also prove that MA condition is equivalent to NI condition. A maximal monotone $T:X{\rightrightarrows}X^*$ is type (NI) [@SimRanJMA96] if $$\inf_{(y,y^*)\in T}{\langle{x^{**}-y},{x^*-y^*}\rangle}\leq 0,
\forall (x^*,x^{**})\in X^*\times X^{**}.$$ For proving this equivalence we will show that if *some* $h\in
{\mathcal{F}}_T$ satisfies condition , then *all* function in Fitzpatrick family of $T$ satisfies condition . Observe again that, for a function in Fitzpatrick family, $h\geq \pi$ holds by definition.
The main results of this work are the two theorems bellow:
\[th:1\] Let $X$ be a real Banach space and $h$ be a convex function on $X\times X^*$. Then $h$ satisfies the condition [@MASvJCA08 eq. (4)] $$\label{eq:t1.a}
h\geq \pi\qquad h^*\geq \pi_*$$ if, and only if, $h$ satisfies the auxiliary condition [@MASvJCA08 eq. immediately bellow eq. (29)], $$\label{eq:th1.b}
\inf_{(x,x^*)\in X\times X^*} h_{(x_0,x_0^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+
\frac{1}{2}
{ {\|x^*\|}^2 }=0,\qquad \forall (x_0,x_0^*)\in X\times X^*.$$
\[th:2\] Let $X$ be a real Banach space and $T:X{\rightrightarrows}X^*$. The following conditions are equivalent:
1. $T$ is type *MA*, that is, $T$ is maximal monotone and there exists some $h\in{\mathcal{F}}_T$ such that $h^*\geq\pi_*$ (and $h\geq\pi$),
2. $T$ is maximal monotone and *all* $h\in {\mathcal{F}}_T$, satisfies the condition $h^*\geq\pi_*$ (and $h\geq\pi$),
3. $T$ is maximal monotone and *some* $h\in {\mathcal{F}}_T$ satisfies the condition $$\inf_{(x,x^*)\in X\times X^*} h_{(x_0,x_0^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+
\frac{1}{2}
{ {\|x^*\|}^2 }=0,\qquad \forall (x_0,x_0^*)\in X\times X^*.$$
4. $T$ is maximal monotone and *all* $h\in {\mathcal{F}}_T$ satisfies the condition $$\inf_{(x,x^*)\in X\times X^*} h_{(x_0,x_0^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+
\frac{1}{2}
{ {\|x^*\|}^2 }=0,\qquad \forall (x_0,x_0^*)\in X\times X^*.$$
5. $T$ is type *NI*: $$\inf_{(y,y^*)\in T}{\langle{x^{**}-y},{x^*-y^*}\rangle}\leq 0,\qquad \forall
(x^*,x^{**})\in X^*\times X^{**}.$$
where $\pi$ and $\pi_*$ are the duality products in $X\times X^*$ and $X^*\times X^{**}$, as described in .
Proof of the main results {#sec:proof}
=========================
Let $\bar h:=\operatorname{cl\,conv}h$. As $h$ is convex, $$\bar h(x,x^*)=\lim\inf_{(y,y^*)\to(x,x^*)} h(y,y^*),$$ and, for any $(x_0,x_0^*)\in X\times X^*$, $$\bar h_{(x_0,x_0^*)}(x,x^*)=\lim\inf_{(y,y^*)\to(x,x^*)}
h_{(x_0,x_0^*)}(y,y^*).$$ As the duality product is continuous and $(\operatorname{cl\,conv}h)^*=h^*$, condition holds for $h$ if, and only if, it holds for $\bar h$. As the norms are continuous (this is indeed trivial), condition holds for $h$ if, and only if, it holds for $\bar h$. So, it suffices to prove the theorem for the case where $h$ is lower semicontinuous, and we assume it from now on in this proof.
For the sake of completeness, we discuss the implication $\Rightarrow$. Take $(x_0,x_0^*)\in X\times X^*$. If condition holds for $h$, then it holds for $h_{(x_0,x_0^*)}$ and using [@MASvJCA08 Theorem 3.1, eq. (12)] we conclude that condition holds.
For proving the implication $\Rightarrow$, first note that, for any $(z,z^*)\in X\times X^*$, $$h_{(z,z^*)}(0,0)\geq\inf_{(x,x^*)}
h_{(z,z^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+ \frac{1}{2}{ {\|x^*\|}^2 }.$$ Therefore, using also we obtain $$h(z,z^*)-{\langle{z},{z^*}\rangle}= h_{(z,z^*)}(0,0)\geq 0.$$ Since $(z,z^*)$ is an arbitrary element of $X\times X^*$ we conclude that $h\geq \pi$.
For proving that, under assumption , $h^*\geq\pi_*$, take some $(y^*,y^{**})\in X^*\times X^{**}$. First, use Fenchel-Young inequality to conclude that for any $(x,x^*), (z,z^*)\in X\times X^*$, $$\begin{aligned}
h_{(z,z^*)}(x,x^*)
\geq&
{\langle{x},{y^*-z^*}\rangle}+{\langle{x^*},{y^{**}-z}\rangle}-\left(h_{(z,z^*)}\right)^*(y^*-z^*,y^{**}-z).
\end{aligned}$$ As $\left(h_{(z,z^*)}\right)^*=(h^*)_{(z^*,z)}$, $$\begin{aligned}
\left(h_{(z,z^*)}\right)^*(y^*-z^*,y^{**}-z)&=
h^*(y^*,y^{**})-{\langle{z},{y^*-z^*}\rangle}-{\langle{z^*},{y^{**}-z}\rangle}-{\langle{z},{z^*}\rangle}\\
&=h^*(y^*,y^{**})-{\langle{y^*},{y^{**}}\rangle}+{\langle{y^*-z^*},{y^{**}-z}\rangle}.
\end{aligned}$$ Combining the two above equations we obtain $$\begin{aligned}
h_{(z,z^*)}(x,x^*)
\geq&
{\langle{x},{y^*-z^*}\rangle}+{\langle{x^*},{y^{**}-z}\rangle}\\
&-{\langle{y^*-z^*},{y^{**}-z}\rangle}+{\langle{y^*},{y^{**}}\rangle}-h^*(y^*,y^{**}).
\end{aligned}$$ Adding $(1/2){ {\|x\|}^2 }+(1/2){ {\|x^*\|}^2 }$ in both sides of the above inequality we have $$\begin{aligned}
h_{(z,z^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+\frac{1}{2}{ {\|x^*\|}^2 }
\geq&
{\langle{x},{y^*-z^*}\rangle}+{\langle{x^*},{y^{**}-z}\rangle}
+\frac{1}{2}{ {\|x\|}^2 }+\frac{1}{2}{ {\|x^*\|}^2 }
\\
&-{\langle{y^*-z^*},{y^{**}-z}\rangle}+{\langle{y^*},{y^{**}}\rangle}-h^*(y^*,y^{**}).
\end{aligned}$$ Note that $${\langle{x},{y^*-z^*}\rangle}+\frac{1}{2}{ {\|x\|}^2 }\geq
-\frac{1}{2}{ {\|y^*-z^*\|}^2 },\qquad
{\langle{x^*},{y^{**}-z}\rangle}+\frac{1}{2}{ {\|x^*\|}^2 }\geq
-\frac{1}{2} { {\|y^{**}-z\|}^2 }.$$ Therefore, for any $(x,x^*), (z,z^*)\in X\times X^*$, $$\begin{aligned}
h_{(z,z^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+\frac{1}{2}{ {\|x^*\|}^2 }
\geq&
-\frac{1}{2}{ {\|y^*-z^*\|}^2 } -\frac{1}{2} { {\|y^{**}-z\|}^2 }
\\
&-{\langle{y^*-z^*},{y^{**}-z}\rangle}+{\langle{y^*},{y^{**}}\rangle}-h^*(y^*,y^{**}).
\end{aligned}$$ Using now assumption we conclude that the infimum, for $(x,x^*)\in X\times X^*$, at the left hand side of the above inequality is $0$. Therefore, taking the infimum on $(x,x^*)\in
X\times X^*$ at the left hand side of the above inequality and rearranging the resulting inequality we have $$\begin{aligned}
h^*(y^*,y^{**})-{\langle{y^*},{y^{**}}\rangle}\geq
-\frac{1}{2}{ {\|y^*-z^*\|}^2 } -\frac{1}{2} { {\|y^{**}-z\|}^2 }
-{\langle{y^*-z^*},{y^{**}-z}\rangle}.
\end{aligned}$$ Note that $$\sup_{z^*\in X^*} -{\langle{y^*-z^*},{y^{**}-z}\rangle}
-\frac{1}{2}{ {\|y^*-z^*\|}^2 }
=\frac{1}{2}{ {\|y^{**}-z\|}^2 }.$$ Hence, taking the sup in $z^*\in X^*$ at the right hand side of the previous inequality we obtain $$h^*(y^*,y^{**})-{\langle{y^*},{y^{**}}\rangle}\geq 0$$ and condition holds.
First use Theorem \[th:1\] to conclude that item 1 and 3 are equivalent. The same theorem also shows that items 2 and 4 are equivalent.
Now assume that item 3 holds, that is, for some $h\in {\mathcal{F}}_T$, $$\inf_{(x,x^*)\in X\times X^*} h_{(x_0,x_0^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+
\frac{1}{2}
{ {\|x^*\|}^2 }=0,\qquad \forall (x_0,x_0^*)\in X\times X^*.$$ Take $g\in {\mathcal{F}}_T$, and $(x_0,x_0^*)\in X\times X^*$. First observe that, for any $(x,x^*)\in X\times X^*$, $
g_{(x_0,x_0^*)}(x,x^*)\geq{\langle{x},{x^*}\rangle}$ and $$g_{(x_0,x_0^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+
\frac{1}{2}{ {\|x^*\|}^2 }\geq
{\langle{x},{x^*}\rangle}+\frac{1}{2}{ {\|x\|}^2 }+
\frac{1}{2}{ {\|x^*\|}^2 }\geq 0.$$ Therefore, $$\label{eq:aux0}
\inf_{(x,x^*)\in X\times X^*} g_{(x_0,x_0^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+
\frac{1}{2}
{ {\|x^*\|}^2 }\geq 0.$$ As the square of the norm is coercive, there exist $M>0$ such that $$\left\{ (x,x^*)\in X\times X^*\;|\; h_{(x_0,x_0^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }
+ \frac{1}{2}{ {\|x^*\|}^2 }<1
\right\}\subset
B_{X\times X^*}(0,M),$$ where $$B_{X\times X^*}(0,M)=\left\{ (x,x^*)\in X\times X^*\;|\;
\sqrt{{ {\|x\|}^2 }+{ {\|x^*\|}^2 }}<M\right\}.$$ For any $\varepsilon>0$, there exists $(\tilde x,\tilde x^*)$ such that $$\min\left\{1,\varepsilon^2\right\}>
h_{(x_0,x_0^*)}(\tilde x,\tilde x^*)+\frac{1}{2}{ {\|\tilde x\|}^2 }
+ \frac{1}{2}{ {\|\tilde x^*\|}^2 }.$$ Therefore $$\label{eq:aux1}
\begin{array}{l}
\varepsilon^2 >
h_{(x_0,x_0^*)}(\tilde x,\tilde x^*)+\frac{1}{2}{ {\|\tilde x\|}^2 }
+\frac{1}{2}{ {\|\tilde x^*\|}^2 }\geq
h_{(x_0,x_0^*)}(\tilde x,\tilde x^*)
-{\langle{\tilde x},{\tilde x^*}\rangle}\geq 0,\\[.5em]
M^2\geq { {\|\tilde x\|}^2 }+{ {\|\tilde x^*\|}^2 }.
\end{array}$$ In particular, $$\varepsilon^2 > h_{(x_0,x_0^*)}(\tilde x,\tilde x^*)
-{\langle{\tilde x},{\tilde x^*}\rangle}.$$ Using now the fact that operators type MA satisfies the restricted Brøndsted-Rockafellar property [@MASvJCA08 Theorem 3.4] we conclude that there exists $(\bar x,\bar x^*)$ such that $$\label{eq:aux2}
h_{(x_0,x_0^*)}(\bar x,\bar x^*)={\langle{\bar x},{\bar x^*}\rangle},\quad
{\|\tilde x-\bar x\|}<\varepsilon, \quad
{\|\tilde x^*-\bar x^*\|}<\varepsilon.$$ Therefore, $$h(\bar x+x_0,\bar x^*+x_0^*)-{\langle{\bar x+x_0},{\bar x^*+x_0^*}\rangle}=
h_{(x_0,x_0^*)}(\bar x,\bar x^*)-{\langle{\bar x},{\bar x^*}\rangle}=0,$$ and $(\bar x+x_0,\bar x^*+x_0^*)\in T$. As $g\in {\mathcal{F}}_T$, $$g(\bar x+x_0,\bar x^*+x_0^*)={\langle{\bar x+x_0},{\bar x^*+x_0^*}\rangle},$$ and $$\label{eq:aux3}
g_{(x_0,x_0^*)}(\bar x,\bar x^*)={\langle{\bar x},{\bar x^*}\rangle}.$$ Using the first line of we have $$\varepsilon^2 >
h_{(x_0,x_0^*)}(\tilde x,\tilde x^*)+
\bigg[\frac{1}{2}{ {\|\tilde x\|}^2 }
+\frac{1}{2}{ {\|\tilde x^*\|}^2 }+
{\langle{\tilde x},{\tilde x^*}\rangle}
\bigg]
-{\langle{\tilde x},{\tilde x^*}\rangle}\geq
\frac{1}{2}{ {\|\tilde x\|}^2 }
+\frac{1}{2}{ {\|\tilde x^*\|}^2 }+
{\langle{\tilde x},{\tilde x^*}\rangle}.$$ Therefore, $$\label{eq:aux4}
\varepsilon^2> \frac{1}{2}{ {\|\tilde x\|}^2 }
+\frac{1}{2}{ {\|\tilde x^*\|}^2 }+
{\langle{\tilde x},{\tilde x^*}\rangle}.$$ Direct use of gives $$\begin{aligned}
{\langle{\bar x},{\bar x^*}\rangle}&={\langle{\tilde x},{\tilde x^*}\rangle}
+{\langle{\bar x-\tilde x},{\tilde x^*}\rangle}
+{\langle{\tilde x},{\bar x^*-\tilde x^*}\rangle}
+{\langle{\bar x-\tilde x},{\bar x^*-\tilde x^*}\rangle}\\
&\leq {\langle{\tilde x},{\tilde x^*}\rangle}
+{\|\bar x-\tilde x\|}\,{\|\tilde x^*\|}
+{\|\tilde x\|}\,{\|\bar x^*-\tilde x^*\|}
+{\|\bar x-\tilde x\|}\,{\|\bar x^*-\tilde x^*\|}\\
&\leq {\langle{\tilde x},{\tilde x^*}\rangle}
+\varepsilon [ {\|\tilde x^*\|}+{\|\tilde x\|}]
+\varepsilon^2
\end{aligned}$$ and $$\begin{aligned}
{ {\|\bar x\|}^2 }+{ {\|\bar x^*\|}^2 }&\leq
\left( {\|\tilde x\|}+{\|\bar x-\tilde x\|}\right)^2
+ \left( {\|\tilde x^*\|}+{\|\bar x^*-\tilde x^*\|}\right)^2\\
&\leq { {\|\tilde x\|}^2 }+ { {\|\tilde x^*\|}^2 }
+2\varepsilon[{\|\tilde x\|}+{\|\tilde x^*\|}]+2\varepsilon^2
\end{aligned}$$ Combining the two above equations with we obtain
$$g_{(x_0,x_0^*)}(\bar x,\bar x^*)
+\frac{1}{2}{ {\|\bar x\|}^2 }
+\frac{1}{2}{ {\|\bar x^*\|}^2 }\leq
{\langle{\tilde x},{\tilde x^*}\rangle}
+\frac{1}{2}{ {\|\tilde x\|}^2 }
+\frac{1}{2}{ {\|\tilde x^*\|}^2 }+2\varepsilon[{\|\tilde x\|}+{\|\tilde x^*\|}]+2\varepsilon^2$$ Using now and the second line of we conclude that $$g_{(x_0,x_0^*)}(\bar x,\bar x^*)
+\frac{1}{2}{ {\|\bar x\|}^2 }
+\frac{1}{2}{ {\|\bar x^*\|}^2 }\leq
2\varepsilon\;M\sqrt{2}+3\varepsilon^2.$$ As $\varepsilon$ is an arbitrary strictly positive number, using also we conclude that $$\inf_{(x,x^*)\in X\times X^*} g_{(x_0,x_0^*)}(x,x^*)+\frac{1}{2}{ {\|x\|}^2 }+
\frac{1}{2}
{ {\|x^*\|}^2 }=0.$$ Altogether, we conclude that if item 3 holds then item 4 holds. The converse item 4$\Rightarrow$ item 3 is trivial to verify. Hence item 3 and item 4 are equivalent. As item 1 is equivalent to 3 and item 2 is equivalent to 4, we conclude that items 1,2,3 and 4 are equivalent.
Now we will deal with item 5. First suppose that item 2 holds. Since $\mathcal{S}_T\in {\mathcal{F}}_T$ $$(\mathcal{S}_T)^*\geq \pi_*.$$ As has already been observed, for any proper function $h$ it holds that $(\mbox{cl\,conv}\,h)^*=h^*$. Therefore $$(\mathcal{S}_T)^*=(\pi+\delta_T)^*\geq \pi_*,$$ that is $$\label{eq20}
\sup_{(y,y^*)\in T}
{\langle{y},{x^*}\rangle}+{\langle{y^*},{x^{**}}\rangle}-{\langle{y},{y^*}\rangle}\geq
{\langle{x^*},{x^{**}}\rangle}, \forall
(x^*,x^{**})\in X^*\times X^{**}$$ After some algebraic manipulations we conclude that is equivalent to $$\inf_{(y,y^*)\in T}{\langle{x^{**}-y},{x^*-y^*}\rangle}\leq 0,\qquad \forall
(x^*,x^{**})\in X^*\times X^{**},$$ that is, $T$ is type (NI). If item 5 holds, by the same reasoning we conclude that holds and therefore $(\mathcal{S}_T)^*\geq \pi_*$. As $\mathcal{S}_T\in {\mathcal{F}}_T$, we conclude that item 5 $\Rightarrow$ item 1. As has been proved previously item 1 $\Rightarrow$ item 2.
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[^1]: IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil ([[email protected]]{})
[^2]: Partially supported by Brazilian CNPq scholarship 140525/2005-0.
[^3]: IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil ([[email protected]]{})
[^4]: Partially supported by CNPq grants 300755/2005-8, 475647/2006-8 and by PRONEX-Optimization
[^5]: <http://www.polyu.edu.hk/~ama/events/conference/EriceItaly-OCA2001/Abstract.html>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
An accurate analytic approximation of the transfer function for the power spectra of primordial density perturbations in mixed dark matter models is presented. The fitting formula in a matter-dominated Universe ($\Omega_0=\Omega_M=1$) is a function of wavenumber $k$, redshift $z$ and four cosmological parameters: the density of massive neutrinos, $\Omega_{\nu}$, the number of massive neutrino species, $N_{\nu}$, the baryon density, $\Omega_{b}$ and the dimensionless Hubble constant, $h$. Our formula is accurate in a broad range of parameters: $k\le 100\;h/Mpc$, $z\le 30$, $\Omega_{\nu}\le 0.5$, $N_{\nu}\le 3$, $\Omega_{b}\le 0.3$, $0.3\le h\le 0.7$. The deviation of the variance of density fluctuations calculated with our formula from numerical results obtained with CMBfast is less than $6\%$ for the entire range of parameters. It increases with $\Omega_bh^{2}$ and is less than $\le 3\%$ for $\Omega_bh^{2}\le 0.05$.
The performance of the analytic approximation of MDM power spectra proposed here is compared with other approximations found in the literature ([@hol89; @pog95; @ma96; @eh3]). Our approximation turns out to be closest to numerical results in the parameter space considered here.
author:
- 'B. Novosyadlyj , R. Durrer , V.N. Lukash'
date: 'Received …; accepted …'
title: An analytic approximation of MDM power spectra in four dimensional parameter space
---
epsf.tex =-1cm
Introduction
============
Finding a viable model for the formation of large scale structure (LSS) is an important problem in cosmology. Models with a minimal number of free parameters, such as standard cold dark matter (sCDM) or standard cold plus hot, mixed dark matter (sMDM) only marginally match observational data. Better agreement between predictions and observational data can be achieved in models with a larger numbers of parameters (CDM or MDM with baryons, tilt of primordial power spectrum, 3D curvature, cosmological constant, see, [*e.g.*]{}, [@vkn98] and refs. therein). In view of the growing amount of observational data, we seriously have to discuss the precise quantitative differences between theory and observations for the whole class of available models by varying all the input parameters such as the tilt of primordial spectrum, $n$, the density of cold dark matter, $\Omega_{CDM}$, hot dark matter, $\Omega_{\nu}$, and baryons, $\Omega_b$, the vacuum energy or cosmological constant, $\Omega_{\Lambda}$, and the Hubble parameter $h$ ($h=H_0/100\;km/s/Mpc$), to find the values which agree best with observations of large scale structure (or even to exclude the whole family of models.).
Publicly available fast codes to calculate the transfer function and power spectrum of fluctuations in the cosmic microwave background (CMB) ([@sz96], CMBfast) are an essential ingredient in this process. But even CMBfast is too bulky and too slow for an effective search of cosmological parameters by means of a $\chi^2$-minimization, like that of Marquardt (see [@nr92]). To solve this problem, analytic approximations of the transfer function are of great value. Recently, such an approximation has been proposed by [@eh3] (this reference is denoted by EH2 in the sequel). Previously, approximations by [@hol89; @pog95; @ma96] have been used.
Holtzman’s approximation is very accurate but it is an approximation for fixed cosmological parameters. Therefore it can not be useful for the purpose mentioned above. The analytic approximation by [@pog95] is valid in the 2-dimensional parameter space $(\Omega_{\nu},
h)$, and $z$ (the redshift). It has the correct asymptotic behavior at small and large $k$, but the systematic error of the transfer function $T(k)$ is relatively large (10%-15%) in the important range of scales $0.3\le k\le 10\;h/$Mpc. This error, however introduces discrepancies of 4% to 10% in $\sigma_R$ which represents an integral over $k$. Ma’s analytic approximation is slightly more accurate in this range, but has an incorrect asymptotic behavior at large $k$, hence it cannot be used for the analysis of the formation of small scale objects (QSO, damped $Ly_\alpha$ systems, $Ly_\alpha$ clouds etc.).
Another weak point of these analytic approximations is their lack of dependence on the baryon density. Sugiyama’s correction of the CDM transfer function in the presence of baryons ([@bar86; @sug95]) works well only for low baryonic content. Recent data on the high-redshift deuterium abundance ([@tyt96]), on clustering at $100$Mpc$/h$ ([@eh4]) and new theoretical interpretations of the $Ly_\alpha$ forest ([@wei97]) suggest that $\Omega_{b}$ may be higher than the standard nucleosynthesis value. Therefore pure CDM and MDM models have to be modified. (Instead of raising $\Omega_b$, one can also look for other solutions, like, [*e.g.*]{} a cosmological constant, see below.)
For CDM this has been achieved by Eisenstein $\&$ Hu (1996, 1997a[^1]) using an analytical approach for the description of small scale cosmological perturbations in the photon-baryon-CDM system. Their analytic approximation for the matter transfer function in 2-dimensional parameter space ($\Omega_{M}h^2$, $\Omega_b/\Omega_{M}$) reproduces acoustic oscillations, and is quite accurate for $z<30$ (the residuals are smaller than 5%) in the range $0.025\le \Omega_{M}h^{2}\le 0.25$, $0\le \Omega_{b}/\Omega_{M}\le 0.5$, where $\Omega_M$ is the matter density parameter.
In EH2 an analytic approximation of the matter transfer function for MDM models is proposed for a wide range of parameters ($0.06\le \Omega_{M}h^{2}\le 0.4$, $\Omega_b/\Omega_{M}\le 0.3$, $\Omega_{\nu}/\Omega_{M}\le 0.3$ and $z\le 30$). It is more accurate than previous approximations by [@pog95; @ma96] but not as precise as the one for the CDM+baryon model. The baryon oscillations are mimicked by a smooth function, therefore the approximation looses accuracy in the important range $0.03\le k\le 0.5~h/$Mpc. For the parameter choice $\Omega_{M}=1$, $\Omega_{\nu}=0.2$, $\Omega_b=0.12$, $h=0.5$, [*e.g.*]{}, the systematic residuals are about 6% on these scales. For higher $\Omega_{\nu}$ and $\Omega_{b}$ they become even larger.
For models with cosmological constant, the motivation to go to high values for ${\Omega}_\nu$ and ${\Omega}_b$ is lost, and the parameter space investigated in EH2 is sufficient. Models without cosmological constant, however, tend to require relatively high baryon or HDM content. In this paper, our goal is thus to construct a very precise analytic approximation for the redshift dependent transfer function in the 4-dimensional space of spatially flat matter dominated MDM models, $T_{MDM}(k;\Omega_{\nu},N_{\nu},\Omega_{b},h;z)$, which is valid for $\Omega_M =1$ and allows for high values of ${\Omega}_\nu$ and ${\Omega}_b$. In order to keep the baryonic features, we will use the EH1 transfer function for the cold particles+baryon system, $T_{CDM+b}(k;\Omega_{b},h)$, and then correct it for the presence of HDM by a function $D(k;\Omega_{\nu},N_{\nu},\Omega_{b},h;z)$, making use of the exact asymptotic solutions. The resulting MDM transfer function is the product $T_{MDM}(k)=T_{CDM+b}(k)D(k)$.
To compare our approximation with the numerical result, we use the publicly available code ’CMBfast’ by Seljak $\&$ Zaldarriaga 1996.
The paper is organized as follows: In Section 2 a short description of the physical parameters which affect the shape of the MDM transfer function is given. In Section 3 we derive the analytic approximation for the function $D(k)$. The precision of our approximation for $T_{MDM}(k)$, the parameter range where it is applicable, and a comparison with the other results are discussed in Sections 4 and 5. In Section 6 we present our conclusions.
Physical scales which determine the form of MDM transfer function
=================================================================
We assume the usual cosmological paradigm: scalar primordial density perturbations which are generated in the early Universe, evolve in a multicomponent medium of relativistic (photons and massless neutrinos) and non-relativistic (baryons, massive neutrinos and CDM) particles. Non-relativistic matter dominates the density today, $\Omega_{M}=\Omega_b+\Omega_{\nu}+\Omega_{CDM}$. This model is usually called ’mixed dark matter’ (MDM). The total energy density may also include a vacuum energy, so that $\Omega_0=\Omega_M+\Omega_\Lambda$. However, for reasons mentioned in the introduction, here we investigate the case of a matter-dominated flat Universe with $\Omega_M=1$ and $\Omega_\Lambda=0$. Even though $\Omega_\Lambda\neq 0 $ seems to be favored by some of the present data, our main point, allowing for high values of $\Omega_b$, is not important in this case and the approximations by EH2 can be used.
Models with hot dark matter or MDM have been described in the literature by [@fan84; @sha84; @vb85; @hol89; @luk91], Davis, Summers $\&$ Schlegel 1992, [@sch92; @van92], Pogosyan $\&$ Starobinsky 1993, 1995, [@nov94], Ma $\&$ Bertschinger 1994, 1995, [@sz96], EH2, [@vkn98] and refs. therein. Below, we simply present the physical parameters which determine the shape of the MDM transfer function and which will be used explicitly in the approximation which we derive here[^2].
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Since cosmological perturbations cannot grow significantly in a radiation dominated universe, an important parameter is the time of equality between the densities of matter and radiation $$z_{eq}=\frac{2.4\times 10^4}{1-N_{\nu}/7.4}h^{2}t_{\gamma}^{-4}-1,\eqno(1)$$ where $t_{\gamma}\equiv T_{\gamma}/2.726 K$ is the CMB temperature today, $N_{\nu}$=1, 2 or 3 is the number of species of massive neutrinos with equal mass (the number of massless neutrino species is then $3-N_{\nu}$). The scale of the particle horizon at this epoch, $$k_{eq}=4.7\times 10^{-4}\sqrt{1+z_{eq}}\;h/Mpc,\eqno(2)$$ is imprinted in the matter transfer function: perturbations on smaller scales ($k>k_{eq}$) can only start growing after $z_{eq}$, while those on larger scales ($k<k_{eq}$) keep growing at any time. This leads to the suppression of the transfer function at $k>k_{eq}$. After $z_{eq}$ the fluctuations in the CDM component are gravitationally unstable on all scales. The scale $k_{eq}$ is thus the single physical parameter which determines the form of the CDM transfer function.
The transfer function for HDM ($\nu$) is more complicated because two more physical scales enter the problem. The time and horizon scale when neutrino become non-relativistic ($m_\nu\simeq 3T_\nu$) are given by $$z_{nr}=x_{\nu}(1+z_{eq})-1\;\;,$$ $$k_{nr}=3.3\times 10^{-4}\sqrt{x_{\nu}(1+x_{\nu})(1+z_{eq}})\;h/Mpc,\eqno(3)$$ where $x_{\nu}\equiv \Omega_{\nu}/\Omega_{\nu\;eq}\;$, $\;\;\Omega_{\nu\;eq}\simeq N_\nu /(7.4-N_\nu )$ is the density parameter for a neutrino component becoming non-relativistic just at $z_{eq}$. The neutrino mass can be expressed in terms of ${\Omega}_\nu$ and $N_\nu$ as ([@pb93]) $m_{\nu}=94\Omega_{\nu}h^{2}N_{\nu}^{-1}t_{\gamma}^{-3}\;$eV.
The neutrino free-streaming (or Jeans[^3]) scale at $z\le z_{nr}$ is $$k_{F}(z)\simeq 59\sqrt{{1\over 1+z_{eq}}+{1\over 1+z}}\;
\Omega_{\nu}N_{\nu}^{-1}t_{\gamma}^{-4}\;h^3/Mpc,\eqno(4)$$ which corresponds to the distance a neutrino travels in one Hubble time, with the characteristic velocity $v_{\nu}\simeq {1\over x_{\nu}}{1+z\over 1+z_{eq}}.$ Obviously, $k_F\ge k_{nr}$, and $k_{nr}{}^{>}_{<}k_{eq}$ for $\Omega_{\nu}{}^{>}_{<}{\Omega}_{\nu\;eq}\;$.
The amplitude of $\nu$-density perturbation on small scales ($k>k_{nr}$) is reduced in comparison with large scales ($k<k_{nr}$). For scales larger than the free-streaming scale ($k<k_F$) the amplitude of density perturbations grows in all components like $(1+z)^{-1}$ after $z_{eq}$. Perturbations on scales below the free-streaming scale ($k>k_F$) are suppressed by free streaming which is imprinted in the transfer function of HDM. Thus the latter should be parameterized by two ratios: $k/k_{nr}$ and $k/k_F$.
The transfer function of the baryon component is determined by the sound horizon and the Silk damping scale at the time of recombination (for details see EH1).
In reality the transfer function of each component is more complicated due to interactions between them. At late time ($z<20$), the baryonic transfer function is practically equal to the one of CDM, for models with $\Omega_{b}<\Omega_{CDM}$ (see Figs. 1,2). After $z_{eq}$, the free-streaming scale decreases with time (neutrino momenta decay with the expansion of the Universe whereas the Hubble time grows only as the square root of the scale factor, see Eq. (4)), and neutrino density perturbations at smaller and smaller scales become gravitationally unstable and cluster with the CDM+baryon component. Today the $\nu$ free-streaming scale may lie in the range of galaxy to clusters scales depending on the $\nu$ mass. On smaller scales the growing mode of perturbation is concentrated in the CDM and baryon components. Matter density perturbation on these scales grow like $\sim t^{\alpha}$, where $\alpha=(\sqrt{25-24\Omega_{\nu}}-1)/6$ ([@dor80]).
An analytic approximation for the MDM transfer function
=======================================================
To construct the MDM transfer function we use the analytic approximation of EH1 for the transfer function of cold particles plus baryons and correct it for the presence of a $\nu$-component like [@pog95] and [@ma96]: $$T_{MDM}(k)=T_{CDM+b}(k)D(k)~.\eqno(5)$$ The function $D(k)$ must have the following asymptotics: $$D(k\ll k_{nr})=1,$$ $$D(k\gg k_F)=(1-\Omega_{\nu})\left({1+z\over
1+z_{eq}}\right)^{\beta},$$ where $\beta=1-1.5\alpha$. After some numerical experimentation we arrive at the following ansatz which satisfies these asymptotics $$D(k)=\left[{1+(1-\Omega_{\nu})^{1/\beta}{1+z\over 1+z_{eq}}
({k_F\over k_{nr}})^{3} \Sigma _{i=1}^{3}
\alpha_{i}\left({k\over k_F}\right)^i
\over 1+(\alpha_{4}k/k_{nr})^{3}}\right]^{\beta}.\eqno(6)$$
We minimize the residuals in intermediate region ($k_{nr}<k<k_F$) by determining $\alpha_{i}$ as best fit coefficients by comparison with the numerical results.
By $\chi^2$ minimization ([@nr92]) we first determine the dependence of the coefficients $\alpha_{i}$ on $\Omega_{\nu}$ keeping all other parameters fixed, to obtain an analytic approximation $\alpha_{i}(\Omega_{\nu},z)$. The main dependence of $T_{MDM}(k)$ on $\Omega_{b},\;N_{\nu},\;h$ and $z$ is taken care of by the dependence of $T_{CDM+b}$, $k_{nr}$, $k_F$ and of the asymptotic solution on these parameters. We then correct $\alpha_{i}$ by minimization of the residuals to include the slight dependence on these parameters.
Finally, the correction coefficients have the following form: $$\alpha_{i}=a_{i}A_{i}(z)B_{i}(\Omega_b)C_{i}(h)D_{i}(N_{\nu}),$$ where $a_{i}=a_{i}(\Omega_{\nu})$, $A_{i}(0)=B_{i}(0.06)=C_{i}(0.5)=D_{i}(1)=1$. The functions $A_{i}$ depend also on $\Omega_{\nu}$.
For all our calculations we assume a CMB temperature of $T_{\gamma}=2.726K$ ([@mat94; @kog96]).
Dependence on $\Omega_{\nu}$ and $z$.
-------------------------------------
We first set $h=0.5$, $\Omega_{b}=0.06$, $N_{\nu}=1$ and determine $\alpha_{i}$ for $\Omega_{\nu}=0.1,\;0.2,\;0.3,\;0.4,\;0.5$ and $z=0,\;10,\;20$. We then approximate $D(k)$ by setting $\alpha_{i}=a_{i}A_{i}(z)$, where $A_i(z)=(1+z)^{b_i+c_i(1+z)}$. The dependences of $a_i$, $b_i$ and $c_i$ on $\Omega_{\nu}$, as well as $B_{i}(\Omega_b)$, $C_{i}(h)$ and $D_{i}(N_{\nu})$ are given in the Appendix. The functions $D(k)$ for different $\Omega_{\nu}$ and its fractional residuals are shown in Figs. 3 and 4.
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We now analyze the accuracy of our analytic approximation for the MDM transfer function $T_{MDM}(k)=T_{CDM+b}D(k)$ which in addition to the errors in $D(k)$ contains also those of $T_{CDM+b}(k)$ (EH1). We define the fractional residuals for $T_{MDM}(k)$ by $(T(k)-T_{CMBfast}(k))/T_{CMBfast}(k)$. In Fig. 5 the numerical result for $T_{MDM}(k)$ (thick solid lines) and the analytic approximations (dotted thin lines) are shown for different $\Omega_{\nu}$. The fractional residuals for the latter are given in Fig. 6. Our analytic approximation of $T_{MDM}(k)$ is sufficiently accurate for a wide range of redshifts (see Fig.7). For $z\le 30$ the fractional residuals do not change by more than 2% and stay small.
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Dependence on $\Omega_{b}$ and $h$.
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We now vary $\Omega_{b}$ fixing different values of $\Omega_{\nu}$ and setting the other parameters $h=0.5$, $N_{\nu}=1$. We analyze the ratio $D(k;\Omega_{\nu},\Omega_{b})/D(k;\Omega_{\nu},\Omega_{b}=0.06)$. Since the dominant dependence of $T_{MDM}(k)$ on $\Omega_{b}$ is already taken care of in $T_{CDM+b}(k)$, $D(k)$ is only slightly corrected for this parameter. Correction factors $B_i(\Omega_b)$ ($\sim 1$) as a second order polynomial dependence on $\Omega_b/0.06$ with best-fit coefficients are presented in the Appendix. The fractional residuals of $T_{MDM}(k)$ for different $\Omega_{b}$ are shown in Fig. 8.
The maximum of the residuals grows for higher baryon fractions. This is due to the acoustic oscillations which become more prominent and their analytic modeling in MDM models is more complicated.
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A similar situation occurs also for the dependence of $T_{MDM}(k)$ on $h$. Since the $h$-dependence is included properly in $k_F$ and $k_{nr}$, $D(k)$ does not require any correction in the asymptotic regimes. Only a tiny correction of $D(k)$ in the intermediate range, $k$ ($0.01<k<1$) is necessary to minimize the residuals. By numerical experiments we find that this can be achieved by multiplying $\alpha_{1},...\alpha_{4}$ by the factors $C_i(h)$ which are approximated by second order polynomial on $h/0.5$ with coefficients determined by $\chi^2$ minimization (see Appendix). The fractional residuals of $D(k)$ for different $h$ are shown in Fig. 9 (top panel), they remain stable in the range $0.3\le h\le 0.7$. But the fractional residuals of $T_{MDM}(k)$ slightly grow (about 2-3%, bottom Fig. 9) in the range $0.1\le k \le 1$ when $h$ grows from 0.3 to 0.7. This is caused by the fractional residuals of $T_{CDM+b}(k)$ (see middle panel).
Dependence on $N_{\nu}$.
------------------------
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The dependence of $D(k)$ on the number of massive neutrino species, $N_{\nu}$, is taken into account in our analytic approximation by the corresponding dependence of the physical parameters $k_{nr}$ and $k_F$ (see Eq.(6)). It has the correct asymptotic behaviour on small and large scales but rather large residuals in the intermediate region $0.01<k<10$. Therefore, the coefficients $\alpha_{i}$ ($i=1,...,4$) must be corrected for $N_{\nu}$. To achieve this, we multiply each $\alpha_{i}$ by a factor $D_{i}(N_{\nu})$ ($\sim 1$) which we determine by $\chi^2$ minimization. These factors depend on $N_{\nu}$ as second order polynomials. They are given in the Appendix. In Fig. 10 we show the fractional residuals of $T_{MDM}(k)$ for different numbers of massive neutrino species, $N_{\nu}$, and several values of the parameters $\Omega_{\nu}$, $\Omega_b$, $h$ and $z$. The performance for $N_{\nu}=2,3$ is approximately the same as for $N_{\nu}=1$.
Performance
===========
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The analytic approximation of $D(k)$ proposed here has maximal fractional residuals of less than $ 5\%$ in the range $0.01\le k \le 1$. It is oscillating around the exact numerical result (see Fig. 4), which essentially reduces the fractional residuals of integral quantities like $\sigma(R)$. Indeed, the mean square density perturbation smoothed by a top-hat filter of radius $R$ $$\sigma^{2}(R)={1\over 2\pi^{2}}\int_{0}^{\infty}k^{2}P(k)W^{2}(kR)dk,$$ where $W(x)=3(\sin x-x \cos x)/x^3$, $P(k)=AkT_{MDM}^{2}(k)$ (Fig.11) has fractional residuals which are only about half the residuals of the transfer function (Fig.12). To normalize the power spectrum to the 4-year COBE data we have used the fitting formula by [@bun97].
The accuracy of $\sigma(R)$ obtained by our analytic approximation is better than $2\%$ for a wide range of $\Omega_{\nu}$ for $\Omega_{b}=0.06$ and $h=0.5$. Increasing $\Omega_{b}$ slightly degrades the approximation for $N_{\nu}> 1$, but even for a baryon content as high as $\Omega_{b} \sim 0.2$, the fractional residuals of $\sigma(R)$ do not exceed $5\%$. Changing $h$ in the range $0.3-0.7$ and $N_{\nu}=1-3$ do also not reduce the accuracy of $\sigma(R)$ beyond this limit.
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We now evaluate the quality and fitness of our approximation in the four dimensional space of parameters. We see in Fig.12 that the largest errors of our approximation for $\sigma(R)$ come from scales of $\sim 5-10h^{-1}Mpc$[^4]. Since these scales are used for the evaluation of the density perturbation amplitude on galaxy cluster scale, it is important to know how accurately we reproduce them. The quantity $\sigma_{8}\equiv
\sigma(8h^{-1}Mpc)$ is actually the most often used value to test models. We calculate it for the set of parameters $0.05\le \Omega_{\nu}\le 0.5$, $0.06\le \Omega_{b} \le 0.3$, $0.3\le h \le 0.7$ and $N_{\nu}=1,\;2,\;3$ by means of our analytic approximation and numerically. The relative deviations of $\sigma_{8}$ calculated with our $T_{MDM}(k)$ from the numerical results are shown in Fig.13-15.
As one can see from Fig. 13, for $0.3\le h\le 0.7$ and $\Omega_{b}h^{2}\le 0.15$ the largest error in $\sigma_8$ for models with one sort of massive neutrinos $N_{\nu}=1$ does not exceed $4.5\%$ for $\Omega_{\nu}\le 0.5$. Thus, for values of $h$ which are followed by direct measurements of the Hubble constant, the range of $\Omega_{b}h^{2}$ where the analytic approximation is very accurate for $\Omega_{\nu}\le 0.5$ is six times as wide as the range given by nucleosynthesis constraints, ($\Omega_{b}h^{2}\le 0.024$, [@tyt96]). This is important if one wants to determine cosmological parameters by the minimization of the difference between the observed and predicted characteristics of the large scale structure of the Universe.
For models with more than one species of massive neutrinos of equal mass ($N_{\nu}=2,3$), the accuracy of our analytic approximation is slightly worse (Fig. 14, 15). But even for extremely high values of parameters $\Omega_{b}=0.3$, $h=0.7$, $N_{\nu}=3$ the error in $\sigma_{8}$ does not exceed $6\%$.
In redshift space the accuracy of our analytic approximation is stable and quite high for redshifts of up to $z= 30$.
Comparison with other analytic approximations
=============================================
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We now compare the accuracy of our analytic approximation with those cited in the introduction. For comparison with Fig. 12 the fractional residuals of $\sigma(R)$ calculated with the analytic approximation of $T_{MDM}(k)$ by EH2 are presented in Fig. 16. Their approximation is only slightly less accurate ($\sim 3\%$) at scales $\ge 10 Mpc/h$. In Fig. 17 the fractional residuals of the EH2 approximation of $T_{MDM}(k)$ are shown for the same cosmological parameters as in Fig. 16. For $\Omega_{\nu}=0.5$ (which is not shown) the deviation from the numerical result is $\ge 50\%$ at $k\ge 1h^{-1}$Mpc, and the EH2 approximation completely breaks down in this region of parameter space.
The analog to Fig. 13 ($\sigma_8$) for the fitting formula of EH2 is shown in Fig. 18 for different values of $\Omega_{b}$, $\Omega_{\nu}$ and $h$. Our analytic approximation of $T_{MDM}(k)$ is more accurate than EH2 in the range $0.3\le \Omega_{\nu}\le 0.5$ for all $\Omega_{b}$ ($\le 0.3$). For $\Omega_{\nu}\le 0.3$ the accuracies of $\sigma_{8}$ are comparable.
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To compare the accuracy of the analytic approximations for $T_{MDM}(k)$ given by [@hol89], Pogosyan $\&$ Starobinsky 1995, [@ma96], EH2 with the one presented here, we determine the transfer functions for the fixed set of parameters ($\Omega_{\nu}=0.3$, $\Omega_{b}=0.1$, $N_{\nu}=1$, $h=0.5$) for which all of them are reasonably accurate. Their deviations (in %) from the numerical transfer function are shown in Fig. 19. The deviation of the variance of density fluctuations for different smoothing scales from the numerical result is shown in Fig. 20. Clearly, our analytic approximation of $T_{MDM}(k)$ opens the possibility to determine the spectrum and its momenta more accurate in wider range of scales and parameters.
Conclusions
===========
We propose an analytic approximation for the linear power spectrum of density perturbations in MDM models based on a correction of the approximation by EH1 for CDM plus baryons. Our formula is more accurate than previous ones ([@pog95; @ma96], EH2) for matter dominated Universes ($\Omega_{M}=1$) in a wide range of parameters: $0\le \Omega_{\nu}\le 0.5$, $0\le \Omega_{b}\le 0.3$, $0.3\le h\le
0.7$ and $N_{\nu}\le 3$. For models with one, two or three flavors of massive neutrinos ($N_{\nu}=1,\;2,\;3$) it is significantly more accurate than the approximation by EH2 and has a relative error $\le 6\%$ in a wider range for $\Omega_{\nu}$ (see Figs. 13, 18).
The analytic formula given in this paper provides an essential tool for testing a broad class of MDM models by comparison with different observations like the galaxy power spectrum, cluster abundances and evolution, clustering properties of Ly-$\alpha$ lines etc. Results of such an analysis are presented elsewhere.
Our analytic approximation for $T_{MDM}(k)$ is available in the form of a FORTRAN code and can be requested at [**[email protected]**]{} or copied from [**http://mykonos.unige.ch/$\boldmath{\sim}$durrer/**]{}
[*Acknowledgments*]{} This work is part of a project supported by the Swiss National Science Foundation (grant NSF 7IP050163). B.N. is also grateful to DAAD for financial support (Ref. 325) and AIP for hospitality, where the bulk of the numerical calculations were performed. V.L. acknowledges a partial support of the Russian Foundation for Basic Research (96-02-16689a).
Bardeen, J.M., Bond, J.R., Kaiser, N., and Szalay, A.S. 1986, [ApJ]{}, 304, 15 Bond, J.R., & Szalay, A.S. 1983, [ApJ]{}, 274, 443 Bunn, E.F., & White, M. 1997, [ApJ]{}, 480, 6 Davis, M., Summers, F.J., & Schlegel, D. 1992, Nature, 359, 393 Doroshkevich, A.G., Zeldovich, Ya.B., Sunyaev, R.A. & Khlopov M.Yu. 1980, Sov.Astron. Lett., 6, 252 Eisenstein, D.J. & Hu, W. 1996, [ApJ]{}, 471, 542 Eisenstein, D.J. & Hu, W. 1997a, astro-ph/9709112 (EH1) Eisenstein, D.J. & Hu, W. 1997b, astro-ph/9710252 (EH2) Eisenstein, D.J., Hu, W., Silk, J., Szalay, A.S. 1997, astro-ph/9710303 Fang, L.Z., Xiang, S.P., & Li, S.X. 1984, [AA]{}140, 77 Holtzman, J.A. 1989, ApJSS, 71, 1 Kogut, A., et al. 1996, ApJ, 470, 653 Lukash, V.N. 1991, Annals New York Acad. of Sci., 647, 659 Ma, C.-P. 1996, [ApJ]{}, 471, 13 (astro-ph/9605198) Ma, C.-P., & Bertschinger, E. 1994, [ApJL]{}, 434, L5 Ma, C.-P. & Bertschinger, E. 1995, [ApJ]{}, 455, p.7 Mather, J.C., et al. 1994, ApJ, 420, 439 Novosyadlyj, B. 1994, Kinematics Phys. Celest. Bodies, 10, N1, 7 Peebles, P.J.E. 1993, Principles of Physical Cosmology (Princeton University Press) Pogosyan, D.Yu. & Starobinsky, A.A. 1993, [MNRAS]{}, 265, 507 Pogosyan, D.Yu. & Starobinsky, A.A. 1995, [ApJ]{}, 447, 465 Press W.H., Flannery B.P., Teukolsky S.A., Vettrling W.T. 1992, Numerical recipes in FORTRAN (New York: Cambridge University Press) Shafi, Q., & Stecker, F.W. 1984, Phys.Rew.Lett., 53, 1292 Schaefer, R.K., & Shafi, Q. 1992, Nature, 359, 199 Seljak, U. & Zaldarriaga, M. 1996, ApJ, 469, 437 (astro-ph/9603033) Sugiyama ApJS, 100, 281 (astro-ph/9412025). Tytler, D., Fan, X.M. & Burles, S. 1996, Nature, 381, 207 Valdarnini, R., & Bonometto, S.A. 1985, [AA]{}, 146, 235 Valdarnini, R., Kahniashvili T. & Novosyadlyj, B. 1998, A&A, 1998, 336, 11 (astro-ph/9804057) Van Dalen , A., & Schaefer, R.K. 1992 ,[ApJ]{}, 398, 33 Weinberg, D.H., Miralda-Escude, J., Hernquist, L. & Katz, N. 1997, astro-ph/9701012
[^1]: This reference is denoted by EH1 in this paper.
[^2]: Recall the definitions and relationship between the MDM and the partial transfer functions $$T_{MDM}=\Omega_{CDM}T_{CDM}+\Omega_\nu T_\nu +\Omega_b T_b\;,$$ $$T(k)\equiv {\delta (k,z)\over \delta (0,z)}
{\delta (0,z_{in})\over \delta (k,z_{in})}\;\;,$$ where $\delta (k,z)$ is the density perturbations in a given component and $z_{in}$ is a very high redshift at which all scales of interest are still super horizon.
[^3]: Formally the Jeans scale is 22.5$\%$ less than the free-streaming scale (Bond $\&$ Szalay 1983, Davis, Summers $\&$ Schlegel 1992), however, $k_F$ is the relevant physical parameter for collisionless neutrini.
[^4]: Actually, the oscillations in the error of $T(k)$ are somewhat misleading: they are mainly due to baryonic oscillations in the numerical $T(k)$ entering the denominator for the error estimate, so that a slight shift of the phase enhances the error artificially. This is why we concentrate on the error of $\sigma (R)$ (otherwise the error estimate of T(k) should be averaged, see [*e.g.*]{} EH2).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Solar thermal fuels (STFs) are an unconventional paradigm for solar energy conversion and storage which is attracting renewed attention. In this concept, a material absorbs sunlight and stores the energy chemically via an induced structural change, which can later be reversed to release the energy as heat. An example is the azobenzene molecule which has a [*cis*]{}-[*trans*]{} photoisomerization with these properties, and can be tuned by chemical substitution and attachment to templates such as carbon nanotubes, small molecules, or polymers. By analogy to the Shockley-Queisser limit for photovoltaics, we analyze the maximum attainable efficiency for STFs from fundamental thermodynamic considerations. Microscopic reversibility provides a bound on the quantum yield of photoisomerization due to fluorescence, regardless of details of photochemistry. We emphasize the importance of analyzing the free energy, not just enthalpy, of the metastable molecules, and find an efficiency limit for conversion to stored chemical energy equal to the Shockley-Queisser limit. STF candidates from a recent high-throughput search are analyzed in light of the efficiency limit.
[*Keywords*]{}: solar energy conversion, thermodynamics, photoisomerization
author:
- 'David A Strubbe [^1]'
- Jeffrey C Grossman
title: Thermodynamic limits to energy conversion in solar thermal fuels
---
Solar thermal fuels (STFs) are an unconventional paradigm for solar-energy harvesting and storage, which provides long-term storage as chemical energy and later release as heat. Unlike in photovoltaics (PV), incident solar photons are not converted to electricity but rather drive a reversible structural change in a material. Molecules that undergo a structural change on absorption of light (photoisomerization) are referred to as “photochromic,” since in general the optical absorption spectrum will change with the new structure. (STFs have also been referred to as “molecular solar thermal” (MOST) [@poulsen_EES].) Various classes of photochromic molecules are known, such as azobenzene, spiropyran/merocyanine, norbornadiene/quadricyclane, and fulvalene(tetracarbonyl)diruthenium [@Kucharski_review]. The basic concept was developed decades ago [@Bolton], but available molecular materials did not have adequate performance to enable applications, with regard to metrics such as cyclability, stored energy density, visible light absorption, and cost. Modern advances in nanoscience and atomistic computation and design have given new approaches and interest in this idea, as molecular and nanoscale templates and functionalization have produced increases in stored energy density and lifetime [@Kolpak_NanoLett; @Kolpak_JCP; @Kucharski_NatChem; @Liu; @Durgun; @Han; @Quant; @Feng], and performance of solution-based [@poulsen_EES; @Wang] and solid-state devices [@Zhitomirsky] have been demonstrated. While current devices deliver stored energy as heat, it may also be possible to use photo-induced mechanical motion [@Bardeen] to convert the stored energy to other forms [@Salzbrenner].
The question of the actual efficiency of STF devices, taking together all the relevant material properties, is a crucial one for assessing the relevance of STFs as an approach for solar-energy conversion, especially by comparison to the more established PV, solar thermal, and solar fuels technologies. However, the efficiency has remained unclear: it has been estimated experimentally in only a few cases [@poulsen_EES], and given only preliminary and somewhat limited theoretical analysis in the literature [@Bren; @Kucharski_NatChem; @Borjesson]. These works have focused primarily on enthalpy but not considered free energy or the key roles of chemical equilibrium, entropy, and temperature, and have relied on idealized or arbitrary parameters for simplicity. Other work has analyzed the photochemistry in detail but not overall device efficiency [@Bolton; @Almgren], or considered schemes more general than STFs [@Ross].
In the field of PV, the well-known work of Shockley and Queisser [@shockley] (hereafter, SQ) bridged the gap between analysis of the specific PV materials, and analysis of general heat engines, to find an efficiency limit for the single-junction PV scheme under sunlight, with constraints not from the properties of current materials but from rigorous thermodynamics. They found that the maximum efficiency attainable for a single-junction cell at 300 K in unconcentrated sunlight is 32%, achieved for a bandgap of 1.27 eV.
In this paper, we follow the SQ analysis to derive formulae for the efficiency of STFs and their limits from rigorous thermodynamic considerations. We underscore the detailed analogy to PV, including $I$-$V$ characteristics, despite the differing device operation; show the importance of the free energy; find a limit to the quantum yield of photoisomerization; and demonstrate the possibility of attaining the same limit as SQ for conversion of solar energy to stored chemical energy in an STF device. (By contrast, previous analyses showed significantly lower limits.) Previous to the SQ work, researchers had found PV efficiency limits based on empirical models, which could only demonstrate where the current approaches to silicon solar cells might lead, but could not show the potential of other ideas that had not yet been considered. Understanding the SQ limit suggested the benefit of new strategies for photovoltaics such as spectrum splitting, multi-junctions, intermediate bands, hot carriers, multiple exciton generation, singlet fission, etc. [@PolmanAtwater], or hybrid devices using conversion to heat as well as electricity [@Branz]. Similarly this analysis can inspire new paradigms for STFs – indeed, upconversion [@Borjesson_JMCA] and hybrid solar thermal devices [@Dreos] have already been examined in the context of STFs – and point the way to overcoming the limits we show here.
![\[fig:cf\_pv\] Comparison of basic processes in a band diagram for photovoltaics and a potential-energy surface for solar thermal fuels. *Photovoltaics*: (a) Photons with energy below the gap $E_g$ are not absorbed. (b) Photons with energy above the gap are absorbed. The resulting carriers thermalize to the band edge and then have energy $E_g$. (c) Radiative recombination of the excited carriers is a loss mechanism. *Solar thermal fuels*: (a) Photons with energy below the gap $E_g$ are not absorbed. (b) Photons with energy above the gap are absorbed by *trans*. The molecule relaxes to the lowest excited state at the *trans* geometry, relaxes on the potential-energy surface of that excited state, drops to the ground state, and further relaxes in the ground state to the *cis* geometry, storing an energy $\Delta H$. (c) Fluorescence from the excited state (quantum yield $< 1$) is a loss mechanism. (d) The reverse photoisomerization process – absorption by *cis* and conversion to *trans* – undoes the energy storage process and is another loss mechanism. ](compare_final.eps)
Photovoltaics Solar Thermal Fuels
----------------------------- -------------------------------
electrical power energy storage
current conversion rate
voltage chemical potential difference
short-circuit condition thermal equilibrium
open-circuit condition photostationary state
radiative recombination fluorescence
non-radiative recombination unproductive relaxation
: \[table:processes\] Comparison of parallel concepts between photovoltaics and solar thermal fuels. Key differences are the possibility of significant depletion of the ground state in STFs but not PV, the new concept of reverse photoisomerization in STFs, and the fact that the independent variable is [*cis*]{} fraction not the voltage.
We begin by reviewing the SQ analysis and showing the analogy between PV and STFs. The basic processes are diagrammed in Figure \[fig:cf\_pv\]. The SQ limit considers that each photon incident on the cell is not absorbed if it is below the band gap (“below-gap losses”); if it is above the band gap, it is absorbed, but the resulting electron and hole quickly relax to the band edges and provide only energy equal to the band gap (“above-gap losses”). These two loss mechanisms are the most important, and certainly apply to STFs. Consider the schematic potential-energy surfaces for azobenzene. Initially light must have energy of at least $E_g$ to be absorbed by [*trans*]{}, and then quickly loses any excess energy beyond that, as in a solar cell. However, after that further losses occur: the excitation relaxes on the excited-state surface to the minimum. De-excitation to the ground state causes a further loss, as does relaxation on the ground-state surface to [*cis*]{}, at an enthalpy $\Delta H$ above [*trans*]{}. We note that a distinction between absorption threshold and useful energy is in common with systems that relax to a dark state, such as an indirect gap in a semiconductor or a triplet molecular state.
The simple model above does not take into account two other important loss mechanisms considered by SQ: radiative recombination, and voltage loss. While non-radiative recombination might be reducible to zero, radiative recombination is absolutely required by detailed balance: if the cell can absorb, it can emit. Thermally excited electron-hole pairs, populated according to the Boltzmann distribution at 300 K, can recombine and emit photons. Moreover, the population is dependent on the voltage, thus defining the $I$-$V$ characteristics of the cell, as a maximum-power point has to be found between the extremes of open circuit with maximum voltage but no current, and short circuit with no voltage and maximum current. The voltage loss is the difference between the open-circuit voltage and the voltage at maximum power.
Since STFs are not electrical devices, these considerations may seem unrelated, but in fact the analogy with photovoltaics can be carried quite far. Corresponding concepts are compared in Table \[table:processes\]. To begin, consider the Gibbs free energy $G = H - TS$ of a solution of an STF molecule. For concreteness, we will refer to the stable isomer as [*trans*]{} and the higher-energy metastable isomer as [*cis*]{}, as for azobenzene, but the analysis is general. The Gibbs free energy is the relevant thermodynamic quantity for determining the heat released in a system at constant pressure and temperature [@McQuarrie_Simon], as in the STF discharge, and its sign determines whether a process is spontaneous or not. Previous STF works have analyzed only the enthalpy $H$, thus working in some sense in a $T \rightarrow 0$ limit.
Let the fraction of molecules which are in the [*cis*]{} isomer be $x$ and in the [*trans*]{} isomer be $1 - x$. (We assume a dilute solution to ensure “ideal solution” behavior; at high concentrations or with strong interactions between solute molecules, different equations than those below, with more parameters, may be required, such as the “regular solution” model [@McQuarrie_Simon].) Then thermal equilibrium in the dark will satisfy $$\begin{aligned}
\frac{x}{1 - x} = K = e^{- \Delta G^{0} / k T}\end{aligned}$$ where $K$ is the equilibrium constant, $\Delta G^{0}$ is the difference in Gibbs free energy per molecule between [*cis*]{} and [*trans*]{} under standard conditions, $k$ is the Boltzmann constant, and $T$ is the temperature of the solution. The Gibbs free energy will vary as a function of the ratio between [*cis*]{}/[*trans*]{} fractions $Q = x/(1-x)$, according to $$\begin{aligned}
\Delta G \left( Q \right) = \Delta G^{0} + k T {\rm\ ln \ } Q\end{aligned}$$ In equilibrium, $Q$ = $K$, and then $$\begin{aligned}
\Delta G \left( K \right) = \Delta G^{0} + k T {\rm\ ln \ } K = 0\end{aligned}$$ From this equation, we can observe that an STF solution in equilibrium irradiated with sunlight has initial energy storage rate of zero, since $\Delta G = 0$, even though the rate of conversion of molecules is maximum. This condition is thus analogous to the short-circuit condition for photovoltaics, since $\Delta G$ corresponds to voltage and conversion rate to current. As $Q$ increases due to the incident light, $\Delta G$ too will grow. This important effect was not considered in previous analyses [@Bren; @Kucharski_NatChem; @Borjesson]. We can integrate to find the total free energy stored, when cycling between two compositions $x_1$ and $x_2$: $$\begin{aligned}
\label{eq:storage}
\Delta G_{\rm tot} = \int_{x_1}^{x_2} \left[ \Delta G^{0} + k T {\rm\ ln \ } \frac{x}{1-x} \right] dx
= \left[ \Delta G^{0} x + kT x \ln x + kT \left(1 - x\right) \ln \left( 1 - x\right) \right]_{x_1}^{x_2}\end{aligned}$$ a familiar expression from entropy of mixing, depending on temperature and fraction $x$ as well as the intrinsic molecular quantity $\Delta G^{0}$.
The rate of conversion of molecules from [*cis*]{} to [*trans*]{}, given rate constants $k_{\rm c}$ and $k_{\rm t}$ under the given illumination conditions, is $$\begin{aligned}
\frac{dx}{dt} = \left( 1 - x \right) k_{\rm t} - x k_{\rm c}\end{aligned}$$ where $$\begin{aligned}
k_{\rm t} = \int I \left( \omega \right) \sigma_{\rm t} \left( \omega \right) \phi_{{\rm t} \rightarrow {\rm c}} \left( \omega \right) d\omega\end{aligned}$$ and similarly for [*cis*]{}. $I$ is the incident solar photon flux (photons per time per area), which we approximate as the blackbody spectrum at 6000 K, as in the SQ analysis. $\sigma_{\rm t}$ is the absorption cross-section, and $\phi_{{\rm t} \rightarrow {\rm c}} $ is the photoisomerization quantum yield from [*trans*]{} to [*cis*]{}. Thus the conversion rate declines over time as $1 - x$ falls and $x$ grows. Eventually a new equilibrium in the presence of the light is established, called the “photostationary state” [@Bandara] in which $dx/dt = 0$, in which case the ratio of fractions must be $$\begin{aligned}
Q_{\rm max} = \left. \frac{x}{1 - x} \right|_{\rm max} = \frac{k_{\rm t}}{k_{\rm c}}\end{aligned}$$ This ratio represents a maximum in the sense that continued irradiation will not result in further conversion of [*trans*]{} to [*cis*]{}. In fact, if the ratio were higher, incident light would actually promote a net conversion the other way, towards the photostationary state. This condition is analogous to open-circuit condition for photovoltaics, since $\Delta G$ is maximum but the conversion rate is zero. The composition of the photostationary state is key for the stored energy density, representing the maximum $x_2$ possible in Equation \[eq:storage\] and is an important target for STF design. The calculation of the constants $k_{\it c}$ and $k_{\it t}$ is complicated: while the absorption cross-section is straightforward, the quantum yield is difficult to measure experimentally, and challenging to obtain theoretically, involving calculation of non-adiabatic excited-state dynamics after light absorption [@Neukirch]. The quantum yield depends sensitively on solvent and excitation energy [@Bandara], and on functionalization, which may cause sensitization, quenching, or modification of potential-energy surfaces [@Ceroni; @Bren]. Adsorption on a metal surface [@Comstock; @Comstock2] or packed templating on carbon nanotubes [@Kucharski_NatChem] can dramatically reduce quantum yields, showing a key role of the environment.
However, we can put a simple limit on the photostationary state ratio, $Q_{\rm max}$, from energy conservation. Incident photons must have at least a threshold energy $E_g$ in order to be absorbed by ${\it trans}$. Therefore, $\Delta G$ cannot exceed this value: $$\begin{aligned}
E_{\rm g} \ge \Delta G \left( Q_{\rm max} \right) = \Delta G^{0} + k T {\rm\ ln \ } Q_{\rm max}\end{aligned}$$ The resulting constraint on the [*cis*]{} fraction in the photostationary state is $$\begin{aligned}
x \le \frac{1}{1+e^{-\left( E_{\rm g} - \Delta G^{0} \right) / k T}}\end{aligned}$$ The difference between $E_g$ and $\Delta H$ appears as a loss in the potential-energy surface, due to contributions including the barrier in the ground state $\Delta H^{\ddagger}$, and was considered as a fundamental constraint in the work of Börjesson *et al.*. However, considering an ensemble at finite temperature, this need not be the case. The population $x$ of products can build up, increasing their free energy, up to the limit just cited, $E_{\rm g} \ge \Delta G$, irrespective of $\Delta H$. Considering specifically the barrier height, we note that transition-state theory [@McQuarrie_Simon] for thermal reversion assumes that the molecules at the barrier are in thermal equilibrium with those in the metastable state, *i.e.* no free-energy difference between the top of the barrier and the product *cis* molecules. As a result, the barrier height $\Delta H^{\ddagger}$ does not necessarily imply any loss of free energy, and need not be considered in our efficiency analysis, although of course it is critical for the storage lifetime [@Kolpak_NanoLett].
We have identified conditions analogous to open circuit and short circuit in photovoltaics. We can continue with an analogy to the $I$-$V$ characteristics of photovoltaics. For STF, this plot is of conversion rate of molecules *vs.* free-energy difference, with the different points on the curve corresponding to different values of $x$. The actual rate of energy storage, like $P = I V$ in an electrical device, is $$\begin{aligned}
P_{\rm storage} = \frac{dx}{dt} \Delta G\end{aligned}$$ We can find the “maximum power point” ($x$ that maximizes $P_{\rm storage}$) by solving $dP_{\rm storage}/dx = 0$. The efficiency $\eta$ is given by $$\begin{aligned}
\eta \left( x \right) = \frac{P_{\rm storage}}{P_{\rm incident}} =
\frac{\left[ \left(1 - x\right)k_t - x k_c \right] \left[\Delta G^0 + kT \ln \frac{x}{1-x}\right]}
{A_{\rm mol} \int \hbar \omega I \left( \omega \right) d \omega}\end{aligned}$$ where $A_{\rm mol}$ is an effective molecular area (which will cancel out in the final result). This equation is not a limit but an actual efficiency (assuming only independent molecules in an ideal solution) which can be computed if the properties involved are known.
Now we will consider bounds on the rate constants $k_{t}$ and $k_{c}$, depending on the photoisomerization quantum yield. We can put a simple bound on the quantum yield via consideration of fluorescence from molecules in the excited state, which is analogous to radiative electron-hole recombination in PV. An excited molecule may relax (radiatively or non-radiatively) to the ground state at any point along its path from [*trans*]{} to [*cis*]{}; at some points this relaxation will produce [*trans*]{} and at others will produce [*cis*]{}. What we can say for certain is that the vibronic states reached by initial excitation from the [*trans*]{} ground state can fluoresce and relax back to a [*trans*]{} structure, and so this process sets an upper bound on the quantum yield.
Let $B_{t}$ be the rate constant for absorption by [*trans*]{}, the same as $k_{t}$ if the quantum yield were unity. $$\begin{aligned}
B_t = \int I \left( \omega \right) \sigma_{t} \left( \omega \right) d\omega\end{aligned}$$ Then the absorption rate is $B_t x_t$. According to SQ’s analysis and detailed balance, a similar quantity will govern radiative recombination back to [*trans*]{}, with the modifications:
1. the solar photon flux is replaced by the blackbody spectral intensity at room temperature ($T$ = 300 K), $$\begin{aligned}
I_{\rm bb} \left( \omega \right) = \frac{2 \omega^2}{\pi c^2} \frac{1}{e^{\hbar \omega/kT} - 1},\end{aligned}$$
2. there is an additional factor of 2 to account for the fact that the device can only absorb from one illuminated side but can radiate from both sides, and
3. the emission probability is given by $\sigma_t$ multiplied by the Boltzmann factor $e^{E_g / kT}$ (using the energy difference between the ground and excited states of [*trans*]{}), since the emission is proportional to the occupation of excited states, which are increased by this factor when the system is driven out of equilibrium under illumination.
This radiative recombination coefficient is $$\begin{aligned}
A_t = 2 e^{E_g / kT} \int I_{\rm bb} \left( \omega \right) \sigma_{t} \left( \omega \right) d\omega\end{aligned}$$ The emission rate then is $A_t x_s$, where $x_s$ is the fraction of molecules in the excited state.
An upper bound on the conversion rate to [*cis*]{} comes from taking this radiative recombination as the only process preventing an excited [*trans*]{} molecules from converting to [*cis*]{}: $$\begin{aligned}
\frac{dx}{dt} \le B_{t} \left( 1 - x \right) - A_{t} x_s - x k_{c}\end{aligned}$$ The first two terms represent the rate due to absorption by [*trans*]{}, $k_t x_t$. $$\begin{aligned}
k_t x_t \le B_t x_t - A_t x_s = \int \left( I \left( \omega \right) x_t - 2 I_{\rm bb} \left( \omega \right) e^{E_g / kT} x_s \right)
\sigma_{t} \left( \omega \right) d\omega
%
%\end{aligned}$$ Comparing to the expression for $k_t$, we find in fact a bound on the quantum yield of photoisomerization across the spectrum: $$\begin{aligned}
\phi_t \left( \omega \right) \le 1 - \frac{2 I_{\rm bb} \left( \omega \right)}{I \left( \omega \right)} e^{\Delta G_{st} / kT}\end{aligned}$$ This is expressed in terms of the free-energy difference between the ground and excited states of [*trans*]{}, which is not a quantity that is easily measured or controlled. Instead, we can use the inequality $\Delta G_{st} \ge \Delta G$, which is required for the excited state to be able to drive the structural change to [*cis*]{}. Then $$\begin{aligned}
\phi_t \left( \omega \right) \le 1 - \frac{2 I_{\rm bb} \left( \omega \right)}{I \left( \omega \right)} e^{\Delta G / kT}\end{aligned}$$ This quantum yield bound decreases as a function of conversion percentage (through $e^{\Delta G / kT} = Q e^{\Delta G^{0} / kT}$), and therefore makes a contribution to the $I-V$ characteristics of the STF. Moreover, we have shown that the quantum yield cannot reach unity even in principle, due to microscopic reversibility. This bound can be used in place of the simple assumption of $\phi_t = 1$ in previous efficiency analyses. Following the SQ approach, we can let the absorption probability for [*trans*]{} be 1 above the band gap and 0 below, which can be approached in practice by making the device thick enough so that all incoming light is absorbed. This is the maximum possible absorption, which will lead to the best efficiency, and implies a cross-section equal to $A_{\rm mol}$. $$\begin{aligned}
\sigma_t \left( \omega \right) = \left\{ \begin{array}{lr}
0 : \hbar \omega < E_g \\
A_{\rm mol} : \hbar \omega > E_g
\end{array}
\right.\end{aligned}$$ On the other hand, absorption by [*cis*]{} reduces the efficiency, and so we will take $\sigma_c \left( \omega \right) = 0$, the lowest possible absorption. (While quantum-mechanical sum rules require some absorption, it can be pushed arbitrarily far out of the solar spectrum to achieve a similar result.) This limit also sets the [*cis*]{} $\rightarrow$ [*trans*]{} reverse photoisomerization to zero, removing this loss from consideration. That could also happen via $\phi_{c \rightarrow t} = 0$, as for the “one-way” photoisomerizable molecules such as dihydroazulene/vinylheptafulvene which do not exhibit a reverse process [@Kucharski_review]. We have now a simplified model giving an upper bound to the efficiency, involving as parameters only $E_g$ and $\Delta G^{0}$, both of which can be straightforwardly measured and computed theoretically: $$\begin{aligned}
\eta \left( x \right) =
\frac{\left[ \left(1 - x\right) \int_{E_g} I \left( \omega \right) d\omega - 2 x e^{\Delta G^{0}/kT} \int_{E_g} I_{\rm bb} \left( \omega \right) d\omega \right] \left[\Delta G^0 + kT \ln \frac{x}{1-x}\right]}
{\int \hbar \omega I \left( \omega \right) d \omega}\end{aligned}$$ The numerical solutions of the conversion rate *vs.* free-energy difference, a curve analogous to $I$-$V$ characteristics, are shown in Figure \[fig:i\_v\_power\] for the case $E_g$ = 1.3 eV and various values of $\Delta G^{0}$. Changing $\Delta G^{0}$ has little effect on the conversion rate at fixed $\Delta G$, but it strongly affects the maximum $\Delta G$ attainable (at the photostationary state). This maximum $\Delta G$ increases with $\Delta G^{0}$ but saturates at 1.1 eV and the curves become indistinguishable beyond that. The power is also plotted as a function of $x$, which shows a slow rise and steep fall. The smallest $x$ where power generation occurs is the thermal population, which of course decreases with increasing $\Delta G^{0}$. The value of $x$ at which the maximum power is attained falls with increasing $\Delta G^0$, showing a trade-off between maximum rate of energy storage and the maximum amount of energy that can be stored (as in equation \[eq:storage\]).
![\[fig:i\_v\_power\] (a) Conversion rate vs free-energy difference, a curve analogous to $I$-$V$ characteristics, but traced out by varying the *cis* fraction. The lines for $\Delta G^0 = \infty$, $\Delta G^0 = $ 1.3 eV, and the Shockley-Queisser $I-V$ characteristic shape are indistinguishable. (b) The power being stored as a function of *cis* fraction. The legend for $\Delta G^0$ applies to both plots, and $E_g$ = 1.3 eV. ](fig2.eps)
In photovoltaics, power electronics can be used to vary the resistive load across the device in order to operate close to the maximum power point. In STF, we need to control the [*cis*]{} fraction to do the equivalent. For example, the rate at which the solution flows through a plate where it is exposed to the sun [@poulsen_EES] can be optimized (given the charging rate) in order to keep the solution near the maximum power point, if one wished to achieve the maximum energy storage rate. Of course, doing so would result in quite a small conversion percentage and thus not be the best choice for energy storage density. An alternate possibility is controlling [*cis*]{} fraction via differential solubility or density of the two isomers in a liquid phase. Consider the case $\Delta G^{0} \rightarrow \infty$. This is consistent with the requirement that free energy decreases, which stipulates only $\Delta G < E_g$ (as we used for the limit on the photostationary state). In this limit, for a given $\Delta G$, $x$ goes to 0, removing the loss of absorption due to depletion of the *trans* molecules, and remarkably reducing the efficiency equation to one equivalent to SQ (following the translation of concepts in Table \[table:processes\]): $$\begin{aligned}
\eta_{\rm \Delta G^{0} \rightarrow \infty} \left( \Delta G \right) =
\frac{\left[ \int_{E_g} I \left( \omega \right) d\omega - 2 e^{\Delta G/kT} \int_{E_g} I_{\rm bb} \left( \omega \right) d\omega \right] \Delta G }
{\int \hbar \omega I \left( \omega \right) d \omega}\end{aligned}$$ For lesser values of $\Delta G^{0}$, the efficiency is reduced due to [*trans*]{} depletion, but $\Delta G^{0} \ge E_g$ is sufficient to obtain almost the maximum efficiencies. These results are plotted in Figure \[fig:efficiency\], exhibiting the maximum of 32% at 1.27 eV for $\Delta G^{0} \rightarrow \infty$. Comparing to the experimentally estimated efficiency of 0.07% for the Ru-dithiafulvalene system [@poulsen_EES], it is clear there is the possibility of great improvement in STFs.
![\[fig:efficiency\] Efficiency limit in converting incident solar energy to stored chemical energy, as a function of band gap, at the optimal [*cis*]{} fraction $x$ for each gap. The ultimate limit is 32% at $E_g = 1.27$ eV, in the limit $\Delta G^0 \rightarrow \infty$ which is identical to the Shockley-Queisser limit. The curve for $\Delta G^0 = E_g$ is almost indistinguishable from these limits, but as $\Delta G^0$ is reduced, the maximum efficiency drops and moves to a higher value of $E_g$. The point at which efficiency rises above zero is approximately $\Delta G^{0}$. ](fig3.eps)
High-throughput screening of molecules for STF applications has already begun, using azobenzene derivatives [@Liu] and later norbornadiene/quadricyclane derivatives [@Kuisma]. These works have assessed their candidates only by considering parameters separately, or with regard to the older attempt at an efficiency limit [@Borjesson]. Our improved and more fundamental limit enables a more powerful screening without unnecessary assumptions. We reassess the azobenzene derivatives of [@Liu] in Figure \[fig:highthroughput\], using the estimated $E_g$ (as PBE Kohn-Sham gap + 0.9 eV, as in that work) and $\Delta G^{0}$ as the total energy difference (neglecting effects of vibrational entropy or volume changes). We observe that several molecules indeed come very close to the maximum efficiency limit. Many of the molecules have a large enough $\Delta G^0$ to reach the maximum efficiency for their $E_g$, but their potential efficiency is limited by having too large $E_g$. Therefore, smaller $E_g$ should be a design goal to find improved STF molecules.
![\[fig:highthroughput\] Efficiency limits for converting incident solar energy to stored chemical energy, for 62 candidate azobenzene derivative molecules identified in a high-throughput search using density-functional theory [@Liu]. The optical gap $E_g$ is the Kohn-Sham band gap + 0.9 eV, and $\Delta G^{0}$ is the total energy difference. The curve shows the limit $\Delta G^{0} \rightarrow \infty$. ](fig4.eps "fig:")\
We underscore numerous important new aspects in our approach to understanding STF efficiency limits. We do not assume a quantum yield $\phi = 1$ but in fact derived an upper bound for $\phi$ dependent on the extent of charging. We showed that the ground-state barrier to thermal reversion, $\Delta G^{\ddagger}$, does not inherently cause a loss and does not enter into our final limit. We demonstrated the critical importance of the free energy, not just the enthalpy, because of the effect of temperature and entropy of mixing. Most importantly, we found that thermodynamic considerations provide the same 32% efficiency limit, as same optimal band gap, as in the Shockley-Queisser analysis.
This analysis of the fundamental limits to STF efficiency helps to benchmark candidate materials and devices against potential performance, identify the weak points that are most important to improve, and focus thinking on applications by showing what is the best performance we can expect. Our results demonstrate that STFs may match the peak efficiencies of PV (despite the much lower performance of current STF devices), although further losses may occur in conversion of the stored chemical energy, into [*e.g.*]{} electricity. This level of performance is very promising for applications where heat will be used directly [@HEATS], and also where the storage feature is particularly valued. We believe this thermodynamic approach to STF efficiency could enable the development of new STF materials and paradigms, and provides new insights into photochemistry generally, for example by demonstrating the existence of limits to quantum yield regardless of details of reaction pathways.
acknowledgments
===============
We would like to acknowledge helpful discussions with Alexie Kolpak, Yun Liu, Tim Kucharski, and Jee Soo Yoo. This work was supported by the U. S. Advanced Research Projects Agency - Energy under Grant DE-AR0000180.
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[^1]: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the interface physics of bipartite magnetic materials deposited on a topological insulator. This comprises antiferromagnets as well as ferrimagnets and ferromagnets with multiple magnetic moments per unit cell. If an energy gap is induced in the Dirac states on the topological surface, a topological magnetoelectric effect has been predicted. Here, we show that this effect can act in opposite directions on the two components of the magnet in a certain parameter region. Consequently, an electric field will mainly generate a staggered field rather than a net magnetization in the plane. This is relevant for the current attempts to detect the magnetoelectric effect experimentally, as well as for possible applications. We take a field-theoretic approach that includes the quantum fluctuations of both the Dirac fermions on the topological surface as well as the fermions in the surface layer of the magnet in an analytically solvable model. The effective Lagrangian and the Landau-Lifshitz equation describing the interfacial magnetization dynamics are derived.'
author:
- Stefan Rex
- 'Flavio S. Nogueira'
- 'Asle Sudb[ø]{}'
title: 'Topological staggered field-electric effect with bipartite magnets'
---
Introduction
============
Since their discovery, topological insulators (TIs) [@HasanKane_RMP; @QiZhang_RMP] have attracted much attention due to their unique surface properties. In three-dimensional TIs, every surface exhibits linearly dispersing conducting states inside the bulk band gap. These can be described as Dirac fermions and exhibit spin-momentum locking. If time-reversal symmetry (TRS) at the surface is broken by an orthogonal net magnetization, the Dirac states become massive, i.e., a gap opens in their energy dispersion. It has been shown that this generates a Chern-Simons (CS) term in the effective field-theory which imposes a topological magnetoelectric (TME) effect [@QiHughesZhang_PRB; @EssinMooreVanderbilt] on the surface, where an electric field induces a net in-plane magnetization. This distinct response to an electromagnetic field is a hallmark of the TI phase.
Magnetic order on the TI surface can be established by doping with $3d$ transition metals [@Hor2010; @Chen659; @KouNanoLett; @KouACSNano; @OkadaPRL; @Chang2013; @LiPRL2015a], proximity coupling to a magnetic insulator in bilayer structures [@WeiPRL2013; @KandalaAPL2013; @YangPRB2013; @MooderaNature2016; @Bi2Se3-YIG], or a combination of both [@LiPRL2015b]. In [@MooderaNature2016], a magnetization orthogonal to the surface was realized even at room-temperature in EuS-Bi$_2$Se$_3$ bilayers. In theoretical works, a broad range of potential applications of such heterojunctions combining ferromagnetic insulators (FMIs) and TIs have been suggested, e.g., related to spintronics [@GarateFranz; @NomuraNagaosa; @YokoyamaZangNagaosa; @TserkovnyakLoss; @FerreirosCortijo; @LinderPRB2014; @FerreirosBuijinstersKatsnelson; @SemenovPRB2012; @SemenovPRB2014; @DuanPRB2015; @FM_trilayer], and several further implications of the TME effect have been discussed, including the formation of magnetic monopoles [@QiScience2009], and the interplay with long-range Coulomb interaction [@Flavio_PRL; @FM_bilayer; @FM_trilayer].
So far, not much focus has been directed at more general classes of magnetic materials. Mostly, it is assumed that the TME effect will occur in the same way as long as a net magnetization is present. However, several technologically relevant materials do not have a simple ferromagnetic (FM) structure, but are instead ferrimagnets (FiMs) or antiferromagnets (AFMs). For instance, one of the most prominent materials for spintronics devices is yttrium iron garnet (YIG), a FiM with a complicated crystal structure [@YIG_Cherepanov; @YIG_Bauer]. In YIG, an enhancement of the magnetization has been recently observed in a bilayer structure YIG-Bi$_2$Se$_3$, where Bi$_2$Se$_3$ is doped with Cr [@Bi2Se3-YIG]. It is thus natural to ask if and how the topological effects will manifest itself in multicomponent FMIs, FiMs or AFM insulators. In AFMs, there is no net magnetization (except in some cases for special surface orientations [@LuoPRB2013]). However, a gap can still be opened at the Dirac points, as in the FM and FiM cases, by means of magnetic doping in the TI. Such a system has recently been realized experimentally [@HKG16].
In the present paper, we study a bilayer heterostructure consisting of a bipartite magnetic insulator (BMI) and a TI. We show that, depending on the microscopic parameters of the BMI, the TME effect can take the opposite sign on the two sublattices, turning the overall electric-field response from a TME effect into a topological staggered-field (TSE) effect. Our calculation is to be understood as a proof of principle, as the model we employ is simplified and may not suffice to make quantitative predictions. On the other hand, we are able to obtain fully analytic solutions within a field-theoretic approach that accounts for the fermionic quantum fluctuations on both the BMI and the TI surfaces. We will derive the effective Lagrangian, revealing the structure of the magnetoelectric response, and the Landau-Lifshitz equation (LLE) of the interfacial magnetization dynamics. We work in Gaussian units and set $\hbar=1$. All calculations are done at zero-temperature. This is justified as long as the Fermi level is tuned to lie in the induced energy gap, for instance by gating of the interface.
The model we use is described in the following section. We discuss the non-topological fluctuation effects originating with the electrons on the BMI surface in section \[Sec:IntegrateChi\], before we move on to the topological effects that are revealed upon integrating out the Dirac states in section \[Sec:TSE\]. We summarize our results in section \[Sec:Conclusion\].
Model system {#SecModel}
============
Describing the heterostructure, one has to account for the contributions from the bulk of the BMI, the surfaces of the BMI and the TI, hopping across the interface due to proximity, and Coulomb interactions between the Dirac electrons at the interface. The bulk of the TI is required to guarantee the existence of the topological surface states, but does not appear explicitly. The model system is illustrated in Fig. \[Fig:Model\].
We start with the surface of the TI which is chosen to be the $(x,y)$-plane and described by the Dirac Lagrangian $$\mathcal{L}_\text{D} = \Psi^\dagger[i\partial_t-iv_F(\sigma_y\partial_x-\sigma_x\partial_y) + e(\varphi+\phi)]\Psi
\label{Dirac_1}
,$$ where $\Psi=[\psi_\uparrow~ \psi_\downarrow]^T$ are the surface Dirac fermions, $v_F$ is the Fermi velocity, $\varphi$ the fluctuating potential of Coulomb interactions among the Dirac fermions, and $\phi$ is any externally applied electric potential. A term quadratic in $\varphi$ describes the Coulomb interaction in the plane [@Flavio_PRL; @FM_bilayer; @FM_trilayer]: $$\mathcal{L}_\text{Cou}(\mathbf{r}) = -\frac{1}{8\pi^2}[{\boldsymbol{\nabla}}_\parallel\varphi(\mathbf{r})]\cdot\int d^2r^\prime\frac{{\boldsymbol{\nabla}}_\parallel^{\prime}\varphi(\mathbf{r}^\prime)}{|\mathbf{r}-\mathbf{r}^\prime|},$$ where ${\boldsymbol{\nabla}}_\parallel=(\partial_x,\partial_y)$ denotes the in-plane gradient operator.
We model the bulk bipartite magnetic material as two interpenetrating FMs (denoted by indices $i=1,2$) that are coupled by an exchange interaction, $\mathcal{L}_\text{bulk} = \mathcal{L}_1 + \mathcal{L}_2 + \mathcal{L}_\text{ex}$, where $$\label{Eq:L-Mag}
\mathcal{L}_i = -\textbf{b}({\mathbf{m}}_i)\cdot\partial_t{\mathbf{m}}_i - \frac{\kappa}{2}({\boldsymbol{\nabla}}{\mathbf{m}}_i)^2$$ and $$\mathcal{L}_\text{ex}(\mathbf{r}) = -\lambda{\mathbf{m}}_1(\mathbf{r})\cdot{\mathbf{m}}_2(\mathbf{r}).\label{Eq:Lexchange}$$ Here, $\mathbf{b}$ is the Berry connection, which satisfies ${\boldsymbol{\nabla}}_{{\mathbf{m}}_i}\times\mathbf{b}({\mathbf{m}}_i) = {\mathbf{m}}_i/m_i^2$, $\kappa>0$ is the FM exchange energy, and $\lambda>0$ ($<0$) for AFM (FM) coupling of the two components. In the bulk model, we ignore anisotropy terms. It will turn out that the system intrinsically contains anisotropy, and additional bulk contributions would not qualitatively alter the physics.
![(Color online) The model system: a) Bilayer heterostructure consisting of a bipartite magnetic insulator (BMI) deposited on a topological insulator (TI). b) By means of the parameter $\mu=\overline{m}_2/\overline{m}_1$, the magnet can be tuned to be in an antiferromagnetic (AFM), ferrimagnetic (FiM), or ferromagnetic (FM) configuration at mean-field. c) The model involves fermionic fields $\Psi$ and $\chi_{1,2}$ on the surfaces of both the TI (blue plane) and the BMI (grey plane), respectively, which are coupled by the amplitudes $h$ (hopping across the interface) and $t$ (local coupling of the two sublattices).[]{data-label="Fig:Model"}](model.png){width="\columnwidth"}
In order to describe the surface Berry phases associated to the two sublattices, we introduce fermionic fields $\chi_i=[\chi_{i\uparrow}~\chi_{i\downarrow}]^T$, $i=1,2$ representing sublattice indices, which when integrated out generate the desired surface Berry phases. This procedure to generate Berry phases is well known in the literature [@Stone-WZW-term; @Fujikawa-2005], and is very useful in our case because it permits coupling the underlying sublattice fermions to the Dirac surface states. The surface layer of the bipartite magnetic insulator is thus described by the Hamiltonian, $$\begin{aligned}
{\cal H}_{\rm surf}=-t(\chi_1^\dagger\chi_2+\chi_2^\dagger\chi_1)-J\sum_{i=1,2}{\bf m}_i\cdot\chi_i^\dagger{\boldsymbol{\sigma}}\chi_i
\label{Chi_Mag_Surface}
,\end{aligned}$$ where $J$ the strength of the exchange coupling to the respective magnetization ${\mathbf{m}}_i(z=0)$, ${\boldsymbol{\sigma}}$ are the Pauli matrices, and $t$ a paramater coupling the surface fermions of the BMI on different sublattices. It will be crucial in obtaining a TSE, and also leads to mixed Berry phase terms originating on the different sublattices. When $t=0$, the surface Berry phases decouple and just correspond to a shift of the Berry phases already present in Eq. (\[Eq:L-Mag\]). Note that the Lagrangian only accounts for coupling of fermions $\chi_1$ and $\chi_2$ within one unit cell, thus being momentum-independent in the continuum limit. Further electron dynamics (gradient terms) is neglected. This rough approximation is valid as long as the magnet is a strong insulator and the gap is much larger than the induced gap in the Dirac states. It does not spoil the generation of the surface Berry phases, however. Furthermore, the lattice model of the surface of the magnet does not explicitly include nearest-neighbor exchange interactions, which are already captured by the Lagrangian of the magnetic bulk. The chemical potential is set to zero for the electrons on both surfaces because the Fermi level is assumed to be tuned to lie in the gap.
If the surfaces of the TI and the AFM or FiM are in proximity to each other, there is also an amplitude $h$ that couples the surface fermions of the magnetic insulator to the surface fermions of the topological insulator, $$\mathcal{L}_\text{int} = h[\Psi^\dagger(\chi_1+\chi_2)+(\chi_1^\dagger+\chi_2^\dagger)\Psi].$$ Our calculation amounts to integrating out all fermionic fields in order to obtain an effective theory of the magnetization.
Quantum fluctuations of the sublattice fermions {#Sec:IntegrateChi}
===============================================
We start by integrating out the fermions $\chi_i$ of the BMI surface to obtain an effective model for the Dirac fermions $\Psi$. We assume that the mean-field direction of the magnetization is orthogonal to the interface, such that a mass in the Dirac states can be induced. We write ${\mathbf{m}}_i^\text{mf}={\overline{m}}_i{\hat{\mathbf{e}}}_z$ and define the dimensionless parameter $\mu={\overline{m}}_2/{\overline{m}}_1$, where without loss of generality $|\mu|\leq 1$. Then, $\mu>0$ describes a FM, $-1<\mu<0$ a FiM, and $\mu=-1$ an AFM (Fig. \[Fig:Model\]b). We also introduce $\tau=t^2/J^2{\overline{m}}_1^2$, which will be useful later. From Eq. \[Chi\_Mag\_Surface\], we define a matrix $$A = \begin{pmatrix}i\partial_t+ J{\mathbf{m}}_1\cdot{\boldsymbol{\sigma}}&t\\t&i\partial_t+ J{\mathbf{m}}_2\cdot{\boldsymbol{\sigma}}\end{pmatrix}$$ such that the action of the surface of the magnetic insulator is symbolically written as $\mathcal{S}_\text{surf} = \chi^\dagger A \chi$, where $\chi^\dagger=(\chi_1^\dagger,\chi_2^\dagger)$. The integral over spacetime is implicit in this symbolic representation. We use a spinor $\tilde \Psi^\dagger=(\Psi^\dagger,\Psi^\dagger)$ that contains the same Dirac fermion twice, to write $\mathcal{L}_\text{int}=h\chi^\dagger \tilde \Psi + \text{h.c.}$ We next proceed by integrating out the magnetic surface fermions $\chi$, $$\begin{aligned}
\mathcal{Z} & = & \int\! D\,[\overline{\chi},\chi]\,e^{i\int dt\int d^2r(\mathcal{L}_\text{surf}+\mathcal{L}_\text{int})} \nonumber \\
& = &\int\! D\,[\overline{\chi},\chi]\,e^{i(\chi^\dagger A \chi-h \chi^\dagger \tilde \Psi -h \tilde \Psi^\dagger \chi)} \nonumber \\
&= & \exp\left(i\text{Tr}\,\ln A+ih^2 \tilde \Psi^\dagger A^{-1} \tilde \Psi\right).
\label{Eq:GaussianIntegral}\end{aligned}$$ Note that the notation $\text{Tr}$ also contains the integration over the quantum numbers besides the matrix trace. We will discuss the two terms in the last line separately in the following subsections.
Surface corrections to the bulk terms {#Sec:NonTop}
-------------------------------------
The term $\text{Tr}\ln\,A$ in Eq. is independent of the topological Dirac states. It leads to the Berry phases mentioned previously and renormalizes the magnetic bulk terms at the surface. Details of the calculation and complete analytical expressions can be found in Appendix \[App:Anisotropy\]. We finally obtain $$\begin{aligned}
&&\delta\mathcal{L}_\text{mag}(\mathbf{r},t) =\notag\\
&&-2J^2{\mathbf{m}}_1\cdot\text{diag}(T^{00}-T^{zz},T^{00}-T^{zz},T^{00}+T^{zz})\cdot{\mathbf{m}}_2\notag\\
&&+2J^2\sum_{i=1,2}\Big\{\left[(D_i^{00}+D_i^{zz}){\overline{m}}_i + (T^{00}+T^{zz}){\overline{m}}_{3-i}\right]m_{iz}\notag\\
&&{}- D_i^{zz}m_{iz}^2 +\mathcal{D}^{0z}_i{\hat{\mathbf{e}}}_z\cdot\left[{\mathbf{m}}_i(\mathbf{r},t)\times\partial_t{\mathbf{m}}_i(\mathbf{r},t)\right]\Big\}\notag\\
&&{}+2J^2\mathcal{T}^{0z}{\hat{\mathbf{e}}}_z\cdot\big[{\mathbf{m}}_1(\mathbf{r},t)\times\partial_t{\mathbf{m}}_2(\mathbf{r},t)\notag\\
&&{}+{\mathbf{m}}_2(\mathbf{r},t)\times\partial_t{\mathbf{m}}_1(\mathbf{r},t)\big]
\label{Eq:MagLagr}
,\end{aligned}$$ where $D_i^{00}, D_i^{zz}, \mathcal{D}^{0z}_i, T^{00}, \mathcal{T}^{0z}$ and $T^{zz}$ are functions of $t,J,{\overline{m}}_i$ and the lattice spacing $a$. The Berry phases are represented by the cross-product terms. The terms proportional to $\mathcal{D}^{0z}_i$ shift the Berry phases introduced in Eq. , while the term proportional to $\mathcal{T}^{0z}$ is a mixed Berry phase term. We remark that $\mathcal{T}^{0z}\propto t$, thus no mixed Berry phase appears if $t=0$.
Furthermore, the coupling of ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ given by Eq. is renormalized by the first line in Eq. and becomes anisotropic. This leads to in-plane and out-of-plane effective exchange couplings given by, $$\lambda_{\rm eff}^\parallel=\lambda+2J^2(T^{00}-T^{zz}),$$ $$\lambda_{\rm eff}^\perp=\lambda+2J^2(T^{00}+T^{zz}).$$ An evaluation of our analytic expressions (Appendix \[App:Anisotropy\]) reveals that the dynamically generated coupling favors AFM alignment of the two magnetic components. Indeed, using Eqs. (\[Eq:T00\]) and (\[Eq:Tzz\]) of Appendix A, we obtain, $$\label{Eq:T00-Tzz}
T^{00}-T^{zz}=\frac{t^2[2|t^2-J^2{\overline{m}}_1{\overline{m}}_2|+2t^2+J^2({\overline{m}}_1^2+{\overline{m}}_2^2)]}{2a^2|t^2-J^2{\overline{m}}_1{\overline{m}}_2|(M_++M_-)^3},$$ $$\label{Eq:T00+Tzz}
T^{00}+T^{zz}=\frac{t^2[1+{\rm sgn}(t^2-J^2{\overline{m}}_1{\overline{m}}_2)]}{a ^2(M_++M_-)^3},$$ where, $$\begin{aligned}
M_\pm^2&=&\frac{J^2}{2}({\overline{m}}_1^2+{\overline{m}}_2^2)+t^2
\nonumber\\
&\pm&\frac{J^2}{2}|{\overline{m}}_1+{\overline{m}}_2|\sqrt{({\overline{m}}_1-{\overline{m}}_2)^2+\left(\frac{2t}{J}\right)^2}.\end{aligned}$$ The coupling constants show a discontinuity at $t^2=J^2{\overline{m}}_1{\overline{m}}_2$, or $\tau=\mu$, as shown in Fig. \[Fig:Discontinuity\]. Indeed, we see that Eq. (\[Eq:T00-Tzz\]) diverges for $t^2=J^2{\overline{m}}_1{\overline{m}}_2$, while (\[Eq:T00+Tzz\]) vanishes when $t^2<J^2{\overline{m}}_1{\overline{m}}_2$. This divergence obviously does not occur when ${\overline{m}}_1{\overline{m}}_2<0$, corresponding to the AFM case, further corroborating the favoring of the AFM alignment. Physically, the divergence for $\tau=\mu$ implies the vanishing of the in-plane susceptibility.
![(Color online) The anisotropic fluctuation-induced antiferromagnetic exchange coupling of ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ at the surface, which renormalizes the exchange coupling induced from the bulk. a) In the component along the mean-field direction, the coupling constant is given by $T^{00}+T^{zz}$ (see main text) and shows a finite discontinuity at $\mu=\tau$ (dashed line). b) In the component orthogonal to the mean-field direction, the AFM coupling $T^{00}-T^{zz}$ diverges at the discontinuity. The color scale is identical in both plots. c) The quantities $T^{00}$ (thin solid blue line), $T^{zz}$ (dash-dotted blue line), $T^{00}+T^{zz}$ (bold solid red line), and $T^{00}-T^{zz}$ (dashed red line) as a function of $\mu$ for a specific value of $\tau$ ($\tau=0.45$), which is indicated by the thin white dotted lines in plots a) and b). d) The anisotropy terms $D_1^{zz}$ (thin solid blue line), $D_2^{zz}$ (dashed blue line), and $D_i^{00}+D_i^{zz}$ (bold solid red line, identical for $i=1,2$) behave similarly, showing a discontinuity at $\tau=\mu$. The vicinity of this line is excluded from the further analysis.[]{data-label="Fig:Discontinuity"}](T_diagrams.png){width="\columnwidth"}
The remaining terms in Eq. describe a z-axis anisotropy in both magnetizations. As we mentioned in section \[SecModel\], our model does not account for possible anisotropy contributions originating with the bulk of the magnet. Such terms would simply be renormalized by the corresponding coefficients in Eq without changing the physical picture.
Our view of the dynamically generated surface terms as corrections to the bulk values will only hold as long as the surface effects are not too large. As can be seen from Fig. \[Fig:Discontinuity\], within our model some surface terms are divergent at the discontinuity at $\mu=\tau$. Therefore, the vicinity of this line in parameter space will be excluded in our further analysis.
As a side remark, the fluctuation effects discussed in this subsection can easily be generalized to account for magnetizations that are, at mean-field, tilted relative to the surface. We have checked that Eq. remains valid when the $z$ components are replaced by mean-field components in an arbitrary direction.
Effective Dirac Lagrangian
--------------------------
The term $h^2 \tilde \Psi^\dagger A^{-1} \tilde \Psi$ in Eq. may now be added to Eq. to yield an effective action for the Dirac fermions $$\mathcal{S}_\text{eff} = \int dt\int d^2 r{\cal L}_{\rm eff}= \int dt\int d^2 r\left(\mathcal{L}_\text{D} + h^2\tilde{\Psi}^\dagger A^{-1} \tilde{\Psi}\right).$$ Multiplying out $\tilde \Psi^\dagger A^{-1}\tilde \Psi$ into single-fermion operators again, we find the effective Lagrangian of the Dirac electrons at the coupled surfaces, $$\begin{aligned}
\mathcal{L}_\text{eff} &=& \mathcal{L}_\text{D} + \gamma\Psi^\dagger\left(\frac{t^2}{J^2}-{\mathbf{m}}_1\cdot{\mathbf{m}}_2\right)\Psi \notag\\
&&{}+\Psi^\dagger\left(J_1{\mathbf{m}}_1\cdot{\boldsymbol{\sigma}}+J_2{\mathbf{m}}_2\cdot{\boldsymbol{\sigma}}\right)\Psi,
\label{Eq:EffDiracLagrangian1}\end{aligned}$$ where we have defined the constant $$\gamma = \frac{2th^2J^2}{\text{det}\,A}$$ and the effective magnetic coupling constants for the two sublattices $$\label{Eq:Jconstant}
J_i = \frac{h^2J}{\text{det}\,A}\left(J^2{\mathbf{m}}_{3-i}^2-t^2\right),$$ where $$\begin{aligned}
\text{det}\,A & = & (-\partial^2_t - t^2)^2 + J^2 \partial^2_t ({\mathbf{m}}_1^2 + {\mathbf{m}}_2^2) \nonumber \\
& + & J^2( J^2 {\mathbf{m}}_1^2 {\mathbf{m}}_2^2 - 2 t^2 {\mathbf{m}}_1\cdot{\mathbf{m}}_2) . \end{aligned}$$ In $\text{det}\,A$, the fluctuations in ${\mathbf{m}}_{1,2}$ are not of leading order. Therefore, we will approximate the determinant in the Dirac Lagrangian by its mean-field value $\text{det}\,A^\text{mf} = t^4 + J^2 \left[J^2 {\overline{m}}_1^2 {\overline{m}}_2^2 - 2 t^2 {\overline{m}}_1 {\overline{m}}_2 \right]$, whereby we also neglected higher-order time derivatives in the low-frequency limit. Furthermore, we assume that the coupling $h$ of the surface fermions $\chi$ and $\Psi$ at the interface is small compared to the internal energy scales of the magnet, $t$ and $J{\overline{m}}_i$. Otherwise, one obtains a renormalization of the time scale. It is interesting to note that the term $\propto\gamma$ in Eq. contributes to the chemical potential of $\Psi$. The chemical potential may be tuned by means by adjusting $\phi$ appearing in Eq. , and the mean-field part of the second term in Eq. may thus always be adjusted away. We will only keep the remainder to linear order in the fluctuations.
Note that the sign of $J_i$ in Eq. depends on the parameter $t$ appearing in Eq. , as well as the magnitude of the magnetic moments. This is a key observation that we will return to when discussing the topological effects in the next section.
Topological magnetoelectric effects {#Sec:TSE}
===================================
Now, we express the effective Lagrangian Eq. as $$\mathcal{L}_\text{eff} = \overline{\Psi} (i\slashed\partial + m_\Psi)\Psi + \overline{\Psi} (\tilde{\sigma}-\slashed{a})\Psi
\label{Eq:EffDiracLagrangian2}
,$$ where the first term is the mean-field part with $\partial=(\partial_t,v_F\nabla_\parallel)$ and $m_\Psi=J_1{\overline{m}}_1+J_2{\overline{m}}_2$, whereas the second term contains the fluctuating fields $\tilde{\sigma}=J_1\tilde{m}_{1z}+J_2\tilde{m_2}_z$ and $$\mathbf{a} = \begin{pmatrix}-e(\varphi+\phi)+\gamma({\overline{m}}_1\tilde{m}_{2z}+{\overline{m}}_2\tilde{m}_{1z})\\J_1\tilde{m}_{1y}+J_2\tilde{m}_{2y}\\-J_1\tilde{m}_{1x}-J_2\tilde{m}_{2x}\end{pmatrix}
.$$ From this representation, one can see that the out-of-plane fluctuations of the magnetization contribute to the effective electric potential at the interface. This is a result of the fluctuations in the chemical potential that we have observed in Eq. . To obtain an effective field theory for the magnetizations that contains the proximity effects induced by the topological insulator, we also have to integrate out the remaining fermions $\Psi$ and the fluctuating Coulomb potential $\varphi$. Equation is formally equivalent to the field theory studied in Refs. , given that the mass term $m_\Psi$ is nonzero. This is naturally the case for FMs and FiMs (except at $\mu=\tau$, which we already excluded), while it might be enforced by doping in the case of an AFM.
Integrating out $\Psi$ yields the fluctuation-induced Lagrangian to one-loop order in the vacuum polarization diagrams [@Flavio_PRL; @FM_bilayer], $$\delta\mathcal{L}_\text{eff} =
\frac{\epsilon_{\mu\nu\lambda}a^\mu\partial^\nu a^\lambda}{8\pi}
-\frac{(\epsilon_{\mu\nu\lambda}\partial^\nu a^\lambda)^2}{24\pi m_\Psi}
-\frac{m_\Psi\tilde{\sigma}^2}{2\pi}
+\frac{(\partial\tilde{\sigma})^2}{24\pi m_\Psi}
\label{EqLagrangianNoFermion}$$ The first term is the CS term that is responsible for all topologically protected contributions to the Lagrangian. The other terms correspond to a Maxwell term and out-of-plane anisotropy.
Besides these dynamical terms, a term describing the energy at mean-field is produced after all fermionic fields have been integrated out. This term can be expanded into a Landau theory for the mean-field magnetizations at the BMI-TI interface. The Landau expansion can be found in App. \[App:LandauExpansion\], where we find that the quadratic term is always negative. This serves as a check that our model, where we treated ${\overline{m}}_{1,2}$ as parameters, is consistent with the existence of a magnetic phase.
Reinserting $\mathbf{a}$, we can separate $\delta\mathcal{L}_\text{eff}$ in a Coulomb-interaction ($\varphi$-dependent) part $\mathcal{L}_\varphi$ and the remaining dynamically generated terms $\mathcal{L}_\text{dyn}$. After integrating out $\varphi$, the Coulomb contributions become $$\mathcal{L}_\varphi(\mathbf{r},t) = 2\rho(\mathbf{r},t)\int\! d^2r^\prime\frac{\rho(\mathbf{r}^\prime,t)}{|\mathbf{r}-\mathbf{r}^\prime|}
,$$ with the charge density $$\begin{aligned}
\rho &=& \frac{e}{8\pi v_F}{\boldsymbol{\nabla}}_\parallel\cdot(J_1{\mathbf{m}}_1+J_2{\mathbf{m}}_2)+\frac{e^2}{24\pi m_\Psi}{\boldsymbol{\nabla}}_\parallel \mathbf{E}_\text{ext}\notag\\
&&{}-\frac{e}{24\pi m_\Psi v_F}\left[{\boldsymbol{\nabla}}_\parallel \times\partial_t(J_1{\mathbf{m}}_1+J_2{\mathbf{m}}_2)\right]\cdot{\hat{\mathbf{e}}}_z\notag\\
&&{}+\frac{\gamma e}{24\pi m_\Psi}\left({\boldsymbol{\nabla}}_\parallel \right)^2({\overline{m}}_1 m_{2z}+{\overline{m}}_2 m_{1z}),
\label{Eq:ChargeDensity}\end{aligned}$$ where $\mathbf{E}_\text{ext}=-{\boldsymbol{\nabla}}\phi$ is the externally applied electric field. We also define the Coulomb field induced by the charge density, $$\mathbf{E}_\text{Cou}(\mathbf{r}) = -\int\!d^2r^\prime\,\frac{\mathbf{r}-\mathbf{r}^\prime}{|\mathbf{r}-\mathbf{r}^\prime|^3}\rho(\mathbf{r}^\prime).$$ For low frequency and momentum, the last two terms in Eq. will be negligible compared to the first two terms.
The part of the Lagrangian that is due to the non-trivial topology (i.e., stemming from the CS term), where we write $\mathbf{M}=J_1{\mathbf{m}}_1+J_2{\mathbf{m}}_2$ for brevity, can be expressed explicitly as $$\begin{aligned}
\mathcal{L}_\text{topol} &=&
\frac{e}{4\pi v_F}\mathbf{M}_\shortparallel\cdot(\mathbf{E}_\text{ext}
+\mathbf{E}_\text{Cou})\notag\\
&&{}- \frac{1}{8\pi v_F^2}\left(\mathbf{M}\times\partial_t\mathbf{M}\right)\cdot{\hat{\mathbf{e}}}_z\notag\\
&&{}+\frac{\gamma}{4\pi v_F}\mathbf{M}\cdot{\boldsymbol{\nabla}}_\parallel\left({\overline{m}}_1 m_{2z}+{\overline{m}}_2 m_{1z}\right).
\label{Eq:LagTopol}\end{aligned}$$ The first term represents the magnetoelectric coupling, involving both the external field and the fluctuation-induced Coulomb field. The second term is a Berry phase. Unlike the Berry phase generated by the fluctuations of $\chi$, this expression always includes mixed terms, regardless of the parameter $t$. Finally we also obtain a topological coupling of the magnetic in-plane and out-of-plane fluctuations.
At this point, we can discuss how the system will respond to an electric field. This is the main result of our paper. As we can see from Eq. , the electric field is coupled to $\mathbf{M}$ in the same way as it couples to the magnetic polarization in the usual TME effect. Now, let us write $\mathbf{M}$ in terms of the net magnetization ${\mathbf{m}}={\mathbf{m}}_1+{\mathbf{m}}_2$ and the staggered field ${\mathbf{l}}={\mathbf{m}}_1-{\mathbf{m}}_2$, $$\mathbf{M} = \frac12(J_1+J_2){\mathbf{m}}+ \frac12(J_1-J_2){\mathbf{l}}.$$ Obviously, if $J_1$ and $J_2$ have the same sign, an electric field will mainly generate a net in-plane magnetization, while the coupling to the staggered field is small. Overall, the system will behave as one would expect for a simple FM. However, if $J_1$ and $J_2$ have opposite signs, an electric field will mainly induce a staggered field in the plane, while the response in the net magnitization will be weak. This is because the usual TME effect takes place on both sublattices, but with opposite direction. Going back to Eq. , it is easy to find the parameter region where this topological staggered field-electric (TSE) effect can be found. In terms of the dimensionless model parameters, the condition for $J_1$ and $J_2$ having opposite signs is $\mu^2<\tau<1$, see figure \[Fig:TME\]. A purely TSE response is expected if $J_1=-J_2$, which is the case if $\tau=\frac{1}{2}(1+\mu^2)$. Remarkably, the predominantly TSE response can appear even in a FM material ($\mu>0$), if it consists of multiple magnetic components per unit cell with different magnitude and a suitable parameter $t$. Thus, it is possible that experiments fail to detect the usual TME effect even when a decent gap opening occurs. In contrast, a purely AFM material ($\mu=-1$) would not show any coupling to the staggered field, even in the presence of a mass term $m_\Psi$ by magnetic doping, because $J_1=J_2$ for equally strong magnetic moments on the two sublattices. Our model of the BMI is quite simple, and for a real material it might be much harder to find the parameter regions that allow for the observation of the TME or TSE effect. However, it is a remarkable finding that the overall topological response in a BMI-TI heterostructure can depend dramatically on microscopic details of the magnet.
![(Color online) Left: parameter regions of the bipartite magnet where the topological response to an electric field has the same (white area) or opposite (grey area) direction on the two sublattices, corresponding to a predominantly magneto-electric (TME) or staggered field-electric (TSE) effect, respectively. Here, $\tau$ is the dimensionless amplitude of the coupling of the fermions on the two sublattices and $\mu$ is the ratio of the mean-field values of the magnetizations on the sublattices. Close to the dashed line at $\mu=\tau$, our results may not be applicable. Right: illustration of the topological effects for a FiM: a) both magnetizations ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ (black) pointing in their mean-field directions. b) TME effect: if the topological response to the electric field $\mathbf{E}$ (red) has the same sign on both sublattices, an in-plane net magnetization ${\mathbf{m}}_\shortparallel$ (blue) is generated, while the induced in-plane staggered field ${\mathbf{l}}_\parallel$ (orange) is small. c) TSE effect: if the topological response to $\mathbf{e}$ has opposite signs for ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$, an in-plane staggered field is generated, while ${\mathbf{m}}_\shortparallel$ is small. The overall sign of these effects depends on the sign of the mass term $m_\Psi$.[]{data-label="Fig:TME"}](TME_diagram.png "fig:"){width="0.4\columnwidth"} ![(Color online) Left: parameter regions of the bipartite magnet where the topological response to an electric field has the same (white area) or opposite (grey area) direction on the two sublattices, corresponding to a predominantly magneto-electric (TME) or staggered field-electric (TSE) effect, respectively. Here, $\tau$ is the dimensionless amplitude of the coupling of the fermions on the two sublattices and $\mu$ is the ratio of the mean-field values of the magnetizations on the sublattices. Close to the dashed line at $\mu=\tau$, our results may not be applicable. Right: illustration of the topological effects for a FiM: a) both magnetizations ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ (black) pointing in their mean-field directions. b) TME effect: if the topological response to the electric field $\mathbf{E}$ (red) has the same sign on both sublattices, an in-plane net magnetization ${\mathbf{m}}_\shortparallel$ (blue) is generated, while the induced in-plane staggered field ${\mathbf{l}}_\parallel$ (orange) is small. c) TSE effect: if the topological response to $\mathbf{e}$ has opposite signs for ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$, an in-plane staggered field is generated, while ${\mathbf{m}}_\shortparallel$ is small. The overall sign of these effects depends on the sign of the mass term $m_\Psi$.[]{data-label="Fig:TME"}](magn_picture.png "fig:"){width="0.58\columnwidth"}
A restriction on our findings is imposed by the discontinuity discussed in the previous section. Due to divergent terms, our results on the TSE effect will not be applicable for parameters in the vicinity of the line $\mu=\tau$ in Fig. \[Fig:TME\].
Previous work has found a Coulomb-mediated magnetic dipolar interaction [@FM_trilayer]. The Coulomb interaction in the present work will lead to the same effect within each sublattice. Moreover, there will be a dipolar interaction between the sublattices. Again, for a system in the TSE regime, we will get an effect in the opposite direction. Thus, the inter-component dipolar interaction will favour counteralignment instead of alignment of ${\mathbf{m}}_{1,\shortparallel}$ and ${\mathbf{m}}_{2,\shortparallel}$.
Our model also reveals a topological coupling of the in-plane components of the magnetic moments and the gradient in the out-of-plane component as described by the last term in Eq. , which can be understood as an anomalous spin-stiffness term. This term has not been considered in previous studies and can lead to a spin canting effect if the magnetization is not homogeneous, as in the presence of spin-waves or domain walls. For the observation of the electromagnetic response it will, however, not be important.
The full Lagrangian describing the magnetic moments in the system is now given by $$\mathcal{L}_\text{tot} = \mathcal{L}_\text{bulk} + \mathcal{L}_\varphi + \mathcal{L}_\text{dyn} + \delta\mathcal{L}_\text{mag},\label{Eq:Lmag_tot}$$ from which the coupled LLEs for the motion of ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ at the interface can be derived. It takes the form $$\partial_t \begin{pmatrix} {\mathbf{m}}_1 \\ {\mathbf{m}}_2 \end{pmatrix} =
\Xi^{-1} \begin{pmatrix} {\mathbf{m}}_1\times\mathbf{d}_1 \\ {\mathbf{m}}_2\times\mathbf{d}_2 \end{pmatrix}
.$$ For details, we refer to Appendix \[App:LLE\]. The $(6\times6)$ matrix $\Xi$ contains all Berry phase terms. In particular, there are off-diagonal terms that stem from the fluctuation-induced mixed Berry phases. Such terms are generated by the fermions $\chi_i$ (if $t\neq0$) as well as the Dirac fermions $\Psi$. The contribution by the fluctuations of $\Psi$ is of topological origin, as it stems from the CS term. The effective fields $\mathbf{d}_i$ contain, besides spin-stiffness and anisotropy terms, a topological part $$\begin{aligned}
\mathbf{d}^i_\text{topol} &=&
\frac{eJ_i}{4\pi v_F}\mathbf{E}_\text{Cou}
+ \frac{eJ_i}{4\pi v_F}\mathbf{E}_\text{ext}
-\frac{\gamma{\overline{m}}_{3-i}}{4\pi v_F}(\nabla_\parallel\cdot\mathbf{M}){\hat{\mathbf{e}}}_z\notag\\
&&{}-\frac{\gamma J_i}{4\pi v_F}\nabla_\parallel({\overline{m}}_1 m_{2z} + {\overline{m}}_2 m_{1z})\end{aligned}$$ corresponding to Eq. . The first two terms show explicitly how the external electric field and the Coulomb field affect the magnetization dynamics as a consequence of the magnetoelectric effects discussed above.
Conclusion {#Sec:Conclusion}
==========
We have studied the topological effects at the interface of a TI and a BMI within an analytically accessible model that accounts for the fermionic quantum fluctuations at the surfaces of both materials. We have demonstrated that the TME effect that is known for magnetic TI surfaces can take the opposite sign for the different magnetic components, depending on microscopic details of the material. This leads to an overall TSE response to an electric field, while the induced net magnetization in the plane can be weak even in the presence of a stable energy gap in the Dirac dispersion. Thus, experiments that aim at detecting the TME effect might also look for a response in the staggered field. A response in the magnetization can be absent even when a FM insulator is used, if there are multiple magnetic components with different magnitude. In addition to the TSE effect, we have derived several dynamically generated Berry phases, including terms mixing ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$. We also found a topological coupling of in-plane and out-of-plane magnetic components which is present for non-homogeneous magnetization. The fluctuations of the fermions on the magnets’ surface cause discontinuity in our model, close to which our results are not applicable.
Acknowledgements {#acknowledgements .unnumbered}
================
S. R. and A. S. acknowledge funding by the Norwegian Research Council, Grants No. 205591/V20 and No. 216700/F20.
Derivation of the surface corrections {#App:Anisotropy}
=====================================
Here, we derive the magnetic surface terms discussed in Sec. \[Sec:NonTop\] that are generated by $\text{Tr}\,\ln A$ in the Gaussian integral, Eq. . Splitting $A=A^\text{mf}+A^\text{fl}$ in the mean-field part and the quantum fluctuations, $$A^\text{mf} = \begin{pmatrix}i\partial_t+J{\overline{m}}_1\sigma_z&t\\t&i\partial_t+J{\overline{m}}_2\sigma_z\end{pmatrix}$$ $$A^\text{fl} = \begin{pmatrix}J{\tilde{\mathbf{m}}}_1\cdot{\boldsymbol{\sigma}}&0\\0&J{\tilde{\mathbf{m}}}_2\cdot{\boldsymbol{\sigma}}\end{pmatrix}
,$$ we obtain the usual expansion $$\text{Tr}\,\ln A = \text{Tr}\,\ln A^\text{mf} - \frac12\text{Tr}\left(GA^\text{fl}\right)^2
\label{Eq:MagTrace}$$ where the first term is a constant corresponding to the ground-state energy that will be dealt with in App. \[App:LandauExpansion\], while the second term describes the dynamics close to equilibrium to leading order. The propagator $G$ is given by $\left(A^\text{mf}\right)^{-1}$. In reciprocal space and imaginary time, $G$ depends only on the frequency $\omega$ but not on momentum, because the hopping terms in our model are momentum-independent. For all momentum integrals, we use $\pi/a$ as a cut-off value, where $a$ is the lattice spacing. The propagator can be written in the form $$G = \frac{1}{\text{det}\,A^{\text{mf}}}\begin{pmatrix}D_1^0+D_1^z\sigma_z&T^0+T^z\sigma_z\\T^0+T^z\sigma_z&D_2^0+D_2^z\sigma_z\end{pmatrix}
,$$ where the components are $$\begin{aligned}
D_1^0(\omega) &=& i\omega^3+i\omega J^2{\overline{m}}_2^2 + i\omega t^2\\
D_2^0(\omega) &=& i\omega^3+i\omega J^2{\overline{m}}_1^2 + i\omega t^2\\
D_1^z(\omega) &=& J{\overline{m}}_1\omega^2 + J^3{\overline{m}}_2^2{\overline{m}}_1 - t^2J{\overline{m}}_2\\
D_2^z(\omega) &=& J{\overline{m}}_2\omega^2 + J^3{\overline{m}}_1^2{\overline{m}}_2 - t^2J{\overline{m}}_1\\
T^0(\omega) &=& t\omega^2 + t^3 - tJ^2{\overline{m}}_1{\overline{m}}_2\\
T^z(\omega) &=& -it\omega J\left({\overline{m}}_1+{\overline{m}}_2\right)\end{aligned}$$ and the determinant is $$\begin{aligned}
\text{det}\,A^\text{mf}(\omega) &=& \left[\omega^2+\frac{J^2}{2}({\overline{m}}_1^2+{\overline{m}}_2^2)+t^2\right]^2 \notag\\
&&{}-\frac{J^4}{4}({\overline{m}}_1^2-{\overline{m}}_2^2)^2 - t^2J^2({\overline{m}}_1+{\overline{m}}_2)^2.\notag\\\end{aligned}$$ Performing the trace in Eq. at $T=0$ then leads to the Lagrangian $$\begin{aligned}
\lefteqn{\delta\mathcal{L}_\text{mag}(\Omega) =}\notag\\
&&-J^2\sum_{i=1,2}\Big[\left(D^{00}_i(\Omega)-D_i^{zz}(\Omega)\right){\tilde{\mathbf{m}}}_i(\Omega)\cdot{\tilde{\mathbf{m}}}_i(-\Omega)\notag\\
&&\hspace{0.5cm}{}+2D^{zz}_i(\Omega)\tilde{m}_{i,z}(\Omega)\tilde{m}_{i,z}(-\Omega)\notag\\
&&\hspace{0.5cm}{}+i\left(D^{z0}_i(\Omega)-D^{0z}_i(\Omega)\right){\hat{\mathbf{e}}}_z\cdot\left({\tilde{\mathbf{m}}}_i(\Omega)\times{\tilde{\mathbf{m}}}_i(-\Omega)\right)\Big]\notag\\
&&{}-J^2\left(T^{00}(\Omega)-T^{zz}(\Omega)\right)\notag\\&&\times\left({\tilde{\mathbf{m}}}_1(\Omega)\cdot{\tilde{\mathbf{m}}}_2(-\Omega)+{\tilde{\mathbf{m}}}_1(-\Omega)\cdot{\tilde{\mathbf{m}}}_2(\Omega)\right)\notag\\
&&{}-iJ^2\left(T^{z0}(\Omega)-T^{0z}(\Omega)\right)\notag\\&&\times{\hat{\mathbf{e}}}_z\cdot\left({\tilde{\mathbf{m}}}_1(\Omega)\times{\tilde{\mathbf{m}}}_2(-\Omega)+{\tilde{\mathbf{m}}}_2(\Omega)\times{\tilde{\mathbf{m}}}_1(-\Omega)\right)\notag\\
&&{}-2J^2T^{zz}(\Omega)\left(\tilde{m}_{1z}(\Omega)\tilde{m}_{2z}(-\Omega)+\tilde{m}_{1z}(-\Omega)\tilde{m}_{2z}(\Omega)\right), \notag\\ \label{Eq:LongMagLagr}\end{aligned}$$ with frequency $\Omega$, containing the integrals $$D^{\alpha\beta}_i(\Omega) = \frac{1}{a^2}\int\!\frac{\text{d}\omega}{2\pi}\,\frac{D^\alpha_i(\omega)D^\beta_i(\omega-\Omega)}{\left[\text{det}\,A^\text{mf}(\omega)\right]\left[\text{det}\,A^\text{mf}(\omega-\Omega)\right]}
\label{Eq:DIntDefinition}$$ and $$T^{\alpha\beta}(\Omega) = \frac{1}{a^2}\int\!\frac{\text{d}\omega}{2\pi}\,\frac{T^\alpha(\omega)T^\beta(\omega-\Omega)}{\left[\text{det}\,A^\text{mf}(\omega)\right]\left[\text{det}\,A^\text{mf}(\omega-\Omega)\right]}.
\label{Eq:TIntDefinition}$$ with $\alpha,\beta\in\{0,z\}$ and $i=1,2$. These integrals can be solved exactly by partial fraction decomposition, since the zeros of the denominator are known: $\text{det}\,A^\text{mf}(\omega)=0$ if $\omega^2=N^\pm$, with $$\begin{aligned}
N^\pm &=& \pm J\sqrt{\frac{J^2}{4}({\overline{m}}_1^2-{\overline{m}}_2^2)^2 + t^2({\overline{m}}_1+{\overline{m}}_2)^2}\notag\\
&&{}-\frac12 J^2({\overline{m}}_1^2+{\overline{m}}_2^2) - t^2,\end{aligned}$$ where $N^-<0$ and $N^+\leq0$. Namely, $N^+=0$ if $t^2=J^2{\overline{m}}_1{\overline{m}}_2$, i.e., in terms of the dimensionless parameters, if $\tau=\mu$. This is where the discontinuity which is discussed in Sec. \[Sec:NonTop\] is located. In the integrals, we neglect terms of order $\Omega^2$ or higher in the long-wavelength limit, and obtain $$\begin{aligned}
D^{00}_1(\Omega) &=& \frac{1}{4a^2\sqrt{-N^+}\left(N^+-N^-\right)^3}\notag\\
&&{}\times\Big[-(N^+)^3+5(N^+)^2N^-\notag\\
&&{}+2\left(J^2{\overline{m}}_2^2+t^2\right)\left((N^+)^2+3N^+N^-\right)\notag\\
&&{}+\left(J^2{\overline{m}}_2^2+t^2\right)^2\left(3N^++N^-\right)\Big]\notag\\
&&{}+(\text{same with }N^+\leftrightarrow N^-) + \mathcal{O}(\Omega^2)
\label{Eq:IntD00}\end{aligned}$$ $$\begin{aligned}
D_i^{0z}(\Omega) &=& \frac{i\Omega}{16a^2N^+\sqrt{-N^+}(N^--N^+)^3}\notag\\
&&{}\times\Big[J{\overline{m}}_1(N^+)^2\left(2N^++9N^-\right)\notag\\
&&{}+J{\overline{m}}_1\left(J^2{\overline{m}}_2^2+t^2\right)N^+(N^+-5N^-)\notag\\
&&{}+J{\overline{m}}_2\left(J^2{\overline{m}}_2^2+t^2\right)\left(J^2{\overline{m}}_1{\overline{m}}_2-t^2\right)\notag\\
&&{}\times(10N^+-2N^-)\Big]\notag\\
&&{}+ (\text{same with }N^+\leftrightarrow N^-)+ \mathcal{O}(\Omega^3)
\label{Eq:IntD0z}\end{aligned}$$ $$\begin{aligned}
D^{zz}_1(\Omega) &=& \frac{-J^2}{4a^2N^+\sqrt{-N^+}\left(N^+-N^-\right)^3}\notag\\
&&\times\Big[{\overline{m}}_1^2(N^+)^2\left(N^++3N^-\right)\notag\\
&&{}+ 2{\overline{m}}_1{\overline{m}}_2\left(J^2{\overline{m}}_1{\overline{m}}_2-t^2\right)N^+(3N^++N^-)\notag\\
&&{}+ {\overline{m}}_2^2\left(J^2{\overline{m}}_1{\overline{m}}_2-t^2\right)^2(5N^+-N^-)\Big]\notag\\
&&{}+(\text{same with }N^+\leftrightarrow N^-) + \mathcal{O}(\Omega^2)
\label{Eq:IntDzz}\end{aligned}$$ $$\begin{aligned}
\label{Eq:T00}
T^{00}(\Omega) &=& \frac{-t^2\left(J^2{\overline{m}}_1{\overline{m}}_2-t^2-N^+\right)}{a^2\sqrt{-N^+}\left(N^+-N^-\right)^2}\notag\\
&&\times\left[1 + \frac{\left(5N^+-N^-\right)\left(J^2{\overline{m}}_1{\overline{m}}_2-t^2-N^+\right)}{4N^+\left(N^+-N^-\right)}\right]\notag\\
&& {}+(\text{same with }N^+\leftrightarrow N^-) + \mathcal{O}(\Omega^2)\end{aligned}$$ $$\begin{aligned}
T^{0z}(\Omega) &=& \frac{i\Omega t^2({\overline{m}}_1+{\overline{m}}_2)}{16a^2N^+\sqrt{-N^+}(N^--N^+)^3}\notag\\
&&{}\times\big[\left(t^2-J^2{\overline{m}}_1{\overline{m}}_2\right)\left(10N^++2N^-\right)\notag\\
&&{}-7(N^+)^2+N^+N^-\big]\notag\\
&&{}+(\text{same with }N^+\leftrightarrow N^-) + \mathcal{O}(\Omega^3)
\label{Eq:IntT0z}\end{aligned}$$ $$\begin{aligned}
\label{Eq:Tzz}
T^{zz}(\Omega) &=&\frac{t^2J^2({\overline{m}}_1+{\overline{m}}_2)^2\left(3N^++N^-\right)}{4a^2\sqrt{-N^+}\left(N^+-N^-\right)^3}\notag\\
&&{}+ (\text{same with }N^+\leftrightarrow N^-) + \mathcal{O}(\Omega^2)\end{aligned}$$ Expressions for $D_2^{00}(\Omega)$, $D^{0z}(\Omega)$, and $D_2^{zz}(\Omega)$ can be obtained from Eqs. , , and , respectively, by exchanging ${\overline{m}}_1\leftrightarrow{\overline{m}}_2$. It turns out that $D^{00}_1+D^{zz}_1=D^{00}_2+D^{zz}_2=-(T^{00}+T^{zz})$. Furthermore, $D_i^{z0}(\Omega) = D_i^{0z}(-\Omega) = -D_i^{0z}(\Omega)$ and $T^{z0}(\Omega)=T^{0z}(-\Omega)=-T^{0z}(\Omega)$. These relations follow by substituting $\omega\rightarrow(\omega+\Omega)$ in Eqs. and and the fact that only odd powers of $\Omega$ appear in Eqs. and . For ease of notation, we write $D^{0z}_i(\Omega)=i\Omega \mathcal{D}^{0z}_i$ and $T^{0z}(\Omega)=i\Omega \mathcal{T}^{0z}$, where $\mathcal{D}^{0z}_i$ and $\mathcal{T}^{0z}_i$ are frequency-independent.
The effective magnetic surface Lagrangian that is evoked by the fermionic fluctuations, Eq. , is in real space and time given by $$\begin{aligned}
\lefteqn{\delta\mathcal{L}_\text{mag}(\mathbf{r},t) =}\notag\\
&&-J^2\sum_{i=1,2}\Big[\left(D^{00}_i-D^{zz}_i\right){\tilde{\mathbf{m}}}_i^2(\mathbf{r},t)+2D^{zz}_i\tilde{m}_{i,z}^2(\mathbf{r},t)\notag\\
&&{}-2\mathcal{D}^{0z}_i{\hat{\mathbf{e}}}_z\cdot\left({\tilde{\mathbf{m}}}_i(\mathbf{r},t)\times\partial_t{\tilde{\mathbf{m}}}_i(\mathbf{r},t)\right)\Big]\notag\\
&&{}-2J^2(T^{00}-T^{zz}){\tilde{\mathbf{m}}}_1(\mathbf{r},t)\cdot{\tilde{\mathbf{m}}}_2(\mathbf{r},t)\notag\\
&&{}+2J^2\mathcal{T}^{0z}{\hat{\mathbf{e}}}_z\cdot\big[{\tilde{\mathbf{m}}}_1(\mathbf{r},t)\times\partial_t{\tilde{\mathbf{m}}}_2(\mathbf{r},t)\notag\\
&&{}+{\tilde{\mathbf{m}}}_2(\mathbf{r},t)\times\partial_t{\tilde{\mathbf{m}}}_1(\mathbf{r},t)\big]\notag\\
&&{}-4J^2T^{zz}\tilde{m}_{1z}(\mathbf{r},t)\tilde{m}_{2z}(\mathbf{r},t).
\label{Eq:dLmag} \end{aligned}$$ Equation in section \[Sec:NonTop\] follows by writing the Lagrangian in terms of ${\mathbf{m}}_i = {\overline{m}}_i{\hat{\mathbf{e}}}_z + {\tilde{\mathbf{m}}}_i$ again, where constant terms are discarded. The meaning of the different contributions is discussed in the main text.
In the special case of a pure AFM, where ${\overline{m}}_1=-{\overline{m}}_2$, a mathematical subtlety arises. Namely, the solution of the integrals $D_i^{\alpha\beta}(\Omega)$ and $T^{\alpha\beta}(\Omega)$ by partial fraction decomposition requires a different ansatz, because the zeros of the denominator are degenerate: $N^+=N^-=-J^2{\overline{m}}_1^2-t^2$. The integrals are notably easier as a consequence of multiple cancellations, and we find, again to leading order in $\Omega$ in the low-frequency regime, $$D^{00}_{1,\text{AFM}}(\Omega) = D^{00}_{2,\text{AFM}}(\Omega) = -\frac{1}{4a^2\sqrt{J^2{\overline{m}}_1^2+t^2}} + \mathcal{O}(\Omega^2)$$ $$D^{0z}_{1,\text{AFM}}(\Omega) = \!-D^{0z}_{2,\text{AFM}}(\Omega)\! = \frac{i\Omega J{\overline{m}}_1}{8a^2(J^2{\overline{m}}_1^2+t^2)^{3/2}} + \mathcal{O}(\Omega^3)$$ $$D^{zz}_{1,\text{AFM}}(\Omega) = D^{zz}_{2,\text{AFM}}(\Omega) = \frac{J^2{\overline{m}}_1^2}{4a^2(J^2{\overline{m}}_1^2+t^2)^{3/2}} + \mathcal{O}(\Omega^2)$$ $$T^{00}_\text{AFM}(\Omega) = \frac{t^2}{4a^2(J^2{\overline{m}}_1^2+t^2)^{3/2}} + \mathcal{O}(\Omega^2)$$ $$T^{0z}_\text{AFM}(\Omega) = T^{zz}_\text{AFM}(\Omega) = 0.$$ We have checked that these expressions are identical to the continuous limit ${\overline{m}}_2\rightarrow -{\overline{m}}_1$ of the integrals in the general case. Notably, no mixed Berry phase term is generated for the AFM. The fluctuation-induced Lagrangian takes the simplified form: $$\begin{aligned}
\lefteqn{\delta\mathcal{L}_\text{mag}^\text{AFM} =}\notag\\
&&\frac{J^2\!\left[t^2{\mathbf{m}}_1\!\cdot{\mathbf{m}}_2
+ 2t^2{\overline{m}}_1(m_{1z}\!-m_{2z})
+ J^2{\overline{m}}_1^2(m_{1z}^2\!+m_{2z}^2)\right]}{-2a^2(J^2{\overline{m}}_1^2+t^2)^{3/2}}\notag\\
&&{}+\frac{J^3{\overline{m}}_1}{4a^2(J^2{\overline{m}}_1^2+t^2)}{\hat{\mathbf{e}}}_z\cdot({\mathbf{m}}_1\times\partial_t{\mathbf{m}}_1 - {\mathbf{m}}_2\times\partial_t{\mathbf{m}}_2)\end{aligned}$$
Fluctuation-induced Landau theory {#App:LandauExpansion}
=================================
In this appendix, we present the Landau expansion of the energy in terms of the mean-field magnetizations at the interface. Here, we allow arbitrary directions of the magnetizations. Thus, the Landau theory is still valid if ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ are not aligned with each other or the $z$ axis at mean-field. For simplicity, we drop the overline-notation indicating mean-field values in this appendix.
The energy contains two contributions, namely (i) from the term $\det A$ in Eq. originating with the quantum fluctuations of the sublattice fermions, and (ii) from a similar term $\det B$ generated by the quantum fluctuations of the Dirac fermions, where $B$ is defined such that Eq. can be written as $\mathcal{L}_\text{eff}=\Psi^\dagger B \Psi$. The energy density is then given by $$\mathcal{E} = -\int\!\frac{d\omega}{2\pi}\int\!\frac{d^2k}{2\pi}(\ln\det A + \ln\det B)
,$$ where we use the cut-off value $\pi/a$ in divergent momentum integrals. We did not include Landau terms for the bulk in Eq. , however, any bulk contributions would simply add up with the interface terms shown here. We obtain the following expansion to fourth order, where $\perp$ indicates the component orthogonal to the interface and $\parallel$ the in-plane component: $$\begin{aligned}
\mathcal{E} &=& J^2\Bigg[\frac{-1}{4a^2|t|}({\mathbf{m}}_1-{\mathbf{m}}_2)^2-t^2K_2({\mathbf{m}}_1+{\mathbf{m}}_2)_\perp^2\notag\\
&&{}-\left(t^2K_2(1-v_F^2)+\frac{5h^4}{128\pi v_F^2|t|^3}\right)({\mathbf{m}}_1+{\mathbf{m}}_2)^2_\shortparallel\Bigg]\notag\\
&&{}+J^4\big[c_1(m_1^4+m_2^4) + c_2m_1^2m_2^2+c_3({\mathbf{m}}_1\cdot{\mathbf{m}}_2)^2\notag\\
&&{}+ c_4(m_1^2+m_2^2){\mathbf{m}}_1\cdot{\mathbf{m}}_2+ c_5(m_1^2m_{1\shortparallel}^2+m_2^2m_{2\shortparallel}^2)\notag\\
&&{}+c_6(m_1^2m_{2\shortparallel}^2+m_2^2m_{1\shortparallel}^2)+c_7({\mathbf{m}}_1^2+{\mathbf{m}}_2^2)({\mathbf{m}}_{1\shortparallel}\!\cdot\!{\mathbf{m}}_{2\shortparallel})\notag\\
&&{}+c_8(m_{1\shortparallel}^2+m_{2\shortparallel}^2){\mathbf{m}}_1\cdot{\mathbf{m}}_2+K_1({\mathbf{m}}_{1\shortparallel}+{\mathbf{m}}_{2\shortparallel})^4\notag\\
&&{}+2c_8({\mathbf{m}}_{1\shortparallel}\cdot{\mathbf{m}}_{2\shortparallel})({\mathbf{m}}_1\cdot{\mathbf{m}}_2)\big].\end{aligned}$$ The coefficients of the fourth-order terms are $$\begin{aligned}
c_1 &=& \frac{1}{64a^2|t|^3} + K_1 +K_3 - K_4\\
c_2 &=& \frac{-7}{64a^2|t|^3} + K_1 + K_2 + K_3 - K_4\\
c_3 &=& \frac{5}{16a^2|t|^3} + 4K_1 - 4K_4\\
c_4 &=& \frac{-1}{16a^2|t|^3} +4K_1 +K_2 +2K_3 - 4K_4\\
c_5 &=& \frac{7h^4}{1024\pi v_F^2|t|^5} - 2K_1 -v_F^2(K_3-K_4)\\
c_6 &=& \frac{237h^4}{1024\pi v_F^2|t|^5} -2K_1 - v_F^2(K_2+K_3-K_4)\\
c_7 &=& \frac{47h^4}{512\pi v_F^2|t|^5} -4K_1 -v_F^2(K_2+2K_3-2K_4)\,\,$$ $$\begin{aligned}
c_8 &=& \frac{-63h^4}{512\pi v_F^2|t|^5} -4K_1 + 2v_F^2K_4\end{aligned}$$ and we have used the constants $$K_1 = \frac{6435h^8}{2^{15}\pi v_F^2|t|^9}$$
$$K_2 = h^4\frac{92\pi^2v_F^2 +108\pi v_Fa|t|+33a^2t^2}{48v_F|t|^5(2\pi v_F + a|t|)^3}+\frac{5h^4\log(1+\frac{2\pi v_F}{a|t|})}{64\pi v_F^2|t|^5}$$
$$K_3 = h^4\frac{1408\pi^3v_F^3 + 2396\pi^2v_F^2a|t|+1392\pi v_Fa^2t^2+279a^3|t|^3}{384v_F|t|^5(2\pi v_F + a|t|)^4}
+\frac{35h^4\log(1+\frac{2\pi v_F}{a|t|})}{512\pi v_F^2|t|^5}$$
$$K_4 = h^4\frac{9008\pi^4v_F^4 + 20000\pi^3v_F^3a|t|+16920\pi^2v_F^2a^2t^2+6500\pi v_Fa^3|t|^3+965a^4t^4}{1280v_F|t|^5(2\pi v_F + a|t|)^5}
+\frac{63h^4\log(1+\frac{2\pi v_F}{a|t|})}{1024\pi v_F^2|t|^5}$$
It turns out that the second-order term is always negative, indicating a stable magnetic phase at the interface.
For the special cases of a FM, with ${\mathbf{m}}_1={\mathbf{m}}_2=\mathbf{n}$, and an AFM, with ${\mathbf{m}}_1=-{\mathbf{m}}_2=\mathbf{n}$, the Landau theory can be simplified: $$\begin{aligned}
\lefteqn{\mathcal{E}_\text{FM}}\notag\\
&=& -4J^2\left[t^2K_2n_\perp^2+\left(t^2K_2(1-v_F^2)+\frac{5h^4}{128\pi v_F^2|t|^3}\right)n_\shortparallel^2\right]\notag\\
&& + J^4\big[(16c_1+c_2+c_3+2c_4)n^4\notag\\
&&{}+ 2(c_5+c_6+c_7+2c_8)n_\shortparallel^2n^2+16K_1n_\shortparallel^4\big]\end{aligned}$$ $$\begin{aligned}
\mathcal{E}_\text{AFM} &=& -\frac{J^2n^2}{a^2|t|}+ J^4\big[(16c_1+c_2+c_3-2c_4)n^4 \notag\\
&&{} + 2(c_5+c_6-c_7)n_\shortparallel^2n^2\notag\\
&&{}+16K_1n_\shortparallel^4\big]\end{aligned}$$
Landau-Lifshitz equation {#App:LLE}
========================
Applying the Euler-Lagrange formalism on the total Lagrangian Eq. leads to the two equations of motion (with $i=1,2$ and $j=3-i$) $$-\frac{{\mathbf{m}}_i}{{\mathbf{m}}_i^2}\times\partial_t{\mathbf{m}}_i + b{\hat{\mathbf{e}}}_z\times\partial_t{\mathbf{m}}_i
+ c{\hat{\mathbf{e}}}_z\times\partial_t{\mathbf{m}}_j = \mathbf{d}_i. \label{Eq:eom}$$ with the coefficients $$b = 4J^2\mathcal{D}^{0z}_i-\frac{J_i^2}{4\pi v_F^2}
,$$ $$c = 4J^2\mathcal{T}^{0z}-\frac{J_1J_2}{4\pi v_F^2}$$ and the effective field $\mathbf{d}_i = \mathbf{d}^i_\text{topol} + \mathbf{d}^i_\text{non-top}$ which consists of a part generated by the CS term, $$\begin{aligned}
\mathbf{d}^i_\text{topol} &=&
\frac{eJ_i}{4\pi v_F}\mathbf{E}_\text{Cou}
+ \frac{eJ_i}{4\pi v_F}\mathbf{E}_\text{ext}
-\frac{\gamma{\overline{m}}_j}{4\pi v_F}(\nabla_\parallel\cdot\mathbf{M}){\hat{\mathbf{e}}}_z\notag\\
&&{}-\frac{\gamma J_i}{4\pi v_F}\nabla_\parallel({\overline{m}}_1 m_{2z} + {\overline{m}}_2 m_{1z})\end{aligned}$$ and the remainder containing various spin-stiffness and anisotropy terms besides the renormalized magnetic coupling of the sublattices $$\begin{aligned}
\lefteqn{\mathbf{d}^i_\text{non-top} =}\notag\\
&&-\kappa\left(\nabla_\parallel\right)^2{\mathbf{m}}_i
- \lambda{\mathbf{m}}_j
- 4J^2 D_1^{zz}m_{1z}{\hat{\mathbf{e}}}_z\notag\\
&&{}- 2J^2\text{diag}(T^{00}-T^{zz},T^{00}-T^{zz},T^{00}+T^{zz})\cdot{\mathbf{m}}_j\notag\\
&&{}+ 2J^2\left[(D^{00}_i+D^{zz}_i){\overline{m}}_i+(T^{00}+T^{zz}){\overline{m}}_j\right]{\hat{\mathbf{e}}}_z\notag\\
&&{}+\frac{m_\Psi J_i}{\pi v_F^2}(J_1{\overline{m}}_1+J_2{\overline{m}}_2-M_z){\hat{\mathbf{e}}}_z
-\frac{J_i}{12\pi m_\Psi v_F^2}\partial_t^2\mathbf{M}\notag\\
&&{}- \frac{J_i}{12\pi m_\Psi v_F}\partial_t\left[\gamma\nabla_\parallel({\overline{m}}_1 m_{2z}+{\overline{m}}_2 m_{1z})-e\mathbf{E}_\text{ext}\right]\times{\hat{\mathbf{e}}}_z\notag\\
&&{}-\frac{J_i}{12\pi m_\Psi}\nabla_\parallel\left(\nabla_\parallel\cdot\mathbf{M}\right)\notag\\
&&{}- \frac{\gamma{\overline{m}}_j}{12\pi m_\Psi v_F}\left[\partial_t(\nabla_\parallel\times\mathbf{M})\cdot{\hat{\mathbf{e}}}_z\right]{\hat{\mathbf{e}}}_z\notag\\
&&{}+\frac{\gamma^2}{12\pi m_\Psi}\left(\nabla_\parallel\right)^2({\overline{m}}_2^2 m_{1z} + {\overline{m}}_1^2 m_{2z}){\hat{\mathbf{e}}}_z\notag\\
&&{}+\frac{\gamma e}{12\pi m_\Psi}(\nabla_\parallel\cdot\mathbf{E}_\text{ext}){\hat{\mathbf{e}}}_z
- \frac{J_i}{12\pi m_\Psi}\left(\nabla_\parallel\right)^2M_z{\hat{\mathbf{e}}}_z\notag\\\end{aligned}$$ with the short-hand notation $\mathbf{M}=J_1{\mathbf{m}}_1+J_2{\mathbf{m}}_2$. The second and third term in Eq. are due to the fluctuation-induced Berry phases. Taking the cross product with $\mathbf{m}_i$ in Eq. , using $\partial_t{\mathbf{m}}_i^2=0$, one obtains $$(1-bm_iz)\partial_t{\mathbf{m}}_i - cm_{iz}\partial_t{\mathbf{m}}_j + c({\mathbf{m}}_i\cdot\partial_t{\mathbf{m}}_j){\hat{\mathbf{e}}}_z = {\mathbf{m}}_i\times\mathbf{d}_i.
\label{Eq:eom2}$$ The equations of motion can now be rewritten in matrix form, $$\Xi\cdot\begin{pmatrix}\partial_t{\mathbf{m}}_1\\\partial_t{\mathbf{m}}_2\end{pmatrix}
= \begin{pmatrix}{\mathbf{m}}_1\times\mathbf{d}_1\\{\mathbf{m}}_2\times\mathbf{d}_2\end{pmatrix},$$ where the entries of the $(6\times6)$ matrix $\Xi$ follow from Eq. : $$\begin{aligned}
\lefteqn{\Xi = \mathbb{1}_{(6\times6)}}\notag\\&&+
\begin{pmatrix}
-bm_{1z} & 0 & 0 & -cm_{1z} & 0 & 0 \\
0 & -bm_{1z} & 0 & 0 & -cm_{1z} & 0 \\
0 & 0 & -bm_{1z} & cm_{1x} & cm_{1y} & 0 \\
-cm_{2z} & 0 & 0 & -bm_{2z} & 0 & 0 \\
0 & -cm_{2z} & 0 & 0 & -bm_{2z} & 0 \\
cm_{1x} & cm_{2y} & 0 & 0 & 0 & -bm_{2z}
\end{pmatrix}\notag\\\end{aligned}$$
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---
abstract: |
Let $(X,T,\mu)$ be a Cantor minimal sytem and $[[T]]$ the associated topological full group. We analyze $C^*_\pi([[T]])$, where $\pi$ is the Koopman representation attached to the action of $[[T]]$ on $(X,\mu)$.
Specifically, we show that $C^*_\pi([[T]])=C^*_\pi([[T]]')$ and that the kernel of the character $\tau$ on $C^*_\pi([[T]])$ coming from containment of the trivial representation is a hereditary $C^*$-subalgebra of $C(X)\rtimes\mathbb{Z}$. Consequently, $\ker\tau$ is stably isomorphic to $C(X)\rtimes\mathbb{Z}$, and $C^*_\pi([[T]]')$ is not AF.
We also prove that if $G$ is a finitely generated, elementary amenable group and $C^ *(G)$ has real rank zero, then $G$ is finite.
address: ' Departamento de Matemática, Universidade Federal de Santa Catarina,Campus Universitário Trindade, 88040-900, Florianópolis - SC, Brazil'
author:
- Eduardo Scarparo
bibliography:
- 'bibliografia.bib'
title: 'On the $C^*$-algebra generated by the Koopman representation of a topological full group'
---
Introduction
============
In this work, we study the real rank zero and AF properties for certain classes of group $C^*$-algebras. The motivations are the classical equivalence between amenability of a group and nuclearity of its $C^*$-algebra, and the equivalence between local finiteness of a group and finiteness of its uniform Roe algebra worked out in [@MR2800923], [@MR3158244] and [@scarparo_2017].
For a compact metric space $X$, both $C(X)$ being AF and having real rank zero are equivalent to total disconnectedness of $X$.
If a group $G$ is countable and locally finite, then $C^*(G)$ is clearly AF (hence it has real rank zero). It is an open problem whether there exists a non-locally finite group $G$ such that $C^*(G)$ is AF.
In [@MR1164146 Theorem 2], Kaniuth proved that if $G$ is a nilpotent group and $C^*(G)$ has real rank zero, then $G$ is locally finite.
In section \[rank\], we show that if $G$ is a finitely generated, elementary amenable group, and $C^*(G)$ has real rank zero, then $G$ is finite (Theorem \[tempo\]). Our proof relies on the fact that infinite, finitely generated, elementary amenable groups virtually map onto $\mathbb{Z}$ [@MR1275829 Chapter I, Lemma 1].
Let $(X,T,\mu)$ be a Cantor minimal system and $\pi$ the Koopman representation associated to the action of the topological full group $[[T]]$ on $(X,\mu)$.
Notice that $C^*([[T]])$ does not have real rank zero, since $[[T]]$ maps onto $\mathbb{Z}$ (by [@MR978619 Theorem 1.1(i)], or [@MR1710743 Proposition 5.5]). On the other hand, by results of Matui, the commutator subgroup $[[T]]'$ is simple ([@MR2205435]) and non-locally finite (this follows from much sharper results from [@MR3103094]). Hence, commutators of topological full groups form a class which is not covered by Theorem \[tempo\].
Futhermore, it was proven by Juschenko and Monod ([@MR3071509]) that $[[T]]$ is amenable. In Section \[main\], we prove that $C^*([[T]]')$ is not AF. This is done by showing that $C^*_\pi([[T]])=C^*_\pi([[T]]')$, and that the kernel of the character $\tau$ on $C^*_\pi([[T]])$ coming from containment of the trivial representation is a hereditary $C^*$-subalgebra of $C(X)\rtimes\mathbb{Z}$. Consequently, $\ker\tau$ is stably isomorphic to $C(X)\rtimes\mathbb{Z}$, and $C^*_\pi([[T]]')$ is not AF and has real rank zero.
In Section \[odom\], we discuss examples coming from odometers.
Elementary amenable groups and real rank zero {#rank}
=============================================
Recall that the class of elementary amenable groups is the smallest class of groups containing all abelian and all finite groups, and closed under taking subgroups, quotients, extensions and inductive limits.
We will use the following fact about elementary amenable groups, due to Hillman ([@MR1275829 Chapter I, Lemma 1]). See also [@MR3586092 Lemma 1] for a slightly different proof.
\[aff\]
If $G$ is an infinite, finitely generated, elementary amenable group, then there is a subgroup of finite index of $G$ which admits a homomorphism onto $\mathbb{Z}$.
A $C^*$-algebra $A$ is said to have real rank zero if every hereditary $C^*$-subalgebra of $A$ has an approximate unit of projections (not necessarily increasing). We refer the reader to, for example, [@MR1402012 Section V.7] for other equivalent definitions of real rank zero.
\[blim\] If $A$ is an infinite-dimensional, real rank zero $C^*$-algebra, then it contains a sequence of non-zero, orthogonal projections.
Since $A$ is infinite-dimensional, there is a sequence $(a_n)_{n\in\mathbb{N}}\subset A$ of non-zero, positive elements such that $a_ja_k=0$ when $j\neq k$ (see, for example, [@MR1468229 Exercise 4.6.13] or [@MR0066569]).
For each $n\in\mathbb{N}$, take a non-zero projection $p_n$ in the hereditary (hence real rank zero) $C^*$-subalgebra $\overline{a_nAa_n}$. By construction, $p_jp_k=0$ when $j\neq k$.
\[tempo\] If $G$ is a finitely generated, elementary amenable group and $C^*(G)$ has real rank zero, then $G$ is finite.
Suppose $G$ is infinite. By Lemma \[aff\], there is a subgroup $H$ of $G$ with finite index $n$, and $\Phi\colon H\to \mathbb{Z}$ a surjective homomorphism. Let $\varphi\colon C^*(H)\to C^*(\mathbb{Z})$ be the $*$-homomorphism induced by $\Phi$, and $\varphi_n\colon M_n(C^*(H))\to M_n(C^*(\mathbb{Z}))$ the inflation of $\varphi$.
There is an injective $*$-homomorphism $\psi\colon C^*(G)\to M_n(C^*(H))$ such that the image of $\varphi_n\circ \psi$ is infinite-dimensional. For the convenience of the reader, we sketch the construction of $\psi$, which is standard.
Let $x_1,\dots,x_n\in G$ be such that $x_1=e$ and $G=\sqcup_{i=1}^nx_iH$. Consider the following unitary defined on canonical basis vectors: $$\begin{aligned}
U\colon\bigoplus_{i=1}^n\ell^2(H)&\to\ell^2(G)\\
\delta_{i,h}&\mapsto\delta_{x_ih}.\end{aligned}$$
Let $S\colon B(\ell^2(G))\to M_n(B(\ell^2(H))$ be the isomorphism induced by $U$.
By using the left regular representations $\lambda_G$ and $\lambda_H$, we see $C^*(G)$ as contained in $B(\ell^2(G))$ and analogously for $C^*(H)$.
It is easy to check that $S(\lambda_G(g))\in M_n(C^*(H))$ for every $g\in G$. Hence, $S(C^*(G))\subset M_n(C^*(H))$. Furthermore, for $h\in H$, we have that $S(\lambda_G(h))_{1,1}=\lambda_H(h)$. Let $\psi:=S|_{C^*(G)}$. Then $\varphi_n( \psi(C^*(G)))$ is infinite-dimensional.
Hence, by Lemma \[blim\], $M_n(C^*(\mathbb{Z}))\simeq M_n(C(\mathbb{T}))$ contains a sequence of non-zero, orthogonal projections. Since $\mathbb{T}$ is connected, we get a contradiction. Hence, $G$ is finite.
Recall that a $C^*$-algebra $A$ is said to have property (SP) if every non-zero hereditary $C^*$-subalgebra of $A$ contains a non-zero projection. Furthermore, $A$ is said to have residual property (SP) if every quotient of $A$ has property (SP) (see [@MR3352760 Section 7] for more details about these properties).
In the proof of Theorem \[tempo\], the only aspects of real rank zero that were used are that it implies property (SP) and that having real rank zero is closed under taking quotients. In particular, Theorem \[tempo\] remains true if one replaces “real rank zero" by “residual property (SP)".
Koopman representation of a topological full group {#main}
==================================================
Given a unitary representation $\pi$ of a group $G$, we denote by $C^*_\pi(G)$ the $C^*$-algebra generated by the image of $\pi$.
We will denote the Cantor set by $X$.
Let $\alpha$ be an action of a group $G$ on $X$ by homeomorphisms. The topological full group associated to $\alpha$, denoted by $[[\alpha]]$, is the group of all homeomorphisms $\gamma$ on $X$ for which there exists a finite partition of $X$ into clopen sets $\{A_i\}_{i=1}^n$ and $g_1,\dots,g_n\in G$ such that $\gamma|{A_i}=\alpha_{g_i}|_{A_i}$ for $1\leq i \leq n$. That is, $[[\alpha]]$ consists of the homeomorphisms on $X$ which are locally given by the action $\alpha$.
Fix $T$ a minimal homeomorphim on $X$. We denote by $[[T]]$ the topological full group associated to the $\mathbb{Z}$-action induced by $T$.
Let $\mu$ be a $T$-invariant probability measure on $X$. Note that $\mu$ is also invariant under the action of $[[T]]$ on $X$. Let $\pi\colon[[T]]\to B(L^2(X,\mu))$ be given by $\pi(g)(f):=f\circ g^{-1}$, for $g\in [[T]]$ and $f\in L^2(X,\mu)$. This $\pi$ is the so called Koopman representation associated to the action of $[[T]]$ on $(X,\mu)$.
We will use the faithful representation of $C(X)\rtimes\mathbb{Z}$ in $B(L^2(X,\mu))$, with $C(X)$ acting by multiplication operators, and, for n$\in\mathbb{Z}$, $\delta_n:=\pi(T^n)$, so that $C(X)\rtimes\mathbb{Z}:=\overline{\operatorname{span}}\{f\delta_n:f\in C(X),n\in\mathbb{Z}\}$.
Given $g\in[[T]]$ and $\{A_i\}_{i=1}^n$ a partition of $X$ into clopen sets such that $g|_{A_i}=T^{n_i}|{A_i}$ for $1\leq i\leq n$, notice that
$$\pi(g)=\sum 1_{T^{n_i}(A_i)}\delta_{n_i}. \label{deta}$$
In particular, $C^*_\pi([[T]])\subset C(X)\rtimes\mathbb{Z}$.
Given $n\in\mathbb{N}$, we say that a subset $A\subset X$ is $n$-disjoint if $$A, T(A), \dots, T^{n-1}(A)$$ are pairwise disjoint.
Suppose $A\subset X$ is a clopen and $n$-disjoint set. Consider the symmetric group $S_n$ acting on $\{0,\dots,n-1\}$. For $\sigma\in S_n$, let $\sigma_A\in[[T]]$ be given by $$\begin{aligned}
\label{sigma}
\sigma_A(x)=\begin{cases}T^{\sigma(i)-i}(x), &\text{if $0\leq i < n$ and $x\in T^i(A)$ }\\
x, & \text{if $x\notin \sqcup_{i=0}^{n-1}T^i(A)$},\end{cases}\quad x\in X.\end{aligned}$$
Note that, for $0\leq i <n$, $\sigma_A(T^i(A))=T^{\sigma(i)}(A)$.
\[sym\] Let $n\geq 4$ and $A\subset X$ be a clopen and $n$-disjoint set. For every $\sigma\in S_n$, it holds that $\pi(\sigma_A)\in C^*_\pi([[T]]')$.
Notice first that $\{1_{T^i(A)}\delta_{i-j}\}_{0\leq i,j< n}$ forms a system of matrix units in $C(X)\rtimes\mathbb{Z}$ of type $M_n(\mathbb{C})$ (we see $M_n(\mathbb{C})$ as matrices indexed by the set $\{0,\dots,n-1\}$).
Let $B:=(\sqcup_{i=0}^{n-1}T^i(A))^\mathrm{c}$ and $\varphi\colon \mathbb{C}\oplus M_n(\mathbb{C})\to C(X)\rtimes\mathbb{Z}$ be the $*$-homomorphism given by $\varphi(\alpha,e_{ij}):=\alpha 1_B+1_{T^i(A)}\delta_{i-j}$, for $\alpha\in\mathbb{C}$ and $0\leq i,j\leq n-1$.
Let $\rho\colon S_n\to \mathbb{C}\oplus M_n(\mathbb{C})$ be the direct sum of the trivial representation and the permutation representation.
Given $\sigma\in S_n$, by and , it holds that $\pi(\sigma_A)=1_B+\sum1_{T^{\sigma(i)}(A)}\delta_{\sigma(i)-i}=\varphi(\rho(\sigma))$.
Furthermore, since $n\geq 4$, the permutation representations of $S_n'$ and $S_n$ decompose into the direct sum of a trivial representation and an irreducible representation of degree $n-1$. Therefore, we have that $C^*_\rho((S_n)')=C^*_\rho(S_n)$.
Hence, $\pi(\sigma_A)\in C^*_\pi([[T]]')$ for any $\sigma\in S_n$.
Given $A\subset X$ clopen, consider the continuous function $$\begin{aligned}
t_A\colon A&\to\mathbb{N}\\
x&\mapsto\min\{k\geq 1:T^k(x)\in A\}.\end{aligned}$$ This is the so called function of first return to $A$.
Notice that, for $j\in\mathbb{Z}$, it holds that $$\label{fac}
t_{T^j(A)}\circ T^j|_A=t_A.$$
Let $T_A\in[[T]]$ be defined by $$\begin{aligned}
\label{time}
T_A(x)=\begin{cases}T^{t_A(x)}(x), &\text{if $x\in A$ }\\
x, & \text{otherwise}.\end{cases},\quad x\in X.\end{aligned}$$
If $B\subset X$ is a clopen set disjoint from $A$, then $T_A$ and $T_B$ commute.
In order to prove Lemma \[tempi\], we will have to analyze the spectrum of $C^*$-algebras generated by certain commuting unitaries, and the next lemma will be useful for this.
We consider the circle $\mathbb{T}$ as a pointed space with basepoint $1$.
\[univ\] The universal $C^*$-algebra generated by commuting unitaries $z_1,\dots,z_n$ subject to the relations $\{(z_i-1)(z_j-1)=0:1\leq i\neq j\leq n\}$ is $C(\bigvee_{k=1}^n \mathbb{T})$, with $z_k$ being given by $$\begin{aligned}
z_k\colon \bigvee_{i=1}^n \mathbb{T}&\to \mathbb{C} \\
(x,i)&\mapsto\begin{cases}x, &\text{if } i=k \\
1, & \text{if } i\neq k. \end{cases}\end{aligned}$$
Consider the embedding $F\colon \bigvee_{i=1}^n \mathbb{T}\to \mathbb{T}^n$ which takes $x$ in the $i$-th copy of $\mathbb{T}$ and sends it into $(F(x)_i)_{1\leq i \leq n}\in\mathbb{T}^n$ such that $F(x)_i:=x$ and $F(x)_j:=1$ if $j\neq i$. Also let $F'\colon C(\mathbb{T}^n)\to C(\bigvee_{i=1}^n \mathbb{T})$ be given by $F'(f):=f\circ F$, for $f\in C(\mathbb{T}^n)$.
For $1\leq i\leq n$, let $w_i\in C(\mathbb{T}^n)$ be given by $w_i(y):=y_i$, for $y\in \mathbb{T}^n$. Then $F'(w_i)=z_i$.
Assume $n>1$. Let $A:=C^*(\{(w_i-1)^k(w_j-1)^l: i\neq j\text{ and }k,l\in\mathbb{N}\})$. We claim that $\ker F'=A$. Clearly, $A\subset \ker F'$.
Let $Y:=\mathbb{T}^n\setminus\operatorname{Im}(F)$. Notice that $\ker F'=\{f\in C(\mathbb{T}^n):f|_{\operatorname{Im}(F)}=0\}\simeq C_0(Y)$. By the Stone-Weierstrass Theorem, in order to show that $A= C_ 0(Y)$, it is sufficient to show that, for every $y\in Y$, there is $f\in A$ such that $f(y)\neq 0 $, and that $A$ separates the points of $Y$. The proof of the former condition is trivial, so we only show that $A$ separates the points of $Y$.
Take $(x_1,\dots,x_n),(y_1,\dots y_n)\in Y$ distinct points. There is $i$ such that $x_i\neq y_i$. Without loss of generality, assume $x_i\neq 1$. Take $j\neq i$ such that $x_j\neq 1$. Then, by choosing $k\in\mathbb{N}$ appropriately, we get $(x_i-1)^k(x_j-1)\neq(y_i-1)^k(y_j-1)$.
Since $C(\mathbb{T}^n)$ is the universal $C^*$-algebra generated by $n$ commuting unitaries and $ C(\bigvee_{i=1}^n \mathbb{T})$ is generated by $\{z_1,\dots,z_n\}$, the result follows.
\[tempi\] Let $A\subset X$ be a clopen and $3$-disjoint set. Then $\pi(T_A)\in C^*_\pi([[T]]')$.
Given $\sigma\in S_3$, $x\in A$ and $0\leq i,j <3$, we have that $\sigma_A T_{T^i(A)}\sigma_A^{-1}(T^j(x))=T^j(x)$ if $j\neq\sigma(i)$ and
$$\begin{aligned}
\sigma_A T_{T^i(A)}\sigma_A^{-1}(T^{\sigma(i)}(x))&=\sigma_AT_{T^i(A)}(T^i(x))\\
&=T^{\sigma(i)-i}T^{t_{T^i(A)}(T^i(x))}(T^i(x))\\
&\stackrel{(*)}=T^{\sigma(i)-i}T^{t_{T^{\sigma(i)}(A)}(T^{\sigma(i)}(x))}(T^i(x))\\
&=T^{t_{T^{\sigma(i)}(A)}(T^{\sigma(i)}(x))}(T^{\sigma(i)}(x))\\
&=T_{T^{\sigma(i)}(A)}(T^{\sigma(i)}(x)),
\end{aligned}$$
where the equality in (\*) is due to . Hence, $\sigma_A T_{T^i(A)}\sigma_A^{-1}=T_{T^{\sigma(i)}(A)}$.
In particular, for $0\leq i,j< 3$, we have that $T_{T^i(A)}(T_{T^j(A)})^{-1}\in [[T]]'$.
If $0\leq i\neq j<3$, then $T^i(A)$ and $T^j(A)$ are disjoint, hence $(\pi(T_{T^i(A)})-1)(\pi(T_{T^j(A)})-1)=0$.
Then, by Lemma \[univ\], there is a $*$-homomorphism
$$\begin{aligned}
\varphi\colon C\left(\bigvee_{i=1}^3 \mathbb{T}\right)&\to C^*(\{\pi(T_{T^i(A)}):0\leq i < 3\}\\
z_i&\mapsto \pi(T_ {T^{i-1}(A)}).\end{aligned}$$
Furthermore, by the Stone-Weierstrass theorem, $C(\bigvee_{i=1}^3 \mathbb{T})$ is generated by $$\{z_iz_j^*:1\leq i,j\leq 3\}.$$ Hence, $\pi(T_A)\in C^*_\pi([[T]]')$.
\[com\] Let $(X,T,\mu)$ be a Cantor minimal system and $\pi$ the Koopman representation associated to the action of $[[T]]$ on $(X,\mu)$. Then $C^*_\pi([[T]])=C^*_\pi([[T]]')$.
By [@MR3241829 Theorem 4.7], given $m\in\mathbb{N}$, \[\[T\]\] is generated by $$\bigcup_{n\geq m}\{T_A,\sigma_A:\sigma\in S_n,\text{$A\subset X$ is clopen and $n$-disjoint}\}.$$
By Lemmas \[sym\] and \[tempi\], the result follows.
Notice that $1_X\in L^2(X,\mu)$ is invariant under $\pi([[T]])$. Therefore, $\pi$ contains the trivial representation.
\[new\] Let $\rho\colon G\to B(H)$ be a unitary representation which weakly contains the trvial representation, and $\tau$ the associated character on $C^*_\rho(G)$. Then $\ker\tau=\overline{\operatorname{span}}\{1-\rho(g):g\in G\}$.
Given $d\in\ker\tau$ and $\epsilon>0$, take $d'\in\operatorname{span}\rho(G)$ such that $\|d-d'\|<\frac{\epsilon}{2}$. Then $\|d-(d'-\tau(d'))\|=\|(d-d')+\tau(d'-d)\|<\epsilon$. Furthermore, $d'-\tau(d')\in\ker\tau\cap\operatorname{span}\rho(G)$.
Since $\ker\tau\cap\operatorname{span}\rho(G)=\operatorname{span}\{1-\rho(g):g\in G\}$, the result follows.
\[espera\] Let $\tau$ be the character on $C^*_\pi([[T]])$ coming from containment of the trivial representation. Then $\ker\tau$ is a hereditary $C^*$-subalgebra of $C(X)\rtimes\mathbb{Z}$.
We are going to show that, for $a\in C(X)\rtimes\mathbb{Z}$ and $b,c\in\ker\tau$, it holds that $bac\in\ker\tau$.
Given $A\subset X$ clopen and $2$-disjoint, notice that $(\delta_0-\delta_1)1_A(\delta_0-\delta_{-1})=\delta_0-(1_{(A\cup T(A))^\text{c}}\delta_0+1_{T(A)}\delta_1+1_A\delta_{-1})\in C^*_\pi([[T]])$.
By using telescoping sums, it follows that, for $n,m\in\mathbb{Z}$ and $A\subset X$ $2$-disjoint and clopen, $(\delta_0-\delta_n)1_A(\delta_0-\delta_m)\in C^*_\pi([[T]])$.
Given $g,h\in[[T]]$, take a basis $\mathcal{B}$ of 2-disjoint, clopen sets for the topology of $X$. Moreover, assume that, for each $A\in\mathcal{B}$, there is $n(A),m(A)\in\mathbb{Z}$ such that $g|_A=T^{n(A)}|_A$ and $h|_{h^{-1}(A)}=T^{m(A)}|_{h^{-1}(A)}$.
Then $$\begin{aligned}
(\delta_0-\pi(g))1_A(\delta_0-\pi(h))&=1_A-\delta_{n(A)}1_A-1_A\delta_{m(A)}+\delta_{n(A)}1_A\delta_{m(A)}\\
&=(\delta_0-\delta_{n(A)})1_A(\delta_0-\delta_{m(A)})\in C^*_\pi([[T]]).\end{aligned}$$
Since $C(X)=\overline{\operatorname{span}}\{1_A:A\in\mathcal{B}\}$, we conclude that, for $g,h\in[[T]]$ and $f\in C(X)$, $(\delta_0-\pi(g))f(\delta_0-\pi(h))\in C^*_\pi([[T]]).$
By Lemma \[new\] and the fact that $C(X)\rtimes\mathbb{Z}=\overline{\operatorname{span}}\{f\delta_n:f\in C(X),n\in\mathbb{Z}\}$, we conclude that, for $b,c\in\ker\tau$ and $a\in C(X)\rtimes\mathbb{Z}$, $bac\in C^*_\pi([[T]])$.
Since $\tau$ is a character, the result follows.
Let $\tau$ be the character on $C^*_\pi([[T]])$ coming from containment of the trivial representation. Then $\ker\tau$ is stably isomorphic to $C(X)\rtimes\mathbb{Z}$. In particular, $C^*_\pi([[T]]')$ has real rank zero and $C^*([[T]]')$ is not AF.
By Theorem \[espera\] and the fact that $C(X)\rtimes\mathbb{Z}$ is simple, it follows that $\ker\tau$ is a full, hereditary $C^*$-subalgebra of $C(X)\rtimes\mathbb{Z}$. Therefore, [@MR0454645 Theorem 2.8] implies that $\ker\tau$ is stably isomorphic to $C(X)\rtimes\mathbb{Z}$.
Furthermore, by Theorem \[com\], $C^*_\pi([[T]])=C^*_\pi([[T]]')$. Since $C(X)\rtimes\mathbb{Z}$ has real rank zero (see, for instance, [@MR2134336] for a proof of this fact), and $K_1(C(X)\rtimes\mathbb{Z})\simeq\mathbb{Z}$, and $K_1(A)=0$ for any AF-algebra $A$, the conclusion follows.
Odometers {#odom}
=========
We start this section by giving a description of $C^*_\pi([[T]])$ when $T$ is an odometer map.
Given $m\in\mathbb{N}$, let $\mathbb{Z}_m:=\mathbb{Z}/m\mathbb{Z}$.
\[odo\]
Let $(n_k)$ be a strictly increasing sequence of natural numbers such that, for every $k$, $n_k|n_{k+1}$. Let $\rho_k\colon \mathbb{Z}_{n_{k+1}}\to\mathbb{Z}_{n_k}$ be the surjective homomorphism such that $\rho_k(1)=1$, and define $$X:=\{(x_k)\in \prod_{k\in\mathbb{N}} \mathbb{Z}_{n_k}: \rho_{k}(x_{k+1})=x_k, \forall k\in\mathbb{N}\}.$$
Consider $$\begin{aligned}
T\colon X&\to X\\
(x_k)&\mapsto (x_k+1).\end{aligned}$$ Then $(X,T)$ is a Cantor minimal system, the so called odometer of type $(n_k)$.
For $k\in\mathbb{N}$ and $l\in\mathbb{Z}_{n_k}$, let $U(k,l):=\{(x_m)\in X:x_k=l\}$.
Using the notation from and , let, for $k\in\mathbb{N}$, $\Gamma_k:=\langle\{ T_{U(k,l)},\sigma_{U(k,0)}\in[[T]]:l\in\mathbb{Z}_{n_k},\sigma\in S_{n_k}\}\rangle$. As proven by Matui in [@MR3103094 Proposition 2.1], $\Gamma_k\subset \Gamma_{k+1}$, $\Gamma_k\simeq\mathbb{Z}^{n_k}\rtimes S_{n_k}$, and $\bigcup_k\Gamma_k=[[T]]$.
For $k\in\mathbb{N}$, let $A_k:=\overline{\operatorname{span}}\{1_{U(k,l)}\delta_m:l\in\mathbb{Z}_{n_k},m\in\mathbb{Z}\}.$ Then $A_k\subset A_{k+1}$, and $C(X)\rtimes \mathbb{Z}=\overline{\bigcup_k A_k}$.
Fix $k\in\mathbb{N}$ and consider the isomorphism $\varphi_k\colon A_k\to C(\mathbb{T},M_{\mathbb{Z}_{n_k}}(\mathbb{C}))$, such that $\varphi_k(1_{U(k,l)})=e_{l,l}$, for $l\in\mathbb{Z}_{n_k}$, and, for $z\in\mathbb{T}$, $$\begin{aligned}
(\varphi_k(\delta_1)(z))_{i,j}:=\begin{cases}1, &\text{if } 0< i\leq n_k-1\text{ and }j=i-1 \\
z, & \text{if } i=0\text{ and }j=n_k-1\\
0, & \text{otherwise}. \end{cases}\end{aligned}$$
Let $\pi\colon[[T]]\to U(C(X)\rtimes\mathbb{Z})$ be the homomorphism coming from the Koopman representation and $B_k:=\{b\in M_{\mathbb{Z}_{n_k}}(\mathbb{C}):\forall i,j\in\mathbb{Z}_{n_k},\sum_r b_{i,r}=\sum_s b_{s,j}\}$.
Then, for $\sigma\in S_{n_k}$, we have that $\varphi_k(\pi(\sigma_{U(k,0)}))=\sum e_{\sigma(i),i}$ and $$C^*(\{\varphi_k(\pi(\sigma_{U(k,0)})):\sigma\in S_{n_k}\})\simeq B_k.$$
Furthermore, $\varphi_k(C^*(\pi(\{ T_{U(k,l)}:l\in\mathbb{Z}_{n_k}\})))\simeq C(\bigvee_{l\in\mathbb{Z}_{n_k}} \mathbb{T})$ and $\varphi_k(C^*_\pi(\Gamma_k))=\{f\in C(\mathbb{T},M_{\mathbb{Z}_{n_k}}(\mathbb{C})):f(1)\in B_k\}$.
In [@MR1759493], Dykema and Rørdam gave examples of non-locally finite groups $G$ such that $C^*_{\mathrm{red}}(G)$ has real rank zero. As far as we are aware, there is no known example of non-locally finite group $G$ such that $C^*(G)$ has real rank zero.
Let $(X,T)$ be an odometer as in Example \[odo\]. Does $C^*([[T]]')$ have real rank zero?
\[sonho\] Let $(X,T)$ be an odometer of type $(n_k)$ as in Example \[odo\]. Consider
$$\begin{aligned}
J\colon X&\to X\\
(x_k)&\mapsto (-x_k).\end{aligned}$$
Then $J$ is an involutive homeomorphism on $X$ such that $JTJ=T^{-1}$. Hence, $T$ and $J$ induce an action $\alpha$ of the infinite dihedral group $\mathbb{Z}\rtimes\mathbb{Z}_2$ on $X$. We will use Matui’s technique ([@MR3103094 Proposition 2.1]) in order to compute $[[\alpha]]$.
For every $\gamma\in(\mathbb{Z}\rtimes\mathbb{Z}_2)\setminus\{e\}$, it holds that $\{x\in X:\alpha_\gamma(x)=x\}$ has empty interior (it consists of at most two elements). Hence, given $g\in[[\alpha]]$, there exists a unique continuous function $c_g\colon X\to\mathbb{Z}\rtimes\mathbb{Z}_2$ such that, for $x\in X$, $g(x)=\alpha_{c(g)}(x)$.
For $k\in\mathbb{N}$ and $l\in\mathbb{Z}_{n_k}$, let $U(k,l)$ be as in Example \[odo\] and $$\Gamma_k:=\{g\in [[\alpha]]:\text{$c_g$ is constant on $U(k,l)$ for $l\in\mathbb{Z}_{n_k}$}\}.$$
Define $J_{k,l}\in[[\alpha]]$ by $$\begin{aligned}
J_{k,l}(x)=\begin{cases}T^{2l}J(x), &\text{if $x\in U(k,l)$ }\\
x, & \text{otherwise},\end{cases}\quad x\in X.\end{aligned}$$
Then $\Gamma_k=\langle \{T_{U(k,l)},J_{k,l},\sigma_{U(k,0)}:l\in\mathbb{Z}_{n_k},\sigma\in S_{n_k}\}\rangle$ and $$\begin{aligned}
\label{nao}
\Gamma_k\simeq(\mathbb{Z}\rtimes\mathbb{Z}_2)^{n_k}\rtimes S_{n_k},\text{ } \Gamma_k\subset\Gamma_{k+1},\text{ and }\bigcup_k\Gamma_k=[[\alpha]].\end{aligned}$$
Notice that the constant sequence $(0)\in X$ is a fixed point for $J$. Hence, [@MR1245825 Theorem 3.5] implies that $C(X)\rtimes(\mathbb{Z}\rtimes\mathbb{Z}_2)$ is AF (see also [@MR962104]). Moreover, it follows from that the abelianization of $[[\alpha]]$ is locally finite.
Therefore, the two obstructions that were used for ruling out the possibility of $C^*([[T]])$ and $C^*([[T]]')$ being AF do not hold for $C^*([[\alpha]])$.
Let $\alpha$ be as in Example \[sonho\]. Is $C^*([[\alpha]])$ AF?.
Acknowledgements {#acknowledgements .unnumbered}
================
Part of this work was carried out while the author was attending the research program *Classification of operator algebras: complexity, rigidity, and dynamics* at the Mittag-Leffler Institute. The author thanks the organizers of the program and the staff of the institute for the excellent work conditions provided.
The author also thanks J. Carrión, T. Giordano, K. Li and J. Rout for helpful conversations related to topics of this work.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A simple model based on the WKB approximation for one-dimensional ballistic multi–gate reconfigurable field–effect transistors (RFETs) with Schottky-Barrier contacts has been developed for the drain current taking into account electron and hole band-to-band tunneling. By using a proper approximation of both the Fermi-Dirac distribution function and transmission probability, an analytical solution for the Landauer integral can be obtained. A comparative analysis of the two-gate and triple-gate RFETs is performed based on the numerical integration of the current integral.'
author:
- 'Igor Bejenari [^1][^2][^3]'
title: 'Drain Current Model of One-Dimensional Ballistic Reconfigurable Transistors'
---
[Bejenari : Analytical Model of One-Dimensional Ballistic Schottky-Barrier Transistors]{}
Carbon-nanotube field-effect transistor (CNTFET), analytical transport model, Schottky barrier (SB), band-to-band tunneling (BTBT), Wentzel-Kramers-Brillouin (WKB) approximation.
Introduction
============
of the recent requirements for CMOS technology listed in the International Roadmap for Devices and Systems (IRDS) [@IRDS] include high–mobility channel materials, gate–all–around (nanowire) structures, scaling down supply voltages lower than 0.6 V, controlling source/drain series resistance within tolerable limits, providing lower Schottky–barrier (SB) height, and fabrication of advanced nonplanar multi–gate and nanowire MOSFETs. Along with FETs based on semiconductor nanowires, carbon-nanotube FETs (CNTFETs) satisfy these requirements [@Guo_IEEE2004; @Franklin_2012; @Peng2017]. Downscaling the transistor dimensions goes along with a transformation of ohmic contacts into Schottky contacts [@Larson_IEEE2006; @Leonard2011]. Due to a possible low channel resistance (or even ballistic conduction), the metal-semiconductor contact resistance can significantly affect or even dominate the performance of SB transistors [@Chen_IEEE2006; @Heinze2003; @Chen2005]. In contrast to conventional FETs, multi-gate reconfigurable field–effect transistors (RFET) can be configured between an n– and p–type by applying an electrical signal, which selectively controls charge carrier injections at each Schottky contact, explicitly avoiding the material doping [@Weber2017; @Mikolajick2017]. RFETs have the potential to enable adaptive and reconfigurable electronics, which can lead to the initiation of radically new circuit paradigms and computing schemes based on the reprogrammable logic with the reduced number of required devices. Along with the electron tunneling through SB, the band-to-band tunneling (BTBT) of electrons has significant effect on RFET characteristics. This leads to an increase of current and decrease of a subthreshold swing, which can be less than its limit value of 60 mV/dec typical for MOSFETs at room temperature [@Zhang_IEEE2006]. For both tunnel- and multi-gate RFETs, it has been experimentally demonstrated, that the subthreshold limit value can be decreased down to 30 and 40 mV/dec, respectively [@Gandhi_IEEE2011; @Jeon2017; @Choi_IEEE2007; @Appenzeller93_2004].
For circuit design, the description of the device behavior based on the nonequilibrium Green’s function (NEGF) method, Wigner transport equation, and Boltzmann equation formalism is unsuitable in terms of memory and time [@Guo_IEEE2004; @Ossaimee_EL2008; @Leonard_Nanotech2006; @Maneux_SSE2013]. To reduce the computation time, TCAD simulation tools have been used to analyze the ${I-V}$ characteristics of RFETs with SB contacts solving the current integral involved in the transport calculations numerically [@Darbandy2016; @Martinie_2012]. For practical circuit design based on simulations in a SPICE-like environment, compact models are required. In the framework of the constant effective SB approximation using an energy independent transmission probability, different simple analytical expressions for the drain current have been reported in the literature for RFETs [@Martinie_2012; @Jeon2017; @Fregonese_IEEE2011; @Weber_IEEE2014]. In these models, the simulated ${I-V}$ characteristics agree with experimental data in a limited bias range [@Maneux_SSE2013]. The reason is that the analytical expression for the drain current corresponding to the thermionic emission with a shifted Fermi level and including energy-independent transmission can be used at small bias, when the contribution of thermally excited electrons in the total current is large enough [@Bejenari_IEEE2017]. The analytical current calculations on the basis of drift-diffusion model do not properly take into account the effect of SB tunneling and BTBT on the electron transport [@Antidormi_IEEE2016; @Zhang_2015]. The empirical continuous compact dc model based on a set of empirical fitting parameters is reliable in the framework of experimental data [@Hasan_2017], but it cannot be used for predictions.
In this paper, we demonstrate the drain current model, which allows to simplify solving of the current integral. It potentially enables to simulate ${I-V}$ characteristics of one-dimensional reconfigurable multil–gate transistors with SB contacts with reduced computation time. We adopt the pseudo-bulk approximation [@Fregonese_IEEE2011] to self–consistently estimate the channel potential variation under applied bias with respect to channel charge. The drain–current model captures a number of features such as ballistic transport, transmission through the SB contacts, band-to-band tunneling and ambipolar conduction. It can be applied to quasi-1D RFETs based on both nanowires and nanotubes at large bias voltages.
Transport Model
===============
Energy Band Model
-----------------
We consider $N$ gates with left- and right-end coordinates ${\left[z_{L,n},z_{R,n}\right]}$ ($n=1,2,\dots,N$) placed along the channel. The given band model was adopted from the evanescent mode analysis approach [@Oh_IEEE2000; @Michetti_IEEE2010; @Jimenez_Nanotech2007; @Zhang_2015]. The electrostatic potential, ${\psi(r)}$, inside a transistor contains a transverse potential ${\psi_{t}(r)}$, which describes the electrostatics perpendicular to the channel and represents a partial solution of Poisson’s equation, as well as a longitudinal potential ${\psi_{l}(r)}$ called evanescent mode, responsible for the potential variation along the channel. The transverse potential inside the channel is reduced to ${\psi_{t}(r)\approx \psi_{cc}}$, where ${\psi_{cc}}$ is the channel (surface) potential at the current control point [@Lundstrom_IEEE2003; @Mothes2015]. The longitudinal solution ${\psi_{l}(r)}$ is obtained solving the Laplace equation along the transport direction. Therefore, near the source and drain contacts, the conduction subband edge is given by exponentially decaying functions. Since electrons with high energy mainly tunnel through the Schottky barrier, the conduction subband edge $E_{\rm{C}}^{\rm{s}}$ ($E_{\rm{C}}^{\rm{d}}$) in the vicinity of the source (drain) contact can be approximated by a linear decaying function $$\begin{aligned}
E_{\rm{C}}^{\rm{s}}(z)=E_{m,0}-q\psi_{cc,1} + E^s_b\left(1-\frac{z}{\lambda_s}\right),
\label{eq:SBHight_1} \\
E_{\rm{C}}^{\rm{d}}(z)=E_{m,0}-q\psi_{cc,N} + E^d_b \left[1+\frac{z-L}{\lambda_{d}}\right],
\label{eq:SBHight_2} \end{aligned}$$ where $L$ is the total length of the channel, $\lambda_{s(d)}$ is a characteristic length of the decaying electrostatic potential that can be interpreted as an effective SB width and ${E^{s(d)}_b=\phi_{b}+q\psi_{cc,1(N)}-E_{m,0}-qV_{s(d)}}$ is the bias dependent potential barrier height with respect to the bottom of the $m$th conduction subband ${E_{m,0}-q\psi_{cc,1(N)}}$ at the source and drain contacts, correspondingly. For cylindrical gate-all-around FETs, the asymptotic value of $\lambda$ is approximately given by ${(2\kappa t_{\rm{ox}}+d_{\rm{ch}})/4.81}$, where ${\kappa=\epsilon_{\rm{ch}}/\epsilon_{\rm{ox}}}$ can be obtained if the oxide thickness, $t_{\rm{ox}}$, is significantly smaller than the channel diameter, ${d_{\rm{ch}}}$ [@Oh_IEEE2000]. For gate-all-around CNTFETs, the CNT diameter, ${d_{\rm{CNT}}}$, is often smaller than the oxide thickness, therefore, the asymptotic value of ${\lambda}$ is slightly modified [@Wong_IEEE2015PI]. In the case of double-gate FETs, the similar approximation of the characteristic length reads ${\lambda\approx (2\kappa t_{\rm{ox}}+t_{\rm{ch}})/\pi}$, where ${t_{\rm{ch}}}$ is the thickness of the channel [@Oh_IEEE2000].
Between two adjacent gates with bias voltages ${V_{g,n-1}}$ and ${V_{g,n}}$, the electrostatic potential is supposed to be linearly dependent on space variable in the inner part of the channel. Hence, the conduction band edge ${E_{\rm{C,n}}^{\rm{in}}}$ in the nth adjacent interval ${(z_{R,n-1},z_{L,n})}$ is defined as $$\begin{gathered}
E_{\rm{C,n}}^{\rm{in}}(z)=E_{m,0}-q\psi_{cc,n-1} \\
+q(\psi_{cc,n}-\psi_{cc,n-1})\left(\frac{z_{R,n-1}-z}{z_{L,n}-z_{R,n-1}}\right), \label{eq:band_inner} \end{gathered}$$ where index $n=2,3,\dots,N$. In the case of mirror-symmetric band structure, the valence subband edge ${E_{\rm{V}}}$ is described as ${E_{\rm{V}}(z)=E_{\rm{C}}(z)-2E_{m,0}}$.
Fig. \[fig1:BandDiagram\] shows the conduction band profile along the channel. The gate length ${L_g}$ of the device coincides with the channel length ${L}$. The metal-semiconductor barrier height referenced to source Fermi level $E_{Fs}$ is described by a bias independent parameter, ${\phi_{b}}$, which is commonly defined by the difference between the metal work function, $\phi_M$, and semiconductor electron affinity, $\chi_{SC}$, i.e., $\phi_{b}\approx\phi_M-\chi_{SC}$ [@Sze_2007; @Svensson2011; @Tung2014]. For holes, the similar parameter ${\phi^{h}_{b}}$ is given by ${\phi^{h}_{b}=E_g-\phi_{b}}$, where ${E_g=2E_{m,0}}$ is the band gap. The source and drain Fermi levels $E_{Fs}$ and $E_{Fd}$, respectively, are related as $E_{Fd}=E_{Fs}-qV_{ds}$, where $V_{ds}=V_{d}-V_{s}$ is the drain–source voltage.
The contribution of electrons injected from the source and drain to the total current depends on both the energy dependent transmission through the channel and electron distribution in the contacts.
Piece-Wise Approximation of Fermi-Dirac Distribution Function
-------------------------------------------------------------
The electron distribution in the source/drain contacts is given by the equilibrium Fermi-Dirac distribution function ${f_{FD}(E-E_F)=1/\left\{\exp\left[\left(E-E_F\right)/{k_BT}\right]+1\right\}}$. To find an analytical expression for the current, we use a piece-wise approximation for $f_{FD}(E)$ given by [@Bejenari_IEEE2017] $${ f_{\rm{app}}(E)= \left\{
\begin{array}{l}
1-\frac{1}{2}\exp\left( \frac{E-E_F}{{c_1 k_BT}}\right), E \leq E_F \\
\frac{1}{2}\exp\left(\frac{E_F-E}{{c_1 k_BT}}\right),~ E_F<E<E_F+c_2 k_BT \\
\exp\left(\frac{E_F-E}{{k_BT}}\right), ~ E\geq E_F+c_2 k_BT
\end{array}
\right.\
\label{eq:Fermi_Dirac}
}$$ where ${c_1=2\ln(2)}$ and ${c_2=2\ln^2(2)/(2\ln(2)-1)\approx 2.49}$. The approximation $f_{\rm{app}}(E)$ provides accurate values of the electron distribution function at different temperatures in the whole energy range with a maximum relative error of about 6-9 percent in the vicinity of Fermi level $E_F$.
Transmission Probability
------------------------
In order to estimate the transparency of the source/drain contacts, we use the transmission probability across each SB obtained in the framework of the Wentzel–Kramers–Brillouin (WKB) approximation. Using the effective mass (parabolic one–band) approach, the probability $T^{s(d)}_{b}(E)$ for electrons to tunnel through a linear decaying potential barrier of the kind ${E^s_b\left(1-z/\lambda\right)}$ or ${E^d_b\left[1+(z-L)/\lambda\right]}$ is given by the following expression [@Sze_2007] $$\begin{aligned}
&T^{s(d)}_{b}(E) =\exp\left\{ -\alpha \sqrt{|E^{s(d)}_{b}|} \gamma \left(E/|E^{s(d)}_{b}|\right) \right\},
\label{eq:Transmission}\\
&\gamma(x) =\left(1-x\right)^{3/2},\end{aligned}$$ where ${\alpha=4\lambda_{s(d)}\sqrt{2m^*}/(3\hbar)}$. For CNTs, the electron effective mass is ${m^*=4E_{m,0} \hbar^2/(3a^2V_{\pi}^2)}$ with $a=\SI{2.49}{\angstrom}$ - carbon–carbon atom distance and $V_{\pi}=\SI{3.033}{\electronvolt}$ – carbon $\pi-\pi$ bond energy in the tight binding model [@Mintmire1998].
To obtain an analytical expression for the current, we use the following approximation for $\gamma(x)$ in (\[eq:Transmission\]) $$\begin{aligned}
&\gamma_{\rm{app}}(x) =(1-x)(1-px),
\label{eq:gama_app}\\
&p=\varphi-\sqrt{\varphi^2-\varphi}\approx 0.618,\end{aligned}$$ where the quantity $\varphi=(1+\sqrt{5})/2$ represents the golden ratio and $x=E/E^{s(d)}_{b}$ is a dimensionless variable. The absolute error of $\gamma_{\rm{app}}(x)$ is less than 0.016 for all ${x\in[0,1]}$. Nevertheless, the implementation of $\gamma_{\rm{app}}(x)$ in (\[eq:Transmission\]) leads to an increase of relative error of the approximate transmission probability $T^{s(d)}_{\rm{app}}(E)$ with gate voltage due to term ${E^{s(d)}_{b}}$. To reduce the relative error, we introduce a correction factor ${\exp\left[\alpha \Delta (E^{s(d)}_{b})^{1/2}\right]}$ with the constant $\Delta < \text{max} \left|\gamma(x)-\gamma_{\rm{app}}(x)\right|$ in the final expression of current. The approximate transmission probability $T^{s}_{\rm{app}}(E)$ based on (\[eq:Transmission\]) and (\[eq:gama\_app\]) is used in region 2 if there is only one potential barrier.
If ${E^{s(d)}_{b}>E_{g}}$, electrons can tunnel through the band gap from valence band to conduction band and vice versa. The probability of such band-to-band (BTB) tunneling for electrons and holes with equal masses is given in the parabolic one-band approximation by $$T^{s(d)}_{\rm{BTB}} =\exp\left\{ -\alpha E^{3/2}_{g}/ E^{s(d)}_{b} \right\}.
\label{eq:BTBT}$$
In the non-parabolic two-band approximation, the energy dispersion for electrons and holes in CNT is ${E_{m,l}=\pm\sqrt{E_{m,0}^2+(\hbar v_F k_l)^2}}$, where ${v_F\approx10^8}$ cm/s is the Fermi velocity. The electron effective mass $m^*$ and ${v_F}$ are related by ${m^*=E_{m,0}/v^2_F}$. In this case, the probability $T^{s(d)}_{b}(E)$ for electrons to tunnel through a linear decaying potential barrier is obtained in the WKB approximation as $$\begin{aligned}
&T^{s(d)}_{b}(E) =\exp\left\{ - \beta \zeta \left(\frac{E+E_{m,0}-|E^{s(d)}_{b}|}{E_{m,0}}\right)/|E^{s(d)}_{b}| \right\},
\label{eq:Transmission_2Band}\\
&\zeta(x) ={\pi}/{2}-x\sqrt{1-x^2} -\arcsin\left(x\right),\end{aligned}$$ where ${\beta=\lambda_{s(d)} E^2_{m,0}/(\hbar v_F)}$ and $x=E/E^{s(d)}_{b}$ is a dimensionless variable. In the non-parabolic two-band approximation, the probability of BTB tunneling reads[@Jena_APL2008] $$T^{s(d)}_{\textrm{BTB}} =\exp\left\{ -\frac{3\pi}{16}\alpha E^{3/2}_{g}/ E^{s(d)}_{b} \right\}.
\label{eq:BTBT_2Band}$$ A comparison of (\[eq:BTBT\]) and (\[eq:BTBT\_2Band\]) shows that the probability of BTB tunneling obtained in the non–parabolic two–band approximation is greater than that obtained in the parabolic one–band approximation. In the inner part of the channel, BTB tunneling ${T^{\rm{in},n}_{\rm{BTB}}}$ of electrons between two adjacent gates is given by (\[eq:BTBT\]) or (\[eq:BTBT\_2Band\]), where ${E^{s(d)}_{b}}$ is replaced by the difference ${q|\psi_{cc,n}-\psi_{cc,n-1}|}$ and the characteristic length $\lambda_{s(d)}$ is replaced by the distance ${z_{L,n}-z_{R,n-1}}$ between two adjacent gates $n-1$ and $n$ ($n=2,3,\dots,N$).
If ${q\psi_{cc,1(N)}=E_{m,0}-\phi_{b}+V_{s(d)}}$, there is no SB located at the source (drain), then the transmission probability of electrons or holes to inject from the source (drain) into the channel is equal to 0 if the electron energy belongs to the band gap (${\phi_b-V_{s(d)} -2 E_{m,0}< E <\phi_b-V_{s(d)}}$) and it is 1 otherwise.
The probability of electron transmission through the potential barrier increases with energy. At a large gate voltage, electrons with high energy or close to the Fermi level tunnel through the thin potential barrier with a rather small reflection probability ${1-T^{s(d)}_{b}(E)}$ and mainly contribute to the current, whereas the contribution of electrons with low energy is not essential due to a small transmission probability ${T^{s(d)}_{b}(E)}$. Hence, the multiple reflections between two potential barriers can be neglected. In this case, the total transmission probability reads $$T_{tun}(E)=T^{s}(E)T^{in}_{BTB}T^{d}(E).
\label{eq:total_Transmission}$$ The approximate total transmission probability ${T^{tun}_{\rm{app}}(E)}$ can be obtained by using (\[eq:Transmission\])–(\[eq:total\_Transmission\]).
The presented approach is valid if electron-phonon scattering is relatively small, i.e., the channel length $L$ is of the order of an electron mean free path $L_{\rm{mfp}}$, such that ${L/L_{\rm{mfp}}<1/\overline{T}_b}$, where ${\overline{T}_b}$ is an average value of the SB transmission probability characterizing a source/drain contact transparency [@Knoch2008]. Depending on the applied bias, the mean free path $L_{\rm{mfp}}$ can vary from 60 to 200 nm [@Fuller2014; @Franklin_2010; @Zhang2008_nl; @Purewal2007; @Yao2000] at room temperature in CNTFETs. Also, the model does not take into account direct source-to-drain tunneling and short–channel effects (e.g., SS degradation and Drain-Induced Barrier Lowering), which are determined purely by electrostatics and essentially affect the current at ${L\approx\lambda}$ [@Knoch2008].
Total Current
-------------
To calculate the total electron current, we use the Landauer-Buttiker approximation for a one-dimensional system [@Datta_1995] $$I = \frac{4q}{h} {\int\limits_{-\infty}^\infty T_{tun}(E)\left[ f_{FD}(E-E_{Fs}) - f_{FD}(E-E_{Fd}) \right]dE},
\label{eq:current}$$ where the product of the spin and electron subband degeneracies gives a factor of 4 in front of the integral (\[eq:current\]) for CNTFETs.
Results
=======
![Total current $I$ calculated numerically as a function of tube potential ${\psi_{cc,1}}$ at the drain–source and program gate voltages ${V_{ds}=V_{pd}}$ equal to 0.1, 0.5, and 1 V. CNT chirality (19, 0), bandgap ${E_g = 0.579}$ eV, CNT diameter ${d_{\rm{CNT}}=1.48}$ nm, SB hight ${\phi_b}=0.1$ eV, characteristic length ${\lambda_{s(d)}=3}$ nm, equal gate lengths ${L_{g} = 45}$ nm, and temperature ${T = 300}$ K for the two-gate CNTFET. []{data-label="fig:I_psi_2G"}](Ids_Vpg_Vds01_05_1.eps){width="3.5in"}
Fig. \[fig:I\_psi\_2G\] shows the total current $I$ calculated numerically in the framework of two-band approximation (\[eq:Transmission\_2Band\])-(\[eq:BTBT\_2Band\]) and (\[eq:current\]) as a function of tube potential ${\psi_{cc,1}}$ at different values of drain–source voltage ${V_{ds}}$ and similar values of the program gate for the two–gate RFET with equal gate lengths of 45 nm. The program gate voltage ${V_{pd}}$ is supposed to be equal to the corresponding tube potential ${\psi_{cc,2}}$.
At a larger drain–source voltage (${V_{ds}=2}$ V), the total current strongly depends on the gate voltage in the whole interval of ${\psi_{cc}}$, because the contribution of electrons injected from the drain into the channel is negligibly small due to the large reflection of such electrons from the potential barrier in the channel. The On/Off ratio is equal to $2.12\cdot10^2$, ${1.08\cdot10^6}$, and ${2.11\cdot10^2}$ at the drain–source and program gate voltages ${V_{ds}=V_{pd}}$ equal to 0.1, 0.5, and 1 V, correspondingly. The subthreshold swing ${SS=\left( {d \log_{10}I}/{d V_{gs}} \right)^{-1}}$ equals 63, 31, and 118 mV/dec, respectively. Therefore, it can be considerably less than 60 mV/dec at ${V_{ds}=V_{pd}=0.5}$ V, when the corresponding On/Off ratio is ${1.08\cdot10^6}$.
![Total current $I$ calculated numerically as a function of tube potential ${\psi_{cc,2}}$ at the drain–source and program gate voltages ${V_{ds}=V_{pd}=V_{ps}}$ equal to 0.1, 0.5, and 1 V. CNT chirality (19, 0), bandgap ${E_g = 0.579}$ eV, CNT diameter ${d_{\rm{CNT}}=1.48}$ nm, SB hight ${\phi_b}=0.1$ eV, characteristic length ${\lambda_{s(d)}=3}$ nm, and temperature ${T = 300}$ K for the triple-gate CNTFET with a length of 25, 20, and 25 nm of the 1st, 2nd, and 3rd gate, respectively. []{data-label="fig:I_psi_3G"}](Ids3G_Vpg_Vds01_05_1.eps){width="3.5in"}
Fig. \[fig:I\_psi\_3G\] depicts the total current $I$ calculated numerically in the framework of two-band approximation (\[eq:Transmission\_2Band\])-(\[eq:BTBT\_2Band\]) and (\[eq:current\]) as a function of tube potential ${\psi_{cc,2}}$ at different values of drain–source voltage ${V_{ds}}$ and similar values of the program gates for the triple–gate RFET. The program gate voltages ${V_{ps}}$ and ${V_{pd}}$ are supposed to be equal to the values of corresponding tube potentials ${\psi_{cc,1}}$ and ${\psi_{cc,3}}$. The On/Off ratio is equal to $2.53\cdot10^2$, ${7.17\cdot10^5}$, and ${4.77\cdot10^3}$ at the drain–source and program gate voltages ${V_{ds}=V_{pd}=V_{ps}}$ equal to 0.1, 0.5, and 1 V, correspondingly. The subthreshold swing ${SS}$ equals 62, 59, and 63 mV/dec, respectively. A comparison of transfer characteristics shown in Fig. \[fig:I\_psi\_2G\] and Fig. \[fig:I\_psi\_3G\] indicates that the contribution of holes to the current of the n–type triple-gate RFET is diminished by an order of magnitude compared to that of the n–type two-gate RFET, i.e. the ambipolarity is greatly reduced. As a result, the On/Off ratio for the triple-gate RFET is greater by an order of magnitude in comparison to that for the two-gate CNTFET at large ${V_{ds}}$. But, the $SS$ is similar to the thermionic limit value of 60 mV/dec, which is about twice greater than ${SS=31}$ mV/dec for the double-gate RFET at ${V_{ds}=V_{pd}=0.5}$ V.
Conclusion
==========
A simple model for ballistic one-dimensional multi-gate transistors with SB contacts taking into account band-to-band tunneling has been developed. The model allows to find an analytical solution of the current integral, therefore, it can significantly decrease the evaluation times and eases the implementation of the model in Verilog-A. We have introduced a piece-wise approximation for Fermi–Dirac distribution function and modified the transmission probability using simple elementary functions, which allow to simplify the current calculations. Our model can be used for the analysis of experimental data as well as for performance predictions for different SB heights, characteristic lengths, gate lengths, and either electron effective mass or band gap of channel material for quasi-1D multi–gate RFETs based on both semiconductor nanowires and nanotubes. A comparative analysis showed, that the ambipolarity in the triple-gate RFETs is strongly suppressed compared to that in the two-gate RFETs. In contrast, the subthreshold swing for two-gate RFETs can reach a minimum value of 31 mV/dec, which is about twice less than 60 mV/dec typical for the triple-gate RFETs.
The author would like to thank Prof. Michael Schröter and Dr. Martin Claus for valuable discussions.
[45]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[^1]: This work was supported in part by DFG project CL384/2 and DFG project SCHR695/6.
[^2]: I. Bejenari is with the Chair for Electron Devices and Integrated Circuits, Department of Electrical and Computer Engineering, Technische Universität Dresden, 01062, Germany.
[^3]: I. Bejenari is also with Institute of Electronic Engineering and Nanotechnologies, Academy of Sciences of Moldova, MD 2028 Chisinau, Moldova (e-mail:[email protected]).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We discuss some mathematical aspects of the problem of inverting gravitational field data to extract the underlying mass distribution. While the forward problem of computing the gravity field from a given mass distribution is mathematically straightforward, the inverse of this forward map has some interesting features that make inversion a difficult problem. In particular, the forward map has an infinite-dimensional kernel which makes the inversion fundamentally non-unique. We characterize completely the kernels of two gravitational forward maps, one mapping mass density to the Newtonian scalar potential, and the other mapping mass density to the gravity gradient tensor, which is the quantity most commonly measured in field observations. In addition, we present some results on unique inversion under constrained conditions, and comment on the roles the kernel of the forward map and non-uniqueness play in discretized approaches to the continuum inverse problem.'
author:
- Ulvi Yurtsever
- Caren Marzban
- Marina Meilǎ
title: On the Gravitational Inverse Problem
---
[^1]
[**Weighing the shape of a gravitating body**]{}
More than thirty years ago, Mark Kac asked “Can you hear the shape of a drum?" meaning: do two distinct planar domains always have distinct spectra of eigenvalues for their respective Laplace operators (acting on functions) with the usual Dirichlet (or Neumann) boundary conditions? If the answer is yes, the shape of a “drum" can be inferred by hearing its spectrum (characteristic sound), if the answer is no, then two distinctly shaped drums may have identical spectra (in which case they are called “isospectral domains") [@Kac1966; @Conway1994].
Kac’s article [@Kac1966] stimulated a long line of research which eventually settled his question in the negative: There do exist isospectral domains (and, more generally, isospectral Riemann surfaces and isospectral Riemannian manifolds in higher dimensions) which are [*not*]{} isometric. In other words, the spectral inverse problem is ill-defined, subject to a fundamental ambiguity which can be precisely characterized [@berger].
A similar ambiguity plagues the gravitational inverse problem, that is, the problem of inferring the precise shape of a mass distribution by observing its distant gravitational field.
The gravitational inverse problem is the problem of inverting the gravitational forward map, which we take to be a map sending a compact supported mass distribution to a gravity observable: in practice, the observable could be either the Newtonian gravitational potential or gravity gradients.
More precisely, and focusing on the gravity potential $\Phi$ for the moment, what we will mean by the gravitational inverse problem is the following: Given a spherical region $B_R=\{\vec{r} : |\vec{r}| < R \}$ of radius $R$ in ${{\mbox{\bbold R}}}^3$, and a solution $\Phi(\vec{r})$ (the gravitational potential in free space) of the Laplace equation ${\nabla}^2 \Phi=0$ outside the region $B_R$ (i.e.for $|\vec{r}|>R$) which vanishes at infinity, find a mass density distribution $\rho(\vec{r}\, ')$ supported inside $B_R$ which gives rise to $\Phi(\vec{r})$ in the exterior region outside $B_R$. In plainer language, find a $\rho(\vec{r}\, ')$ with support inside $B_R$ such that $$\Phi(\vec{r})= - \, G \int_{B_R} \frac{\rho(\vec{r}\, ')}{|\vec{r} - \vec{r}\, '|}
\, d^3 r' \; \; \; \; \; \; \; \; \; {\rm for} \; r > R \; .$$
[**Kernel of the forward map onto the gravitational potential**]{}
Equation (1) of course represents the unique solution to the “forward problem" where one searches for a solution $\Phi$ to ${\nabla}^2 \Phi = 4 \pi G \rho$ with vanishing boundary conditions at infinity. Of key interest is the “kernel" of this (linear) forward map, i.e. the set of mass distributions $\rho$ supported inside $B_R$ that are mapped to a potential $\Phi$ via Eq.(1) which identically vanishes outside $B_R$.
[****]{}: The kernel of the forward map Eq.(1) mapping mass distributions $\rho$ supported in $B_R$ to solutions of Laplace equation outside the region $B_R$ (i.e.for $|\vec{r}|>R$) is precisely functions $\rho$ satisfying $$\rho = {\nabla}^2 \chi \; ,$$ where $\chi(\vec{r})$ is any (sufficiently smooth) function on ${\mbox{\bbold R}}^3$ with support inside $B_R$ (i.e.$\chi(\vec{r})=0$ for $r>R$). In other words, if $\rho$ is a solution of the inverse problem for a given exterior potential $\Phi$, then $\rho + \nabla^2 \chi$ is also a solution for any $\chi \in {C_0}^{\! \alpha}(B_R)$, where $\alpha$ is a sufficiently large integer. Normally, $\alpha \geq 2$ should be sufficient, but smoothness is not a key issue; in particular, $\chi$ can even be a distribution if point-mass (delta-function) singularities need to be allowed in the problem.
[**Proof**]{} in one direction is easy: Every function in the kernel is given by the forward image of a function of the kind Eq.(2). To prove this, let $\Phi$ be a function belonging to the kernel, i.e. let $\Phi$ vanish outside $B_R$. Put $$\chi \equiv \frac{1}{4 \pi G} \Phi \; .$$ Then $\chi \in {C_0}^{\! \alpha}(B_R)$ and $\rho \equiv \nabla^2
\chi$ satisfies the Laplace equation ${\nabla}^2 \Phi = 4 \pi G \rho$ everywhere (with vanishing boundary conditions at infinity). Therefore, $\Phi$ satisfies Eq.(1) with this $\rho$, which is what we needed to prove.
Conversely, let $\rho$ be a density distribution supported inside $B_R$ such that $\rho = \nabla^2 \chi$ for some $\chi \in {C_0}^{\! \alpha}(B_R)$. Then, according to Eq.(1), the gravitational potential $\Phi$ which the forward map sends $\rho$ onto satisfies $$\Phi (\vec{r}) =
- \, G \int_{B_R} \frac{\nabla^2 \chi (\vec{r}\, ' )}{|\vec{r} - \vec{r}\, '|}
\, d^3 r' \; \; \; \; \; \; \; \; \; {\rm for} \; r > R \; .$$ To show that the right hand side of Eq.(3) is in the kernel of the forward map, i.e., that it vanishes for $r>R$, use Green’s identity: $$\int_{B} (U \nabla^2 V - V \nabla^2 U) \, d^3 r
=\int_{\partial B} \left( U \frac{\partial V}{\partial n}
- V \frac{\partial U}{\partial n} \right) d \sigma \; ,$$ where $B$ is any region bounded by the surface $\partial B$, and $U, \;
V$ are arbitrary functions on ${\mbox{\bbold R}}^3$. Applying Eq.(4) with $B$ taken as the region $B_R$, $U(\vec{r}\, ') \equiv
1/|\vec{r}-\vec{r}\, '|$, and $V (\vec{r}\, ' ) \equiv \chi (\vec{r}\, ' )$, and noting that $\nabla^2 (1/|\vec{r}-\vec{r}\, '|) =0 $ when $r>R$ and $r' < R$, it immediately follows that the right hand side of Eq.(3) vanishes outside $B_R$ (i.e. for $r>R$). This completes the proof of Theorem 1.
[**A geometric interpretation of the kernel:**]{}
One way to conceptualize the kernel of the gravitational (potential) forward map is to note that the geometric freedom of choice in the inverse “datum" $\Phi(\vec{r})$ is that of choosing an arbitrary function on the two-sphere $S^2$:
[****]{}: Let $\Phi(\vec{r})$ be any solution of the free-space Laplace equation ${\nabla}^2 \Phi=0$ outside the region $B_R$ which vanishes at infinity, as in the formulation of the gravitational inverse problem. Then $\Phi$ is completely determined outside $B_R$ by its values on any two-sphere $S_{R_1}$ of radius $R_1 >R$ (or, more generally, by its values on any closed surface which encloses $B_R$).
[**Proof:**]{} This is really a restatement of a standard result in potential theory (uniqueness of solutions to the Dirichlet problem): There exists a unique Green’s function $G(\vec{r}, \vec{r}_0 )$, defined for $\vec{r}$, $\vec{r}_0$ outside $B_{R_1}$, such that $G$ satisfies $\nabla^2 G(\vec{r},
\vec{r}_0 ) = \delta(\vec{r} - \vec{r}_0 )$ and vanishes for $\vec{r} \in S_{R_1}$ and for $\vec{r} \in S_{\infty}$ \[as discussed, e.g., in [@Jackson], for the two-sphere $S_{R_1}$ $G(\vec{r}, \vec{r}_0 )$ can be constructed explicitly using the classic “method of images"\]. Plugging such a $G(\vec{r}, \vec{r}_0)$ into Eq.(4) as $V$ and taking $\Phi(\vec{r})$ as $U$ and the region $B$ as the region [*outside*]{} $B_{R_1}$ we obtain, by virtue of the vanishing boundary conditions at infinity, $$\Phi(\vec{r}_0 ) = \int_{S_{R_1}} \Phi(\vec{r}) \frac{\partial G(\vec{r},
\vec{r}_0)}{\partial n} \, d \sigma \; .$$ Therefore $\Phi$ everywhere outside $B_{R_1}$ is determined uniquely by its values on the two-sphere $S_{R_1}$.
We can now understand the kernel Eq.(2) in the following way: Since the data for $\Phi$ consist of the values of a function defined on a two-surface $S_{R_1}$, we can infer from these data uniquely at the most another function of two variables, and not the full three-dimensional density field $\rho(\vec{r})$. In fact, the forward kernel (or the ambiguity in the corresponding inversion) as described by Eq.(2) corresponds precisely to this geometric statement.
[**Weighing the shape of a body of known radial density**]{}
One might hope that practical (physical) prior constraints on the three-dimensional density distribution $\rho(\vec{r})$ might make it uniquely recoverable from its far-zone gravity field despite the fundamental non-uniqueness of the inverse problem. For example, we want the density to be positive everywhere, which is a requirement that constrains the ambiguity Eq.(2) to some extent. However, simple spherically-symmetric counterexamples show that positivity is not a sufficiently strong constraint to help provide us with a unique inversion. As the next step in a series of physically-reasonable constraints on $\rho$, we might assume a known positive radial density distribution with profile $\rho(\vec{r}) \equiv \rho_0 (r) > 0$ distributed on some arbitrary compact three-dimensional region $D$ in ${\mbox{\bbold R}}^3$. Put another way, such a density profile represents a body of arbitrary shape carved out of a spherically symmetric (hence spherical) mass distribution. Again, counterexamples based on hollow spherical shells show that this is not quite enough for unique inversion. Nevertheless, it turns out that if we further constrain the region $D$ such that it is connected and has no “holes" (i.e., if $D$ is topologically a ball), and, furthermore, if $D$ is “radially convex" in a sense made precise below, then unique inversion is possible:
[****]{}: Let $D$ be compact region in ${\mbox{\bbold R}}^3$ such that its boundary $\partial D$ is a connected and simply-connected surface (in other words, $\partial D$ is a topological two-sphere) which is [*radially convex*]{} in the sense that any straight line in ${\mbox{\bbold R}}^3$ passing through the center-of-mass of the volume $D$ intersects $\partial D$ at precisely two points. Assume that $D$ is filled with material of a known non-negative mass density $\rho(\vec{r})$ which, when it is nonzero, is distributed spherically-symmetrically with respect to the coordinate origin given by the center of mass. That is, if $\vec{r}$ lies inside $D$, then $\rho(\vec{r}) = \rho_0 (r) > 0$, and if $\vec{r}$ is outside $D$, then $\rho(\vec{r}) = 0$. Under these conditions, $D$ itself (or, equivalently, its boundary $\partial D$) is uniquely recoverable from the far-zone gravity field of this radial density distribution.[^2]
This result is not too surprising in view of Theorem 2, since the specification of $\partial D$ entails just a single real function on the two-sphere $S^2$ (measuring just how much we need to deform $S^2$ in order to stretch it onto $\partial D$). The forward map Eq.(1) can then be interpreted as a nonlinear map from real functions on $S^2$ (representing the deformations of $S^2$ needed to obtain $\partial D$) to real functions on $S^2$ (representing the values of the potential $\Phi$ on $S_R$), and we will now show that this map is locally one-to-one.
[**Proof of Theorem 3:**]{} The main idea of the proof is simple: explicitly write down, in terms of spherical-harmonic coefficients, the forward transform mapping the “shape function" of $\partial D$ to the exterior potential $\Phi$, and show that the derivative of this nonlinear forward map is nonsingular. The result then follows from the inverse function theorem as generalized to infinite-dimensional spaces [@Marsden]. In this paper we will give a detailed proof that the forward map has nonsingular derivative at the point (shape) which corresponds to a perfect sphere, so the result holds for shapes which are nearby distortions of a perfect sphere (in other words, we will explicitly prove that the forward map is invertible in some open neighborhood of the perfect sphere in the space of all shapes $D$ which satisfy the conditions of the theorem). This case covers most planetary bodies at the levels of resolution we are interested in. Nevertheless, the statement that the forward map is nonsingular everywhere remains valid, although we are not going to give an explicit proof of it here. The proof of this more general case is substantially similar apart from requiring more careful estimates.
To proceed with the proof, introduce a spherical coordinate system $( r, \theta , \phi )$ centered at the center of mass of the volume $D$. In this coordinate system, let ${\bf n}(\theta , \phi )$ denote the half-line which starts at the origin and expands outward in the direction $(\theta , \phi )$. Let $\psi (\theta , \phi )$ be the (positive) function which gives the length of the radial vector which starts at the origin and ends at the intersection point of the line ${\bf n}(\theta , \phi )$ with the boundary $\partial D$ as $(\theta , \phi )$ ranges over the unit two-sphere $S^2$ of all possible directions \[by the radial convexity assumption, there exists a unique such intersection point for each direction $(\theta , \phi )$\]. The function $\psi (\theta , \phi )$ can then be taken to be the “shape function" which specifies $D$, and, explicitly, we can write $$D = \{ (r, \Omega) \; | \; r \leq \psi ( \Omega ) \} \; ,$$ and $$\partial D = \{ (r, \Omega ) \; | \; r = \psi ( \Omega ) \} \; ,$$ where we introduced the short-hand notation $\Omega \equiv (\theta ,
\phi )$ for the angular coordinates. The exterior gravitational potential $\Phi$ can be expanded in spherical harmonics[@Jackson]: $$\Phi(\vec{r}) = \sum_{l,m} d_{lm} \frac{Y_{lm} (\Omega )}{r^{l+1}} \;
\; \; \; \; \; \; \; {\rm for} \; r>R> \max_{\Omega} \psi(\Omega ) \; ,$$ where we can regard $\{ d_{lm} \} \equiv {\bf D}$ as an infinite sequence (vector) of “observables" which completely describes the data for the inverse problem in view of Theorem 2. On the other hand, according to Eq.(1), for $r>R> \max_{\Omega} \psi(\Omega )$ we have $$\begin{aligned}
\Phi(\vec{r}) & = & G \int_{r' < \psi(\Omega ' )} \rho_0(r') \, \frac{d^3 r'}
{| \vec{r} - \vec{r}\, ' |} \nonumber \\
& = & G \int_{0}^{\psi( \Omega ')} \rho_0(r') \, r'^2 \, dr' \int_{S^2}
\sum_{l,m} \frac{r'^l}{r^{l+1}} Y_{lm}(\Omega) Y^{\ast}_{lm}(\Omega ')
\, d \Omega ' \nonumber \\
& = & G
\sum_{l,m} \frac{Y_{lm}(\Omega)}{r^{l+1}} \int_{S^2} Y^{\ast}_{lm}(\Omega ')
\; \mu_{l+2} \left( \psi(\Omega ') \right)
\, d \Omega ' \nonumber \\
& = & G \sum_{l,m} f_{lm}[\psi ] \frac{Y_{lm} (\Omega )}{r^{l+1}} \; ,\end{aligned}$$ where $$\mu_n (w) \equiv \int_{0}^{w} \rho_0 (r) \, r^n \, dr \; , \tag{10a}$$ and $$f_{lm} [\psi ] \equiv \int_{S^2}
Y^{\ast}_{lm}(\Omega ) \; \mu_{l+2} \left( \psi(\Omega ) \right) \, d \Omega \;$$ is a vector functional of the shape function $\psi$ which represents the forward map in the same way as ${\bf D} = \{ d_{lm} \}$ represents the data. In fact, introducing the notation ${\bf F} [\psi ] \equiv \{ f_{lm} [ \psi ] \}$ and combining Eqs.(8) and (9), the forward equation for the shape function $\psi$ takes the simple form $${\bf F} [\psi ] = \frac{1}{G} \; {\bf D} \; .$$ (Due to our choice of origin as the center of mass, both $f_{lm}$ and $d_{lm}$ vanish for $l=1$, but this fact will not be of any consequence in what follows.) It is also convenient to introduce a coordinatization of the space of shape functions via a spherical harmonic expansion $$\psi(\Omega ) \equiv \sum_{l,m} s_{lm} Y_{lm}(\Omega ) \; ,$$ and consider the coordinate vector ${\bf S} \equiv \{ s_{lm} \}$ as the representation of the function $\psi (\Omega )$. In this coordinate system the forward map Eq.(11) takes the form $${\bf F} [{\bf S} ] = \frac{1}{G} \; {\bf D} \; ,$$ where $$f_{lm} [{\bf S}] \equiv \int_{S^2}
Y^{\ast}_{lm}(\Omega ) \; \mu_{l+2} \left(
\sum_{p,q} s_{pq} Y_{pq}(\Omega ) \right) d \Omega \; .$$ Assume now, contrary to the conclusion of Theorem 3, that two distinct domains $D_1$ and $D_2$ constrained as in the statement of the theorem give rise to identical external gravitational potentials when filled with the given radial density distribution $\rho_0 (r)$. First of all, since the monopole and dipole moments of the two mass distributions must agree, they must have the same center of mass, therefore we can set up a common spherical coordinate system for both volumes with their shared center of mass chosen as the origin of coordinates. It then follows that there exist two distinct shape functions $\psi_1$ and $\psi_2$, corresponding to the two distinct volumes $D_1$ and $D_2$, which satisfy Eq.(11) with the same data $\bf D$; in other words $${\bf F} [\psi_2 ] = {\bf F} [\psi_2 ] \; .$$ We will now show that Eq.(15) is impossible as long as $\psi_1$ and $\psi_2$ belong to some fixed open neighborhood of a perfect sphere $\{ \psi (\Omega ) \equiv a_0
= {\rm const} \}$ in the infinite-dimensional nonlinear function space of all $\psi$’s. Using the inverse function theorem as generalized to such infinite-dimensional manifolds [@Marsden], it is sufficient to show that the derivative of the map $\bf F$ at the point $\psi(\Omega )
\equiv a_0 $ is a nonsingular linear map. In general, at an arbitrary point $\psi = \psi_0 (\Omega)$, this derivative is given by $$({\bf F}' [\psi_0(\Omega)] \cdot \delta \psi)_{lm} = \int_{S^2}
Y^{\ast}_{lm}(\Omega ) \; \rho_0 \left(
\psi_0 (\Omega ) \right) \;
\psi_0 (\Omega )^{l+2} \; \delta \psi
(\Omega ) \, d \Omega \; ,$$ where ${\bf F}' [\psi_0(\Omega)]$ denotes the derivative evaluated at the point $\psi = \psi_0$, acting (as a linear map) on the tangent vector (linear perturbation) $\delta \psi$, and we have used Eq.(10) to derive this explicit form. Specializing to the perfect sphere $\psi_0(\Omega ) = a_0 (= {\rm const})$ and using the coordinate representation \[cf. Eq.(12)\] $$\delta \psi(\Omega ) \equiv \sum_{p,q} \delta s_{pq} Y_{pq}(\Omega ) \; ,
\; \; \; \; \; \; \delta {\bf S} \equiv \{ \delta s_{pq} \} \; ,$$ Eq.(16) takes the form $$\begin{aligned}
({\bf F}' [a_0] \cdot \delta {\bf S})_{lm} & = & \rho_0 (a_0 ) \, {a_0}^{l+2} \int_{S^2}
Y^{\ast}_{lm}(\Omega ) \left[ \sum_{p,q} \delta s_{pq} Y_{pq}(\Omega )
\right] \, d \Omega \; \nonumber \\
& = & \rho_0 (a_0 ) \, {a_0}^{l+2} \delta s_{lm} \; ,\end{aligned}$$ where we made use of the fact that the $Y_{lm}$’s form an orthonormal basis for $L^2 (S^2)$. According to Eq.(18), the derivative ${\bf F}' [a_0]$ is a [*diagonal*]{} linear map with only nonzero entries (eigenvalues) on the diagonal; therefore, ${\bf F}' [a_0]$ is clearly nonsingular. This completes the proof of Theorem 3.
[**The kernel of the forward map onto gravity gradient observables**]{}
The gravitational gradient tensor is (apart from a minus sign) simply the (symmetric) tensor of second derivatives of the potential $\Phi$ in a cartesian coordinate system: $$T_{ij} \equiv - \frac{\partial^2 \Phi}{\partial x^i \partial x^j} \; .$$ So, for example, we have $$T_{xx} = - \frac{\partial^2 \Phi}{\partial x^2} \; ,
\; \; \; \; \; \; \; T_{yz} = -
\frac{\partial^2 \Phi}{\partial y \, \partial z} \;$$ etc. Independently of coordinates, the gradient tensor can be defined as the double covariant derivative $\nabla \nabla \Phi$ (in general relativity, $T_{ij}$ corresponds to the Riemann curvature tensor $R_{0i0j}$ describing tidal gravitational forces). One can also define the gradient tensor explicitly in terms of the source mass distribution as: $$T_{ij} (\vec{x}) = G \int \frac{\rho(\vec{y}) \,
[\, 3 \, (x^i - y^i ) (x^j - y^j ) - \delta_{ij} \, |\vec{x} - \vec{y}|^2 \, ]}
{|\vec{x} - \vec{y}|^5}
\; d^3 y \; ,$$ where all coordinates are cartesian. The gradient tensor is a particularly useful observable in precision gravimetry since it is better isolated from local non-gravitational acceleration noise compared to other observables, and a large roster of instruments (gradiometers) are available for measuring it.
In practical applications, one often works with a coordinate system where $z$ is the vertical coordinate pointing up from the Earth’s center, and the observable of interest is the $x$—$y$ projection of the gradient tensor in an infinitesimally small neighborhood (tangent plane to the Earth’s spherical surface) around $x=y=0$: $$T \equiv
\begin{pmatrix}
T_{xx} & T_{xy} \cr
T_{xy} & T_{yy}
\end{pmatrix}$$ Typically, a gradiometer takes two kinds of measurements: the component $M_{\times} \equiv 2 T_{xy}$ (“crossline"), and the combination $M_{+} \equiv T_{xx} - T_{yy}$ (inline). The choice of cartesian $x,y$ coordinates is arbitrary upto a rotation $R$, and $T$ transforms under rotations as $$T \longrightarrow R \, T R^{t} \; .$$ Neither $T$ nor its crossline or inline components are invariant under rotations, but $\rm{Tr} (T)$ and $\rm{Det} (T)$ are invariants. In particular, the Euclidean norm of $(M_{+}, M_{\times})$ $$\sqrt{M_{+}^2+M_{\times}^2} = \sqrt{ \mbox{Tr} (T)^2 - 4 \, \mbox{Det} (T) } \;$$ is an invariant. More specifically, it is easy to compute that $(M_{+}, M_{\times})$ transforms under a rotation $$R_{\theta} =
\begin{pmatrix}
\cos \theta & - \sin \theta \cr
\sin \theta & \cos \theta
\end{pmatrix}$$ according to the rule $$\begin{pmatrix}
M_{+} \cr
M_{\times}
\end{pmatrix}
\longrightarrow R_{2 \theta}
\begin{pmatrix}
M_{+} \cr
M_{\times}
\end{pmatrix} \; .$$ Now, define a new observable $$V \equiv M_{+}^2+M_{\times}^2 \; .$$ In other words, $V$ is defined at every point of ${{\mbox{\bbold R}}}^3$ by setting up local coordinates $x,y,z$ such that the $z$-axis goes through the origin, computing the gradient tensor $T$ as in Eqs.(19) and (22) at that point, and then calculating the norm square of the observable $(M_{+}, M_{\times})$. The key features of $V$ are that (i) it is invariant under rotations, therefore uniquely defined independently of the choice of coordinates, and (ii) it obeys the following lemma:
[**Lemma:**]{} $V$ vanishes at a point if and only if $(M_{+}, M_{\times})$ vanishes there for any allowed choice of coordinates $x,y,z$.
[**Proof:**]{} By Eq.(11), $V$ is invariant and equal to the norm-square of $(M_{+}, M_{\times})$ for any choice of coordinates.
Therefore the kernel of the coordinate-dependent observable $(M_{+}, M_{\times})$ is precisely the kernel of the nonlinear but coordinate-independent observable $V$.
[****]{} Let $F[\Phi ]$ be any analytic functional (linear or nonlinear) of the gravity potential $\Phi $ (such as $V$), and let $S$ be any analytic 2-surface lying outside the spherical region $B_R=\{\vec{r} : |\vec{r}| < R \}$ in ${{\mbox{\bbold R}}}^3$ (such as a sphere of radius $> R$). Then, if $F[\Phi ]$ vanishes in any open (two-dimensional) neighborhood on $S$, it vanishes identicallly on all of $S$.
[**Proof:**]{} This just follows from analyticity: $\Phi$ is an analytic function outside the spherical region $B_R=\{\vec{r} : |\vec{r}| < R \}$ in ${{\mbox{\bbold R}}}^3$ since it satisfies the homogeneous Laplace equation there (analyticity follows from standard elliptic regularity theorems [@ellipreg]). Therefore, $F[\Phi ]$ is analytic there and so is its restriction to $S$ since $S$ is analytic. Thus vanishing on any open subset is equivalent to vanishing identically on $S$.
[**Corollary:**]{} If $V$ vanishes in any two-dimensional patch, no matter how small, on any analytic observation surface $S$ lying outside the spherical region $B_R=\{\vec{r} : |\vec{r}| < R \}$, then it vanishes identicallly on all of $S$.
This corollary further illustrates the fact that the kernel of the gravity-gradient observable $(M_{+}, M_{\times})$ is precisely the kernel of the observable $V$, not only locally but also globally.
[****]{} Let $S$ be a sphere of radius $> R$. Then $V$ vanishes on $S$ if and only if $\Phi$ is spherically symmetric (a function of the radius $r$ only), and hence $V$ vanishes identically everywhere outside the spherical region $B_R=\{\vec{r} : |\vec{r}| < R \}$ in ${{\mbox{\bbold R}}}^3$.
[**Proof:**]{} The if part is a simple calculation: It is straightforward to compute that both $M_{+}$ and $M_{\times}$ vanish for a radial (monopole) potential function $\Phi(r)$. For the converse, it is easy to see that if $V$ vanishes on a sphere $S$ then this implies that $\Phi$ is constant on $S$. But this implies, according to Eq.(5) (or just by the uniqueness of solutions to the Dirichlet problem), that $\Phi$ is a radial function of monopole type: $$\Phi (\vec{r} ) = \frac{C}{r} \; ,$$ where $C$ is a constant.
We can now completely characterize the kernel of the forward map from the mass density to the gravity gradient observables $(M_{+}, M_{\times})$:
[****]{}: The kernel of the forward map mapping mass distributions $\rho$ supported in $B_R$ to gravity gradient observables $(M_{+}, M_{\times})$ outside the region $B_R$ (i.e.for $|\vec{r}|>R$) is precisely functions $\rho$ supported in $B_R$ satisfying $$\rho = \rho_0 \, + \, {\nabla}^2 \chi \; ,$$ where $\chi(\vec{r})$ is any (sufficiently smooth) function on ${\mbox{\bbold R}}^3$, and $\rho_0$ is any [*spherically symmetric*]{} function, both supported inside $B_R$ (i.e., both $\chi(\vec{r})$ and $\rho_0 (\vec{r})$ vanish for $r>R$).
[**Proof:**]{} By Theorem 5, the kernel of the map from mass distributions to outside gradients $(M_{+}, M_{\times})$ consists of those mass distributions that give rise to spherically symmetric potentials $\Phi$ outside $B_R$. If $\Phi$ is spherically symmetric outside $B_R$, then consider $\Phi_S$, the spherical average of $\Phi$ (i.e. $\Phi_S (\vec{x})$ = the average of $\Phi$ on the sphere centered at 0 and passing through $\vec{x}$). Then $\Phi_S$ is spherically symmetric everywhere and coincides with $\Phi$ outside $B_R$. Therefore $\chi \equiv \Phi - \Phi_S$ vanishes outside $B_R$, and thus $$\nabla^2 \Phi = \nabla^2 \chi + \nabla^2 \Phi_S \; .$$ Since $\Phi_S$ is everywhere spherically symmetric, so is $\nabla^2 \Phi_S$, and Theorem 6 follows.
[**Kernel of the gravitational forward map and discretization**]{}
In practice, gravity inversion is a discrete problem because (i) the measurements of $\Phi$ (or of the gradients) are finitely many and discretely distributed in space, and (ii) more problematically, the model for the mass distribution $\rho$ is some discretized approximation to a continuous distribution. In all contexts, the characterization of $\rho$ would be a finite list of parameters which uniquely specify $\rho$ in some generally non-linear fashion. For example, these $\rho$ parameters could be the masses, locations, and shape parameters of a finite number of tectonic plates in a geophysical model of the Earth’s crust. Or, as discussed above, they could be masses of finite blocks into which we divide the source distribution in discretizing it. Or, more straightforwardly, they could be the masses of $N$ point-mass centers distributed throughout the source region, approximating with a discrete configuration the true mass distribution in the limit $N \rightarrow \infty$.
In general, the practical, the discretized gravitational inverse problem is the problem of inverting some (generally nonlinear) forward map: $$F: \; \{ p_j \} \longmapsto \{ \Phi_i \} \; , \;
\; \; \; \; \; \Phi_i = F_i [p_j ] \; ,$$ where $p_j$ are finitely many parameters specifying the mass distribution, and $\Phi_i$ are the measurements. It would be conceptually salutary to have the fundamental non-uniqueness in the gravitational inverse problem (the kernel of the froward map) described by Theorem 1 to fall out of the formulation Eq.(31 ) in a natural way. For example, when we simulate a slab of soil using some large number of mass centers regularly placed at fixed lattice points inside the slab, the parameters $p_j$ are simply the point masses $m_j$ assigned to each center at lattice location $j$. In this case, the forward map $F$ is in fact linear: $$\Phi_i = \sum_j F_{ij} m_j \; ,$$ where the matrix $F_{ij}$ is the Green’s function in Eq.(1) in discretized form: $$F_{ij} = - G \frac{1}{|\vec{r_i} - \vec{r_j}|} \; ,$$ with $\vec{r_i}$ being the locations where the measurements $\Phi_i
\equiv \Phi (\vec{r_i})$ are collected. Consider first, for simplicity, a scenario in which we are sampling $\Phi$ at the same number of points $N$ as the number of mass centers in the discretization. In other words, $F$ is now a square $N \times N$ matrix. In view of Theorem 1 characterizing the kernel of the forward map, one might expect $F$ to be singular, with the null space corresponding to a discretized version of the kernel, i.e., a discrete approximation to functions of the form $\nabla^2 \chi$ with $\chi$ supported inside the slab. It turns out, however, that the matrix $F$ given by Eq.(33) is in fact generically nonsingular. Moreover, the $M \times N$ matrix $F$ with $M$ measurement locations and $N$ mass centers is also nonsingular, in the sense that generically it has maximal rank (i.e. trivial null space).
It is in fact easy to see why this is so, because of the following result:
: Given a solution $\Phi(\vec{r})$ of the Laplace equation ${\nabla}^2 \Phi=0$ vanishing at infinity and defined for $r>R$, there exists [*at most*]{} one configuration $\{ m_j , \vec{r_j} \}$ of finitely many point masses placed inside $B_R = \{ r \leq R \}$ (i.e.with $m_j \in {\mbox{\bbold R}}$ and $| \vec{r_j} | \leq R $) that can give rise to this $\Phi$ for $r>R$.
[**Proof**]{}: Suppose, on the contrary, that there are two configurations of point masses, $\{ m_j , \vec{r_j} \}$ and $\{ m_k ' , \vec{r_k}' \}$, that produce the same $\Phi$ for $r>R$. Subtract the second configuration from the first, and correspondingly subtract the $\Phi$ fields that they produce. Since the gravity field depends linearly on the mass distribution, what we obtain is a new configuration $\{ m_1, \cdots , m_N, -m_1 ' , \cdots -m_{N'} ' , \vec{r_1},
\cdots , \vec{r_N}, \vec{r_1}' , \cdots , \vec{r_{N'}} ' \}$ of point masses inside $B_R$ (unless there are some coincident point masses in the two collections, in which case one would simply subtract the corresponding masses and list the location only once), which produces a field $\Phi$ that vanishes identically for $r>R$. Could this actually happen? It turns out the answer is no, unless $\Phi$ is identically zero everywhere (and therefore the two original point-mass configurations are in fact identical). To see this, observe that $\Phi$ produced by a finite set of point masses is a real-analytic function in ${\mbox{\bbold R}}^3$ except at the locations of the point masses where it has singularities. Since $\Phi$ vanishes for $r>R$ and is analytic, it must vanish everywhere in ${\mbox{\bbold R}}^3$ except possibly at the mass centers. But if any of the mass centers had non-zero mass, we could choose points so close to that center that the contribution to $\Phi$ from that center would overwhelm the contributions from any other centers (which are discretely spaced since there are finitely many). This clearly contradicts the fact that $\Phi$ is identically zero in any small neighborhood of the chosen mass-center. Therefore, none of the mass centers can have nonzero mass; the two original configurations of point masses must be identical, and Theorem 7 is proved.
Here is one way to understand the apparent conflict between Theorem 1 and Theorem 7: Consider the two spaces between which the forward map $F$ acts: the space of density distributions $\rho$ and the space of potentials $\Phi$. Any discretization is an attempt to approximate these spaces via a sequence of finite-dimensional subspaces. For example, when we use $N$ point masses, we have an $N$-dimensional subspace of the space of all $\rho$, and as $N$ gets larger and larger this subspace approximates the full space arbitrarily closely, in the sense that for any $\rho_0$, we can find a configuration of $N$ point masses (with large enough $N$) which comes as close as we want to $\rho_0$ (in some locally averaged sense). The same goes for the corresponding potentials $\Phi$: given any solution $\Phi_0$, we can find potentials produced by $N$ point masses that get arbitrarily close to $\Phi_0$ as $N \rightarrow \infty$. But the problem is that these approximating subspaces completely miss the kernel of the true forward map, which is the subspace of mass distributions (and corresponding potentials) given by $\{ \nabla^2 \chi \; | \; \chi \in C_0(B_R ) \}$. The intersection of the approximating subspaces with this kernel subspace is the zero vector, for any finite $N$. This is (mathematically) the explanation for the apparent contradiction between Theorem 1 and Theorem 7.
To resolve this apparent conceptual paradox, one might argue that we must choose the approximating finite dimensional subspaces in such a way that they fully intersect the kernel. But in practice, there is no feasible way to discretize the problem that makes sure this property holds. There is, however, a much simpler practical strategy out of this apparent paradox, and this is the strategy we advocate: Realize that true measurements in the real world always have instrumental noise. What this means is that two potentials are indistinguishable in practice if they differ everywhere by less than, say 1$\sigma$ worth (in some arbitrary units) of instrumental noise. Therefore, e.g. when we look for the intersection between the kernel and our discretized $\rho$-subspace with $N$ point masses, what we are really looking for are all $N$-point-mass configurations that produce a potential $\Phi$ that differs from zero by less than $1\sigma$ throughout the exterior region $r>R$. And in general there are many such configurations. We can see this, for example, in the matrix $F_{ij}$ of Eq.(33): in general this matrix turns out to be highly ill-conditioned (with very small determinant) with lots of eigenvalues close to zero, even though it has no exactly-zero eigenvalues. And the “approximately null" subspace spanned by the small-eigenvalued eigenspaces is precisely the discrete analogue of the kernel of the forward map; it is what corresponds to the subspace $\{ \nabla^2 \chi \; | \; \chi \in C_0(B_R ) \}$ in this discretization. We expect a similar description for the discrete analogue of the forward map’s kernel in any other practical discretization scenario.
[5]{}
M. Kac, Amer. Math. Monthly [**73**]{}, 1966. P. Buser, [*Isospectral Riemann Surfaces*]{}, Ann. Inst. Fourier (Grenoble), [**36:**]{}167, 1986. M. Berger, Sect. 9.12 in [*A Panoramic View of Riemannian Geometry*]{}, Springer-Verlag, 2002. J. D. Jackson, [*Classical Electrodynamics*]{}, John Wiley & Sons, New York 1998. R. Abraham and J. E. Marsden, [*Foundations of Mechanics: A Mathematical Exposition of Classical Mechanics With an Introduction to the Qualitative Theory of Dynamical Systems*]{}, Perseus Books, San Francisco 1994. D. Gilbarg and N. S. Trudinger, [*Elliptic Differential Equations of Second Order*]{}, Springer-Verlag, Berlin, 2001 (reprint of the third 1998 edition).
[^1]: Corresponding Author: [email protected]
[^2]: The assumption that $\partial D$ is a topological two-sphere is redundant since it follows from the assumption of radial convexity as formulated in the theorem. However, it is perhaps useful to emphasize this assumption in a redundant statement since the theorem is certainly false without it.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Mauna Loa Observatory record of direct-beam solar irradiance measurements for the years 1958–2010 is analysed to investigate the variation of clear-sky terrestrial insolation with solar activity over more than four solar cycles. The raw irradiance data exhibit a marked seasonal cycle, extended periods of lower irradiance due to emissions of volcanic aerosols, and a long-term decrease in atmospheric transmission independent of solar activity. After correcting for these effects, it is found that clear-sky terrestrial irradiance typically varies by $\approx 0.2 \pm 0.1$% over the course of the solar cycle, a change of the same order of magnitude as the variations of the total solar irradiance above the atmosphere. An investigation of changes in the clear-sky atmospheric transmission fails to find a significant trend with sunspot number. Hence there is no evidence for a yet unknown effect amplifying variations of clear-sky irradiance with solar activity.'
author:
- 'G. '
nocite:
- '[@Feulner2011d]'
- '[@Feulner2011d]'
title: 'On the Relation between Solar Activity and Clear-Sky Terrestrial Irradiance'
---
Introduction {#s:intro}
============
Long-term records of clear-sky terrestrial solar irradiance are very useful to study changes in atmospheric transmission [*e.g.*, @Hoyt1983]. It is more difficult to derive changes in insolation due to solar variability from these data because the variations related to solar activity are much smaller than the changes in atmospheric transmission of different origin and the systematic and random errors in the data. Nevertheless, such an approach has been tried many times in the past.
The Smithsonian Astrophysical Observatory (SAO) observations from the first half of the 20th century are an important historical example for terrestrial irradiance measurements with the aim to detect solar variability from the ground [@Abbot1932; @Abbot1942; @Aldrich1954]. The SAO observing campaigns were designed to derive a time series of the solar constant or total solar irradiance (TSI) above the atmosphere, but calibration issues [@Allen1958], biases due to strong seasonal variations (@Feulner2011ba), and changes in atmospheric transmission independent of solar activity [@Angstrom1970] – mostly related to volcanic aerosols and local pollution [@Hoyt1979; @Roosen1984] – can pose serious difficulties for investigations of irradiance changes related to solar activity if not properly accounted for.
Despite these problems, the SAO data continue to attract attention. @Weber2010 [@Weber2011] correlated the ground-based irradiance measurements from the SAO data with sunspot numbers and claimed to have found a strong variation of terrestrial insolation with solar activity, with differences of the order of 1% between solar maxima and minima (*i.e.*, one order of magnitude larger than the variations in the TSI on top of the atmosphere). These findings were criticised by @Feulner2011b (2011a, b) who could show that they were due to seasonal bias and the effects of volcanic aerosols and local pollution. The seasonal bias arises from the fact that, by coincidence, days with large sunspot numbers in the SAO data predominantly lie in months with seasonally high atmospheric transmission, making irradiance values at times of high solar activity appear higher. The second bias is introduced because two out of three solar minima in the SAO data covering solar cycles 16, 17, and 18 are affected by reduced atmospheric transmission due to volcanic aerosols and local pollution. This effect makes solar irradiance on the ground during those two minima appear lower, thus again suggesting a stronger variation of terrestrial irradiance with solar activity. After correcting for the seasonal cycle and excluding periods of time affected by aerosols from volcanic eruptions and local pollution, the variations of terrestrial insolation are of the same order of magnitude as the TSI variations (@Feulner2011b, 2011a, b).
More recently, @Hempelmann2012 used terrestrial insolation data taken at Mauna Loa Observatory (MLO) and reported variations of the solar irradiance on the ground a factor of 10 larger than above the atmosphere. Here an improved analysis of these data is presented to independently quantify how much clear-sky terrestrial irradiance varies between solar maxima and minima.
This paper is organised as follows. Section \[s:data\] describes the Mauna Loa terrestrial insolation data and corrections of the effects of the seasonal cycle, aerosols from volcanic eruptions, and a linear long-term trend independent of solar activity. In Section \[s:correlation\] the correlation of the thus corrected irradiance with sunspot number is presented. Section \[s:atf\] analyses the atmospheric transmission factor and its changes with sunspot number, before the results are discussed in the context of previous work in Section \[s:disc\]. Finally, Section \[s:concl\] summarises and concludes this paper.
Data Analysis {#s:data}
=============
A First Look at the Raw Irradiance Data
---------------------------------------
For the analysis in this paper, the ground-based direct-beam solar irradiance data taken since 1958 at Mauna Loa Observatory (MLO, latitude 19.533$^\circ$ N, longitude 155.578$^\circ$ W, elevation 3400 m) are used [@Ellis1971; @Dutton1985; @Dutton1994; @Dutton2001]. These measurements have been obtained at local noon as well as at airmass values of 2, 3, 4, and 5 (both in the morning and in the afternoon). In the following, results for the morning irradiance data at airmass 2 are presented. Airmass 2 is chosen since smaller airmass values correspond to smaller zenith angles and thus shorter paths through the atmosphere. Furthermore, as in most previous studies based on these data, the morning measurements are preferred over the afternoon data or a combined record of the two due to the absence of local influences on the atmospheric transmission because of downslope winds in the morning hours [@Mendonca1969]. Indeed, a comparison of irradiance morning and afternoon data at airmass two for days where both measurements are available shows that morning irradiance values are larger on average and thus apparently less affected by local disturbances in atmospheric transmission (see Figure \[f:daydiff\]).
![Difference of morning and afternoon irradiance measurements at MLO and airmass 2 for days where both are available (red histogram), showing that morning values are on average larger than afternoon values. The blue curve shows a Gaussian function fit to the data with a peak at 12.9 W m$^{-2}$ and a width $\sigma =
17.2$9 W m$^{-2}$.[]{data-label="f:daydiff"}](mlo_daydiff_20.eps){width="70.00000%"}
As a consistency check, the full analysis has been carried out for a combined record of morning and afternoon data and for the other airmass values as well, however, yielding very similar results which are not shown in this paper.
![The MLO terrestrial irradiance data for the time period 1958–2010 in the context of solar activity and volcanic eruptions during this time interval. (a) Raw irradiance measurements for morning measurements at airmass 2. (b) Same as (a), but zooming into the data. (c) Global average optical depth $\tau_{550}$ of volcanic aerosols at $\lambda = 550$ nm from @Sato1993 until the end of 1993 and from @Solomon2011 since the beginning of 1994. Prominent volcanic eruptions are marked. (c) Daily international sunspot numbers $R$ during this time interval, spanning a large part of solar cycle 19, cycles 20, 21, 22, and 23 as well as the beginning of the current cycle 24.[]{data-label="f:mlotime"}](mlo_time_20.eps){width="95.00000%"}
The raw irradiance data at morning airmass 2, for historic reasons given in units of Langley min$^{-1}$ or cal cm$^{-2}$ min$^{-1}$, are shown in Figures \[f:mlotime\]a and \[f:mlotime\]b. In this figure, they are compared to the globally averaged optical depth $\tau_{550}$ of volcanic aerosols at a wavelength of $\lambda =
550$ nm as compiled by @Sato1993 for the years 1959–1993 and by @Solomon2011 for 1994–2010 (Figure \[f:mlotime\]b) and to daily international (also called Wolf or Zürich) sunspot numbers[^1] $R$ (Figure \[f:mlotime\]c).
Several interesting observations can be made in the raw data which are relevant for a proper analysis of correlations with solar activity:
1. The irradiance data exhibit a pronounced seasonal cycle. As shown in @Feulner2011b (2011a), not correcting for seasonal variations can affect correlations with solar activity if different phases of solar activity are not evenly distributed over the seasons in the data.
2. There is a cluster of irradiance data around 0.5 cal cm$^{-2}$ min$^{-1}$ in the years 1975–1976. As one can clearly see the seasonal cycle and as these values are offset by 1 from the majority of the data; these are most likely due to typos in the record where values have a leading digit of 0 instead of 1. These typos can either be corrected or masked out; the latter option has been chosen in this paper.
3. The dimming effects of volcanic aerosols in the atmosphere can be seen directly in the data. These are most pronounced for the largest eruptions, *i.e.*, El Chichón in 1982 and Pinatubo in 1991, of course, but irradiance values are also lower during volcanic episodes in the 1960s and 70s. This can be best seen in the expanded view presented in Figure \[f:mlotime\]b.
4. Looking at periods only weakly affected by volcanic aerosols over the entire record, one can see a slight decrease of terrestrial irradiance with time previously described in @Solomon2011. It will be shown below that this long-term trend is well approximated by a linear fit and not connected to changes in sunspot number.
All these issues have to be taken care of before correlating the MLO terrestrial irradiance data with sunspot number in order to ensure an unbiased estimate of the influence of solar activity on clear-sky irradiance on the ground. In addition, the MLO data are also affected by changes in instrumentation and offsets in calibration which can also influence any trend analysis [*e.g.*, @Dutton2001]. These effects can be eliminated, however, by directly investigating atmospheric transmission factors [@Ellis1971] as shown in Section \[s:atf\].
Corrections Applied to the Irradiance Data {#s:corrections}
------------------------------------------
![Corrections applied to the MLO terrestrial irradiance data at airmass 2. The left-hand panels show the irradiance time series, the right-hand panels scatter plots of irradiance vs. sunspot number. (a) Raw irradiance data as shown in Figures \[f:mlotime\]a and \[f:mlotime\]b. The grey shading indicates the portion of the data used to derive the average seasonal cycle in terrestrial irradiance. (b) The raw data with the seasonal cycle subtracted. (c) The data corrected for both seasonal cycle and for the effects of volcanic aerosols using the globally averaged optical depth shown in Figure \[f:mlotime\]b. The two-year time periods with residuals after the El Chichón and Pinatubo eruptions excluded in the analysis below are indicated by the grey shading. The red line shows the linear long-term trend derived from a fit to the data. (d) Same as (c), but with the long-term trend subtracted.[]{data-label="f:mlotimecorr"}](mlo_time_corr_20.eps){height="95.00000%"}
As a first step, the raw MLO irradiance data for morning airmass 2 as shown in Figures \[f:mlotime\]a and \[f:mlotime\]b are converted to SI units (W m$^{-2}$) by multiplying by a factor of 697.4. These converted irradiance data are shown both as a time series and as a scatter plot vs. sunspot number in Figure \[f:mlotimecorr\]a. Furthermore, for each day with observations the MLO irradiance data are correlated with daily international sunspot numbers and interpolated volcanic aerosol optical depth data from the same day.
In the discussion above, three sources of systematic bias of the irradiance data were identified: the seasonal cycle, attenuation by volcanic aerosols, and a possible long-term trend. These effects will be discussed and, if possible, corrected in the following. The individual corrections and their effect on the time series as well as the correlation between irradiance and sunspot number are shown in Figure \[f:mlotimecorr\].
### Seasonal Cycle
The pronounced seasonal cycle is one of the most obvious features of the raw irradiance time series shown in Figure \[f:mlotimecorr\]a. One way to correct for this effect is to take data from several years only little affected by volcanic aerosols and combine them to compute median monthly values for the irradiance (@Feulner2011b, 2011a). Daily values of this average seasonal irradiance cycle can then be computed from a cubic-spline fit through the data with periodic boundary conditions.
For the MLO data the years from 2000 to 2009 have been chosen to compute the average seasonal cycle (indicated by the grey shading in Figure \[f:mlotimecorr\]a) as this period of time is only weakly affected by volcanic aerosols. Furthermore, a linear trend with time has been subtracted before computing the monthly median irradiance values to remove any long-term changes in the irradiance baseline (be it from solar activity changes, trends in volcanic or other aerosols, or instrumental drift). The resulting cycle of seasonal irradiance anomalies is shown in Figure \[f:seasons\].
![Average seasonal cycle in the terrestrial irradiance at MLO and airmass 2 computed from the irradiance data in the years 2000-2009 (small black squares) and after removing a linear long-term trend. The filled red circles are the monthly medians while the red line shows a fit of cubic splines with periodic boundary conditions through the monthly medians. The blue line indicates the expected seasonal irradiance variations due to Earth’s orbital motion.[]{data-label="f:seasons"}](mlo_irrad_seasons_20.eps){width="70.00000%"}
The irradiance data with the seasonal cycle anomaly removed are shown in Figure \[f:mlotimecorr\]b. The irradiance variations through the year are clearly dominated by the changes in the distance to the Sun on Earth’s elliptical orbit. In addition there are minor, but statistically insignificant, seasonal anomalies, in particular in spring and summer.
Note that there is quite a bit of interannual variation in the seasonal irradiance cycle; therefore, a perfect correction of the effect cannot be expected. Comparing the irradiance time series with the seasonal cycle subtracted shown in Figure \[f:mlotimecorr\]b to the raw irradiance data shows, however, that the seasonal cycle is effectively removed. In particular, the sequence in the irradiance vs. sunspot number scatter plot shown in the right-hand panel of Figure \[f:mlotimecorr\]b exhibits less scatter and visually appears considerably better defined.
### Effects of Volcanic Aerosols
The second atmospheric effect which has to be taken into account when investigating the correlation between solar activity and terrestrial insolation are the attenuating effects of volcanic aerosols. One way of dealing with episodes of volcanic eruptions is to exclude these periods from further analysis as has been done for the SAO terrestrial irradiance data in @Feulner2011b (2011a). This is not a viable option for the frequent volcanic eruptions affecting the MLO data, however. The expected irradiance variations due to solar activity are below 1% or $\Delta I \lesssim 10$ W m$^{-2}$ corresponding to an attenuation by volcanic aerosols with an optical depth of $\tau \lesssim 0.01$. Such a cut in aerosol optical depth would mean that more than half of the MLO irradiance record would have to be excluded from the analysis, and even then the remaining record would suffer from inhomogeneous attenuation by volcanic aerosols resulting in irradiance variations at least of the order of the changes expected from solar variability.
A preferable option would be to correct for the effects of volcanic eruptions using an independent dataset of the optical depth of volcanic aerosols. For the analysis in this paper, the @Sato1993 compilation of global average optical depth $\tau_{550}$ of volcanic aerosols at wavelength $\lambda = 550$ nm is used for the years 1958–1993, extended with the @Solomon2011 data until March 2010 and with a constant value $\tau_{550} = 0.007$ until the end of 2010. The irradiance $I_\mathrm{sv}$ corrected for the seasonal cycle and the effects of volcanic aerosols is then calculated from the irradiance $I_\mathrm{s}$, which is corrected for the seasonal cycle only, using $I_\mathrm{sv} = I_\mathrm{s} \exp (X
\, c_X \, \tau_{550})$. In this formula, $X$ denotes the airmass, and $c_X$ is a correction factor converting the optical depth at 550 nm to the optical depth integrated over the whole visible solar spectrum. The correction factors $c_X$ for the four airmass values are determined empirically, finding $c_2 = 0.50 \pm 0.01$, $c_3 = 0.45 \pm
0.01$, $c_4 = 0.42 \pm 0.01$ and $c_5 = 0.40 \pm 0.01$.
The time series and scatter plot for the irradiance $I_\mathrm{sv}$ corrected for seasons and volcanic aerosols is shown in Figure \[f:mlotimecorr\]c. Overall the correction seems to work well for the late stages of volcanic eruptions. There are, however, considerable residuals during times of rapidly changing volcanic aerosol load, in particular for the Pinatubo and El Chichón eruptions. This is not surprising since the global average aerosol optical depth may not be fully representative for the local aerosol load above Mauna Loa, in particular in the period following the eruptions. Furthermore, the @Sato1993 compilation provides monthly values which are not able to trace the fast changes visible in the daily irradiance record. Two-year periods after the four strongest eruptions (Agung, Fernadina, El Chichón, and Pinatubo) will be masked out for the analysis of the correlation between terrestrial irradiance and sunspot number presented below. Note that there is also a dip in the thus corrected irradiance around the year 1967, corresponding to a decrease in volcanic aerosol optical depth in the @Sato1993 record. As the origin of this period of lower irradiance remains unclear, it will not be excluded from the analysis.
### Long-Term Trend
Looking at the scatter plot of irradiance values corrected for the seasonal cycle and volcanic aerosols with sunspot number shown in the right-hand panel of Figure \[f:mlotimecorr\]c one can indeed see a marked increase of terrestrial solar irradiance with sunspot number. At least part of this trend is clearly driven by what appears to be a separate sequence of measurements at high irradiance ($I
\simeq 1100$ W m$^{-2}$) at relatively high sunspot numbers ($100
\lesssim R \lesssim 300$).
A comparison with the irradiance time series shows that the vast majority of these measurements are from the very first years of the record (years 1958–1961) which, according to Figure \[f:mlotime\], are unaffected by volcanic aerosols and coincide with the particularly strong maximum of solar cycle 19.
These observations are important since the irradiance data shown in Figure \[f:mlotimecorr\]c exhibit a slowly decreasing trend over time. This general decrease in irradiance can be clearly seen during the last decade, for example, or by comparing the beginning of the record and the end. This trend in atmospheric transmission has been noted before and attributed to changes in the atmospheric background aerosol load [@Solomon2011].
It will be shown below that this long-term trend can be detected in the data spanning almost five solar cycles. Note, for example, that the decrease can be seen over the entire solar cycle 23 with a constant slope despite large variations in sunspot number. The trend can thus be regarded as independent of solar activity and has to be subtracted to ensure an accurate analysis of the correlation between solar activity and terrestrial irradiance. To this end a linear trend $I_\mathrm{t} \, (t) \, = \, I_0 + a_I \, t$ is approximated to the entire record of irradiance data corrected for the seasonal cycle and volcanic aerosols, but excluding the two-year time intervals after the El Chichón and Pinatubo eruptions. Furthermore, the data used to fit the linear trend are restricted to the range $0 < R < 200$ to ensure that the fit of the long-term trend is not driven by changes in solar activity between the strong solar maximum of cycle 19 and the considerably less pronounced maximum of cycle 23. Fitting a linear trend using these data yields a slope of $a_I = -0.38 \pm
0.02$ W m$^{-2}$ yr$^{-1}$. This is very similar to the trend found in @Solomon2011. As will be shown below, a multivariate regression has been performed on $I_\mathrm{sv}$ to simultaneously analyse its linear trends with sunspot number and time as an additional test whether the trend in time reported here is indeed independent of solar activity, finding a very similar value for the long-term decrease in terrestrial irradiance with time.
Thus there is indeed a significant long-term decrease in terrestrial irradiance which is independent of solar activity and therefore has to be subtracted from the data before analysing any correlation of irradiance with sunspot number. The resulting irradiance time series of $I_\mathrm{svt} (t) \, = \, I_\mathrm{sv} (t) - I_\mathrm{t} (t)$ and its scatter plot are shown in Figure \[f:mlotimecorr\]d. It should be noted that the conspicuous second sequence at large irradiance values in the scatter plot has now disappeared and that the overall correlation shows a considerably smaller slope than before the subtraction of the linear long-term trend.
Correlation of Terrestrial Irradiance with Sunspot Number {#s:correlation}
=========================================================
After applying the corrections for the seasonal cycle, for the attenuating effects of volcanic aerosols, and for the linear long-term trend, the MLO terrestrial irradiance data at morning airmass 2 can now be correlated with the sunspot number to investigate changes of terrestrial irradiance with solar activity.
![Scatter plot of MLO terrestrial irradiance vs. sunspot number (black squares). The irradiance data are corrected for the seasonal cycle, volcanic aerosols, and a long-term decrease in atmospheric transmission independent of solar activity. The red line shows a linear fit to the correlation.[]{data-label="f:correlation"}](mlo_irrad_r_corr_20.eps){width="70.00000%"}
The corrected irradiance data $I_\mathrm{svt}$ as a function of sunspot number $R$ are presented in Figure \[f:correlation\]. Fitting a line to this correlations yields a slope
$$\mathrm{d} I_\mathrm{svt} / \mathrm{d}R \, = \, + (0.019 \pm 0.007)
\, \mathrm{W\,m}^{-2} .$$
The error has been computed from 10000 bootstrapping simulations where the sample is first duplicated and added to the original sample, then modified by random errors, before half of the enlarged sample is randomly selected to perform the linear regression (see also @Feulner2011b, 2011a). A random error of 0.5% is assumed for the individual MLO irradiance measurements [@Dutton2001] during these bootstrapping simulations.
In the analysis so far, the long-term trend with time $t$ of the irradiance $I_\mathrm{sv}$ corrected for the seasonal cycles and the effects of volcanic aerosols has been removed before correlating with sunspot number $R$. While this is highly illustrative, it is statistically more appropriate to perform a simultaneous multivariate linear regression of $I_\mathrm{sv}$ with respect to trends with $t$ and $R$. Thus one can also independently test whether the linear trend in time reported in Section \[s:corrections\] is indeed independent of solar activity as traced by the sunspot number $R$. This test yields a linear trend of terrestrial irradiance with time of $\partial
I_\mathrm{sv}/\partial t = -0.45 \pm 0.02$ W m$^{-2}$ yr$^{-1}$, which is slightly larger than the value reported above, but compatible within the errors. The slope of the correlation with sunspot number derived in this way is
$$\partial I_\mathrm{sv} / \partial R \, = \, + (0.015 \pm 0.006) \,
\mathrm{W\,m}^{-2} ,$$
again in excellent agreement with the value from Equation (1).
Thus a simultaneous regression of a linear long-term trend with time and a linear trend with sunspot number to the terrestrial irradiance data not yet corrected for the linear trend yields results which are very similar to the ones described above. This is important because if the long-term trend in terrestrial irradiance had been caused by changes in solar activity, this long-term trend would be reflected in the sunspot numbers as well, and the regression would have yielded an insignificant trend with time. This clearly shows that the linear long-term trend described above is indeed not caused by solar activity.
As a test whether this result depends on the choice of sunspot number as indicator of solar activity, the simultaneous regression has been repeated for annual means of the open solar flux $F_\mathrm{S}$ [@Lockwood2009b], finding consistent values for the trends, albeit with larger errors ($\partial I_\mathrm{sv}/\partial t = -0.47
\pm 0.15$ W m$^{-2}$ yr$^{-1}$ and $\partial I_\mathrm{sv}/\partial
F_\mathrm{S} = (1.4 \pm 2.6) \times 10^{-14}$ W m$^{-2}$ Wb$^{-1}$).
In summary, a positive trend of the corrected terrestrial irradiance data with sunspot number can be found in the MLO data which is significant on the $\simeq 2\sigma$ level. The magnitude of these changes of solar irradiance on the ground with respect to changes in irradiance on top of the atmosphere and previous studies on correlations of terrestrial irradiance with solar activity will be discussed in Section \[s:disc\].
Correlation of Atmospheric Transmission with Sunspot Number {#s:atf}
===========================================================
![Same as Figure \[f:mlotimecorr\], but for the atmospheric transmission factor (ATF) and the corrections applied to this quantity.[]{data-label="f:mlotimeatf"}](mlo_time_corr_atf.eps){height="95.00000%"}
The analysis above investigated the correlation of terrestrial insolation with sunspot number to investigate the possibility of a so-far unknown amplification effect produced by changes in clear-sky atmospheric transmission. Alternatively, one can directly study the atmospheric transmission itself. The advantage of this approach is that one can make use of the rationing technique which eliminates both the variations in TSI and any instrumental calibration differences [@Ellis1971]. In this method, atmospheric transmission is analysed in terms of the atmospheric transmission factor (ATF) defined as follows:
$$\mathrm{ATF} \: = \: \frac{1}{3} \left( \frac{I_3}{I_2} \, + \,
\frac{I_4}{I_3} \, + \, \frac{I_5}{I_4} \right) ,$$
where $I_X$ are the morning irradiance values at the different airmass values $X$ of 2, 3, 4, and 5. The ATF for the Mauna Loa data is shown in Figure \[f:mlotimeatf\]a.
Before correlating the ATF with sunspot number $R$, the same corrections have to be applied as for the irradiance data (see Section \[s:corrections\]). The seasonal cycle in the ATF is computed as for the irradiance, but note that the dominant effect caused by the orbital motion of the Earth is already taken care of by the rationing technique. The remaining corrections for the effects of volcanic aerosols and the long-term trend (with a best-fitting value of $\mathrm{d(ATF)} / \mathrm{d} t = -(8.2 \pm 0.4) \times
10^{-5}$ yr$^{-1}$) also follow the procedure discussed in detail for the irradiance. The stepwise corrections and their effect both on the time series and for the correlation with sunspot number are illustrated in Figure \[f:mlotimeatf\].
The correlation between the corrected atmospheric transmission factor ATF and sunspot number $R$ is shown in Figure \[f:atfcorrelation\]. A linear fit to this correlation yields
$$\mathrm{d(ATF)} / \mathrm{d}R \, = \, - (0.7 \pm 1.4) \times 10^{-6} .$$
As for the irradiance data, an improved analysis is performed in which both the long-term trend of the ATF with time and its correlation with sunspot number are fit simultaneously. This analysis yields a value of $\partial \mathrm{(ATF)} / \partial t = -(10.5\pm 0.4) \times
10^{-5}$ yr$^{-1}$ for the long-term trend, in good agreement with the value derived above. The fact that the simultaneous linear regression of the ATF finds a significant long-term trend with time again confirms that this long-term trend is independent of solar activity. The linear correlation of the ATF with sunspot number has a slope of $\partial \mathrm{(ATF)} / \partial R = -(2.0 \pm 1.3) \times
10^{-6}$, again in agreement with Equation (4). Thus no significant change of atmospheric transmission with sunspot number can be detected in the Mauna Loa data.
![Scatter plot of MLO atmospheric transmission factor (ATF) vs. sunspot number (black squares). The ATF data are corrected for the seasonal cycle, volcanic aerosols, and a long-term decrease in atmospheric transmission independent of solar activity. The red line shows a linear fit to the correlation.[]{data-label="f:atfcorrelation"}](mlo_atf_r_corr.eps){width="70.00000%"}
To test for a possible dependence of this result on the choice of solar-activity indicator, the bivariate regression has been repeated replacing the sunspot number $R$ by the open solar flux $F_\mathrm{S}$ (using annual averages of the data as for the analysis of the irradiance data). Again, the trends derived for the open solar flux are in good agreement with the trends computed using sunspot numbers, although the errors are larger ($\partial \mathrm{(ATF)}/\partial t =
(-10.7 \pm 2.7)$ yr$^{-1}$ and $\partial \mathrm{(ATF)}/\partial
F_\mathrm{S} = (-1.6 \pm 4.8) \times 10^{-18}$ Wb$^{-1}$).
Discussion {#s:disc}
==========
The analysis in Section \[s:correlation\] shows that the corrected terrestrial insolation $I_\mathrm{sv}$ linearly increases with sunspot number $R$ with a slope $\partial I_\mathrm{sv} / \partial R \, = \, +
(0.015 \pm 0.006)$ W m$^{-2}$. With typical differences in sunspot number of $\Delta R \approx 150$ between solar maxima and minima, this translates into irradiance variations of $\Delta I \approx 2 \pm
1$ W m$^{-2}$ or $\sim 0.2 \pm 0.1$% over the solar cycle. This is of the same order of magnitude as the variations in total solar irradiance (TSI) observed from satellites [*e.g.*, @Frohlich2004; @Gray2010] which suggests that clear-sky terrestrial irradiance variations with solar activity are not strongly amplified by any hitherto unknown atmospheric feedback processes.
The conflicting results in @Hempelmann2012 who find a one order of magnitude larger variation in terrestrial irradiance with solar activity ($dI/dR \, = \, (0.10-0.18)$ W m$^{-2}$ depending on the way the effects of volcanic eruptions are accounted for) can be understood from the fact that the authors have not corrected the terrestrial irradiance data for the slow long-term decrease in atmospheric transmission not related to changes in solar activity. The combination of large irradiance values at the beginning of the record with large sunspot numbers during the strong maximum of solar cycle 19 then results in a spuriously large slope of the irradiance vs. sunspot number correlation. In addition, excluding periods of up to three years after volcanic eruptions as done in @Hempelmann2012 is insufficient as attenuation effects at the percent level are visible in the data even after this time, resulting in changes of at least the order of magnitude of the changes related to solar variability (see the discussion in Section \[s:corrections\]).
The lack of evidence for any unknown effect amplifying changes of terrestrial irradiance with solar activity in the MLO data is confirmed by an analysis of the correlation of the atmospheric transmission factor with sunspot number which failed to find a significant change of clear-sky atmospheric transmission with solar activity.
Conclusions {#s:concl}
===========
In this paper, the Mauna Loa Observatory (MLO) direct-beam solar irradiance record has been analysed to investigate the correlation between solar activity and terrestrial insolation. After correcting for the seasonal cycle, the attenuating effects of volcanic aerosols, and a long-term decrease in atmospheric transmission independent of solar activity, clear-sky terrestrial irradiance is found to vary by $\Delta I \approx 0.2 \pm 0.1$% between maxima and minima of the 11-year solar activity cycle. These variations are of the same order of magnitude as the changes in total solar irradiance on top of the atmosphere over the solar activity cycle. An investigation of the atmospheric transmission shows that there is no significant trend of clear-sky atmospheric transmission with sunspot number. Thus there is no evidence for an unknown amplification effect of solar activity in terms of strong changes in clear-sky atmospheric transmission.
I would like to thank Ellsworth G. Dutton for providing me with the latest version of the Mauna Loa Observatory terrestrial solar irradiance data and for many helpful comments on these data. I am grateful to the anonymous referee for a thorough review. This research has made use of NASA’s Astrophysics Data System Bibliographic Services.
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[^1]: Available at <http://sidc.oma.be/DATA/dayssn_import.dat>, date of access: 22 December 2011.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Including finite-temperature effects from the electronic degrees of freedom in electronic structure calculations of semiconductors and metals is desired; however, in practice it remains exceedingly difficult when using zero-temperature methods, since these methods require an explicit evaluation of multiple excited states in order to account for any finite-temperature effects. Using a Matsubara Green’s function formalism remains a viable alternative, since in this formalism it is easier to include thermal effects and to connect the dynamic quantities such as the self-energy with static thermodynamic quantities such as the Helmholtz energy, entropy, and internal energy. However, despite the promising properties of this formalism, little is know about the multiple solutions of the non-linear equations present in the self-consistent Matsubara formalism and only a few cases involving a full Coulomb Hamiltonian were investigated in the past. Here, to shed some light onto the iterative nature of the Green’s function solutions, we self-consistently evaluate the thermodynamic quantities for a one-dimensional (1D) hydrogen solid at various interatomic separations and temperatures using the self-energy approximated to second-order (GF2). At many points in the phase diagram of this system, multiple phases such as a metal and an insulator exist, and we are able to determine the most stable phase from the analysis of Helmholtz energies. Additionally, we show the evolution of the spectrum of 1D boron nitride (BN) to demonstrate that GF2 is capable of qualitatively describing the temperature effects influencing the size of the band gap.'
author:
- Alicia Rae Welden
- 'Alexander A. Rusakov'
- Dominika Zgid
title: 'Exploring connections between statistical mechanics and Green’s functions for realistic systems. Temperature dependent electronic entropy and internal energy from a self-consistent second-order Green’s function.'
---
Introduction
============
In molecular quantum-chemical calculations of thermodymanic properties such as Gibbs energy [@mcquarrie1999molecular; @g09thermo], the temperature dependent component is usually dominated by vibrational contributions. This is due to the large gaps between electronic states, which ensure that the excited state populations will be negligible. Consequently, common molecular calculations do not explicitly include temperature effects on the electronic structure.
However, for materials such as doped semiconductors [@guidebookworld] the magnitude of the electronic band gap can be relatively small, or for metals nonexistent altogether, allowing electronic states other than the ground state to be accessible at low temperatures. Thus, it is necessary to include temperature effects into the electronic description. Even though for most materials the vibrational contribution to the specific heat is much larger than the electronic one, there are cases when the incorporation of the electronic contribution is necessary. The electronic contribution to specific heat is important for [**(i)**]{} heavy fermion materials at low temperatures [@metalsfreeenergy], [**(ii)**]{} materials that do not undergo a structural transition that changes the vibrational contribution, causing the stability of phases to primarily depend on the relative electronic contribution to Gibbs energy, [**(iii)**]{} materials with a structural transition where the difference between vibrational contributions of the phases is of the same order as the difference between electronic contributions [@benzenetuckerman; @resorcinol]. Thus, modern materials calculations can benefit from computational tools that provide access to the temperature dependent electronic contribution to the specific heat, Gibbs energy, entropy, or electronic part of the partition function.
While in traditional quantum chemical calculations evaluating the electronic contribution to temperature dependent quantities is certainly not wide spread, a number of such methods exist, most notably, the finite temperature Hartree-Fock (HF) [@mermin1963stability], density functional theory (DFT) [@mermin1965thermal; @dharma2016current; @smith2015thermal; @trickey], Møller-Plesset second order perturbation theory (MP2) [@hirata2013kohn], coupled-cluster (CC) [@altenbokum1987temperature; @mandal2003finite; @sohiratacc], and Lanczos method [@jaklivc1994lanczos; @ftlanczos] for finite temperature configuration interaction (CI) calculations. However, these methods are usually quite difficult to implement and costly to use since they rely on the modification of the parent zero-temperature method to the finite temperature formalism by adapting it to work in the canonical or grand canonical ensemble. For example, to carry out such calculations in the CI formalism, one must obtain the excited states and corresponding Boltzmann factors in order to evaluate the partition function and thermodynamic averages, making application of finite temperature variants of these methods quite cumbersome.
Conversely, for the Green’s function formalism the connection to thermodynamics arises in a straightforward manner and was derived in numerous books in the past [@vanleeuwentext; @mattuck2012guide; @fetter2003quantum; @dzyaloshinski1975methods]. While this theoretical connection is well understood, the actual numerical calculations of the thermodynamic quantities still remain quite challenging since a fully self-consistent imaginary axis (Matsubara) Green’s function calculation is desired. A non-self-consistent Green’s function can result in non-unique thermodynamic quantities.
The fully self-consistent Green’s function calculations are challenging for multiple reasons, such as large imaginary time and frequency grids required for convergence or multiple solutions that can be present due to the non-linear nature of the equations. It is for reasons such as these that in recent years calculations that capitalized on Green’s function language and yield thermodynamic quantities where mostly done for model systems [@potthoff1; @potthoff2; @potthoff2003Hubbard; @PotthoffSE; @potthoff2003self].
Here, we would like to stress that while multiple large scale real axis Green’s function calculations are performed at present for the single shot $G_0W_0$, $GW_0$, or semi-self consistent $GW$ for large realistic systems [@neuhauser2014breaking; @hung2016excitation; @van2006quasiparticle; @caruso2013self; @faleev2004all; @rostgaard2010fully; @govoni2015large], currently, only a few research groups have managed to rigorously generalize the self-consistent finite temperature (imaginary axis) Green’s function formalism to deal with a general Hamiltonian containing all the realistic interactions [@van2006total; @dahlen2006variational; @GF2; @rusakov; @spline; @AlexeiSEET]. Thus, any insight gained from studying even simple periodic systems and analyzing the possible self-consistent solutions of the Matsubara formalism remains valuable.
To the best of our knowledge, here, we present the first application of the fully self-consistent finite temperature Green’s function formalism to evaluate thermodynamic quantities and phase stability for a periodic system described by a full quantum chemical Hamiltonian. We demonstrate that Green’s function formalism leads to a simple calculation of the electronic contribution to the Helmholtz or Gibbs energy, entropy, grand potential, and partition function without explicitly performing any excited state calculations. The presented formalism is exact at the infinite temperature limit since the perturbative Green’s function formulation is a perturbation that contains the inverse temperature as small parameter.
This paper is organized as follows. In Sec. \[sec\_thermo\_con\], we introduce the imaginary Green’s function formalism and its connection to thermodynamics. In Sec. \[sec\_phi\] and \[sec\_eval\_phi\_gf2\], we list properties of the Luttinger-Ward functional which is our main computational object and we explain its evaluation within a self-consistent Green’s function second-order (GF2) periodic implementation. In Sec. \[sec\_results\], we present numerical results first for a benchmark molecular problem and then for two 1D systems: periodic hydrogen as well as boron nitride. Finally, we form conclusions in Sec. \[sec\_conclusions\].
Connection with thermodynamics {#sec_thermo_con}
==============================
One of the first descriptions of the connection between the Green’s function formalism and thermodynamics was presented in the book by Abrikosov, Gorkov, and Dzyaloshinski [@dzyaloshinski1975methods]. Since then multiple texts have appeared that discuss this connection [@vanleeuwentext; @fetter2003quantum; @vanneck]. For an excellent, detailed derivation, we encourage the reader to follow Ref. ; here, we will only mention few basic Green’s function equations for the sake of completeness.
The one-body Green’s function is defined as $$\begin{aligned}
\label{GF_general}
G_{ji}(z_1,z_2)\equiv &
\frac{{\rm Tr}[e^{-\beta \hat{H}^{M}} \hat{G}_{ji}(z_1,z_2)]}{{\rm Tr}[e^{-\beta \hat{H}^{M}}]}\\
= &\frac{1}{i} \frac{{\rm Tr} [\tau \{e^{-i\int_{\gamma }d\bar{z}\hat{H}(\bar{z})}\hat{d}_{j,H}(z_1)\hat{d}^{\dagger}_{i,H}(z_2)\}]}
{{\rm Tr}[\tau \{e^{-i \int_{\gamma } d\bar{z}\hat{H}(\bar{z})} \} ]} \nonumber,
\end{aligned}$$ where $\hat{d}_{j,H}(z_1)$ and $\hat{d}^{\dagger}_{i,H}(z_2)$ are the second-quantized annihilation and creation operators in the Heisenberg representations and $\beta=1/(k_B T)$ is the inverse temperature while $T$ is the actual temperature and $k_B$ is the Boltzmann constant. This Green’s function, depending on how the contour $\gamma$ in the complex plane is closed, can be used to describe system’s time evolution (when $z_1$ and $z_2$ are set to the real-time variables), zero temperature phenomena, or equilibrium phenomena at finite temperature.
Here, we are interested in a formalism used to calculate the initial ensemble average that is applied to systems in thermodynamic equilibrium at finite temperature. This approach is called Matsubara formalism or the “finite-temperature formalism". For this reason, in the Green’s function from Eq. \[GF\_general\] we set $z_1=t_0-i\tau_1$ and $z_2=t_0-i\tau_2$. Consequently, the Green’s function $$\begin{aligned}
\label{GF_Matsubara}
G_{ji}^{M}(\tau_1,\tau_2& )
=
\frac{1}{i}
\big\{
\theta ( \tau_1-\tau_2 )
\frac{ {\rm Tr} [e^{(\tau_1-\tau_2-\beta)\hat{H}^{M}} \hat{d}_{j}e^{(\tau_2-\tau_1) \hat{H}^{M}}\hat{d}^{\dagger}_{i} ] } { {\rm Tr}[e^{-\beta \hat{H}^{M}}]} \nonumber \\
& \pm \theta ( \tau_2-\tau_1 )
\frac{ {\rm Tr} [e^{(\tau_2-\tau_1-\beta)\hat{H}^{M}} \hat{d}^{\dagger}_{i}e^{(\tau_1-\tau_2) \hat{H}^{M}}\hat{d}_{j} ] } { {\rm Tr}[e^{-\beta\hat{H}^{M}}]}
\big\}
\end{aligned}$$ does not describe any time evolution of the system under study. Instead, in this Green’s function the initial state of the system can be the thermodynamic state corresponding to a Hamiltonian, $\hat{H}^{M}=\hat{H}(t_0)-\mu \hat{N}$, where $\hat{N}$ is the particle number operator. Thus, from this Matsubara Green’s function, the initial ensemble average of any one-body operator $\hat{O}=\sum_{ij} O_{ij} \hat{d}^{\dagger}_{i} \hat{d}_{j}$, can be evaluated simply as $$\begin{aligned}
O=\frac{{\rm Tr} [ e^{-\beta\hat{H}^{M}}\hat{O}]} {{\rm Tr} [e^{-\beta\hat{H}^{M}}]}=\pm i \sum_{ij}O_{ij}G^{M}_{ji}(\tau),
\end{aligned}$$ where $\tau=\tau_1-\tau_2$ since the one-body Green’s function matrix elements depend only on the difference of the imaginary time variables. Furthermore, the imaginary time Green’s function $G(\tau)$ can be Fourier transformed to the imaginary frequency Green’s function $G(i\omega_n)$ where for fermions the frequency grid is given by $\omega_n=\frac{(2n+1)\pi}{\beta}$ with $n$ defined here as a positive integer. For simplicity, in the remainder of this paper, we will drop subscript $M$ denoting Matsubara Green’s functions since from now on we will only discuss this finite temperature formalism.
The discussion above shows that at finite temperature one can get grand canonical ensemble averages of one-body operators using the Matsubara formalism. However, let us ask one more question: Can we get a system’s static thermodynamic variables such as electronic Gibbs or Helmholtz energy, internal energy, and electronic entropy from dynamic (frequency dependent) variables such as Green’s function and self-energy?
The thermodynamics of a system can be described by a thermodynamical potential, such as the grand potential $\Omega$ [@vanleeuwentext] $$\label{LW_functional}
\Omega = \frac{1}{\beta}\{ \Phi -{\rm Tr}[\Sigma G+{\rm ln}(\Sigma-G^{-1}_0)]\},$$ where the self-energy $\Sigma=\Sigma(i\omega_n)$ describes all the frequency dependent correlational effects present in the system and $G_0=G_0(i\omega_n)$ is the reference (usually non-interacting) system’s Green’s function and $G=G(i\omega_n)$ is the interacting Green’s function. The interacting and non-interacting Green’s functions are connected through the Dyson equation, $\Sigma=G^{-1}_0-G^{-1}$. The $\Phi$ functional from Eq. \[LW\_functional\] is called the Luttinger-Ward functional[@LW1960] [@footnote2] and is defined as
$$\Phi= \sum_{m=1}^{\infty} \frac{1}{2m}{\rm Tr} [\sum_{n}\Sigma^{(m)}(i\omega_n) G(i\omega_n) ],$$
where $\Sigma^{(m)}(i\omega_n)$ is a self-energy containing all irreducible and topologically inequivalent diagrams of order $m$. The detailed derivation of Eq. \[LW\_functional\] is presented in Ref. .
Thus, the computational object that provides a connection between the static and dynamic quantities is the Luttinger-Ward functional. This scalar functional $\Phi=\hat{\Phi}[G]$ depends on the Green’s function and has multiple important properties for Green’s function theory.
Properties of the Luttinger-Ward Functional {#sec_phi}
===========================================
The formal properties of Luttinger-Ward functional have been discussed extensively before. The functional has previously been applied to calculate energies for atoms and molecules [@van2006total; @dahlen2006variational], the electron gas [@VE], as well as for the Hubbard lattice [@potthoff2003Hubbard; @Janis; @Janis2; @potthoff2]. In the following section, we will only outline some of the most salient properties of the Luttinger-Ward functional [ to]{} benefit the reader.
Self-energy as a functional derivative
--------------------------------------
The self-energy $\Sigma=\Sigma(i\omega_n)$ can be obtained as a functional derivative of the Luttinger-Ward functional $$\beta \frac{\delta \hat{\Phi}[G]}{\delta G} = \hat{\Sigma}[G].$$ Here, the self-energy is defined as a functional of Green’s function that is evaluated independent of the Dyson equation. Consequently, in the non-interacting limit, where $\Sigma=0$, it follows that the Luttinger-Ward functional is zero itself, $\hat{\Phi}[G]=0$.
Connection with the grand potential $\Omega$
--------------------------------------------
The grand potential is a number but the mathematical object defined in Eq. \[LW\_functional\] can be viewed more generally as a functional of Green’s functions $\Omega[G]$. When a Green’s function is a self-consistent solution of the Dyson equation then the functional derivative $\frac{\delta \Omega[G]}{\delta G}=0$ since $\delta \Phi={\rm Tr}[\Sigma \delta G]$ and $\delta \Omega[G]= \frac{1}{\beta} \{ \delta \Phi - {\rm Tr}[\delta \Sigma G + \Sigma \delta G - G \delta \Sigma] \}$. Consequently, we can conclude that the functional $\Omega[G]$ at the stationary point is equal to the grand potential. Having grand potential one gains access to the partition function (Z) since $$\label{eqn:gc_pt}
\Omega=-\frac{1}{\beta}{\rm ln} Z.$$ The Helmholtz energy, $A=E-TS$, where $E$ is the internal energy and $S$ is the entropy of a system at a given temperature $T$, is connected to the grand potential as $A=\Omega+\mu N$, where $N$ is the number of electrons in the system. Thus, knowing the grand potential, we can easily calculate the electronic entropy as $$S=\frac{E-\Omega-\mu N}{T}$$ as long as we have access to the internal energy of a system. The internal energy at a given temperature $T$ can be evaluated using the Galitskii-Migdal formula $$E=\frac{1}{2}{\rm {Tr}}\left[\left(h+F\right)\gamma\right]+\frac{2}{\beta}\sum_{n}^{N_\omega}{\rm {Re}\big(\rm Tr}[G(i\omega_n)\Sigma(i\omega_n)]\big)\label{eq:Etot},$$ where $\gamma$ is the one-body density matrix, $h$ is the one-body Hamiltonian, $F$ is the Fock matrix of a system, and $N_\omega$ is the size of the imaginary grid. Consequently, having access to the Luttinger-Ward functional of a system yields multiple electronic thermodynamic quantities.
Universality
------------
Given two systems $A$ and $B$ at the same physical temperature $T$ and the same chemical potential $\mu$ described by two Hamiltonians $\hat{H}_A=\sum_{ij}t^A_{ij}a_{i}^{\dagger}a_{j}+\sum_{ijkl}v_{ijkl}a_{i}^{\dagger}a_{j}^{\dagger}a_{l}a_{k}$ and $\hat{H}_B=\sum_{ij}t^B_{ij}a_{i}^{\dagger}a_{j}+\sum_{ijkl}v_{ijkl}a_{i}^{\dagger}a_{j}^{\dagger}a_{l}a_{k}$ such that they have the same two-body integrals $v_{ijkl}$ but different one-body integrals $t^{A}\ne t^{b}$ the Luttinger-Ward functional is same (universal) for both of them. Since $\hat{\Phi}^A=\hat{\Phi}^B$, then it must also hold that $\hat{\Sigma}^A(G)=\hat{\Sigma}^B(G)$. In other words, two systems described by different $G_0$ but having the same two-body interactions are described by the same Luttinger-Ward functional.
Evaluation of Luttinger-Ward Functional Within GF2 {#sec_eval_phi_gf2}
==================================================
Description of the GF2 algorithm
--------------------------------
The self-energy, which we evaluate self-consistently in this work, is computed perturbatively at the second-order (GF2) level. GF2 is advantageous for many reasons, namely, among others it behaves qualitatively correct for moderately strongly correlated systems [@GF2], unlike methods such as MP2 or CCSD which tend to diverge in these cases. GF2 has small fractional charge and fractional spin errors [@fractional]. GF2 is carried out self-consistently on imaginary time $\tau$ and imaginary frequency $i\omega_n$ axes with a computational scaling of $\mathcal{O} (n_\tau N^5$) for molecular cases, where $n_\tau$ is a prefactor that depends on the size of the imaginary time grid and $N$ is the number of orbitals present in the problem. We build the Green’s function using the following expression $${G}(i\omega_n)=[(i\omega_n + \mu){S} - {F} - {\Sigma}(i\omega_n)]^{-1}$$ where ${F}$ and ${S}$ are the Fock and overlap matrices in the atomic orbital (AO) basis, respectively, and $\mu$ is the chemical potential, which guarantees a correct particle number. To obtain $\Sigma(i\omega_n)$, we solve the Dyson equation given as $${\Sigma}(i\omega_n) = {G_0}(i\omega_n)^{-1} - {G}(i\omega_n)^{-1}$$ where ${G_0}(i\omega_n)=[(i\omega_n + \mu){S} - {F}]^{-1}$ is the non-interacting Green’s function while $G(i\omega_n)$ is the interacting Green’s function since it contains the self-energy. To reduce the number of necessary grid points, we employ a spline interpolation method to evaluate the Green’s function [@spline] and Legendre orthogonal polynomials to expand the $\Sigma(\tau)$ matrix [@legendre]. As pointed out previously, this self-consistent evaluation guarantees that both the Galitskii-Migdal and Luttinger-Ward energies are stationary with respect to the Green’s function. For a full discussion of the algorithm and implementation details of GF2, we refer the reader to Refs. . The main computational object in our evaluation is the self-energy in the AO basis, which can be expressed as $$\label{se}
\begin{split}
\Sigma_{ij}(\tau) = -\sum_{klmnpq}^{ }G_{kl}(\tau)G_{mn}(\tau)G_{pq}(\tau) \times \\
\times v_{ikmq}(2v_{ljpn} - v_{pjln}).
\end{split}$$
where $v_{ijkl}$ are the two-electron integrals in the AO basis in the chemist’s notation. Eq. \[se\] is evaluated in a Legendre polynomial basis [@legendre] to accelerate the calculation.
The details of a periodic GF2 implementation have been reported in Ref. [@rusakov]. The basic differences from the molecular version include solving the Dyson equation in $\textbf{k}$-space via $$\label{GFk}
{G}^{k}(i\omega_n) = \left[(i\omega_n+\mu){S}^{k} - {F}^{k} - {\Sigma}^{k}(i\omega_n)\right]^{-1}$$ and complicating the real-space self-energy, ${\Sigma}^{\mathbf{0g}}(\tau)$, evaluation by additional cell index summations: $$\label{GF2_pbc_real}
\begin{split}
\Sigma_{ij}^{\mathbf{0g}}(\tau) = -\sum_{\mathbf{g_1,\ldots,g_6}}\sum_{klmnpq}G_{k~l}^{\mathbf{g_3g_6}}(\tau)G_{m~n}^{\mathbf{g_1g_4}}(\tau)G_{p~q}^{\mathbf{g_5g_2}}(-\tau) \times \\ \times v_{i~m~q~k}^{\mathbf{0g_1g_2g_3}}(2v_{j~n~p~l}^{\mathbf{gg_4g_5g_6}}-v_{j~l~p~n}^{\mathbf{gg_6g_5g_4}}).
\end{split}$$ ${\Sigma}^{k}(i\omega_n)$ in Eq. \[GFk\] and ${\Sigma}^{\mathbf{0g}}(\tau)$ in Eq. \[GF2\_pbc\_real\] are interconvertible via corresponding Fourier transforms from the $\mathbf k$- to real-space and between the imaginary time and imaginary frequency domains also via Fourier transform. The self-energy calculation according to Eq. \[GF2\_pbc\_real\] results in $\mathcal{O}(N^5N_{cell}^4n_{\tau})$ formal scaling of the computation cost with $N$ the number of orbitals in the unit cell and $N_{cell}$ the number of real space cells. This is typically a computational bottleneck of the GF2 self-consistency procedure.
The level of self-consistency at which the correlated Green’s function equation should be iterated can depend on particular phases present in the phase diagram, such as Mott (see Ref. [@Kotliar]). In particular, the non-interacting Green’s function $G_{0}(i\omega_n)$ build using updated Fock matrix or the correlated Green’s function $G(i \omega_n)$ can re-enter the evaluation of the self-energy. We observe that the use of $G_{0}(i \omega_n)$ with the updated Fock matrix in the self-consistent evaluation of the self-energy is a well-behaved procedure when the strong correlations and the Mott phases emerge. The full self-consistent cycle, with $G(i \omega_n)$ re-entering the evaluation of the self-energy becomes ill behaved for these cases and we experienced difficulty converging it. We therefore use the former “partial” self-consistency ($G_{0}(i \omega_n)$ with the updated Fock matrix in the self-consistent evaluation of the self-energy) for the Mott regime. Such scheme is not uncommon in DMFT type calculations [@Kotliar].
The expression for the total energy likewise acquires a cell summation according to Eqs. 13 and 14 in Ref. [@rusakov]: $$\begin{aligned}
&E_{tot} =E_{1b}+E_{2b}\\ \nonumber
&=\frac{1}{2} \sum_{\mathbf{g}, i, j}\gamma^{\mathbf{0g}}_{ij}(2h^{\mathbf{0g}}_{ij}+\left[\Sigma_{ \infty}\right]^{\mathbf{0g}}_{ij}) + \\
&+ \frac{2}{\beta}\sum_{\mathbf{g}, i, j}\operatorname{Re}\left[\sum_{n} G^{\mathbf{0g}}_{ij}(i\omega_n)\Sigma^{\mathbf{0g}}_{ij}(i\omega_n) \right]. \nonumber\end{aligned}$$
Evaluation of the grand potential in the $\mathbf k$-space
----------------------------------------------------------
The evaluation of the Luttinger-Ward functional in conserving approximations [@baymkadanoff] such as the self-consistent second-order Green’s function (GF2)[@GF2; @fractional; @legendre] is quite straightforward and was originally derived by Luttinger and Ward [@LW1960] as
$$\Phi^{(2)} = \frac{1}{4}{\rm Tr}[\sum_{n}\Sigma^{(2)}(i\omega_n)G(i\omega_n) ],$$
where $\Sigma^{(2)}(i\omega_n)$ is the frequency dependent part of the second-order self-energy. Since both $G(i\omega_n)$ and $\Sigma^{(2)}(i\omega_n)$ are readily available from a GF2 calculation, we are able to easily evaluate all terms of the functional.
The expression for grand potential from Eq. \[LW\_functional\] can be conveniently reformulated as $$\begin{split}
\Omega_{LW} = \frac{1}{2} {\rm Tr}[\gamma \Sigma_{\infty}] + {\rm Tr}[G \Sigma]
\\ +{\rm Tr}[{\rm ln} \{ 1 - G \Sigma \} ] + {\rm Tr}[{\rm ln} \{ G_0^{-1} \}].
\end{split}
\label{eqn:LW_reform}$$
An excellent derivation of the above equation is given in Ref. for molecular systems. However, as we mentioned before, for molecular systems the changes due to the electronic contributions of the Gibbs or Helmholtz energy are negligible due to the size of the gap.
Here, we list detailed steps that need to be executed when dealing with crystalline systems where the electronic effects influencing Helmholtz energy can be significant. In the periodic implementation, we evaluate the grand potential per unit cell, $\Omega^{\mathbf{00}}$. The overall expression is given as a sum of all the components $$\begin{split}
\Omega^{\mathbf{00}} = \Omega_{ \rm Tr[G \Sigma]}^{\mathbf{00}} + \Omega_{ \rm Tr[\rm ln \{ 1-G \Sigma \} ]}^{\mathbf{00}}
\\ + \Omega_{ \rm Tr[\rm ln \{ G_0^{-1} \} ]}^{\mathbf{00}} + \Omega_{\frac{1}{2} \rm Tr[\gamma \Sigma_{\infty}]}^{\mathbf{00}}
\end{split}
\label{eqn:LW_central}$$
where we sum the contribution per unit cell for each term present in Eq. \[eqn:LW\_reform\]. Due to the crystalline symmetry, it is often more convenient to calculate these quantities in $\mathbf k$-space and transform the resulting quantity to the real space rather than using an explicit real space representation of all the quantities involved. All terms in Eq. \[eqn:LW\_central\] containing the Green’s function $G$, or self-energy $\Sigma$ were computed in $\mathbf k$-space and then Fourier transformed to real space. For example, to calculate the $ \Omega_{ \rm Tr[G \Sigma]}^{\mathbf{00}}$ contribution, we can first evaluate the $\mathbf k$-dependent quantity
$$\Omega^k_{\rm Tr[G \Sigma]} = \frac{2}{\beta} \sum_{i,j,n}^{N_{\omega}} G^k(i\omega_n)_{ij} \Sigma^k(i\omega_n)_{ji}
\label{eqn:Trace}$$
where the indices $i$ and $j$ run over all atomic orbitals in a given $k-$block. Subsequently, we perform Fourier transform of $\Omega^k_{\rm Tr[G \Sigma]}$ to yield the real space $\Omega^{\mathbf{00}}_{\rm Tr[G \Sigma]}$ contribution.
The most cumbersome evaluation is of the term $ \Omega_{ \rm Tr[\rm ln \{ 1-G \Sigma \} ]}^{\mathbf{00}} $, which requires diagonalization of the matrix $ G^k \Sigma^k + (G^k \Sigma^k)^{\dagger} - G^k \Sigma^k (G^k \Sigma^k)^{\dagger}$, where the matrices $G^k$ and $\Sigma^k$ are understood to be dependent on $i\omega_n$. Once the eigenvalues of this matrix, $\epsilon^{k}_{i}$, have been computed for each imaginary frequency point, $i\omega_n$, we have
$$\Omega_{ \rm Tr[\rm ln \{ 1-G \Sigma \} ]}^{k} = \frac{2}{ \beta } \sum_{n, i}^{N_{\omega}} \rm ln \{ 1 - \epsilon^{k}_{i} \},
\label{eqn:logterm}$$
which can be Fourier transformed to the real space.
To include the contribution from the Fock matrix, $\Omega_{ \rm Tr[\rm ln \{ G_0^{-1} \} ]}^{\mathbf {00}}$, we calculate
$$\Omega_{ \rm Tr[\rm ln \{ G_0^{-1} \} ]}^{k}=\begin{cases}
\frac{1}{\beta} \: \sum_{i}\rm ln(1 + e^{\beta(\epsilon^{k}_{i} - \mu)}) + \epsilon^{k}_{i}, & \text{if $\epsilon^{k}_{i}-\mu <0$}\\
\frac{1}{\beta} \: \sum_{i}\rm ln(1 + e^{-\beta(\epsilon^{k}_{i} - \mu)}), & \text{otherwise}.
\end{cases}
\label{eqn: Fock}$$
where by analyzing if the term $\epsilon^{k}_{i}-\mu$ is smaller or greater than zero we account for the cases where the absolute value of the Fock matrix eigenvalue can be large leading to numerical problems if only one branch of the above expression is used. This term accounts for occupation changes with temperature. For high $\beta$ values (low values of the actual temperature $T$), the expression reduces to
$$\Omega_{ \rm Tr[\rm ln \{ G_0^{-1} \} ]}^{k}=\sum_{N_{occ}}\epsilon^{k}_{i},$$
which allows electrons to occupy only the lowest available state, as is expected at a very low temperature.
We calculate the term $\Omega_{\frac{1}{2} \rm Tr[\gamma \Sigma_{\infty}]}^{\mathbf{00}}$ directly in the real space as $\frac{1}{2} {\rm Tr}[\gamma \Sigma_\infty]$, since in the AO basis, the decay of both $\gamma$ which is the density matrix and $\Sigma_{\infty}$ which is the frequency independent part of the self-energy is rapid enough for a relatively few number of cells to assure a converged value of $\Omega_{\frac{1}{2} \rm Tr[\gamma \Sigma_{\infty}]}^{\mathbf{00}}$.
Results {#sec_results}
=======
HF molecule
-----------
In this subsection, for a simple molecular example, a hydrogen fluoride molecule, we provide a calibration of the thermodynamic quantities such as internal energy $E$, Helmholtz energy $A$, and entropy $S$, which are evaluated at the GF2 level and compared to the full configuration interaction (FCI) calculation. The evaluation of the thermodynamic quantities at the FCI level can be done only for very small molecular examples, such as HF, since such a system has only 10 electrons and 6 basis functions in the STO-3G basis, resulting in a small number of possible configurations necessary to evaluate the FCI grand potential. Note that to describe a true physical system at very high temperature we would require a very large basis set. Here, we use HF as a model molecular system that is calculated in a minimal basis set solely to enable comparison of GF2 with FCI. The full configuration interaction (FCI) quantities for HF molecule were calculated previously by Kou and Hirata [@FCISoHirata]. In addition, we show the same system calculated with finite-temperature at the Hartree-Fock level. We would like to emphasize that we provide this molecular example as a benchmark only, in order to compare our method with highly accurate FCI quantum chemical data. Typically electronic contributions to thermodynamics are not considered for molecular systems.
Let us first note that in order to calculate the FCI partition function and subsequently grand potential in the grand canonical ensemble, we need to evaluate $$Z^{GC}=\sum_{N=0}^{2n}\sum_{S_z}\sum_{i}\langle \Phi_{i}^{(N,S_z)}|exp\{-\beta(\hat{H}-\mu\hat{N})\}| \Phi_{i}^{(N,S_z)}\rangle,$$ where the $\Phi_{i}^{(N,S_z)}$ is the FCI wave function with $N$ electrons and $S_z$ quantum number and the number of possible occupation runs from 0 to $2n$, where $n$ is the number of orbitals. Consequently, we need to explicitly obtain the information about every possible excited state present in the system with different number of electrons. Such a task quickly becomes impossible for any larger systems. In contrast, in Green’s function methods, we never need to explicitly evaluate any information concerning specific excited states. It is sufficient to evaluate Eq. \[eqn:LW\_reform\] and then Eq. \[eqn:gc\_pt\] to obtain the grand canonical partition function. Thus, even for relatively large systems such calculations remain feasible.
Results from our calibration are shown in Fig. \[fig:FCI\_GF2\_HF\]. The numerical values used in the plots are tabularized in the Supplemental Information. The detailed description of the grids on which we evaluate $\Sigma(\tau)$ and $G(i\omega_n)$ can be found in Ref. [@footnote1]. For the hydrogen fluoride molecule the temperature range is huge due to the size of the Hartree-Fock HOMO-LUMO gap in this system, which is around 1.0 a.u. corresponding to a temperature of around $3.0 {\ensuremath{\times 10^{5}}}$ K. Consequently, to make every state accessible to the electrons in this system, we require extremely high temperatures, as indicated by our results.
In the very high temperature limit, both finite temperature Hartee-Fock and GF2 yield the internal energy (E), Helmholtz energy (A), and entropy (S) in excellent agreement with FCI results. This is of course expected since at very high temperatures the electronic behavior is well described by mean field theories. For the intermediate temperatures, GF2 thermodynamic quantities (E, S, and A) are closer to FCI than thermodynamic quantities obtained in finite temperature Hartee-Fock. For this system at intermediate temperatures, the GF2 thermodynamic quantities are always overestimated while the finite temperature Hartree-Fock always underestimate them in comparison to FCI. The GF2 and Hartree-Fock entropies for low temperatures are well recovered and comparable. As expected, the low temperature GF2 internal energy is closer to FCI than the one evaluated using finite temperature Hartree-Fock. For our very lowest temperature ($10^3$ K), we recover a small negative entropy. This is a numerical artifact brought about from the level of convergence of the internal energies (1.0[$\times 10^{-5}$]{} a.u.) and the expression for entropy which requires multiplication by $\beta$, $S=\beta (E-\Omega-\mu N)$. Thus, the smallest error $\approx 10^{-5}$ in the energy will result in $ \approx 10^{-3}$ error in the entropy due to multiplication by $\beta=100$.
{width="2.25in"} {width="2.25in"} {width="2.25in"}
Periodic calculation of 1D hydrogen
-----------------------------------
In our previous work, GF2 was implemented for periodic systems and applied to a 1D hydrogen solid [@rusakov] in the mini-Huzinaga [@huzinaga1985basis] basis set. We consider the same system in this work, where we have used 5,000 Matsubara frequencies to discretize the Green’s function in the frequency domain, 353 imaginary-time points, and 27 Legendre polynomials. We have found this grid size is sufficient to evaluate energy differences between the systems at various temperatures. Up to 73 real space unit cells appear in the self-energy evaluation (Eq. \[GF2\_pbc\_real\]).
This system is simple enough to be a test bed for self-consistent Green’s function theory; however, it displays a phase diagram that is characteristic of realistic solids. At different internuclear separations, corresponding to different pressures, we were able to recover multiple solutions. Although yielding different electronic energies and different spectra, these solutions can be mathematical artifacts of the nonlinear self-consistency procedure present in GF2; however, they can also have physical meaning corresponding to different solid phases.
To decide which phase is more stable at a given temperature, it is necessary to consider the Helmholtz energies that we are able to obtain from the Luttinger-Ward functional for every solution. Previously, for the inverse temperature of $\beta=100$, at most of the geometry points, we have identified two possible phases with different internal energies, $E$, that were obtained starting the iterative GF2 procedure either from an insulating or metallic solution. The results of our investigation can be found in Fig. 4 of Ref. . Currently, to analyze the stability of the solutions, for a range of inverse temperatures $\beta=25,75,100$, we discuss internal energy $E$, Helmholtz energy $A$, and the entropic contribution $TS$ to the Helmholtz energy. We also improved our convergence criteria not only converging the internal energy $E=E_{1b}+E_{2b}$ (as we have done in the previous work) but also converging both the $E_{1b}$, $E_{2b}$, and the Helmholtz energy separately. This much more stringent procedure to analyze convergence of GF2 leads us to slightly revised solutions for the 1D hydrogen solid which we discuss in the subsequent sections.
The spectra are produced from analytical continuation of the imaginary axis Green’s function $G(\mathbf{k}, i\omega_n)$ to the real axis $G(\mathbf{k}, \omega_n)$ [@ALPS]. The spectral weight is proportional to $\text{Im}G(\mathbf{k}, \omega_n)$. A 2D color projection of the spectral function on the $(\mathbf{k},\omega_n)$ plane can be viewed as a “correlated band structure” analogous to the conventional band structure within effective one-electron models such as Hartree–Fock and DFT. As in one-electron models, zero spectral weight at the Fermi energy $\omega_F$ is indicative of a gapped system. The peaks emerging immediately below and above $\omega_F$ for a given $\mathbf{k}$ correspond to the energies of the highest occupied (HOCO) and lowest unoccupied (LUCO) crystalline orbitals, respectively. We should stress, however, that since $G(\mathbf{k}, \omega_n)$ is a many-body correlated Green’s function, such correspondence is not rigorous and merely serves as a convenient analogy.
### Short bond length/high pressure
At the interatomic separation of 0.75 Å, we recovered only one gapless, metallic solution for all the values of inverse temperature ($\beta=25$, 75, 100). We established that starting from two different initial guesses leads in both cases to two final solutions that were different in internal energy and Helmoltz energy by less than $10^{-4}$ a.u. Consequently, we deemed that we obtained the same metallic solution in both cases. The spectral functions and spectral projections for this metallic solution at different values of inverse temperature are shown in Fig. \[fig:075\]. For all the temperatures examined in this short bond length regime, the self-energy displays a Fermi liquid character.
{width="6.69in"}\
In our previous work [@rusakov], we observed a small internal energy difference between the solutions obtained using different starting point at $\beta=100$. We currently observe that this difference can be eliminated if we assure that the convergence criteria are not only fulfilled for the total energy $E_{tot}=E_{1b}+E_{2b}$ but also both the $E_{1b}$ and $E_{2b}$ components separately.
### Intermediate bond length/intermediate pressure
Spectral functions and projections for the 1D hydrogen solid with an interatomic separation of 1.75 Å are presented in Fig. \[fig:175\]. We have displayed differences in internal energy (E), entropy (written as -T$\Delta$S), and Helmholtz energy (A) in Table \[tab:1.75data\]. For this system at the inverse temperatures of $\beta=100$ and $75$ we obtained two solutions from two different initial guesses. We are able to characterize the first solution (“solution 1") as a band insulator since the spectral function shows a gap and the self-energy displays a Fermi liquid profile. The second solution (“solution 2") is gapless and therefore a metal. At an inverse temperature of $\beta=25$, both “solution 1" and “solution 2" obtained from two different starting guesses have the same spectra and identical Helmholtz energy, indicating that there is only one stable solution — a single phase. Note that in Fig. \[fig:175\] for $\beta=25$ both the spectral functions for “solution 1" and “solution 2" seem to have different heights; however, it is an illusion since both spectral functions are plotted with a different z-axis range. We deem that the Helmholtz energy is identical for both these solutions at $\beta=25$ since the obtained differences are below our convergence threshold which is $1\times 10^{-5}$ a.u.
For two lower temperatures ($\beta=100$ and $75$) by comparing thermodynamic quantities, we are able to determine which phase, “solution 1" or “solution 2", is the most thermodynamically stable. From the data in Table \[tab:1.75data\], we are able to determine that “solution 1" is the most stable phase from the positive value of $\Delta$A at both $\beta$=100 and $\beta$=75. It is also interesting to note, that this solution is stable due to the entropic factor not due to the difference in the internal energy.
$\beta$ $\Delta$ E -T $\Delta$ S $\Delta$ A
--------- -------------------------- --------------------------- ---------------------------
100 -0.00370 0.10854 0.10484
75 -0.00528 0.09971 0.09443
25 1.36[$\times 10^{-8}$]{} -2.94[$\times 10^{-6}$]{} -2.93[$\times 10^{-6}$]{}
: Thermodynamic data for a 1D periodic hydrogen solid with separation R=1.75 Å. The units for $\beta$ are 1/a.u. The units for all other quantities are a.u. All values are obtained by subtracting “solution 1" from “solution 2". $\Delta$ E =E$_{sol2}$-E$_{sol1}$, $\Delta$ A =A$_{sol2}$-A$_{sol1}$, $\Delta$ S =S$_{sol2}$-S$_{sol1}$. Note that the quantities at $\beta$=25 are below the precision of convergence (1 [$\times 10^{-5}$]{}).[]{data-label="tab:1.75data"}
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### Long bond length/low pressure
Similar to the R=1.75 Å regime, for an interatomic separation of 2.5 Å at inverse temperatures $\beta=100$ and $75$, we recover two solutions from the two different initial guesses, see Fig. \[fig:2.5\]. At a high temperature $\beta$=25 we observe only one solution independent of the initial guess, and thus only a single phase is present. From the positive value of $\Delta$A we are able to see that “solution 1" is the most stable phase at $\beta$=75 and $\beta$=100 (Table \[tab:2.5data\]). This solution is a band insulator since it has a Fermi liquid self-energy profile at these temperatures. The other solution, denoted as “solution 2" at low temperature is metallic and changes into a Mott insulator at high temperatures. The internuclear distance of R=2.5 Å is close to the region where the phase transition occurs, thus results obtained from GF2 which is a low order perturbation expansion may not be reliable. The low level perturbation theories such as GF2 are known to be more accurate deep within a phase and can experience problems near the phase transition point.
$\beta$ $\Delta$ E -T $\Delta$ S $\Delta$ A
--------- -------------------------- -------------------------- ---------------------------
100 0.08489 0.16257 0.24746
75 0.07103 0.16155 0.23259
25 6.09[$\times 10^{-8}$]{} 2.96[$\times 10^{-5}$]{} 2.97 [$\times 10^{-5}$]{}
: Thermodynamic data for a 1D periodic hydrogen solid with separation R=2.5 Å. The units for $\beta$ are 1/a.u. The units for all other quantities are a.u. All values are obtained by subtracting “solution 1" from “solution 2". $\Delta$ E =E$_{sol2}$-E$_{sol1}$, $\Delta$ A =A$_{sol2}$-A$_{sol1}$, $\Delta$ S =S$_{sol2}$-S$_{sol1}$. Note that the quantities at $\beta$=25 are below or comparable with the precision of convergence (1 [$\times 10^{-5}$]{}).[]{data-label="tab:2.5data"}
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Finally, at an interatomic separation of R=4.0 Å, for all the temperatures, both initial guesses yield the same converged GF2 result - a single phase. The spectra as a function of inverse temperature can be seen in Fig. \[fig:4.0\]. This single solution is a Mott insulator as confirmed by the divergent imaginary part of the self-energy.
{width="6.69in"}
### GF2 phase diagram for 1D hydrogen solid
It is instructive now to collect all the data and construct a simple phase diagram for the 1D hydrogen solid as a function of inverse temperature $\beta$ and intermolecular distance R. We have plotted the phase diagram in Fig. \[fig:phase\]. On this diagram, for these regions where two phases coexist, we denoted the most stable phase according to the Helmholtz energy $A$ by framing its symbol using a black line.
For the shortest bond length (R=0.75 Å), the 1D hydrogen solid remains metallic at all temperatures considered and only one phase is present. At intermediate bond lengths (R=1.75, 2.0 Å), multiple phases coexist. At lower temperatures, we recover both a metal and band insulator as possible solutions, with the band insulator being the most stable phase according to the Helmholtz energies. This is in line with physical intuition that we should recover an insulator rather than a metal at low temperature. For R=1.75 Å, at temperatures higher than $\beta=50$, we recover only the metallic phase.
For R=2.0 Å at high temperature ($\beta=25$), we see both a metal and a Mott insulator coexisting, with the Mott insulator being the most stable phase. At lower temperatures, we see the coexistance of a metallic and band insulator solution. The Helmholtz energy indicates that band insulator is the most stable phase.
At a longer bond length of (R=2.5 Å), multiple phases are still present. As in the cases of intermediate bond length, we recover both a band insulator and a metal solution, with Helmholtz energy favoring the band solution. We would like to reiterate that a phase transition occurs somewhere in this intermediate region, and it is likely that the results of a second-order perturbation theory may not be accurate enough. At higher temperatures, we recover a Mott insulator as the only phase present.
For the largest separation (R=4.0 Å), the system remains a Mott insulator at all studied temperatures.
![A phase diagram containing all distances and temperatures calculated for a 1D hydrogen solid. Where multiple phases exist, the most stable phase is outlined in black. $\beta$ is inverse temperature in units 1/a.u. and R is the separation between hydrogens in Å.[]{data-label="fig:phase"}](phase_diagram.pdf){width="3.375in"}
Periodic calculation of 1D boron nitride
----------------------------------------
In this section we present a periodic calculation of 1D boron nitride (BN) at inverse temperatures $\beta=$70, 75, and 100. The B–N distance is set to 1.445 Å, the bond length in the corresponding 2D system [@C4CS00102H]. For these calculations we used a modification of the ANO-pVDZ Gaussian basis set [@doi:10.1021/ct100396y] from which, in order to avoid linear dependencies, we removed the diffuse (below 0.1) exponents and polarization functions. Larger grids of 30000 Matsubara frequencies and 100 Legendre polynomials were found necessary for an adequate discretization for the self-energy and Green’s function. The self-energy is evaluated encompassing 39 cells — as many as needed for a converged Hartree–Fock exchange.
![Spectral projection in the vicinity of the Fermi energy at $\beta$=100 of periodic 1D boron nitride solid. Only the “correlated bands” corresponding to the conventional HOCO’s and LUCO’s (see Sec. V. B) are displayed. Note that the energy scale is in eV. []{data-label="fig:BN_proj"}](proj_BN_b100.pdf){width="3.375in"}
![Periodic 1D boron nitride solid spectrum for the “correlated HOCO and LUCO” (see Sec. V. B) for several temperatures at k=$-\pi$. The chemical potential was adjusted for $\omega=0$ to fall in the middle of the band gap. []{data-label="fig:BN_3"}](b_100_75_70.pdf){width="3.375in"}
Shown in Fig. \[fig:BN\_proj\] is the spectral projection of the real-frequency correlated Green’s function at $\beta$=100. Using the analogy previously discussed at the end of Section V. B, we plot the “highest occupied” and “lowest unoccupied correlated bands” of 1D BN separated by a sizable gap of approximately 3.5 eV at k=-$\pi$. This magnitude of the band gap is expected based on the properties of its 2D counterpart [@C4CS00102H].
Once again, we treat such a simple system as a benchmark example and we are interested in the evolution of the 1D BN band gap as the temperature changes and influences electronic degrees of freedom. Let us stress that while an evolution of the spectrum as a function of temperature is expected, in order to reproduce it reliably, the computational procedure must be very robust. We not only must be able to iteratively converge Green’s function and self-energy yielding different Green’s functions at different temperatures, we also must continue the results obtained on the imaginary axis to the real axis risking that the continuation will obscure the spectral features and the gap temperature dependence will no longer be visible.
In Fig. \[fig:BN\_3\] we plot the spectrum at k=-$\pi$ for $\beta$=100, 75, and 70. As expected, with the increase of temperature (decrease of $\beta$) the spectral peaks corresponding to the bands broaden and the band gap reduces. Note that both Fig. \[fig:BN\_proj\] and \[fig:BN\_3\] use eV as energy units for clarity.
Conclusion {#sec_conclusions}
==========
While the theory describing the connection between the Matsubara Green’s function formalism and thermodynamics has been known since the 1960s, few computational methods are currently capable of employing the Matsubara formalism for realistic systems. This is due to its computationally demanding nature that requires complicated time and frequency grids as well as the iterative nature of the equations. Moreover, little is know about obtaining different solutions corresponding to different phases or non-physical solutions that can appear as the result of the non-linear procedure used when Green’s functions are constructed iteratively on the imaginary axis.
In our last work [@rusakov], we have demonstrated one of the first applications of the fully self-consistent Matsubara Green’s function formalism to a benchmark periodic problem with realistic interactions, that is, a 1D hydrogen solid. In the current work, we have shown that it is possible to evaluate temperature dependent thermodynamic quantities using a self-consistent second-order Green’s function method. Using the self-consistent Green’s function and self-energy, we were able to evaluate the Luttinger-Ward functional at various temperatures and obtain static thermodynamic quantities such as Helmholtz energy, internal energy, and entropy. Evaluation of these quantities gives us access to the partition function and any thermodynamic quantity that can be derived from it. Unlike the FCI case, in GF2 we do not need to explicitly calculate excited state energies or Boltzmann factors, making calculation of thermodynamic quantities for larger systems feasible.
To calibrate the thermodynamic data obtained from GF2 against FCI, we have illustrated that for a hydrogen fluoride molecule at high temperature, we are able to obtain energies in excellent agreement with finite temperature FCI. In the lower temperature regime, we observed, as expected, some deviation from the FCI answer but the overall quality of the results still remained high.
Finally, since the thermodynamic data can be used to construct phase diagrams, we used a simple 1D hydrogen solid to investigate possible phases at different temperatures and interatomic distances. We obtained different phases such as a band insulator, Mott insulator, or metal, and we were able to distinguish which phase is more stable at various temperatures. Determining the stability is possible due to our ability to calculate not only the internal energy but also the entropic contribution at each temperature. Consequently, based on the difference of the Helmholtz energy, we are able to determine the most stable phase for each of the temperature points.
Additionally, we have performed a calculation of 1D periodic BN solid demonstrating that GF2 can reproduce the gap deviation as a function of temperature. Thus, for higher temperatures, we have observed the narrowing of the electronic bandgap.
While we acknowledge that GF2 is a low order perturbative method and as such can deliver results that are inaccurate, we believe that for most weakly and moderately correlated systems such as semiconductors with small band gaps, it can be used in the future to provide accurate thermodynamic data and phase diagrams. For systems where the correlations are strong, GF2 can be combined with methods such as self-energy embedding theory (SEET) [@AlexeiSEET; @LAN; @nguyen2016rigorous] to provide accurate answers concerning thermodynamics.
D.Z. and A.R.W. would like to acknowledge a National Science Foundation (NSF) grant No. CHE-1453894. A.A.R. acknowledges a Department of Energy (DOE) grant No. ER16391 and Prof. Emanuel Gull for helpful discussions.
Appendix: High-frequency expansion of the Green’s function for evaluating the Luttinger-Ward functional {#Appendix}
=======================================================================================================
Here, we consider the contribution to the Luttinger-Ward functional in the high frequency limit. In the Matsubara formalism for large frequencies, the Green’s function and self-energy can be expressed as a series $$G(i\omega_n) = \frac{G_1}{i\omega_n} + \frac{G_2}{(i\omega_n)^2}+\mathcal{O}(\frac{1}{(i\omega_n)^3}),$$ $$\Sigma(i\omega_n) = \frac{\Sigma_1}{i\omega_n} + \frac{\Sigma_2}{(i\omega_n)^2}+\mathcal{O}(\frac{1}{(i\omega_n)^3}).$$
The coefficients for the Green’s function expansion for non-orthogonal orbitals with a quantum chemistry Hamiltonian [@rusakov2014local] are given as
$$\begin{split}
G_1 &= S^{-1}, \\
G_2&= S^{-1}(F-\mu S)S^{-1}.
\end{split}$$
The $\Sigma_1$ and $\Sigma_2$ coefficient of the self-energy has a complicated explicit form as demonstrated in Ref. and is evaluated numerically as $$\begin{split}
\Sigma_1 &= {\rm Re}\left(\Sigma(i\omega_{max})\times i\omega_{max}\right) \\
\Sigma_2 &= \left(\Sigma(i\omega_{max})-\frac{\Sigma_1}{i\omega_{max}}\right)\times (i\omega_{max})^2 .
\end{split}$$
For the molecular example used in this work, the high frequency contribution to ${\rm Tr}[G \Sigma]$ was evaluated in the same manner as previously discussed in Ref. . At the very high temperatures considered for molecular examples, the high frequency contribution becomes negligible. However, it is necessary to include at the lower temperatures considered. The high-frequency contribution to the $\Omega^k_{Tr[G \Sigma]}$ term of Eq. \[eqn:Trace\] can be evaluated in a way analogous to the molecular case.
To evaluate the high frequency contribution to the logarithm term ${\rm Tr}[{\rm ln}\{ 1 - G\Sigma \}]$ in Eq. \[eqn:LW\_reform\], we expanded the logarithm as a Taylor series ${\rm ln}(1-x) = x - \frac{x^2}{2} + \dots$ to yield
$${\rm ln}\Big(1-\frac{G_1 \Sigma_1}{(i\omega_n)^2}\Big) = \frac{G_{1}\Sigma_1}{(i\omega_n)^2} + \mathcal{O}\Big(\frac{1}{(i\omega_n)^3} \Big).$$
We exclude in this expansion all terms that are equal or smaller in magnitude than $\mathcal{O}\Big(\frac{1}{(i\omega_n)^3} \Big)$. Thus, it is only necessary practically to evaluate the first term in the expansion to capture the important contribution from the high frequency limit. We would like to emphasize that although we do not present results including the high frequency contribution to the logarithm term ${\rm Tr}[{\rm ln}\{ 1 - G\Sigma \}]$ in Eq. \[eqn:LW\_reform\] for the periodic hydrogen solid, this term was evaluated and was found to be minuscule compared to the magnitude of the other energies. However, we would like stress that it is possible that this contribution may be substantial for other systems.
Supplemental Information
========================
---------- --------- --------- --------- ----------- ----------- ----------- ------- -------- -------
$10^{3}$ -98.571 -98.588 -98.597 -98.571 -98.585 -99.845 0.0 -0.003 0.0
$10^{4}$ -98.571 -98.588 -98.597 -98.571 -98.621 -99.944 0.0 0.0 0.0
$10^{5}$ -97.944 -98.135 -98.049 -101.021 -103.067 -102.107 3.175 3.566 3.475
$10^{6}$ -96.794 -96.988 -96.945 -150.563 -151.410 -151.244 4.979 4.949 4.958
$10^{7}$ -92.028 -92.057 -92.056 -729.937 -730.100 -730.095 5.348 5.348 5.348
$10^{8}$ -88.483 -88.487 -88.487 -6846.975 -6847.001 -6847.003 5.406 5.406 5.406
---------- --------- --------- --------- ----------- ----------- ----------- ------- -------- -------
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| {
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---
abstract: 'Let $\field$ be a perfect field and $\sml{F}$ the nonassociative simple Moufang loop consisting of the units in the (unique) split octonion algebra $\octo{\field}$ modulo the center. Then $\aut{\sml{F}}$ is equal to $G_2(\field) \rtimes \aut{\field}$. In particular, every automorphism of $\sml{\field}$ is induced by a semilinear automorphism of $\octo{\field}$. The proof combines results and methods from geometrical loop theory, groups of Lie type and composition algebras; its gist being an identification of the automorphism group of a Moufang loop with a subgroup of the automorphism group of the associated group with triality.'
author:
- |
<span style="font-variant:small-caps;">By GÁBOR P. NAGY</span> [^1]\
*SZTE Bolyai Institute*\
*Aradi vértanúk tere $1$, H-$6720$ Szeged, Hungary*\
*e-mail*: `[email protected]`
- |
<span style="font-variant:small-caps;">PETR VOJTĚCHOVSKÝ</span> [^2]\
*Department of Mathematics, Iowa State University*\
*$400$ Carver Hall, Ames, Iowa, $50011$, USA*\
*e-mail*: `[email protected]`
title:
---
Introduction
============
As we hope to attract the attention of both group- and loop-theorists, we take the risk of being trivial at times and introduce most of the background material carefully, although briefly. We refer the reader to [@SprVel], [@Pflugfelder], [@Carter] and [@HallNagy] for a more systematic exposition.
A groupoid $Q$ is a *quasigroup* if the equation $xy=z$ has a unique solution in $Q$ whenever two of the three elements $x$, $y$, $z\in Q$ are known. A *loop* is a quasigroup with a neutral element, denoted by $\neutral$ in the sequel. *Moufang loop* is a loop satisfying one of the (equivalent) *Moufang identities*, for instance the identity $((xy)x)z=x(y(xz))$. The *multiplication group* $\mlt{L}$ of a loop $L$ is the group generated by all left and right translations $x\mapsto ax$, $x\mapsto
xa$, where $a\in L$.
Let $C$ be a vector space over a field $\field$, and $N:C\to F$ a nondegenerate quadratic form. Define multiplication $\cdot$ on $C$ so that $(C,\,+,\,\cdot)$ becomes a not necessarily associative ring. Then $C=(C,\,N)$ is a *composition algebra* if $N(u\cdot v)=N(u)\cdot N(v)$ holds for every $u$, $v\in C$. Composition algebras exist only in dimensions $1$, $2$, $4$ and $8$, and we speak of an *octonion algebra* when $\dim C=8$. A composition algebra is called *split* when it has nontrivial zero divisors. By [@SprVel Theorem 1.8.1], there is a unique split octonion algebra $\octo{\field}$ over any field $\field$.
Write $\uocto{\field}$ for the set of all elements of unit norm in $\octo{\field}$, and let $\sml{\field}$ be the quotient of $\uocto{\field}$ by its center $\centre{\uocto{\field}}=\{\pm 1\}$. Since every composition algebra satisfies all Moufang identities, both $\uocto{\field}$ and $\sml{\field}$ are Moufang loops. Paige proved [@Paige] that $\sml{\field}$ is nonassociative and simple (as a loop). Liebeck [@Liebeck] used the classification of finite simple groups to conclude that there are no other nonassociative finite simple Moufang loops besides $\sml{\field}$, $\field$ finite.
Liebeck’s proof relies heavily on results of Doro [@Doro], that relate Moufang loops to groups with triality. Before we define these groups, allow us to say a few words about the (standard) notation. Let $G$ be a group. Working in $G\rtimes \aut{G}$, when $g\in G$ and $\alpha\in\aut{G}$, we write $g^\alpha$ for the image of $g$ under $\alpha$, and $\comm{g}{\alpha}$ for $g^{-1}g^{\alpha}$. Appealing to this convention, we say that $\alpha$ *centralizes* $g$ if $g^\alpha=g$. Now, the pair $(G,\,S)$ is said to be a *group with triality* if $S\le \aut{G}$, $S=\spn{\sigma,\,\rho} \cong
S_3$, $\sigma$ is an involution, $\rho$ is of order $3$, $G=\comm{G}{S}$, $\centre{GS}=\{1\}$, and the triality equation $$\comm{g}{\sigma}\comm{g}{\sigma}^\rho\comm{g}{\sigma}^{\rho^{2}}=1$$ holds for every $g\in G$.
We now turn to geometrical loop theory. A *$3$-net* is an incidence structure $\net=(\points,\,\lines)$ with point set $\points$ and line set $\lines$, where $\lines$ is a disjoint union of $3$ classes $\lines_i$ ($i=1$, $2$, $3$) such that two distinct lines from the same class have no point in common, and any two lines from distinct classes intersect in exactly one point. A line from the class $\lines_i$ is usually referred to as an *$i$-line*. A permutation on $\points$ is a *collineation* of $\net$ if it maps lines to lines. We speak of a *direction preserving* collineation if the line classes $\lines_i$ are invariant under the induced permutation of lines.
There is a canonical correspondence between loops and $3$-nets. Any loop $L$ determines a $3$-net when we let $\points=L\times L$, $\lines_1=\{\{ (c,\,y)|y
\in L\}|c\in L\}$, $\lines_2=\{\{(x,\,c)|x\in L\}|c\in L\}$, $\lines_3=\{\{(x,\,y)|x,\,y\in L$, $xy=c\}|c\in L\}$. Conversely, given a $3$-net $\net=(\points,\,\lines)$ and the origin $\origin\in\points$, we can introduce multiplication on the $1$-line $\ell$ through $\origin$ that turns $\ell$ into a loop, called the *coordinate loop* of $\net$. Since the details of this construction are not essential for what follows, we omit them.
Let $\net$ be a $3$-net and $\ell_i\in\lines_i$, for some $i$. We define a certain permutation $\bol{\ell_i}$ on the point set $\points$ (cf. Figure \[Fg:Bol\]). For $P\in\points$, let $a_j$ and $a_k$ be the lines through $P$ such that $a_j\in\lines_j$, $a_k\in\lines_k$, and $\{i,\,j,\,k\}=\{1,\,2,\,3\}$. Then there are unique intersection points $Q_j=a_j\cap \ell_i$, $Q_k=a_k\cap \ell_i$. We define $\bol{\ell_i}(P)=b_j\cap
b_k$, where $b_j$ is the unique $j$-line through $Q_k$, and $b_k$ the unique $k$-line through $Q_j$. The permutation $\bol{\ell_i}$ is clearly an involution satisfying $\bol{\ell_i}(\lines_j)=\lines_k$, $\bol{\ell_i}(\lines_k)=\lines_j$. If it happens to be the case that $\bol{\ell_i}$ is a collineation, we call it the *Bol reflection with axis $\ell_i$*.
(120,60)(-3,0) (50,10)[(0,1)[35]{}]{} (25,10)[(0,1)[20]{}]{} (75,25)(0,4)[5]{}[(0,1)[3]{}]{} (25,20) (50,20) (50,35) (75,35) (22,18)[(5,3)[31]{}]{} (78,37)[(-5,-3)[31]{}]{} (22,20)[(1,0)[31]{}]{} (47,35)[(1,0)[31]{}]{} (20,21)[$P$]{} (79,37)[$P^\prime = {\sigma_{\ell_1}}(P)$]{} (51,10)[$\ell_1$]{} (40,22)[$a_3$]{} (35,29)[$a_2$]{} (60,37)[$b_3$]{} (65,25)[$b_2$]{} (53,17)[$Q_3$]{} (53,38)[$Q_2$]{}
It is clear that for any collineation $\gamma$ of $\net$ and any line $\ell$ we have $\bol{\gamma(\ell)}=\gamma\bol{\ell}\gamma^{-1}$. Hence the set of Bol reflections of $\net$ is invariant under conjugations by elements of the collineation group $\coll{\net}$ of $\net$. A $3$-net $\net$ is called a *Moufang $3$-net* if $\bol{\ell}$ is a Bol reflection for every line $\ell$. Bol proved that $\net$ is a Moufang $3$-net if and only if all coordinate loops of $\net$ are Moufang (cf. [@Bruck p. 120]).
We are now coming to the crucial idea of this paper. For a Moufang $3$-net $\net$ with origin $\origin$, denote by $\ell_i$ ($i=1$, $2$, $3$) the three lines through $\origin$. As in [@HallNagy], we write $\Gamma_0$ for the subgroup of $\coll{N}$ generated by all Bol reflections of $\net$, and $\Gamma$ for the direction preserving part of $\Gamma_0$. Also, let $S$ be the subgroup generated by $\bol{\ell_1}$, $\bol{\ell_2}$ and $\bol{\ell_3}$. According to [@HallNagy], $\Gamma$ is a normal subgroup of index $6$ in $\Gamma_0$, $\Gamma_0=\Gamma S$, and $(\Gamma,\,S)$ is a group with triality. (Here, $S$ is understood as a subgroup of $\aut{\Gamma}$ by identifying $\sigma\in S$ with the map $\tau\mapsto \sigma\tau\sigma^{-1}$.) We will always fix $\sigma=\bol{\ell_1}$ and $\rho=\bol{\ell_1}\bol{\ell_2}$ in such a situation, to obtain $S=\spn{\sigma,\,\rho}$ as in the definition of a group with triality.
The Automorphisms
=================
Let $C$ be a composition algebra over $\field$. A map $\alpha:C\to C$ is a *linear automorphism* (resp.*semilinear automorphism*) of $C$ if it is a bijective $\field$-linear (resp. $\field$-semilinear) map preserving the multiplication, i.e., satisfying $\alpha(uv)=\alpha(u)\alpha(v)$ for every $u$, $v\in C$. It is well known that the group of linear automorphisms of $\octo{\field}$ is isomorphic to the Chevalley group $G_2(\field)$, cf. [@Freudenthal Section 3], [@SprVel Chapter 2]. The group of semilinear automorphisms of $\octo{\field}$ is therefore isomorphic to $G_2(\field)\rtimes\aut{\field}$.
Since every linear automorphism of a composition algebra is an isometry [@SprVel Section 1.7], it induces an automorphisms of the loop $\sml{\field}$. By [@Vojtech Theorem 3.3], every element of $\octo{\field}$ is a sum of two elements of norm one. Consequently, $\aut{\octo{\field}}\le\aut{\sml{\field}}$.
An automorphism $f\in\aut{\sml{\field}}$ will be called *$($semi$)$linear* if it is induced by a (semi)linear automorphism of $\octo{\field}$. By considering extensions of automorphisms of $\sml{\field}$, it was proved in [@Vojtech] that $\aut{\sml{\mathbb F_2}}$ is isomorphic to $G_2(\mathbb
F_2)$, where $\mathbb F_2$ is the two-element field. The aim of this paper is to generalize this result (although using different techniques) and prove that every automorphism of $\aut{\sml{\field}}$ is semilinear, provided $\field$ is perfect. We reach this aim by identifying $\aut{\sml{\field}}$ with a certain subgroup of the automorphism group of the group with triality associated with $\sml{\field}$.
To begin with, we recall the geometrical characterization of automorphisms of a loop.
\[Lm:GeomChar\] Let $L$ be a loop and $\net$ its associated $3$-net. Any direction preserving collineation which fixes the origin of $\net$ is of the form $(x,\,y)\mapsto
(x^\alpha,\,y^\alpha)$ for some $\alpha\in\aut{L}$. Conversely, the map $\alpha:L\to L$ is an automorphism of $L$ if and only if $(x,\,y)\mapsto
(x^\alpha,\,y^\alpha)$ is a direction preserving collineation of $\net$.
We will denote the map $(x,\,y)\mapsto (x^\alpha,\,y^\alpha)$ by $\sqmap{\alpha}$.
By [@HallNagy Propositions 3.3 and 3.4], $\net$ is embedded in $\Gamma_0=\Gamma S$ as follows. The lines of $\net$ correspond to the conjugacy classes of $\sigma$ in $\Gamma_0$, two lines are parallel if and only if the corresponding involutions are $\Gamma$-conjugate, and three pairwise non-parallel lines have a point in common if and only if they generate a subgroup isomorphic to $S_3$. In particular, the three lines through the origin of $\net$ correspond to the three involutions of $S$.
As the set of Bol reflections of $\net$ is invariant under conjugations by collineations, every element $\varphi\in\coll{\net}$ normalizes the group $\Gamma$ and induces an automorphism $\indmap{\varphi}$ of $\Gamma$. It is not difficult to see that $\varphi$ fixes the three lines through the origin of $\net$ if and only if $\indmap{\varphi}$ centralizes (the involutions of) $S$.
\[Pr:Centralizer\] Let $L$ be a Moufang loop and $\net$ its associated $3$-net. Let $\Gamma_0$ be the group of collineations generated by the Bol reflections of $\net$, $\Gamma$ the direction preserving part of $\Gamma_0$, and $S\cong S_3$ the group generated by the Bol reflections whose axis contains the origin of $\net$. Then $\aut{L}\cong\centralizer{\aut{\Gamma}}{S}$.
Pick $\alpha\in\aut{L}$, and let $\indmap{\sqmap{\alpha}}$ be the automorphism of $\Gamma$ induced by the collineation $\sqmap{\alpha}$. As $\sqmap{\alpha}$ fixes the three lines through the origin, $\indmap{\sqmap{\alpha}}$ belongs to $\centralizer{\aut{\Gamma}}{S}$.
Conversely, an element $\psi\in\centralizer{\aut{\Gamma}}{S}$ normalizes the conjugacy class of $\sigma$ in $\Gamma S$ and preserves the incidence structure defined by the embedding of $\net$. This means that $\psi=\indmap{\varphi}$ for some collineation $\varphi\in\coll{\net}$. Now, $\psi$ centralizes $S$, therefore $\varphi$ fixes the three lines through the origin. Thus $\varphi$ must be direction preserving, and there is $\alpha\in\aut{L}$ such that $\varphi=\sqmap{\alpha}$, by Lemma \[Lm:GeomChar\].
It remains to add the last ingredient—groups of Lie type.
\[Th:Main\] Let $\field$ be a perfect field. Then the automorphism group of the nonassociative simple Moufang loop $\sml{\field}$ constructed over $\field$ is isomorphic to the semidirect product $G_2(\field)\rtimes\aut{\field}$. Every automorphism of $\sml{\field}$ is induced by a semilinear automorphism of the split octonion algebra $\octo{\field}$.
We fix a perfect field $\field$, and assume that all simple Moufang loops and Lie groups mentioned below are constructed over $\field$.
The group with triality associated with $M$ turns out to be its multiplicative group $\mlt{M}\cong D_4$, and the graph automorphisms of $D_4$ are exactly the triality automorphisms of $M$ (cf. [@Freudenthal], [@Doro]). To be more precise, Freudenthal proved this for the reals and Doro for finite fields, however they based their arguments only on the root system and parabolic subgroups, and that is why their result is valid over any field.
By [@Freudenthal], $\centralizer{D_4}{\sigma} = B_3$, and by [@Liebeck Lemmas 4.9, 4.10 and 4.3], $\centralizer{D_4}{\rho} = G_2$. As $G_2<B_3$, by [@Gorenstein p. 28], we have $\centralizer{D_4}{S_3} = G_2$.
Since $\field$ is perfect, $\aut{D_4}$ is isomorphic to $\Delta \rtimes
(\aut{\field} \times S_3)$, by a result of Steinberg (cf. [@Carter Chapter 12]). Here, $\Delta$ is the group of the inner and diagonal automorphisms of $D_4$, and $S_3$ is the group of graph automorphisms of $D_4$. When $\chr{\field}=2$ then no diagonal automorphisms exist, and $\Delta=\inn{D_4}$. When $\chr{\field}\neq 2$ then $S_3$ acts faithfully on $\Delta / \inn{D_4} \cong C_2 \times C_2$. Hence, in any case, $\centralizer{\Delta}{S_3} = \centralizer{D_4}{S_3}$. Moreover, for the field and graph automorphisms commute, we have $\centralizer{\aut{D_4}}{S_3} =
\centralizer{D_4}{S_3} \rtimes \aut{\field}$.
We have proved $\aut{M}\cong G_2\rtimes \aut{\field}$. The last statement follows from the fact that the group of linear automorphisms of the split octonion algebra is isomorphic to $G_2$.
One of the open questions in loop theory is to decide which groups can be obtained as multiplication groups of loops. Thinking along these lines we ask: *Which groups can be obtained as automorphism groups of loops*? Theorem \[Th:Main\] yields a partial answer. Namely, every Lie group of type $G_2$ over a perfect field can be obtained in this way.
Finally, the former author asked the latter one at the Loops ’99 conference whether $\aut{\sml{\field}}$ is simple when $F$ is finite. We now know that this happens if and only if $\field$ is a finite prime field of odd characteristic.
[99]{}
. The geometry of binary systems. *Advances Math.* [**49**]{} (1983), 1–105.
. [*A survey of binary systems.*]{} (Springer-Verlag, Berlin, 1958).
. [*Simple groups of Lie type.*]{} (Wiley Interscience, 1972).
. Simple Moufang loops. [*Math. Proc. Cambridge Philos. Soc.*]{} [**83**]{} (1978), 377–392.
. Oktaven, Ausnahmegruppen und Oktavengeometrie. [*Geometria Dedicata*]{} [**19**]{} (1985), 1–63.
and [R. Solomon]{}. [*The classification of the finite simple groups.*]{} No. 3. Part I, Mathematical Surveys and Monographs [**40**]{}(3) (Providence, R.I., AMS, 1998).
and [G. P. Nagy]{}. On Moufang 3-nets and groups with triality. [*Acta Sci. Math. (Szeged)*]{} [**67**]{} (2001), 675–685.
. The classification of finite simple Moufang loops. [*Math. Proc. Cambridge Philos. Soc.*]{} [**102**]{} (1987), 33–47.
. A class of simple Moufang loops. [*Proc. Amer. Math. Soc.*]{} [**7**]{} (1956), 471–482.
. [*Quasigroups and Loops: Introduction*]{}, Sigma series in pure mathematics; [**7**]{} (Heldermann Verlag Berlin, 1990).
and [F. D. Veldkamp]{}, [*Octonions, Jordan Algebras, and Exceptional Groups*]{}, Springer Monographs in Mathematics (Springer Verlag, 2000).
. Finite simple Moufang loops. PhD Thesis, Iowa State University, 2001.
[^1]: Supported by the “János Bolyai Fellowship” of the Hungarian Academy of Sciences, and by the grants OTKA T029849 and FKFP 0063/2001.
[^2]: Partially supported by the Grant Agency of Charles University, grant no. $269$/$2001$/B-MAT/MFF, and by research assistantship at Iowa State University.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The ATLAS Transition Radiation Tracker (TRT) is the outermost of the three sub-systems of the ATLAS Inner Detector at the Large Hadron Collider at CERN. In addition to its tracking capabilities, the TRT provides particle identification (PID) ability through the detection of transition radiation X-ray photons. The latter functionality provides substantial discriminating power between electrons and hadrons in the momentum range from 1 to 200 GeV. In addition, the measurement of an enhancement of signal time length, which is related to high specific energy deposition (dE/dx), can be used to identify highly ionizing particles, increasing the electron identification capabilities at low momentum and improving the sensitivity of searches for new physics. This talk presents the commissioning of TRT PID during early 2010 7 TeV data taking. Performance in 2010 and 2011 demonstrating the TRT’s ability to identify electrons, complementary to calorimeter based identification methods, will also be shown.'
author:
- Elizabeth Hines on behalf of the ATLAS collaboration
title: Performance of Particle Identification with the ATLAS Transition Radiation Tracker
---
Introduction
============
The ATLAS Inner Detector (ID) is composed of three detector sub-systems: the silicon-based Pixel and SemiConductor Tracker (SCT) detectors, and the gaseous drift tube Transition Radiation Tracker (TRT) [@DetectorPaper]. The TRT is the outermost of the three sub-systems. It employs a unique design which combines tracking measurements with particle identification based on detection of transition radiation (TR). The detection of TR allows for discrimination between electrons and pions over the energy range 1-200 GeV and is a crucial component of the electron selection criteria in ATLAS [@ElectronPaper]. These proceedings present the particle identification (PID) performance of the TRT observed in $\sqrt{s} = 7$ TeV proton-proton collision data collected with the ATLAS detector at the Large Hadron Collider (LHC) in 2010.
Transition radiation is emitted when a highly-relativistic charged particle with a Lorentz factor $\gamma \gtrsim 10^{3}$ traverses boundaries between materials of differing dielectric constants. The active region of the TRT detector contains almost 300,000 straw drift tubes of 4mm diameter. The space between the straws is filled with radiator material. The TR photons (soft X-rays) emitted in the radiator are absorbed in the active gas inside the straw tubes, which serve as detecting elements both for tracking and for particle identification.
Particle identification in the TRT can be further augmented at momenta p $\lesssim$ 10 GeV through measurements of the time-over-threshold (ToT) of the straw signals, which vary as a function of energy deposition (dE/dx) in the straws. To achieve the best electron-pion separation, TR and dE/dx-based measurements are combined in a single likelihood function for a particle type.
Transition Radiation Tracker
============================
The TRT is a straw tracker composed of 298,304 carbon-fiber reinforced Kapton straws, arranged in a barrel and two symmetrical end-cap configurations [@StrawPaper]. The barrel section covers $560 < R <
1080$ mm and $|z| < 720$ mm and has the straws aligned with the direction of the beam axis [@BarrelPaper].[^1] The two end-cap sections cover $827 < |z| < 2744$ mm and $617 < R < 1106$ mm and have the straws arranged in planes composing wheels, aligned perpendicular to the beam axis, pointing outwards in the radial direction [@EndcapPaper]. The TRT extends to pseudo-rapidity $|\eta| = 2$. The average number of TRT hits per track is around 34, except in the transition region between barrel and end-caps and at the edge of the acceptance ($|\eta| \lesssim 1.7$) where it is reduced to approximately 25 hits. Polypropylene fibers interwoven between straw layers are used in the barrel for radiator material and regular polypropylene foils in the end-caps. The straws are filled with a gas mixture of 70% Xe, 27% CO$_2$ and 3% O$_2$. Xenon was chosen for its high efficiency to absorb TR photons of typical energy 6$-$15 keV.
The TRT operates as a drift chamber: when a charged particle traverses the straw, it ionizes the gas, creating about 5-6 primary ionization clusters per mm of path length. The straw wall is held at a potential of about $-$1530V with respect to a 31 $\mu$m diameter gold-plated tungsten wire at the center that is referenced to ground. The electrons drift toward the wire and cascade in the strong electric field, producing a detectable signal. On each wire the signal is amplified, shaped and discriminated against two adjustable thresholds, a low-threshold (LT) of about 300 eV and a high-threshold (HT) of about 6-7 keV [@ElectronicsPaper]. The two thresholds allow for simultaneous measurement of tracking information and identification of characteristic large energy deposits due to the absorption of TR photons. For any triggered event, the TRT reads out data over three nominal bunch crossing periods, 3x25 ns.[^2] The measured drift times are at most $\sim$ 50 ns. Low threshold information is read-out separately in time intervals of 3.12 ns length so that each bunch-crossing is divided into eight time bins. The first low threshold 0 $\rightarrow$ 1 transition marks the leading edge (LE) of the signal (hit), and the leading edge time t$_{LE}$ is defined as the center of the first bin set to 1. Similarly, the last 1 $\rightarrow$ 0 transition is called the trailing edge (TE) of the hit. High-threshold information is recorded at a coarser granularity, every 25 ns (once per bunch-crossing), giving three HT bits per hit for a triggered event. A hit is said to be a HT hit if any of the three HT bits is high. The leading (trailing) edge time depends on the time when the closest (furthest) ionization electron cluster arrives at the wire at the center of the straw. The leading edge time is thus directly related to the track-to-wire distance r$_{track}$. If the furthest electrons were always produced exactly at the straw wall and drifted for the full straw radius of 2mm, t$_{TE}$ time would be independent of r$_{track}$. Due to the finite interaction length (and thus the limited number of primary ionization clusters) and signal shaping effects, this is not always the case. A particle that deposits more energy will, on average, have a higher signal, exceed the threshold sooner, and fall back below threshold later. Thus, larger energy deposits result in an earlier LE, later TE and longer ToT on average. This correlation can be used to obtain a ToT-based dE/dx estimate.
Data samples and trigger requirements
=====================================
Data from proton-proton collisions at the LHC at $\sqrt{s}$ = 7 TeV recorded by the ATLAS detector in 2010 were used for the studies reported in these proceedings. The detector response for electrons was studied with samples of reconstructed photon conversions and Z boson decays, in order to explore two different momentum ranges and exploit the abundance of photon conversions in early data. The detector response to pions was studied using the same minimum-bias data set as for photon conversions.
A minimum bias trigger was used to record the data set used for the reconstruction of photon conversions and pion candidates. During the initial low-luminosity running period from April 15 to June 5, 2010, the events were collected in the minimum bias trigger stream at a rate that was typically between 40Hz and 200Hz, providing a high statistics sample of electrons from photon conversions. This data set corresponds to an integrated luminosity of approximately $\mathcal{L}$ = 9 nb$^{-1}$.
Data recorded June 24, 2010 - October 29, 2010, corresponding to an integrated luminosity of $\mathcal{L}$ = 35 pb$^{-1}$, was used to reconstruct electron candidates from Z boson decays. Events were required to be triggered by an electron trigger that has close to 100% efficiency for electrons from Z boson decays selected in this analysis.
The LHC bunch spacing during both running periods was 150 ns or greater. Pile-up from multiple interactions per bunch crossing was small. The average number of minimum bias interactions per beam crossing was less than 0.2 in the data set used for photon conversions, and about three in the data set used to reconstruct the sample of Z bosons.
The results observed in data were compared to Monte Carlo (MC) simulations [@SimulationPaper]. The detector response to electrons from photon conversions and pions in data were compared to [ Pythia]{} non-diffractive minimum bias MC simulation. The electrons from Z boson decays were compared to [Pythia]{} $Z \to e^{+}e^{-}$ MC simulation.
Electron candidates
-------------------
Photon conversions to electron-positron pairs were used to reconstruct a pure sample of electrons in early data. The photon conversion candidates [@ConvConfNote] are required to have two tracks, each with a minimum of 20 TRT hits and four silicon (SCT and Pixel) hits. The conversion vertex is required to be well reconstructed and to be at least 60 mm away from the primary vertex in the radial direction. To improve the sample purity, a tag and probe method is applied to the two tracks of the selected photon conversion candidates. The tag leg is required to have a ratio of the number of TRT high-threshold hits to total TRT hits of at least 0.12, which corresponds to at least three high-threshold hits on a track with the minimum total number of 20 TRT hits. For a conversion candidate passing these requirements, the probe leg is declared to be an electron candidate. The two tracks are treated independently; if both of the tracks pass the tag requirement, each is also used as a probe. Over 500,000 electron candidates satisfy these selection criteria, providing a high statistics sample of electron candidates at the early stages of collision data-taking.
A second sample of electron candidates is obtained from the reconstruction of $Z \to e^{+}e^{-}$ decays. Electrons from this sample have higher momenta, and can thus be used to probe the TR performance at higher values of $\gamma$. Electron candidates are required to pass the calorimeter based “medium” electron selection criteria [@ElectronPaper], and to have an innermost Pixel layer (b-layer) hit. Candidate events are required to have two such electrons, with a reconstructed di-lepton invariant mass in the range 75 $-$ 105 GeV, based on measurements from their calorimeter clusters. Electrons from Z boson decays are treated in the same way as those from photon conversions. The tag leg is required to have a TRT high-threshold ratio greater than 0.12, and both legs are required to have at least 20 TRT hits.
Pion candidates
---------------
Pion candidates are selected from reconstructed particle tracks that have a minimum of 20 TRT and four silicon hits. Further selection criteria are applied to reject electrons, protons and kaons. Any track that does not have a hit in the innermost Pixel layer or that is reconstructed as a part of a photon conversion candidate is excluded. These two requirements reduce electron contamination from photon conversions, which is the dominant source of electrons in the minimum bias data. In addition, any track with a measured dE/dx above 1.6 MeVg$^{-1}$cm$^2$ in the Pixel detector is excluded in order to reduce the contamination from protons (and to a lesser extent kaons) at low momentum. A track passing these requirements is declared to be a pion candidate.
Transition radiation and high-threshold hits
============================================
This section presents the results of HT studies in electron from photon conversions and pion samples. Figure \[fig:Separation\] shows the HT fraction distributions for electron and pion candidates. The HT fraction is defined as the ratio hits on track that exceed the high threshold to the total number of hits on track. The distribution for electrons is shown in Fig. \[fig:Separation\] is clearly shifted to higher values. This value is defined on a per track basis, whereas the high-threshold probability is defined as the total number of high-threshold hits summed over all candidates divided by the total number of hits summed over all candidates. The following sections show the HT dependence on the $\gamma$ factor, and performance of a requirement on the HT fraction, in terms of the electron efficiency and pion misidentification probability. Finally, validation of hardware settings with 7 TeV collision data are presented.
Transition radiation onset
--------------------------
The first step towards establishing electron identification with the TRT is to observe the expected increase in the average number of HT hits with $\gamma$. The increase has been observed in 2004 test-beam data \[13\],cosmic-ray data \[14\] and for collision data at $\sqrt{s} =
900$ GeV \[15\]. The HT probability observed in 7 TeV collision data is shown in Fig. \[fig:TurnOnCurves\] and is consistent with earlier measurements. The results are shown separately for five intervals in pseudo-rapidity $\eta$ reflecting different detector regions. The errors shown are statistical only. The average HT fraction was evaluated for tracks in bins of the Lorentz factor $\gamma$. The pions, electrons from photon conversions and electrons from Z boson decays cover different $\gamma$ ranges. For the electron candidates, the sharp turn-on of the transition radiation can be seen, with the HT probability increasing rapidly from 0.05 to a plateau of 0.2 − 0.3 depending on $\eta$ region. The HT plateau level in the end-cap region is higher than in the barrel. Electrons from the reconstructed Z decays allow studies of HT probability at $\gamma \sim 10^5$, which can not be accessed with electrons from photon conversions. Small differences in the HT probability for the electrons from conversions and $Z \rightarrow ee$ decays in the overlapping $\gamma$ range can not be resolved at the current statistical uncertainty.
The pion candidates shown in Fig. \[fig:TurnOnCurves\] populate the region $\gamma < 10^3$. In this $\gamma$ range, HT hits are caused by large ionization energy deposits due to Landau dE/dx fluctuations. The HT probability for pion candidates increases gradually from about 0.04 at $\gamma \sim 1$ to about 0.07 at $\gamma \sim$ 700 (p $\sim$ 100 GeV) due to the rise of $<dE/dx>$ with increasing track momentum. This behavior was cross-checked with a sample of pion candidates from [$K^{0}_{\mathrm{s}} $ ]{}decays that has higher pion purity, and the results were in good agreement.
Electron efficiency and pion misidentification probability
----------------------------------------------------------
The HT-based electron-pion separation demonstrated in Fig. \[fig:Separation\] is utilized by a requirement of a minimum HT fraction for electron identification. Figure \[fig:EffVsHTEl\] shows the fraction of electron candidates that pass a HT fraction selection requirement, in bins of $|\eta|$. The pion misidentification probability p$_{\pi\rightarrow e}$ is the probability for a pion to pass an electron HT fraction selection criteria and is shown in Fig. \[fig:EffVsHTPion\]. The pion rejection power is $1/p_{\pi\rightarrow e}$. A direct comparison of the electron efficiency and pion misidentification probability is shown in Fig. \[fig:PionRej\]. A benchmark point of a cut on high-threshold fraction that has a 90% electron efficiency is used. The uncertainty on the pion misidentification probability shown in Fig. \[fig:PionRej\] was estimated by varying the selection criteria such that the electron efficiency changed by $\pm$2%. The range of $\pm$2% is sufficiently big to include the uncertainties due to hadron contamination in the electron sample of about 1%.
The minimum HT fraction that is required for an electron to pass the ATLAS “tight” electron selection requirement [@ElectronPaper], the corresponding efficiency for electrons to pass this criterion as well as the pion misidentification probability are summarized in Table \[tab:tightEff\]. The current HT fraction selection criteria were determined based on MC studies prior to the start of collision data-taking, and was chosen such that a pion rejection factor of at least 10 would be achieved after applying the HT fraction electron selection criteria [@DetectorPaper]. In the range 0.625 $< |\eta|
< $1.07, only a factor of four was achieved due to fewer hits on track in the transition region and a relatively large HT hit probability for pions for geometric reasons. In the highest $\eta$ bin, the pion rejection factor is almost 100.
![Pion mis-ID probability for the HT faction criteria that gives 90% electron efficiency, []{data-label="fig:PionRej"}](figure4.eps){width="80mm"}
**$\eta$ Range** **Minimum HT fraction** **Electron Efficiency** **Pion misidentification probability**
--------------------------- ------------------------- ------------------------- ----------------------------------------
$0.0 \rightarrow 0.625$ 0.085 0.953 $\pm$ 0.004 0.1268 $\pm$ 0.0003
$0.625 \rightarrow 1.07$ 0.085 0.961 $\pm$ 0.005 0.2420 $\pm$ 0.0004
$1.07 \rightarrow 1.304$ 0.115 0.921 $\pm$ 0.005 0.0473 $\pm$ 0.0001
$1.304 \rightarrow 1.752$ 0.13 0.919 $\pm$ 0.002 0.0174 $\pm$ 0.0001
$1.752 \rightarrow 2.0$ 0.155 0.882 $\pm$ 0.002 0.0109 $\pm$ 0.0001
: Fraction of electron and pion candidates that pass the HT fraction cut used in “tight” electron identification for each $\eta$ range. Errors given are statistical only.
\[tab:tightEff\]
Validation of hardware settings
-------------------------------
In order to determine the optimal average high threshold setting, data corresponding to an integrated luminosity of 20 nb$^{-1}$ was taken with different HT settings in July 2010, and the results of the pion rejection study with different settings are reported in this section. An electron trigger that maximized the number of reconstructed photon conversion candidates was used to record these data.
The value of the high threshold can be varied by changing the Digital to Analogue Converter setting (DAC counts) on the Amplification, Shaping, Discrimination, and Base-Line Restoration (ASDBLR) chip [@ElectronicsPaper], in steps of about 60 eV. Prior to the start of collision data-taking, the average HT was adjusted to the setting that gave the best performance at the test beam. Results from electronics noise scans were used to correct for the large variations in response due to variations in ground offsets.
Validation of the overall average setting for the full detector is reported here. To validate the average HT setting, data were recorded with six different HT settings: nominal settings, $\pm$15 DAC counts from nominal, $\pm$25 DAC counts from nominal, and $+$8 DAC counts from nominal. The high-threshold settings were varied uniformly across the entire detector.
As the threshold is decreased, the HT probability increases for both electron and pion candidates. The optimal average HT setting is determined based on the pion rejection power. The HT fraction selection criteria that gives 90% electron efficiency was determined for different values of high threshold settings and for different $\eta$ bins. Figure \[fig:SpecialRunSummary\] shows the efficiency for a pion candidate to pass the selection criteria as a function of the high threshold setting difference. The selection criteria at 90% electron efficiency was used as a reference for this study. As in the previous section, the uncertainties were estimated by varying the selection criteria such that the electron efficiency changed by $\pm$2%. For all regions, the pion misidentification probability p$_{\pi\rightarrow e}$ is independent of the HT setting in the range of -25 to nominal DAC count. For settings higher than nominal, p$_{\pi\rightarrow e}$ increases. Based on these results, the high-threshold was lowered by eight DAC counts across the detector for 2011 data-taking. The primary reason for lowering the thresholds was to operate at stable settings, where the performance does not vary much if the HT is slightly above or below the nominal.
![Pion mis-ID probability at 90% electron efficiency as a function of hardware settings for different $\eta$ ranges.[]{data-label="fig:SpecialRunSummary"}](figure5.eps){width="80mm"}
Time over threshold based particle identification
=================================================
The measured time over threshold is correlated with the ionization deposit within the straw, and can thus be used to better distinguish between electrons and pions based on their expected dE/dx. For the purpose of the dE/dx measurement, the ToT is defined as the number of bits above threshold in the largest single group of bits above threshold, multiplied by the bin width. This method has a similar performance to a method that uses all bits above threshold, and a better performance than a method that uses t$_{TE}-$t$_{LE}$.
The ToT is subject to several systematic effects that are not related to dE/dx. The t$_{LE}$ depends on the track-to-wire distance due to the drift time. Due to the limited number of primary ionization clusters, the t$_{TE}$ also depends on the track-to-wire distance. The track-to-wire distance related variation in the measured ToT is about 10 ns. Other smaller effects that can cause variations of a few ns along the wire length are signal attenuation (attenuation length $\lambda$ = 4m [@StrawPaper]), signal reflection from the end of the wire that is not read out, signal delay due to the propagation along the wire and signal shaping. These effects are taken into account by corrections that vary with the track-to-wire distance and distance along the straw. The track-to-wire distance dependent corrections also take into account the dependence of the total energy deposit within the straw on the track length.
The ionization loss for electrons and pions differs the most for particles of low momentum, p $<$ 10 GeV. To achieve the best sensitivity, all systematic effects discussed are taken into account. Corrections are made for z dependence in the barrel and R dependence in the end-caps. To take into account the track-to-wire distance dependence, the average corrected ToT measurement is divided by the average track-to-wire distance. The track level ToT-based discriminator is obtained by averaging corrected ToT measurements for all hits on track that do not exceed the HT. The HT hits are not used in order to avoid the correlation between the ToT-based variable and the HT fraction. Figure \[fig:ToTSep\] shows the corrected ToT distributions for the electron and the pion candidates.
Combination of HT and ToT measurements
======================================
The HT fraction and the ToT measurements can be combined to achieve the best electron identification performance. To combine the HT and ToT measurements, two likelihood functions are first formed based on the discriminating variables: one for HT, and one for ToT. Since the HT hits are not used for ToT discriminator, the two likelihoods are assumed to be independent, and are multiplied to form a single combined likelihood. The electrons are then selected by applying a cut on the combined likelihood. Again, a cut value that gives a 90% electron efficiency was determined in different momentum bins, and applied to the pion sample to determine the efficiency for pions to pass the same criterion. Figure \[fig:Comb\] shows the pion misidentification probability p$_{\pi\rightarrow e}$ at 90% electron efficiency as a function of momentum. The uncertainties are again estimated by varying the selection criteria such that the electron efficiency changed by $\pm$2%. It should be noted that any contamination of the pion sample with electrons above the TR threshold will systematically bias the estimate of p$_{\pi\rightarrow e}$ by roughly the same amount. As can be seen in the figure, the ToT-based selection improves the pion rejection at p $<$ 10 GeV, where the discriminating power of the HT is lower than at high momenta.
Summary
=======
Studies in the early collision data collected with the ATLAS detector have confirmed that electron identification based on transition radiation measured by the TRT is performing well, and in some detector regions even exceeds the performance obtained from the current detector simulation. The pion misidentification probability for selection criteria that give 90% electron efficiency is about 5% (rejection factor 20) for the majority of the detector and as low as 1-2% in the best performing detector regions. Analysis of data from a dedicated run with different hardware settings confirmed that the thresholds were close to their optimal value, and only small adjustments were made in order to ensure stable performance under a wide range of operating conditions. The transition radiation measurement was used to identify electrons for the first W boson production cross section measurement by ATLAS \[17\], as well as for the W$^{+}$W$^{-}$ cross section measurement \[18\] and other analyses such as a search for supersymmetry \[19\]. Time over threshold measurements can be used to further improve the electron identification, in particular for tracks with momentum less than 10 GeV.
[9]{} ATLAS Collaboration, [*The ATLAS Experiment at the CERN Large Hadron Collider*]{}, JINST [**3**]{}, S08003 (2008).
ATLAS Collaboration, [*Electron performance measurements with the ATLAS detector using 2010 LHC proton-proton collision data*]{}, ATLAS note CERN-PH-EP-2011-117
E. Abat et al., [*The ATLAS Transition Radiation Tracker (TRT) proportional drift tube: design and performance*]{}, JINST [**3**]{}, P02013 (2008).
E. Abat et al., [*ATLAS TRT Barrel Detector*]{}, JINST [**3**]{}, P02014 (2008).
E. Abat et al., [*The ATLAS TRT end-cap detectors*]{}, JINST [**3**]{}, P10003 (2008).
E. Abat et al., [*The ATLAS TRT electronics*]{}, JINST [**3**]{}, P06007 (2008).
ATLAS Collaboration, [*The ATLAS Simulation Infrastructure*]{}, Eur. Phys. J. [**C70**]{}, 823-874 (2010).
ATLAS collaboration, [*Photon Conversions at $\sqrt{s} = 900$ GeV measured with the ATLAS Detector*]{}, ATLAS note ATLAS-CONF-2010-007 http://cdsweb.cern.ch/record/1274001.
[^1]: ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the center of the detector and the z-axis coinciding with the axis of the beam pipe. The x-axis points from the IP to the center of the LHC ring, and the y-axis points upward. Cylindrical coordinates (R, $\phi$) are used in the transverse plane,$\phi$ being the azimuthal angle around the beam pipe and R, the distance from the IP in the radial direction. The track pseudo-rapidity is defined as $\eta = - \ln(\theta/2)$, where the polar angle $\theta$ is the angle between the track direction and the z axis
[^2]: During 2010 running used in these proceedings, the spacing between bunches was 150 ns or greater.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The structure of the Euler-Lagrange equations for a general Lagrangian theory (e.g. singular, with higher derivatives) is studied. For these equations we present a reduction procedure to the so-called canonical form. In the canonical form the equations are solved with respect to highest-order derivatives of nongauge coordinates, whereas gauge coordinates and their derivatives enter in the right hand sides of the equations as arbitrary functions of time. The reduction procedure reveals constraints in the Lagrangian formulation of singular systems and, in that respect, is similar to the Dirac procedure in the Hamiltonian formulation. Moreover, the reduction procedure allows one to reveal the gauge identities between the Euler-Lagrange equations. Thus, a constructive way of finding all the gauge generators within the Lagrangian formulation is presented. At the same time, it is proven that for local theories all the gauge generators are local in time operators.'
author:
- 'B. Geyer[^1], D.M. Gitman[^2], and I.V. Tyutin[^3]'
title: '[Canonical form of Euler-Lagrange equations and gauge symmetries]{}'
---
Introduction
============
At present increasingly complicated gauge models are used in field and string theory. Generally a comprehensive analysis of their structure is not a simple task. In the Lagrangian formulation the problem includes, in particular, finding generators of gauge symmetries and their algebra, revealing the hidden structure of the equations of motion and so on. One ought to say that in the Hamiltonian formulation there exists a relatively well-developed scheme of constraint finding (Dirac procedure [@Dirac64]) and reorganization [@Dirac64; @GitTy90; @HenTe92; @GitTy01]. The constraint structure can be, in principle, related to the symmetry properties of the initial gauge theory in the Lagrangian formulation [@BorTy98]. However, in the general case, this relation cannot be considered as a constructive method to study the Lagrangian symmetries (it is indirect and complicated). Moreover, the modern tendency is to avoid the non-covariant hamiltonization step and to use the Lagrangian quantization [@BV] for constructing quantum theory. Such an approach incorporates all the Lagrangian structures (in particular, the total gauge algebra).[ ]{}That is why it seems important to develop a reduction procedure within the Lagrangian formulation – in a sense similar to the Dirac procedure in the Hamiltonian formulation – that may allow one in a constructive manner to reveal the hidden structure of the Euler-Lagrange equations (ELE) of motion and to find all the gauge identities and therefore the generators of all the gauge transformations. An idea of such a procedure was first mentioned in publications of the authors (D.G and I.T) [@GitTy83; @GitTy87] (see also Appendix C in [@GitTy90]), but was not appropriately elaborated and some important points where not revealed.
In the present paper we return to this idea studying the structure of the ELE for a general Lagrangian theory (singular, with higher derivatives, and with external fields). In Sect. II we introduce some notation and definitions. In Sect. III, we reduce the ELE of nonsingular theories to the so called canonical form (in the canonical form the equations are solved with respect to highest-order derivatives of nongauge coordinates, whereas, gauge coordinates and their derivatives enter in the right hand sides of the equations as arbitrary functions of time, see below). In Sect. IV we formulate the reduction procedure for the singular case. In a sense, the reduction procedure reveals constraints in the Lagrangian formulation of singular systems and, in that respect, is similar to the Dirac procedure in the Hamiltonian formulation. In Sect. V we demonstrate how the reduction procedure reveals the gauge identities between the ELE. Thus, a constructive way of finding all the gauge generators within the Lagrangian formulation is presented. At the same time it is proven that for local theories all the gauge generators are local in time operators. In the Appendix we collect some Lemmas being useful for our consideration.
General ELE
===========
Notation, definitions, and conventions
--------------------------------------
We consider a system with finite degrees of freedom (classical mechanics). These degrees of freedom are described by the generalized coordinates $q^{a}, $ $a=1,...,n,$ which depend on the time $t.$ The following notation is used:$$\dot{q}^{a}=\frac{dq^{a}}{dt}\,,\;\ddot{q}^{a}=\frac{d^{2}q^{a}}{dt^{2}}\,,\;\cdots \,,\;\;\mathrm{or\;\;}q^{a\left[ l\right] }=\frac{d^{l}q^{a}}{dt^{l}}\,,\;\;l=0,1,...,\;\;\left( q^{a\left[ 0\right] }=q^{a}\right) \,.
\label{2.}$$The coordinates $q^{a}=q^{a\left[ 0\right] }$ are called sometimes velocities of zeroth order; the velocities $\dot{q}^{a}=q^{a\left[ 1\right]
} $ are called velocities of the first order; the accelerations $\ddot{q}^{a}=q^{a\left[ 2\right] }$ are called velocities of second order,$\,$and so on. The space of all the velocities is often called the jet space, see [Kuper79]{}.
As local functions (LF) we call those functions that are defined on the jet space and depend on the velocities $q^{a\left[ l\right] }\;$up to some finite orders $N_{a}\geq 0\;(l\leq N_{a})$. Further, we call $N_{a}$ theorder of the coordinate $q^{a}$ in the LF. For the LF we use the following notation[^4]:$$\begin{aligned}
&&F\left( q^{a},\dot{q}^{a},\ddot{q}^{a},...\right) =F\left( q^{a\left[ 0\right] },q^{a\left[ 1\right] },q^{a\left[ 2\right] },...\right) =F\left( q^{\left[ l\right] }\right) \,,\;q^{\left[ l\right] }=(q^{a\left[ l\right]
},\;0\leq l\leq N_{a}), \notag \\
&&\mathrm{or\;sometimes}:\;F\left( q^{\left[ l\right] }\right) =F\left(
\cdots q^{a\left[ N_{a}\right] }\right) \,. \label{2.a}\end{aligned}$$In the latter form, we indicate only the highest-order derivatives in the arguments of the LF.
The following notation is often used: $\left[ a\right] $ is the number of the indices $a,$ namely, if $\ a=1,...n,$ then $\left[ a\right] =n.$ Similarly, suppose $F_{a}\left( \eta \right) ,\;a=1,...,n$ are some functions, then $\left[ F\right] $ is the number of these functions, $\left[
F\right] =n$, etc. . However differently, writing $q^{a\left[ l\right] }$ we denote by $\left[ l\right] $ the order of the time derivatives, see (\[2.\]).
On the jet space, we define local operators (LO) to be matrix operators $\hat{U}$ of the form$$\hat{U}_{Aa}=\sum_{k=0}^{K}u_{Aa}^{k}\left( \frac{d}{dt}\right) ^{k}\,,
\label{2.10}$$where $K$ is a finite number and $u_{Aa}^{k}$ are some LF. The LO act on columns of LF $f_{a}$ producing columns of LF $F_{A}=\hat{U}_{Aa}f_{a}\,$as well. We define the transposed operator* *to $\hat{U}$ as$$\left( \hat{U}^{T}\right) _{aA}=\sum_{k=0}^{K}\left( -\frac{d}{dt}\right)
^{k}u_{Aa}^{k}=\sum_{k=0}^{K}\left( -1\right) ^{k}\sum_{l=0}^{k}\binom{k}{l}u_{Aa}^{k\left[ l\right] }\left( \frac{d}{dt}\right) ^{k-l}\,. \label{2.11}$$Then the following relation holds true$$F_{A}\hat{U}_{Aa}f_{a}=\left[ \left( \hat{U}^{T}\right) _{aA}F_{A}\right]
f_{a}+\frac{d}{dt}Q\,, \label{2.12}$$where $F_{A}$ , $f_{a}$, and $Q$ are LF. The LO $\hat{U}_{ab}$ is symmetric ($+$) or skewsymmetric ($-$) whenever the relation $\left( \hat{U}^{T}\right) _{ab}=\pm \hat{U}_{ab}\,$ holds true.
Suppose a set of LF $F_{A}\left( \cdots q^{a\left[ N_{a}^{A}\right] }\right)
$, or a set of equations $F_{A}\left( \cdots q^{a\left[ N_{a}^{A}\right]
}\right) =0\,$, be given. In the general case the orders $N_{a}^{A}\ $of the coordinates $q^{a}$ in the functions $F_{A}$ (in the equations $F_{A}=0$) are different, i.e. these orders depend both on $a$ and $A$. The number $\mathcal{N}_{a}=\max_{A}N_{a}^{A}$ is called the order of the coordinate $q^{a}$ in the set of the functions $F_{A}$ (in the set of the equations $F_{A}=0$).
Whenever, a subset $F_{A^{\prime }}=0,\;A^{\prime }\subset A$ has orders $\mathcal{N}_{a}^{\prime }$ of the coordinates less than the corresponding orders of the complete set, namely, $\forall a:$ $\mathcal{N}_{a}^{\prime }<\mathcal{N}_{a}$ , we call this subset the constraint equations.
Generally two sets of equations, $F_{A}\left( q^{\left[ l\right] }\right) =0$ and $f_{\alpha }\left( q^{\left[ l\right] }\right) =0$ are equivalent whenever they have the same set of solutions. In what follows we denote this fact as: $F=0\Longleftrightarrow f=0.$
Suppose that two sets of LF $F_{A}\left( q^{\left[ l\right] }\right) $ and $\chi _{A}\left( q^{\left[ l\right] }\right) $ , $\left[ F\right] =\left[
\chi \right] \,$, are related by some LO,$$F=\hat{U}\chi \,,\;\chi =\hat{V}F\,,\;\hat{U}\hat{V}=1\,. \label{2.9}$$Then we call such functions equivalent and denote this fact as: $F\sim \chi $ . Obviously,$$F\sim \chi \Longrightarrow F=0\Longleftrightarrow \chi =0\,. \label{2.9c}$$If (\[2.9c\]) holds true, we will call the equations $F_{A}=0$ and $f_{\alpha }=0$ strong equivalent.
In what follows we often meet the case where$$\chi _{A}=\left(
\begin{array}{c}
f_{\alpha } \\
0_{G}\end{array}\right) ,\;A=\left( \alpha ,G\right) \,;\;\forall G:\,0_{G}\equiv 0\,.
\label{2.9a}$$Here the equivalence $F\sim \chi $ implies the equivalence of the equations $F=0$ and $f=0$ and the existence of the identities $\hat{V}_{GA}F_{A}\equiv
0.$ Namely,$$F\sim \chi \,\Longrightarrow \left\{
\begin{array}{c}
F=0\Longleftrightarrow f=0 \\
\hat{V}_{GA}F_{A}\equiv 0\end{array}\right. \,. \label{2.9b}$$
ELE
---
Below we restrict our consideration to the Lagrange functions $L$ that are LF on the jet space, and depend on some external coordinates (fields) $u^{\mu }$ (we call the coordinates $u^{\mu }$ external ones in contrast to the coordinates $q^{a}$ , which we call inner coordinates) which are some given functions of time. Thus, $$L=L\left( \cdots q^{a\left[ N_{a}\right] };u^{\mu }\right)
\,,\;\;\,a=1,...,n,\;\;N_{a}\geq 0. \label{2.1}$$The orders $N_{a}$ of the inner coordinates $q^{a}$ in the Lagrange function will be called further the proper orders of the coordinates. Coordinates $q^{a}$ with the proper orders $N_{a}=0$, we call the degenerate coordinates [@GitTy02].
Equations of motion of a Lagrangian theory (the ELE) follow from the action principle $\delta S=0,$ $\ S=\int Ldt\,$, and have the form (merely the inner coordinates have to be varied):$$\frac{\delta S}{\delta q^{a}}=\sum_{l=0}^{N_{a}}\left( -\frac{d}{dt}\right)
^{l}\frac{\partial L}{\partial q^{a\left[ l\right] }}=0\,,\;\;a=1,...,n\,.
\label{2.2}$$
Following [@GitTy02], we classify the Lagrangian theories as nonsingular ($M\neq 0$) and singular ($M=0$) ones by the help of the generalized Hessian $M=\det \left| \left| M_{a\,b}\right| \right| $, where$$M_{a\,b}=\frac{\partial ^{2}L}{\partial q^{a\left[ N_{a}\right] }\partial
q^{b\left[ N_{b}\right] }}\,\, \label{2.3}$$is the generalized Hessian matrix.
In what follows the ELE of a nonsingular (singular) theory will be called the nonsingular (singular) ELE.
Sometimes, it is convenient to enumerate the inner coordinates and organize them into groups such that $q^{a}=\left( q^{a_{0}},...,q^{a_{I}}\right) \,,$ where $a_{i}$ are groups of indices that enumerate coordinates having the same proper orders, $N_{a_{k}}=n_{k}\,$. Besides, we organize these groups such that $n_{I}>n_{I-1}\cdots >n_{0}=0$ $\ (\max \,N_{a}=N_{a_{I}}=n_{I}\,$, and $q^{a_{0}}$ are the degenerate coordinates, $N_{a_{0}}=n_{0}=0\,).$ Thus,$$a=\left( a_{k}\,,\;k=0,1,...,I\right) \,,\ \;\left[ a\right] =\sum_{i}\left[
a_{i}\right] \,,\;\ \left[ a_{i}\right] \geq 0\,,\;n_{I}>n_{I-1}\cdots
>n_{0}=0\,. \label{2.3a}$$Taking into account the notation (\[2.3a\]), we may write the Lagrange function and the ELE as: $$\begin{aligned}
&&L=L\left( \cdots q^{a_{k}\left[ n_{k}\right] };u^{\mu }\right)
\,,\;\;k=0,1,...,I\,; \label{2.4} \\
&&F_{a_{k}}\left( \cdots q^{b\left[ N_{b}+n_{k}\right] };\cdots u^{\mu \left[
n_{k}\right] }\right) =0\,, \label{2.5} \\
&&F_{a_{k}}=\left\{
\begin{array}{c}
M_{a_{k}\,b}q^{b\left[ N_{b}+n_{k}\right] }+K_{a_{k}}\left( \cdots q^{b\left[
N_{b}+n_{k}-1\right] };\cdots u^{\mu \left[ n_{k}\right] }\right)
\,,\;k=1,...,I\, \\
M_{a_{0}}\left( \cdots q^{b\left[ N_{b}\right] };u^{\mu }\right) =\partial
L/\partial q^{a_{0}}\end{array}\right. \;. \label{2.6}\end{aligned}$$Here $M_{a_{k}\,b}\;$is the generalized Hessian matrix and $K_{a_{k}}$ and $M_{a_{0}}$ are some LF of the indicated arguments.
Consider the orders of the inner coordinates in the complete set of the ELE. These orders are $\mathcal{N}_{a}=N_{a}+n_{I}$ . One can see that these orders are, in fact, defined by a subset of (\[2.5\]) with $k=I\,.$ In any subset of the equations (\[2.5\]) with $k<I\,\,$the orders of the coordinates are less than in the complete set. Then according to the above definition, all the ELE with $k<I$ are constraints. The set (\[2.5\]) has the following specific structure: In each equation of the complete set the order of a coordinate $q^{a}$ is the sum of the proper order $N_{a}$ and of the order $n_{k}$ . The latter is the same for all the coordinates and depends only on the number $a_{k}$ of the equation.
Canonical form
--------------
Let a set of equations$$F_{A}\left( \cdots q^{a\left[ \mathcal{N}_{a}\right] }\right) =0\,,
\label{2.8a}$$be given. Suppose that these equations can be transformed to the following equivalent form:$$q^{\alpha \left[ l_{\alpha }\right] }=\varphi ^{\alpha }\left( \cdots
q^{\alpha \left[ l_{\alpha }-1\right] };\cdots q^{g\left[ l_{g}\right]
}\right) \,,\;\;q^{a}=\left( q^{\alpha },q^{g}\right) \,,\ a=\left( \alpha
,g\right) \,,\;\;l_{a}\leq \mathcal{N}_{a}\;. \label{2.8b}$$The equation (\[2.8b\]) present the canonical form of the initial set ([2.8a]{}).* *In the canonical form the equations are solved with respect to the highest-order time derivatives $q^{\alpha \left[ l_{\alpha }\right] }$ of the coordinates $q^{\alpha }$. The coordinates $q^{g}$ (if they exist) and their derivatives $q^{g\left[ l_{g}\right] }$ enter into the set ([2.8b]{}) as arbitrary functions of time. In fact, there are no equations for these coordinates. In what follows we call these coordinates the$\emph{\;}$gauge coordinates whereas $q^{\alpha }$ we call the* *nongauge coordinates. The orders of the coordinates in the canonical forms may be less than those in the initial set. In the general case, one and the same set of equations can have different canonical forms. Generally there are many canonical form of the same set of equations.
Below, we are going to formulate a general procedure of reducing the ELE to the canonical form (in what follows it is called the* *reduction procedure). Our consideration is always local in a vicinity of a given consideration point $q_{0}^{a\left[ l\right] }$ (in the jet space), which is on shell w.r.t. the ELE. We consider theories and coordinates where the consideration point could be selected as zero point. Thus, we suppose that the zero point is on shell. Further we always suppose that the ranks of the encountered Jacobi matrices[^5] are constant in a vicinity of the consideration point. Such suppositions we call ”suppositions of the ranks”. Saying that some suppositions hold true in the consideration point, we always suppose that they hold true in a vicinity of the consideration point. In course of the reduction procedure we perform several typical transformations with LF or with the corresponding equations. Each of such transformations lead to equivalent sets of equations or to equivalent sets of LF (definitions of such equivalences are given above). The proof of these equivalences is based on two Lemmas which are presented in the Appendix. Any statement of the form ”the following equivalence holds true” can be easily justified by these Lemmas.
Canonical form of nonsingular ELE
=================================
A particular case
-----------------
Consider theories without external coordinates and with only two different proper orders of the inner coordinates. In such a case all the indices $a$ can be divided into two groups: $a=\left( a_{1},a_{2}\right) ,$ such that $N_{a_{2}}=n_{2}>N_{a_{1}}=n_{1}\,,\;L=L\left( \cdots q^{a_{2}\left[ n_{2}\right] },\cdots q^{a_{1}\left[ n_{1}\right] }\right) .$ Consider first the case $n_{1}>0.$ Then Eqs. (\[2.5\]) can be written as: $$\begin{aligned}
&&F_{a_{2}}=M_{a_{2}\,a}q^{a\left[ N_{a}+n_{2}\right] }+K_{a_{2}}\left(
\cdots q^{b\left[ N_{b}+n_{2}-1\right] }\right) =0\,, \label{2.15} \\
&&F_{a_{1}}=M_{a_{1}\,a}q^{a\left[ N_{a}+n_{1}\right] }+K_{a_{1}}\left(
\cdots q^{b\left[ N_{b}+n_{1}-1\right] }\right) =0\,. \label{2.16}\end{aligned}$$The equations (\[2.16\]) are constraints. Consider the set$$M_{a_{1}\,a}q^{a\left[ N_{a}+n_{2}\right] }+K_{a_{1}}^{\left( 1\right)
}\left( \cdots q^{b\left[ N_{b}+n_{2}-1\right] }\right) =0\,, \label{2.18}$$obtained from the constraints after being $n_{2}-n_{1}$ times differentiated with respect to the time $t$. Since $M\neq 0,$ the rectangular matrix $M_{a_{1}\,a}$ has a maximal rank, that is why there exists another division of the indices:$$a=\left( a_{|i}\right) ,\;\left[ a_{|i}\right] =\left[ a_{i}\right]
\,,\;i=1,2\,,\;\;\det \,M_{a_{1}b_{|1}}\neq 0\,. \label{2.18a}$$Remark that $$a_{i}=\left( a_{i|1},a_{i|2}\right) \,,\;a_{|i}=\left(
a_{1|i},a_{2|i}\right) \,,\;\left[ a_{1|2}\right] =\left[ a_{2|1}\right] \,.
\label{2.18b}$$The set (\[2.18\]) can be solved with respect to the derivatives $q^{a_{|1}\left[ N_{a_{|1}}+n_{2}\right] }$ as follows: $$q^{a_{|1}\left[ N_{a_{|1}}+n_{2}\right] }=-\left( M_{1}^{-1}\right)
^{a_{|1}a_{1}}\left[ \left( M_{3}\right) _{a_{1}\,a_{|2}}q^{a_{|2}\left[
N_{a_{|2}}+n_{2}\right] }+K_{a_{1}}^{\left( 1\right) }\left( \cdots q^{b\left[ N_{b}+n_{2}-1\right] }\right) \right] . \label{2.20}$$Here the matrices $M_{1}$ and $M_{3}$ are defined by the following block representation of the matrix $M$:$$M_{ab}=\left(
\begin{array}{cc}
\left( M_{2}\right) _{a_{2}\,b_{|1}} & \left( M_{4}\right) _{a_{2}\,b_{|2}}
\\
\left( M_{1}\right) _{a_{1}\,b_{|1}} & \left( M_{3}\right) _{a_{1}\,b_{|2}}\end{array}\right) \,,\;\;\det \,M_{a_{1}b_{|1}}\neq 0\Longrightarrow \det \,M_{1}\neq
0\,.$$Excluding the derivatives $q^{a_{|1}\left[ N_{a_{|1}}+n_{2}\right] }$ from Eqs. (\[2.15\]) by the help of (\[2.20\]), we get the equations$$\left( M_{5}\right) _{a_{2}b_{|2}}q^{b_{|2}\left[ N_{b_{|2}}+n_{2}\right]
}+K_{a_{2}}^{\left( 2\right) }\left( \cdots q^{b\left[ N_{b}+n_{2}-1\right]
}\right) =0\,. \label{2.21}$$Taking into account an useful relation $$\begin{aligned}
\det M &=&\det \left(
\begin{array}{cc}
M_{2} & M_{4} \\
M_{1} & M_{3}\end{array}\right) =\det \left(
\begin{array}{cc}
0 & M_{4}-M_{2}M_{1}^{-1}M_{3} \\
M_{1} & M_{3}\end{array}\right) \label{2.19a} \\
&=&\det M_{1}\det (M_{4}-M_{2}M_{1}^{-1}M_{3})\;, \notag\end{aligned}$$which is related to the Gaussian reduction of matrices [@Gantm59], we get:$$\left.
\begin{array}{c}
\det M\neq 0 \\
\det M_{1}\neq 0\end{array}\right\} \Longrightarrow \det \,M_{5}\neq
0,\;\;M_{5}=M_{4}-M_{2}M_{1}^{-1}M_{3}\;. \label{2.19}$$Therefore, (\[2.21\]) can be solved with respect to the highest-order derivatives $q^{a_{|2}\left[ N_{a_{|2}}+n_{2}\right] }$ as: $$\begin{aligned}
q^{a_{|2}\left[ N_{a_{|2}}+n_{2}\right] } &=&-\left( M_{5}^{-1}\right)
^{a_{|2}a_{2}}\left[ K_{a_{2}}\left( \cdots q^{b\left[ N_{b}+n_{2}-1\right]
}\right) -\left( M_{2}\;M_{1}^{-1}\right)
_{a_{2}}^{a_{1}}K_{a_{1}}^{(1)}\left( \cdots q^{b\left[ N_{b}+n_{2}-1\right]
}\right) \right] \notag \\
&\equiv &\varphi ^{a_{|2}}\left( \cdots q^{b_{|2}\left[ N_{b_{|2}}+n_{2}-1\right] },\cdots q^{b_{|1}\left[ N_{b_{|1}}+n_{2}-1\right] }\right) \,.
\label{2.22}\end{aligned}$$Thus, we get a set$$\begin{aligned}
&&\psi ^{a_{|2}}=q^{a_{|2}\left[ N_{a_{|2}}+n_{2}\right] }-\varphi
^{a_{|2}}\left( \cdots q^{b_{|2}\left[ N_{b_{|2}}+n_{2}-1\right] },\cdots
q^{b_{|1}\left[ N_{b_{|1}}+n_{2}-1\right] }\right) =0\,, \label{2.23} \\
&&F_{a_{1}}=\left( M_{1}\right) _{a_{1}\,a_{|1}}q^{a_{|1}\left[
N_{a_{|1}}+n_{1}\right] }-K_{a_{1}}^{\left( 3\right) }\left( \cdots q^{b_{|2}\left[ N_{b_{|2}}+n_{1}\right] },\cdots q^{b_{|1}\left[ N_{b_{|1}}+n_{1}-1\right] }\right) =0\,, \label{2.24}\end{aligned}$$which is strong equivalent to the initial ELE by virtue of the Lemma 1.
Due to the condition $\det M_{1}\neq 0,$ the equations (\[2.24\]) can be solved with respect to $q^{a_{|1}\left[ N_{a_{|1}}+n_{1}\right] }$ and we obtain: $$\begin{aligned}
q^{a_{|1}\left[ N_{a_{|1}}+n_{1}\right] } &=&-\left( M_{1}^{-1}\right)
^{a_{|1}a_{1}}\left[ \left( M_{3}\right) _{a_{1}\,a_{|2}}q^{a_{|2}\left[
N_{a_{|2}}+n_{1}\right] }+K_{a_{1}}\left( \cdots q^{b\left[ N_{b}+n_{2}-1\right] }\right) \right] \notag \\
&\equiv &f^{a_{|1}}\left( \cdots q^{b_{|2}\left[ N_{b_{|2}}+n_{1}\right]
},\cdots q^{b_{|1}\left[ N_{b_{|1}}+n_{1}-1\right] }\right) \,. \label{2.26}\end{aligned}$$Eqs. (\[2.23\]) and (\[2.26\]) are not of canonical form since the functions $\varphi ^{a_{|2}}$ contain derivatives $q^{b_{|1}\left[
N_{b_{|1}}+n_{2}-1\right] }$ exceeding the ”allowed” order $\left[
N_{b_{|1}}+n_{1}-1\right] $. Now we exclude all the surplus derivatives $q^{a_{|1}\left[ N_{a_{|1}}+n_{1}\right] },...,q^{a_{|1}\left[
N_{a_{|1}}+n_{2}-1\right] }$ from the right hand side of (\[2.23\]) by the help of (\[2.26\]) and corresponding derivatives of it. To this end we need to differentiate (\[2.26\]) not more than $n_{2}-n_{1}-1$ times. Finally, we obtain the following strong equivalent form (the equivalence is justified by the Lemma 1) of the ELE: $$\begin{aligned}
&&q^{a_{|2}\left[ N_{a_{|2}}+n_{2}\right] }=f^{a_{|2}}\left( \cdots q^{b_{|2}\left[ N_{b_{|2}}+n_{2}-1\right] },\cdots q^{b_{|1}\left[ N_{b_{|1}}+n_{1}-1\right] }\right) , \notag \\
&&q^{a_{|1}\left[ N_{a_{|1}}+n_{1}\right] }=f^{a_{|1}}\left( \cdots q^{b_{|2}\left[ N_{b_{|2}}+n_{1}\right] },\cdots q^{b_{|1}\left[ N_{b_{|1}}+n_{1}-1\right] }\right) \,. \label{2.27}\end{aligned}$$It is just the canonical form. Taking into account the division of the indices w.r.t. proper orders of the coordinates, one gets: $$\begin{aligned}
&&q^{a_{2|2}\left[ 2n_{2}\right] }=f^{a_{2|2}}\left( \cdots q^{b_{2|2}\left[
2n_{2}-1\right] },\cdots q^{b_{1|2}\left[ n_{1}+n_{2}-1\right] },\cdots
q^{b_{2|2}\left[ n_{2}+n_{1}-1\right] },\cdots q^{b_{1|1}\left[ 2n_{1}-1\right] }\right) \,, \notag \\
&&q^{a_{1|2}\left[ n_{1}+n_{2}\right] }=f^{a_{1|2}}\left( \cdots q^{b_{2|2}\left[ 2n_{2}-1\right] },\cdots q^{b_{1|2}\left[ n_{1}+n_{2}-1\right]
},\cdots q^{b_{2|1}\left[ n_{2}+n_{1}-1\right] },\cdots q^{b_{1|1}\left[
2n_{1}-1\right] }\right) \,, \notag \\
&&q^{a_{2|1}\left[ n_{2}+n_{1}\right] }=f^{a_{2|1}}\left( \cdots q^{b_{2|2}\left[ n_{1}+n_{2}\right] },\cdots q^{b_{1|2}\left[ 2n_{1}\right] },\cdots
q^{b_{2|1}\left[ n_{2}+n_{1}-1\right] },\cdots q^{b_{1|1}\left[ 2n_{1}-1\right] }\right) \,, \notag \\
&&q^{a_{1|1}\left[ 2n_{1}\right] }=f^{a_{1|1}}\left( \cdots q^{b_{2|2}\left[
n_{1}+n_{2}\right] },\cdots q^{b_{1|2}\left[ 2n_{1}\right] },\cdots
q^{b_{2|1}\left[ n_{2}+n_{1}-1\right] },\cdots q^{b_{1|1}\left[ 2n_{1}-1\right] }\right) \,. \label{2.27a}\end{aligned}$$
Remark that the number of the initial data is equal to $2\sum_{a}N_{a}\,.$ Indeed,$$\begin{aligned}
&&\left[ a_{2|2}\right] \left( n_{2}+n_{2}\right) +\left[ a_{1|2}\right]
\left( n_{1}+n_{2}\right) +\left[ a_{2|1}\right] \left( n_{2}+n_{1}\right) +\left[ a_{1|1}\right] \left( n_{1}+n_{1}\right) \\
&&\,=2\left[ a_{2}\right] n_{2}+2\left[ a_{1}\right] n_{1}=2\sum_{a}N_{a}\,.\end{aligned}$$
One ought to mention that the canonical form (\[2.27a\]) was obtained in [@GitTyL85]. However, the procedure that was used for that purpose did not provide the proof of the equivalence between the initial ELE and the form (\[2.27a\]).
Suppose now that the Lagrange function contains degenerate coordinates $q^{a_{0}}$, $a=\left( a_{0},a_{1}\right) .$ Thus, $L=L\left(
q^{a_{0}},\cdots q^{a_{_{1}}\left[ n_{1}\right] }\right) $ and the ELE read: $$\begin{aligned}
&&F_{a_{1}}\equiv M_{a_{1}\,a}q^{a\left[ N_{a}+n_{1}\right]
}+K_{a_{1}}\left( \cdots q^{b\left[ N_{b}+n_{1}-1\right] }\right) =0\,,
\label{2.15c} \\
&&F_{a_{0}}\equiv \frac{\partial L}{\partial q^{a_{0}}}=M_{a_{0}}\left(
\cdots q^{b\left[ N_{b}\right] }\right) =0\,. \label{2.16c}\end{aligned}$$Despite these equations are formally different from the above case, the whole procedure of reductions goes through without any essential change. In fact, differentiating Eq. (\[2.16c\]) $n_{1}$ times, one obtains $$M_{a_{0}\,a}q^{a\left[ N_{a}+n_{1}\right] }+K_{a_{0}}^{(1)}\left( \cdots q^{b\left[ N_{b}+n_{1}-1\right] }\right) =0, \label{2.18c}$$and all the previous steps may be done as before. Namely, one obtains $$q^{a_{|1}\left[ N_{a_{|1}}+n_{1}\right] }=\varphi ^{a_{|1}}\left( \cdots
q^{b_{|1}\left[ N_{b_{|1}}+n_{1}-1\right] },\cdots q^{b_{|0}\left[
N_{b_{|0}}+n_{0}-1\right] }\right) \,, \label{2.22d}$$and, since $\det ||(M_{1})_{a_{0}\,a_{|0}}||\neq 0$, Eqs. (\[2.16c\]) can be solved with respect to the variable $q^{a_{|0}\left[ N_{a_{|0}}\right] }$ as follows: $$q^{a_{|0}\left[ N_{a_{|0}}\right] }=f^{a_{|0}}\left( \cdots q^{b_{|1}\left[
N_{b_{|1}}\right] },\cdots q^{b_{|0}\left[ N_{b_{|0}}-1\right] }\right) \,.$$Finally, after eliminating the “bad” derivatives in the right hand side of (\[2.22d\]) for $q^{a_{|1}\left[ N_{a_{|1}}+n_{1}\right] }$ one ends up again with Eqs. (\[2.27a\]) but now with $n_{2}\rightarrow
n_{1},\;n_{1}\rightarrow 0$ (by convention: $q^{b_{1|1}\left[ -1\right]
}\equiv 0$).
General nonsingular ELE
-----------------------
Consider the general nonsingular ELE. Here the Lagrange function may contain some degenerate inner coordinates, higher derivatives of some inner coordinates, and, moreover, may depend on some external coordinates, $L=L\left( \cdots q^{a\left[ N_{a}\right] };u^{\mu }\right) \,,\;N_{a}\geq
0\,.$ Thus, we are going to deal with the nonsingular ELE of the form ([2.5]{}). Our aim is to present these equations in an equivalent canonical form.
[Theorem 1: ]{}[The nonsingular ELE]{}* *[(\[2.5\]) can be transformed to the following equivalent canonical form:]{}$$\begin{aligned}
&&f^{a_{i|k}}=q^{a_{i|k}\left[ n_{i}+n_{k}\right] }-\varphi ^{a_{i|k}}\left(
\cdots q^{b_{j|k_{-}}\left[ n_{j}+n_{k_{-}}-1\right] },\cdots q^{b_{j|k_{+}}\left[ n_{j}+n_{k}\right] };\cdots u^{\mu \left[ n_{k}\right] }\right) =0\,,
\notag \\
&&I\geq k_{+}\geq k+1,\,\;k\geq k_{-}\geq 0,\;i,j,k=0,1,...,I\,,
\label{2.28b}\end{aligned}$$[where the indices of the coordinates are divided into groups as follows: ]{}$a=\left( a_{i}\right) $[ is the division of the indices w.r.t. the proper orders of the coordinates, and besides]{} $$a_{i}=\left( a_{i|k}\,,\;i,k=0,1,...,I\,\right) \,,\;\;\left[ a_{i|k}\right]
\geq 0,\;\;\sum_{k}\left[ a_{i|k}\right] =\sum_{k}\left[ a_{k|i}\right] =\left[ a_{i}\right] =\left[ a_{|i}\right] \,.$$[Moreover, the equivalence ]{}$F\sim f$ [between the corresponding LF holds true.]{} [That implies]{}$$F_{a}=\hat{U}_{ab}f^{b}\,,\;f^{b}=\hat{V}^{ba}F_{a}\,,\;\;\hat{U}_{ab}\hat{V}^{bc}=\delta _{a}^{c}\,,$$$\;$ [where]{} $\hat{U}$ [and]{} $\hat{V}$ [are LO]{}. Besides, that implies the strong equivalence between the ELE and their canonical form (\[2.28b\]).
The proof of the Theorem 1 may be considered, in fact, as the general reduction procedure to the canonical form for the nonsingular ELE.
It is reasonable to divide the reduction procedure into two parts. These parts may be called conditionally ”the preliminary resolution”, and ”the subordination procedure”.$${\Large Preliminary\ resolution}$$
Let us introduce the notation $a=\left( \underline{a},a_{I}\right) \,,\;\underline{a}=\left( a_{k}\,,\;k=0,1,...,I\,-1\right) \,,\;N_{\underline{a}}<n_{I}\,,$ such that the ELE read: $$\begin{aligned}
\hspace{-1cm} &&F_{a_{I}}\left( \cdots q^{b\left[ N_{b}+n_{I}\right]
};\cdots u^{\mu \left[ n_{I}\right] }\right) =M_{a_{I}\,b}q^{b\left[
N_{b}+n_{I}\right] }+K_{a_{I}}\left( \cdots q^{b\left[ N_{b}+n_{I}-1\right]
};\cdots u^{\mu \left[ n_{I}\right] }\right) =0\,, \label{2.28} \\
\hspace{-1cm} &&F_{\underline{a}}\left( \cdots q^{b\left[ N_{b}+N_{\underline{a}}\right] };\cdots u^{\mu \left[ N_{\underline{a}}\right]
}\right) =0\,. \label{2.29c}\end{aligned}$$Recall that the equations (\[2.29c\]) can be considered as constraints.
The first step of the procedure is the following: We consider the consistency conditions of the constraints. Namely, we consider the equations that are obtained from the constraints by differentiating them $n_{I}-n_{\underline{a}}$ times, $$F_{\underline{a}}^{\left[ n_{I}-N_{\underline{a}}\right] }=M_{\underline{a}\,b}q^{b\left[ N_{b}+n_{I}\right] }+K_{\underline{a}}^{\left( 1\right)
}\left( \cdots q^{b\left[ N_{b}+n_{I}-1\right] };\cdots u^{\mu \left[ n_{I}\right] }\right) =0\,. \label{2.30}$$Here $K_{\underline{a}}^{\left( 1\right) }$ are some LF of the indicated arguments. Remark that the orders of all the coordinates in the set ([2.30]{}) coincide with the ones in the complete set. For $M\neq 0$, the matrix $$\frac{\partial F_{a}^{\left[ n_{I}-N_{a}\right] }}{\partial q^{b\left[
N_{b}+n_{I}\right] }}=\frac{\partial ^{2}L}{\partial q^{a\left[ N_{a}\right]
}\partial q^{b\left[ N_{b}\right] }}=M_{a\,b} \label{2.29a}$$is invertible. At the same time, the rectangular matrix $M_{\underline{a}\,a} $ has the maximal rank $\left[ \underline{a}\right] $. Therefore, there exists a division of the indices $a$ such that: $$a=\left( \bar{a},a_{|I}\right) ,\;\left[ \bar{a}\right] =\left[ \underline{a}\right] ,\;\left[ a_{|I}\right] =\left[ a_{I}\right] \,,\;\det \,M_{\underline{a}\bar{a}}\neq 0\,. \label{2.32a}$$Thus, the division (\[2.3a\]) of the indices $a$ w.r.t. the coordinate proper orders becomes more detailed,$$\begin{aligned}
&&a_{i}=\left( \bar{a}_{i},a_{i|I}\right) \,,\;\bar{a}=\left( \bar{a}_{i}\right) ,\;a_{|I}=\left( a_{i|I}\right) ,\;\left[ a_{i|I}\right] \geq
0\,, \\
&&\sum_{i}\left[ a_{i|I}\right] =\left[ a_{|I}\right] =\left[ a_{I}\right]
,\;\sum_{i}\left[ \bar{a}_{i}\right] =\left[ \bar{a}\right] =\left[
\underline{a}\right] \,.\end{aligned}$$
Due to (\[2.32a\]), the set (\[2.30\]) can be solved with respect to the derivatives $q^{\bar{a}\left[ N_{\bar{a}}+n_{I}\right] }$ as: $$q^{\bar{a}\left[ N_{\bar{a}}+n_{I}\right] }=-\left( M_{1}^{-1}\right) ^{\bar{a}\underline{a}}\left[ \left( M_{3}\right) _{\underline{a}\,b_{|I}}q^{b_{|I}\,\left[ N_{b_{|I}}+n_{I}\right] }+K_{\underline{a}}^{\left( 1\right) }\left( \cdots q^{b\left[ N_{b}+n_{I}-1\right] };\cdots
u^{\mu \left[ n_{I}\right] }\right) \right] \,, \label{2.32}$$where $$M_{ab}=\left(
\begin{array}{cc}
\left( M_{2}\right) _{a_{I}\,\bar{b}} & \left( M_{4}\right) _{a_{I}\,b_{|I}}
\\
\left( M_{1}\right) _{\underline{a}\,\bar{b}} & \left( M_{3}\right) _{\underline{a}\,b_{|I}}\end{array}\right) \;.$$
Excluding the derivatives $q^{\bar{a}\left[ N_{\bar{a}}+n_{I}\right] }$ from Eqs. (\[2.28\]) by the help of (\[2.32\]), we get the set: $$\begin{aligned}
&&\left( M_{5}\right) _{a_{I}b_{|I}}q^{b_{|I}\left[ N_{b_{|I}}+n_{I}\right]
}+K_{a_{I}}^{\left( 2\right) }\left( \cdots q^{b\left[ N_{b}+n_{I}-1\right]
};\cdots u^{\mu \left[ n_{I}\right] }\right) =0\,, \notag \\
&&M_{5}=M_{4}-M_{2}M_{1}^{-1}M_{3}\,,\;\det M_{5}\neq 0\,, \label{2.34}\end{aligned}$$where $K_{a_{I}}^{\left( 2\right) }$ are some LF of the indicated arguments. The set (\[2.34\]) can be solved with respect to its highest-order derivatives $q^{a_{|I}\left[ N_{a_{|I}}+n_{I}\right] }$ as: $$q^{a_{|I}\left[ N_{a_{|I}}+n_{I}\right] }=\phi ^{a_{|I}}\left( \cdots
q^{b_{|I}\left[ N_{b_{|I}}+n_{I}-1\right] },\cdots q^{\bar{b}\left[ N_{\bar{b}}+n_{I}-1\right] };\cdots u^{\mu \left[ n_{I}\right] }\right) \,,
\label{2.33}$$where $\varphi ^{a_{|I}}$ are some LF. Thus, after the first step we get a set of equations $$\begin{aligned}
&&\psi ^{a_{|I}}=q^{a_{|I}\left[ N_{a_{|I}}+n_{I}\right] }-\phi
^{a_{|I}}\left( \cdots q^{b_{|I}\left[ N_{b_{|I}}+n_{I}-1\right] },\cdots q^{\bar{b}\left[ N_{\bar{b}}+n_{I}-1\right] };\cdots u^{\mu \left[ n_{I}\right]
}\right) =0\,, \label{2.35} \\
&&F_{\underline{a}}\left( \cdots q^{b\left[ N_{b}+N_{\underline{a}}\right]
};\cdots u^{\mu \left[ N_{\underline{a}}\right] }\right) =0\,,\;\underline{a}=\left( a_{k}\,,\;k=0,1,...,I\,-1\right) \,,\;N_{\underline{a}}<N_{I}\,\,,
\label{2.36}\end{aligned}$$ which are strong equivalent to the initial ELE by virtue of the Lemma 1 from the Appendix.
At the second step we turn to the subset (\[2.36\]). We remark that this subset has the same structure as the complete initial set of the ELE if one considers the coordinates $q^{\bar{a}}$ as inner ones and the variables $q^{a_{|I}}\,$as external ones. Namely, let us denote $$\begin{aligned}
&&\overset{1}{F}_{\underline{a}}\left( \cdots q^{\bar{b}\left[ N_{\bar{b}}+N_{\underline{a}}\right] };\cdots u^{\mu _{1}\left[ N_{\underline{a}}\right] }\right) =F_{\underline{a}}\left( \cdots q^{b\left[ N_{b}+N_{\underline{a}}\right] };\cdots u^{\mu \left[ N_{\underline{a}}\right]
}\right) \,, \\
&&u^{\mu _{1}}=\left( u^{\mu },\cdots q^{a_{|I}\left[ N_{a_{|I}}\right]
}\right) ,\;\mu _{1}=\left( \mu ,a_{|I}\right) \,.\end{aligned}$$Then the set (\[2.36\]) can be written as:$$\overset{1}{F}_{\underline{a}}\left( \cdots q^{\bar{b}\left[ N_{\bar{b}}+N_{\underline{a}}\right] };\cdots u^{\mu _{1}\left[ N_{\underline{a}}\right]
}\right) =0\,,\;\underline{a}=\left( a_{k}\,,\;k=0,1,...,I\,-1\right)
\,,\;N_{\underline{a}}<N_{I}\,\,, \label{2.36a}$$where$$\overset{1}{F}_{a_{k}}=\left\{
\begin{array}{c}
M_{a_{k}\,\bar{b}}q^{\bar{b}\left[ N_{\bar{b}}+n_{k}\right] }+\underline{K}_{a_{k}}\left( \cdots q^{\bar{b}\left[ N_{\bar{b}}+n_{k}-1\right] };\cdots
u^{\mu _{1}\left[ n_{k}\right] }\right) ,\;k=1,...,I\,-1 \\
\underline{M}_{a_{0}}\left( \cdots q^{\bar{b}\left[ N_{\bar{b}}\right]
};u^{\mu _{1}}\right) =\partial L/\partial q^{a_{0}}\end{array}\right. \;.$$Here $q^{\bar{a}}$ are the inner coordinates, and $u^{\mu _{1}}$ are the external coordinates. The order of the set (\[2.36a\]) is $2n_{I-1}.$ Furthermore, by virtue of (\[2.32a\]), the matrix$$\frac{\partial \overset{1}{F}_{\underline{a}}^{\left[ n_{I-1}-N_{\underline{a}}\right] }}{\partial q^{\bar{b}\left[ N_{\bar{b}}+n_{I-1}\right] }}=M_{\underline{a}\,\bar{b}} \label{2.37}$$is invertible. Thus, the structure (\[2.5\],\[2.6\]) is repeated completely.
At the same time, the number of the inner variables, the number of the equations, and the order of the set (\[2.36a\]) are less than those of the initial set of the ELE (\[2.5\],\[2.6\]).
Now, we apply the same procedure as in the first step to the reduced set (\[2.36a\]). That will be the second step of the reduction procedure. It will produce equations of similar structure with less inner variables and of lower order. After the last $(I+1)$-th step the ELE (\[2.5\]) may be written in the following strong equivalent form:$$\begin{aligned}
&&q^{a_{i|k}\left[ n_{i}+n_{k}\right] }=\phi ^{a_{i|k}}\left( \cdots
q^{b_{j|k_{+}}\left[ n_{j}+n_{k}\right] },\cdots q^{b_{j|k_{-}}\left[
n_{j}+n_{k}-1\right] };\cdots u^{\mu \left[ n_{k}\right] },\right) , \notag
\\
&&I\geq k_{+}\geq k+1\,,\;k\geq k_{-}\geq 0\,\,,I\geq i,j\geq 0\,,
\label{2.39}\end{aligned}$$where $\phi ^{a_{i|k}}$ are some LF of the indicated arguments (the arguments $\cdots q^{b_{j|k_{+}}\left[ n_{j}+n_{k}\right] }$ result from those coordinates that intermediately have been considered as external ones), and the indices $a_{i}$ are divided into the following groups:$$a_{i}=\left( a_{i|k}\right) \,,\;\;\left[ a_{i|k}\right] \geq 0,\;\;\sum_{k}\left[ a_{i|k}\right] =\sum_{k}\left[ a_{k|i}\right] =\left[ a_{i}\right]
\,,\;\;i,k=0,1,...,I\,.$$
The set (\[2.39\]) is still not the canonical form of the ELE. The reason is that the right hand sides of the set contain derivatives of orders that may exceed the orders $n_{i}+n_{k}$ of the (highest) derivatives $q^{a_{i|k}\left[ n_{i}+n_{k}\right] }$ appearing on the left hand side of the set. We recall that by the definition in the canonical form there is a subordination of derivative orders, namely, the orders of all the derivatives in the right hand sides have to be less than the ones on the left hand side. Explicitly, this subordination would require that the following inequalities should hold: $$\begin{aligned}
&&n_{j}+n_{k_{+}}>n_{j}+n_{k}\,, \notag \\
&&n_{j}+n_{k_{-}}>n_{j}+n_{k}-1\,, \notag\end{aligned}$$which, because of the inequalities $n_{I}>n_{I-1}>\cdots n_{1}>n_{0}$ , is true for the first line and the case $k_{-}=k$ of the second line, and it is definitely not true for the cases $k_{-}<k$. Arranging the equations ([2.39]{}) (for fixed value of $i$) in descending order w.r.t. $k$, and the arguments in the functions $\varphi $ (for fixed value of $j$) also in descending order w.r.t. the value of $k_{+}$ and $k_{-}$, we get, when disregarding the common value $n_{j}$, a quadratic matrix whose main diagonal (i.e. elements with $k=k_{-}$) contains the entries $n_{k}-1$, whereas the entries to the left of that diagonal are equal to $n_{k}$, and to the right of that diagonal are equal to $n_{k}-1$. Therefore, below the main diagonal occur “good” derivatives, and above it occur “bad” derivatives not obeying the subordination requirement.$${\Large Subordination\ procedure}$$
One can see that these ”bad” derivatives can be excluded from the right hand sides by the help of corresponding ”lower” equations of the set and their differential consequences (compare Eqs. (\[2.23\]) and (\[2.26\]) for the simple case $I=2$). In what follows we call such an exclusion the subordination procedure.
In order to be more definite let us write down two arbitrary lines, $\ell >k$, of the right hand sides of the set of equations (\[2.39\]) (for the highest derivatives only):$$\begin{aligned}
\hspace{-1cm} &&\phi ^{a_{i|\ell }}\left( q^{b_{j|I}\left[ n_{j}+n_{\ell }\right] },...,q^{b_{j|\ell +1}\left[ n_{j}+n_{\ell }\right] },q^{b_{j|\ell }\left[ n_{j}+n_{\ell }-1\right] },...,q^{b_{j|k+1}\left[ n_{j}+n_{\ell }-1\right] },q^{b_{j|k}\left[ n_{j}+n_{\ell }-1\right] },...,q^{b_{j|0}\left[
n_{j}+n_{\ell }-1\right] }\right) , \\
\hspace{-1cm} &&\vdots \\
\hspace{-1cm} &&\phi ^{a_{i|k}}\left( q^{b_{j|I}\left[ n_{j}+n_{k}\right]
},...,q^{b_{j|\ell +1}\left[ n_{j}+n_{k}\right] },\;\;q^{b_{j|\ell }\left[
n_{j}+n_{k}\right] },...,q^{b_{j|k+1}\left[ n_{j}+n_{k}\right]
},\;\;q^{b_{j|k}\left[ n_{j}+n_{k}-1\right] },...,q^{b_{j|0}\left[
n_{j}+n_{k}-1\right] }\right) .\end{aligned}$$
Obviously, because $n_{\ell }>n_{k}$ all the derivatives of the equation for $q^{a_{i|\ell }\left[ n_{i}+n_{\ell }\right] }$ with $k\geq \ell _{-}\geq 0$ are “bad” with respect to the derivatives $q^{a_{i|k}\left[ n_{i}+n_{k}\right] }$ (remind $\ell \geq \ell _{-}\geq 0$). However, these “bad” derivatives can be eliminated by the equations for the latter ones, $q^{a_{i|k}\left[ n_{i}+n_{k}\right] }$, and their differential consequences up to the order $n_{\ell }-n_{k}-1$. Thereby, the function $\phi ^{a_{i|\ell
}}$ changes into some new function $\tilde{\phi}^{a_{i|\ell }}$. One can see that doing this we do not change the highest orders of derivatives of the other coordinates, both proper and external ones, in the right hand side of the equation for $q^{a_{i|\ell }\left[ n_{i}+n_{\ell }\right] }$. (Remind, that the derivatives of the external coordinates are $u^{\mu \left[ n_{\ell }\right] }$ and $u^{\mu \left[ n_{k}\right] }$, respectively.)
This subordination procedure, starting with $\ell = I$ may be done for any $k < I$, thereby “cleaning” every entry on the right hand side of equations for $q^{ a_{i|I}\left[n_{i}+n_{I}\right] }$. Namely, the highest orders of derivatives on the r.h.s. become $q^{ b_{j|k_-}\left[n_{i}+n_{k_-} - 1\right]
}$ with $I \geq k_- \geq 0$ (for the case $\ell = I$ no $k_+$ appears). Then the procedure will be applied to the equations for $q^{ a_{i|I-1}\left[n_{i}+n_{I-1}\right] }$, and so forth, up to $q^{ a_{i|0}\left[n_{i}+n_{0}\right] }$, where nothing is to be changed.
After having eliminated all the “bad” derivatives, we transformed the set (\[2.39\]), and therefore the initial ELE, to the following strong equivalent (the equivalence is justified by the Lemma 1) canonical form$$\begin{aligned}
&&q^{a_{i|k}\left[ n_{i}+n_{k}\right] }=\varphi ^{a_{i|k}}\left( \cdots
q^{b_{j|k_{+}}\left[ n_{j}+n_{k}\right] },\cdots q^{b_{j|k_{-}}\left[
n_{j}+n_{k_{-}}-1\right] };\cdots u^{\mu \left[ n_{k}\right] }\right) \,, \\
&&I\geq k_{+}\geq k+1,\;k\geq k_{-}\geq 0,\;i,j,k=0,1,...,I\,,\end{aligned}$$where $\varphi ^{a_{i|k}}$ are some LF of the indicated arguments. This proves the Theorem 1.
We see that there are no gauge coordinates in the nonsingular ELE.
The number of the initial data is equal to $2\sum_{a}N_{a}\,.$ Indeed,$$\sum_{i,k}\left[ a_{i|k}\right] \left( N_{i}+N_{k}\right) =\sum_{i}\left(
N_{i}\sum_{k}\left[ a_{i|k}\right] \right) +\sum_{k}\left( N_{k}\sum_{i}\left[ a_{i|k}\right] \right) =2\sum_{a}N_{a}.$$
One ought to remark that in the general case there exist many different canonical forms of the nonsingular ELE. This uncertainty is related to the possibility of different choices of nonzero minors of a matrix with a given rank (different divisions of the indices $a_{i}$ in course of the reduction procedure). However, as it was demonstrated above, the number of the equations in the canonical form (which is equal to the number of the ELE in the nonsingular case) and the number of the initial data is the same for all possible canonical forms.
Canonical form of singular ELE
==============================
Studying the canonical form of nonsingular ELE, we have demonstrated that the equations in the canonical form are solved with respect to the highest-order derivatives $q^{a_{i|k}\left[ n_{i}+n_{k}\right] },$ where $n_{i}$ are the proper orders of the coordinates $q^{a_{i}}$ . However, considering specific examples, one can see that this is not always true for singular ELE. Namely, in the canonical form of the latter case, the highest orders of the derivatives $q^{a_{i}\left[ l\right] }$ may take on all the values from zero to $n_{i}+I$ . The reduction procedure to the canonical form for the general singular ELE is considered below. In the singular case, already after the first step of the reduction procedure, the ELE cease to have their initial specific structure (\[2.5\],\[2.6\]). Namely, the simple structure of terms with highest-order derivatives in the equations may be lost. That is why in the singular case it is more convenient to formulate the reduction procedure for a more general set of ordinary differential equations, which contains the ELE as a particular case. Namely, further we are going to consider a set of the form[^6]: $$F_{A_{\mu }}\left( \cdots q^{a_{i}\left[ i+\mu \right] }\right)
=0\,;\;\;i=0,1,...,I\,,\;\mu =0,...,J\,. \label{3.1}$$Here $F_{A_{\mu }}\left( \cdots q^{a_{i}\left[ i+\mu \right] }\right) $ are some LF. Via $a_{i}$ and $A_{\mu }$ are denoted sets of indices, $\left[
a_{i}\right] \geq 0,\;\left[ A_{\mu }\right] \geq 0\,$, and the complete set of the inner coordinates in Eqs. (\[3.1\]) is $q^{a}=\left(
q^{a_{0}},...,q^{a_{I}}\right) \,,$ $a=\left(
a_{i}\,,\;\;i=0,1,...,I\right) .$ The indices $A=\left( A_{\mu }\right) $ enumerate the equations. In the general case the number of the indices $A$ (the number of all the equations) is not equal to the number of the indices $a$ (the number of the coordinates).[ ]{}The division of the indices $A$ into the groups is not related to the division of the indices $a$ into the groups. The orders of the coordinates $q^{a_{i}}$ in the complete set ([3.1]{}) are: $\mathcal{N}_{a_{i}}=i+J\,.$ In fact, these orders are defined by a subset of (\[3.1\]) with $\mu =J.$ In all the other equations with $\mu <J\,\ $the coordinates $q^{a_{i}}$ have the orders less than $i+J$. Thus, the latter equations are constraints.
Similar to the ELE (\[2.5\]), the set (\[3.1\]) has the following specific structure: In each equation of the set the order of a coordinate $q^{a_{i}}$ is the sum of the proper order $i$ and of the order $\mu $. The latter is the same for all the coordinates and is related to the number of the equation in the set.
Below we consider the reduction procedure to the canonical form for the equations (\[3.1\]). In fact, this reduction procedure is formulated in the proof of the Theorem 2 given below. The Theorem 2 holds true under certain suppositions of the structure of the functions $F_{A_{\mu }}\,.$ These suppositions are formulated as suppositions of the ranks of some Jacobi matrices involving the functions $F_{A_{\mu }}\,.$ First of all, the complete matrix$$M_{A_{\mu }\,a_{i}}=\frac{\partial F_{A_{\mu }}}{\partial q^{a_{i}\left[
i+\mu \right] }}=\frac{\partial F_{A_{\mu }}^{\left[ J-\mu \right] }}{\partial q^{a_{i}\left[ i+J\right] }}\,, \label{3.2}$$has to have a constant rank in a vicinity of the consideration point (one can see that the matrix $M_{A_{\mu }\,a_{i}}$ coincides with generalized Hessian matrix if the set (\[3.1\]) is the Lagrangian one).
[Theorem 2]{}: [Under certain suppositions of the ranks, the equations (\[3.1\]) can be transformed to the following equivalent canonical form:]{}$$\begin{aligned}
&&f^{a_{i|\sigma }}=q^{a_{i|\sigma }\left[ i+\sigma \right] }-\varphi
^{a_{i|\sigma }}\left( \cdots q^{a_{j|\sigma _{-}}\left[ j+\sigma _{-}-1\right] },\cdots q^{a_{j|\sigma _{+}}\left[ j+\sigma \right] }\right) =0\,,
\notag \\
&&i,j=0,1,...,I\,,\;\;\sigma =-I,...,J\,,\;\;-I\leq \sigma _{-}\leq \sigma
\,,\;\sigma +1\leq \sigma _{+}\leq J+1\,,\; \label{3.3}\end{aligned}$$[where all the indices ]{}$a$[ are divided into groups as follows:]{} $$a_{i}=\left( a_{i|\sigma }\right) \,,\;\;\left[ a_{i|\sigma }\right] \geq
0\,,\;\;\sigma =-I,...,J+1,\;\;\left( \left[ a_{i|\sigma }\right] =0\;\;\mathrm{if}\;\;i+\sigma <0\right) \,, \label{3.4}$$[and it is thought that negative powers of the time derivatives do not exist, that is: ]{}$\left[ q^{a\left[ p\right] }\right] =0$[ for]{} $p<0$[.]{}
[Moreover, the following equivalence between the corresponding LF holds true:]{}$$\begin{aligned}
&&F_{A}\sim \bar{F}_{A}=\left(
\begin{array}{c}
f^{a_{i|\sigma }} \\
0_{G}\end{array}\right) \,,\;\;A=\left( a_{i|\sigma }\,,G\right) \,,\;i=0,1,...,I\,,\;\sigma
=-I,...,J\,, \notag \\
&&0_{G}\equiv 0\,\;\forall G\,,\;\;\left[ G\right] =\left[ A\right] -\left[ a\right] +\sum_{i}\left[ a_{i|J+1}\right] \,. \label{3.3a}\end{aligned}$$[That implies]{}$$F_{A}=\hat{U}_{A}^{B}\bar{F}_{B}\,,\;\bar{F}_{B}=\hat{V}_{B}^{A}F_{A}\,,\;\;\hat{U}_{A}^{B}\hat{V}_{B}^{C}=\delta _{A}^{C}\,, \label{3.3b}$$$\;$ [where]{} $\hat{U}$ [and]{} $\hat{V}$ [are LO]{}.
Let us make some comments to the Theorem 2. The canonical form (\[3.3\]) of the singular ELE differs from that (\[2.28b\]) of the nonsingular ELE. As was demonstrated in the previous Sect., in the latter case the spectrum of the orders of the variables $q^{a_{i}}$ in the canonical form extends from $i+\mu _{\min }$ to $i+J\,.$ In the singular case, we have to admit (and one can see this on specific examples) the spectrum extends from $0$ to $i+J\,.$ Under such a supposition we can justify by the induction the structure (\[3.3\]) of the canonical form. One can see from (\[3.4\]) that each group of the indices $a_{i}$ is divided in subgroups $a_{i}\rightarrow a_{i|\sigma }\,,$ $\sigma =-I,...,J+1$. In the canonical form the singular ELE are solved with respect to the highest-order derivatives $q^{a_{i|\sigma }\left[ i+\sigma \right] }\,,\;\sigma =-I,...,J$, ($\left[ a_{i|\sigma }\right] =0\;$for $i+\sigma <0)$. There are no equations for the coordinates $q^{a_{i|J+1}}\,$. These coordinates enter the set (\[3.3\]) as arbitrary functions of time. They are gauge coordinates according to the general definition. As in the nonsingular case, it is supposed that no coordinate $q^{a_{k|\sigma }}$ in the function $\varphi ^{a_{i|\sigma }}$ has an order greater than $k+\sigma $ (the proper order plus $\sigma $). Besides, the order of the coordinates $q^{b_{k|\sigma
_{-}}}$ in the function $\varphi ^{a_{i|\sigma }}$ has to be less than $k+\sigma _{-}$ .
We are going to prove the Theorem 2 by induction w.r.t. $\mathcal{N}=I+J$. To this end, we consider first equations of lower orders, then we use an induction to prove the general case.
Equations of lower orders
-------------------------
Remark that the case$\ \mathcal{N}=0\,$implies$\;I=J=0$ and the set ([3.1]{}) is reduced to form$$F_{A}\left( q\right) =0,\;\;q=\left( q^{a}\right) \,. \label{3.5b}$$Here the Theorem 2 holds true by virtue of the Lemma 3 from the Appendix.
Let $\mathcal{N}=1.$ That implies either $I=1,$ $J=0$ or $I=0,\;$ $J=1$ . Consider, for example, the first case. Here $\left( i=0,1,\;\mu =0\right) $ and the set (\[3.1\]) reads $$F_{A}\left( q^{a_{0}},q^{a_{1}},\dot{q}^{a_{1}}\right) =0\,,\;\left[ a_{1}\right] >0,\;\,\left[ a_{0}\right] \geq 0\,. \label{3.26}$$In the case under consideration the supposition (\[3.2\]) reads: $$\mathrm{rank}\,\frac{\partial F_{A}}{\partial q^{a_{i}\left[ i\right] }}=r\,.
\label{3.26a}$$Then there exists a division of the indices: $\;A=\left(
A_{/1},A_{/2}\right) $, $a_{i}=\left( a_{i/1},a_{i/2}\right) $, $\left[
A_{/1}\right] =\left[ a_{0/1}\right] +\left[ a_{0/1}\right] =r$, such that
$$\det \,\left| \frac{\partial F_{A_{/1}}}{\partial q^{a_{i/1}\left[ i\right] }}\right| \neq 0\,.$$
Thus, we may solve the equations $F_{A_{/1}}=0$ with respect to $q^{a_{i/1}\left[ i\right] }$,$$F_{A_{/1}}=0\Longleftrightarrow q^{a_{i/1}\left[ i\right] }=\phi
^{a_{i/1}}\left( q^{b_{i/1}\left[ i-1\right] },q^{b_{i/2}\left[ i-1\right]
},q^{b_{i/2}\left[ i\right] }\right) \,. \label{3.27}$$Then we exclude the arguments $q^{a_{i/1}\left[ i\right] }$ from the functions $F_{A_{/2}}$ by the help of (\[3.27\]),$$\bar{F}_{A_{/2}}=\left. F_{A_{/2}}\right| _{^{_{q^{a_{i/1}\left[ i\right]
}=\phi ^{a_{i/1}}}}}=\bar{F}_{A_{/2}}\left( q^{a_{1}}\right) \,.$$By virtue of the Lemma 2 from the Appendix, the functions $\bar{F}_{A_{/2}}$ depend on $q^{a_{1}}$ only . Thus, we have the equivalence[^7]$$F_{A}\sim \bar{F}_{A}=\left(
\begin{array}{l}
F_{A_{/1}}\left( q^{a_{0}},q^{a_{1}},\dot{q}^{a_{1}}\right) \\
\bar{F}_{A_{/2}}\left( q^{a_{1}}\right)\end{array}\right) . \label{3.27a}$$Now we suppose that the matrix $\partial \bar{F}_{A_{/2}}/\partial q^{a_{1}}$ has a constant rank. Therefore (see Lemma 3)$$\bar{F}_{A_{/2}}\sim \left(
\begin{array}{c}
q^{\underline{a}_{1}}-\varphi ^{\underline{a}_{1}}\left( q^{\bar{a}_{1}}\right) \\
0_{G_{1}}\end{array}\right) \,,\;\;a_{1}=\left( \underline{a}_{1},\bar{a}_{1}\right) \,.$$Let us exclude the arguments $q^{\underline{a}_{1}}$, $\dot{q}^{\underline{a}_{1}}$ from the functions $F_{A_{/1}}$, by the help of the equations $q^{\underline{a}_{1}}=\varphi ^{\underline{a}_{1}}\left( q^{\bar{a}_{1}}\right)
,$$$\overset{1}{F}_{A_{/1}}\left( q^{a_{0}},q^{\bar{a}_{1}},\dot{q}^{\bar{a}_{1}}\right) =\left. F_{A_{/1}}\right| _{q^{\underline{a}_{1}}=\varphi ^{\underline{a}_{1}}}\,.$$Then the equivalence holds true: $$F\sim \overset{1}{F}=\left(
\begin{array}{c}
\overset{1}{F}_{A_{/1}}\left( q^{a_{0}},q^{\bar{a}_{1}},\dot{q}^{\bar{a}_{1}}\right) \\
q^{\underline{a}_{1}}-\varphi ^{\underline{a}_{1}}\left( q^{\bar{a}_{1}}\right) \\
0_{G_{1}}\end{array}\right) ,\;\;a=\left( a_{0},a_{1}\right) \,,\;\;a_{1}=\left( \underline{a}_{1},\bar{a}_{1}\right) \,. \label{3.28}$$The set of functions $\overset{1}{F}$ has the same structure as the initial set $F$. However, the number of the nonzero functions $\overset{1}{F}$ is less than the number of the functions $F.$ Moreover, some of the functions $\overset{1}{F}$ depend linearly on a part of the variables. That is why the supposition of the type (\[3.26a\]) for the functions $F^{\left( 1\right)
} $ is reduced to the supposition about the rank of the matrix $\partial
\overset{1}{F}_{A_{/1}}/\partial \left( q^{a_{0}},\dot{q}^{\bar{a}_{1}}\right) $ . Accepting the latter supposition we apply the above reduction procedure to the functions $\overset{1}{F}$ and so on. After the $i $-th stage we have the following equivalence: $$F\sim \overset{i}{F}=\left\{
\begin{array}{c}
\overset{i}{F}_{A_{/i}}\left( q^{a_{0}},q^{\bar{a}_{i}},\dot{q}^{\bar{a}_{i}},\right) \\
q^{\underline{a}_{i}}-\varphi ^{\underline{a}_{i}}\left( q^{\bar{a}_{i}}\right) \\
0_{G_{i}}\end{array}\right. ,\;a=\left( a_{0},a_{1}\right) \,,\;\;a_{1}=\left( \underline{a}_{i},\bar{a}_{i}\right) \,.$$The procedure ends at a $k$-th stage when$$\mathrm{rank\,}\frac{\partial \overset{k}{F}_{A_{/k}}}{\partial \left( \dot{q}^{\bar{a}_{k}},q^{a_{0}}\right) }=\left[ A_{/k}\right] \,.$$Then there exists a division of the indices $\bar{a}_{k}=\left(
a_{1|0},a_{g_{1}}\right) $, $a_{0}=\left( a_{0|0},a_{g_{0}}\right) $, $\left[ a_{1|0}\right] +\left[ a_{0|0}\right] =\left[ A_{/k}\right] $, such that$$\det \,\frac{\partial \overset{k}{F}_{A_{/k}}}{\partial (\dot{q}^{a_{1|0}},q^{a_{0|0}})}\neq 0\,\Longrightarrow \overset{k}{F}_{A_{/k}}\sim
\left(
\begin{array}{c}
\dot{q}^{a_{1|0}}-\varphi ^{a_{1|0}}\left(
q^{a_{1|0}},q^{a_{g_{0}}},q^{a_{g_{1}}},\dot{q}^{a_{g_{1}}}\right) \\
q^{a_{0|0}}-\varphi ^{a_{0|0}}\left( q^{a_{1|0}},q^{a_{g_{0}}},q^{a_{g_{1}}},\dot{q}^{a_{g_{1}}}\right)\end{array}\right) \,.$$Denoting $\underline{a}_{k}\equiv a_{1|-1}\,$, $G=G_{k}$, such that $a=\left( a_{1|-1},a_{0|0},a_{1|0},a_{g}\right) $, and $a_{g}=\left(
a_{g_{0}},a_{g_{1}}\right) $, $\left[ G\right] =\left[ A\right] -\left[ a\right] +\left[ a_{g}\right] $, we get finally the equivalence: $$F\sim \left(
\begin{array}{c}
\dot{q}^{a_{1|0}}-\varphi ^{a_{1|0}}\left(
q^{a_{1|0}},q^{a_{g_{0}}},q^{a_{g_{1}}},\dot{q}^{a_{g_{1}}}\right) \\
q^{a_{0|0}}-\varphi ^{a_{0|0}}\left( q^{a_{1|0}},q^{a_{g_{0}}},q^{a_{g_{1}}},\dot{q}^{a_{g_{1}}}\right) \\
q^{a_{1|-1}}-\varphi ^{a_{1|-1}}\left( q^{a_{1|0}},q^{a_{g_{1}}}\right) \\
0_{G}\end{array}\right) \,. \label{3.29}$$Here $q^{a_{g}}=\left( q^{a_{g_{0}}},q^{a_{g_{1}}}\right) $ are gauge coordinates. Thus, the Theorem 2 holds true in this case.
The case $I=0,\;$ $J=1$ $\left( i=0,\;\mu =0,1\right) $ corresponds to the equations of the form $$F_{A_{1}}\left( q^{a_{1}},\dot{q}^{a_{1}}\right) =0\,,\;F_{A_{0}}\left(
q^{a_{1}}\right) =0\,. \label{3.15}$$Such equations present a particular case ($\left[ a_{0}\right] =0$) of the equations $\bar{F}_{A}=0$ with the LF $\bar{F}_{A}$ defined in (\[3.27a\]). The reduction procedure for the latter case was considered above. It leads to the following equivalence$$F\sim \left(
\begin{array}{c}
\dot{q}^{a_{|1}}-\varphi ^{a_{|1}}\left( q^{a_{|1}},q^{a_{g}},\dot{q}^{a_{g}}\right) \\
q^{a_{|0}}-\varphi ^{a_{|0}}(q^{a_{|1}},q^{a_{g}}) \\
0_{G}\end{array}\right) \,,\;\;a=\left( a_{|0},a_{|1},a_{g}\right) \,,\;\left[ G\right] =\left[ A\right] -\left[ a\right] +\left[ a_{g}\right] .$$Here $q^{a_{g}}$ are the gauge coordinates. Thus, the Theorem holds true in this case as well.
Equations of arbitrary orders
-----------------------------
We have verified that the Theorem 2 holds true for $\mathcal{N}=0,1$. Now we are going to prove the theorem for $\mathcal{N}=I+J=K$ (where $K$ is some fixed number) supposing that the theorem holds true for any $\mathcal{N}<K$ .
At the first step we consider the set $$F_{A_{\mu }}^{\left[ J-\mu \right] }\left( \cdots q^{a_{i}\left[ i+J\right]
}\right) =0\,,\;\;i=0,1,...,I\,,\;\mu =0,...,J\,, \label{3.6}$$which is obtained from the initial set (\[3.1\]) by substituting the constraints by the corresponding consistency conditions (conditions obtained from the constraints $F_{A_{\mu }}$ by $J-\mu $ time differentiations). According to the supposition (\[3.2\]), there exists a division of the indices $A_{\mu }$ and $a_{i}\;$as:$\;A_{\mu }=\left( A_{\mu /1},A_{\mu
/2}\right) ,$ $a_{i}=\left( a_{i/1},a_{i/2}\right) ,\;$ $\sum_{\mu }\left[
A_{\mu /1}\right] =\sum_{i}\left[ a_{i/1}\right] =r$, such that:
$$\det \,\left| \frac{\partial F_{A_{\mu /1}}^{\left[ J-\mu \right] }}{\partial q^{a_{i/1}\left[ i+J\right] }}\right| \neq 0\,. \label{3.5}$$
Thus, we may solve the equations $F_{A_{\mu /1}}^{\left[ J-\mu \right] }=0$ with respect to the derivatives $q^{a_{i/1}\left[ i+J\right] }\,.$ Namely, $$\;F_{A_{\mu /1}}^{\left[ J-\mu \right] }=0\Longleftrightarrow q^{a_{i/1}\left[ i+J\right] }=\varphi ^{a_{i/1}}\left( \cdots q^{b_{j/1}\left[ j+J-1\right] },\cdots q^{b_{j/2}\left[ j+J\right] }\right) \,. \label{3.7a}$$Now we pass from the functions $F_{A_{J/2}}$ to the ones $\bar{F}_{A_{J/2}}$ excluding the arguments $q^{b_{i/1}\left[ i+J\right] }$ from the former, $$\bar{F}_{A_{J/2}}=\left. F_{A_{J/2}}\right| _{f=0}\,=\bar{F}_{A_{J/2}}\left(
\cdots q^{b_{i}\left[ i+J-1\right] }\right) \,. \label{3.8c}$$The fact that the functions $\bar{F}_{A_{J/2}}$ do not depend on both $q^{b_{i/1}\left[ i+J\right] }$ and $q^{b_{i/2}\left[ i+J\right] }\,$ is based on the Lemma 2 from the Appendix. Thus, we have the equivalence (see Lemma 1 from the Appendix) $$F_{A}\sim \left(
\begin{array}{c}
F_{A_{J/1}} \\
\bar{F}_{A_{J/2}} \\
F_{A_{\nu }}\,,\;\nu =0,...,J-1\end{array}\right) \sim \left(
\begin{array}{c}
F_{A_{J/1}} \\
F_{A^{\prime }}^{\prime }\,,\;\end{array}\right) \,, \label{3.9}$$where $$F_{A^{\prime }}^{\prime }=\left( F_{A_{\nu }^{\prime }}^{\prime }\,,\;\nu
=0,...,J-1\right) =\left\{
\begin{array}{c}
F_{A_{\nu }}\left( \cdots q^{b_{i}\left[ i+\nu \right] }\right) \,,\;\nu
=0,...,J-2 \\
F_{A_{J-1}^{\prime }}^{\prime }\left( \cdots q^{b_{i}\left[ i+J-1\right]
}\right) =\left\{
\begin{array}{c}
F_{A_{J-1}} \\
\bar{F}_{A_{J/2}}\end{array}\right.\end{array}\right. \,\,. \label{3.10a}$$
Let us turn to the functions $F_{A_{\nu }^{\prime }}^{\prime }$ . They have the same structure as in (\[3.1\]) and correspond to the case $\mathcal{N}=I+J<K.$ In accordance with the induction hypothesis, supposing, in particular, that the matrix $$M_{A_{\nu }^{\prime }\,a_{i}}^{\prime }=\frac{\partial F_{A_{\nu }^{\prime
}}^{\prime }}{\partial q^{a_{i}\left[ i+\nu \right] }}$$has a constant rank in the consideration point the following equivalence holds true:$$\begin{aligned}
&&F_{A^{\prime }}^{\prime }\sim \left(
\begin{array}{c}
q^{a_{i|\sigma }\left[ i+\sigma \right] }-\varphi ^{a_{i|\sigma }}\left(
\cdots q^{b_{j|\sigma _{-}}\left[ j+\sigma _{-}-1\right] },\cdots
q^{b_{j|\sigma _{+}}\left[ j+\sigma \right] },\cdots q^{b_{j|J}\left[
j+\sigma \right] }\right) \\
0_{G^{\prime }}\end{array}\right) \,, \notag \\
&&i,j=0,1,...,I\;,\;\;\left[ G^{\prime }\right] =\left[ A^{\prime }\right] -\left[ a\right] +\sum_{i}\left[ a_{i|J}\right] \;, \notag \\
&&\sigma =-I,...,J-1,\;\;-I\leq \sigma _{-}\leq \sigma \,,\;\;\sigma +1\leq
\sigma _{+}\leq J-1\,. \label{3.10b}\end{aligned}$$Taking into account (\[3.9\]), we obtain $$\begin{aligned}
&&F\sim \left(
\begin{array}{c}
F_{A_{J/1}}\left( \cdots q^{b_{i}\left[ i+J\right] }\right) \\
q^{a_{i|\sigma }\left[ i+\sigma \right] }-\varphi ^{a_{i|\sigma }}\left(
\cdots q^{b_{j|\sigma _{-}}\left[ j+\sigma _{-}-1\right] },\cdots
q^{b_{j|\sigma _{+}}\left[ j+\sigma \right] },\cdots q^{b_{j|J}\left[
j+\sigma \right] }\right) \\
0_{G^{\prime }}\end{array}\right) \,, \notag \\
&&i,j=0,1,...,I\;,\;\sigma =-I,...,J-1,\;-I\leq \sigma _{-}\leq \sigma
\,,\;\sigma +1\leq \sigma _{+}\leq J-1\,. \label{3.11a}\end{aligned}$$Now we pass from the functions $F_{A_{J/1}}$ to the ones $\bar{F}_{A_{J/1}}$ excluding the arguments $q^{a_{i|\sigma }\left[ p_{i}\right] },\;$ $p_{i}\geq i+\sigma ,$ $\sigma =-I,...,J-1$ from the former. As a result, the following equivalence takes place: $$F\sim \tilde{F}=\left(
\begin{array}{c}
\bar{F}_{A_{J/1}}\left( \cdots q^{b_{i|J}\left[ i+J\right] },\cdots
q^{b_{i|\sigma }\left[ i+\sigma -1\right] }\right) \\
q^{a_{i|\sigma }\left[ i+\sigma \right] }-\varphi ^{a_{i|\sigma }}\left(
\cdots q^{b_{j|\sigma _{-}}\left[ j+\sigma _{-}-1\right] },\cdots
q^{b_{j|\sigma _{+}}\left[ j+\sigma \right] },\cdots q^{b_{j|J}\left[
j+\sigma \right] }\right) \\
0_{G^{\prime }}\end{array}\right) \,. \label{3.11b}$$
The functions $\tilde{F}$ have the same structure as in (\[3.1\]), however, they depend linearly on a part of highest-order derivatives. Here the supposition of the rank for the matrix $$\frac{\partial \tilde{F}_{A}}{\partial (q^{a_{i|J}\left[ i+J\right]
},q^{a_{i|\sigma }\left[ i+\sigma \right] })}\,,\;A=\left(
A_{J/1}\,,a_{i|\sigma }\,,G^{\prime }\right) \label{3.12}$$is equivalent to the same supposition for the matrix $$\frac{\partial \bar{F}_{A_{J/1}}}{\partial q^{b_{i|J}\left[ i+J\right] }}\,.
\label{3.13}$$Let this rank be equal to $\left[ {A_{J/1}}\right] $. In this case there exists a final division of indices, $$a_{i|J}\rightarrow \left( a_{i|J},a_{i|J+1}\right) \,\;\mathrm{with}\;\;[a_{i|J}]=[A_{J/1}]\,,$$such that the equations $\bar{F}_{A_{J/1}}=0$ can be solved with respect to the derivatives $q^{a_{i|J}\left[ i+J\right] }$ and we obtain, instead of the two first lines of (\[3.11b\]), the following expressions: $$\begin{aligned}
&&q^{a_{i|J}\left[ i+J\right] }-\varphi ^{a_{i|J}}\left( \cdots q^{b_{j|J}\left[ j+J-1\right] },\cdots q^{b_{j|\sigma }\left[ j+\sigma -1\right]
},\cdots q^{b_{j|J+1}\left[ j+J\right] }\right) , \notag \\
&&q^{a_{i|\sigma }\left[ i+\sigma \right] }-\varphi ^{a_{i|\sigma }}\left(
\cdots q^{b_{j|\sigma _{-}}\left[ j+\sigma _{-}-1\right] },\cdots
q^{b_{j|\sigma _{+}}\left[ j+\sigma \right] },\cdots q^{b_{j|J}\left[
j+\sigma \right] }\right) , \notag \\
&&i,j=0,1,...,I\;,\;\sigma =-I,...,J-1,\;-I\leq \sigma _{-}\leq \sigma
\,,\;\sigma +1\leq \sigma _{+}\leq J-1\,. \notag\end{aligned}$$Now, let us put together the first two entries of $\varphi ^{a_{i|J}}$ as $\cdots q^{b_{j|\sigma }\left[ j+\sigma -1\right] },\;-I\leq \sigma \leq J$ and remind that for $\sigma =J$ no corresponding $\sigma _{+}$ occurs. Furthermore, let us replace the last entry of $\varphi ^{a_{i|\sigma }}$ as follows: $\cdots q^{b_{j|J}\left[ j+\sigma \right] }\rightarrow \cdots
q^{b_{j|J}\left[ j+\sigma \right] }\cdots q^{b_{j|J+1}\left[ j+\sigma \right]
},\;\;-I\leq \sigma \leq J-1$, then we get the missing contribution to $\sigma _{+}$ for the case under consideration. So, we end up exactly with Eq. (\[3.3\]) and the Theorem 2 is proved.
If the rank is less than $\left[ A_{J/1}\right] $ then the above procedure is applied to the functions $\bar{F}_{A_{J/1}}$. Doing that we lower the number of the equations that are not yet reduced to the canonical form (the equations of the type $\bar{F}_{A_{J/1}}=0$ ). Remark that such a diminution does not happen at the first stage if $\left[ A_{J/2}\right] =0$ . At a certain stage the procedure does not lower the number of the above mentioned equations. This can happen when the rank of the matrix of the type ([3.13]{}) is maximal, i.e. is equal to the number of the functions of the type $\bar{F}_{A_{J/1}}\,$. In such a case we may reduce them to the canonical form as was said above. This can also happen when we do not obtain the functions of the type $\bar{F}_{A_{J/1}}$ in the reduction procedure. That means that already at the previous step the set is reduced to the case $\mathcal{N}=K-1,$ i.e. the possibility of the reduction to the canonical form is proved.
Finally we stress that the reduction procedure is formulated for sets of equations of the type (\[3.1\]) (the ELE are a particular case of such sets). The procedure holds true under certain suppositions of ranks. These suppositions demand various Jacobi matrices of the type $\partial
F_{s}/\partial q^{a\left[ l\right] }$ to have constant ranks in the vicinity of the consideration point. Here $F_{s}=0$ are equations obtained to a given stage of the procedure and $q^{a\left[ l\right] }$ are highest-order derivatives in these equations. It is important to realize that proving the equivalence (\[3.3a\]) we prove at the same time the locality of the operators $\hat{U}$ and $\hat{V}$ from (\[3.3b\]). In fact, the latter proof is provided by the applicability of the Lemmas from the Appendix.
Gauge identities and action symmetries
======================================
It was demonstrated above that in the general case of singular ELE the number of the equations in the canonical form is less than the number of the equations in the initial set of the differential equations. This reduction is related to the fact that in the canonical form we retain the independent equations only, whereas the initial equations may be dependent. The dependence of the equations in the initial set may be treated as the existence of some identities between the initial equations. The identities between the ELE imply the existence of gauge transformations of the corresponding action. Below we discuss this interrelationship in detail.
First, we introduce some relevant definitions: The relation of the form $$\hat{R}^{a}F_{a}\equiv 0\,, \label{4.1}$$where $\hat{R}^{a}$ are some LO, and $F_{a}\left( q^{\left[ l\right]
}\right) $ are some LF, is called the* *identity between the equations $F_{a}\left( q^{\left[ l\right] }\right) =0.$ The identity sign $\equiv $ means that the left hand side of (\[4.1\]) is zero for any arguments $q^{\left[ l\right] }\,.$
Any set $\mathbf{\hat{R}}=\left( \hat{R}^{a}\right) $ of LO that obeys the relation (\[4.1\]) is called the* *generator of an identity.* *Whenever $\mathbf{\hat{R}}$ is a generator than $\hat{n}\mathbf{\hat{R}}$ with some LO $\hat{n}$ is a generator as well. Any linear combination $\hat{n}^{i}\mathbf{\hat{R}}_{i}$ of some generators $\mathbf{\hat{R}}_{i}\,\ $with operator coefficients $\hat{n}^{i}$ is a generator.
A generator $\mathbf{\hat{R}}$ will be called nontrivial if the relation[^8] $\hat{n}\mathbf{\hat{R}}=\hat{O}\left( F\right) $ can only be provided by a LO $\hat{n}$ of the form $\hat{n}=\hat{O}\left( F\right) \,$.
A set of generators $\mathbf{\hat{R}}_{i}$ will be called independent if the relation $\hat{n}^{i}\mathbf{\hat{R}}_{i}=\hat{O}\left( F\right) $ can only be provided by $\hat{n}^{i}$ of the form $\hat{n}^{i}=\hat{O}\left( F\right)
\,.$ Identities generated by independent generators will be called independent.
Note that for any set of LF $F_{a}$ , there always exist trivial generators. Namely, the generators $\mathbf{\hat{R}}_{\mathrm{triv}}=\left( \hat{R}_{\mathrm{triv}}^{a}\right) =\hat{O}\left( F\right) $ of the form $$\hat{R}_{\mathrm{triv}}^{a}=\sum_{k,l}F_{b}^{\left[ k\right] }u^{bk|al}\frac{d^{l}}{dt^{l}}\,,\;u^{bk|al}=-u^{al|bk}\,, \label{4.4}$$with arbitrary antisymmetric LF $u^{bk|al}\;$obviously lead to the identities (\[4.1\]). These identities are not, however, connected to the mutual dependence of the functions $F_{a}$ $.$
An independent set of generators $\mathbf{\hat{R}}_{g}$ is complete whenever any generator $\mathbf{\hat{R}}$ can be represented in the form $\mathbf{\hat{R}}=\hat{\lambda}^{g}\mathbf{\hat{R}}_{g}+\mathbf{\hat{R}}_{\mathrm{triv}}$ with some LO $\hat{\lambda}^{g}.$ Any two complete sets of independent generators $\mathbf{\hat{R}}_{g}$ and $\mathbf{\hat{R}}_{g}^{\prime }$ are related as $\mathbf{\hat{R}}_{g}^{\prime }=\hat{U}_{g}^{g^{\prime }}\mathbf{\hat{R}}_{g^{\prime }}+\mathbf{\hat{R}}_{\mathrm{triv}}\,,$where $\hat{U}$ is an invertible LO.
Supposing now that $F_{a}$ in Eq. (\[4.1\]) are functional derivatives of an action, $F_{a}=\delta S/\delta q^{a},$ such that $F_{a}=0$ are ELE. Let the functions $F_{a}$ obey all the necessary suppositions of ranks such that ELE can be reduced to the canonical form (\[3.3\]). Let us write here this canonical form as follows[^9],$$f^{\alpha }=q^{\alpha \left[ l_{\alpha }\right] }-\varphi ^{\alpha }\left(
\cdots q^{\alpha \left[ l_{\alpha }-1\right] };\cdots q^{g\left[ l_{g}\right]
}\right) =0\,,\;a=\left( \alpha ,g\right) \,, \label{4.5a}$$where $q^{g}$ are gauge coordinates. Moreover, according to the Theorem 2, there exists the equivalence$$F_{a}\sim \bar{F}_{a}=\left(
\begin{array}{l}
f^{\alpha } \\
0_{g}\end{array}\right) \Longrightarrow F_{a}=\hat{U}_{a}^{b}\bar{F}_{b}\,,\;\bar{F}_{a}=\hat{V}_{a}^{b}F_{b}\,,\;\hat{U}_{a}^{b}\hat{V}_{b}^{c}=\delta _{a}^{c}\,,
\label{4.5}$$where $\hat{U}$ and $\hat{V}$ are LO. Now we may consider the identity ([4.1]{}) as an equation for finding the general form for the generator $\mathbf{\hat{R}}$ . Using (\[4.5\]) we transform this problem to the one for finding the operators $\hat{\xi}^{a},$$$\hat{\xi}^{a}\bar{F}_{a}\equiv 0\,,\;\;\hat{R}^{a}=\hat{\xi}^{b}\hat{V}_{b}^{a}\,. \label{4.6}$$Using the explicit form (\[4.5\]) of the functions $\bar{F}_{a}$ , we get $\hat{\xi}^{a}=\left(
\begin{array}{cc}
\hat{\xi}^{\alpha } & \hat{\xi}^{g}\end{array}\right) ,\;a=\left( \alpha ,g\right) $ , where $\hat{\xi}^{\alpha }$ obey the equation $$\hat{\xi}^{\alpha }f^{\alpha }\equiv 0\,, \label{4.7}$$and $\hat{\xi}^{g}$ is a set of arbitrary LO. Since the functions $f$ have the canonical form (\[4.5\]), any solution of the equation (\[4.7\]) is presented by trivial generators of the form $$\hat{\xi}^{\alpha }=\hat{\xi}_{\mathrm{triv}}^{\alpha }=\sum_{k,l}\left(
\frac{d^{l}}{dt^{l}}f_{\alpha ^{\prime }}\right) u^{l\alpha ^{\prime
}|k\alpha }\frac{d^{k}}{dt^{k}}\,,\;\;u^{l\alpha ^{\prime }|k\alpha
}=-u^{k\alpha |l\alpha ^{\prime }}\,, \label{4.8}$$where $u^{l\alpha ^{\prime }|k\alpha }$ are arbitrary antisymmetric LF. To demonstrate that we present the generators $\hat{\xi}^{\alpha }$ as $\hat{\xi}^{\alpha }=\sum_{k=0}^{K}\xi ^{\alpha k}d^{k}/dt^{k}\,,$ where $\xi
^{\alpha k}$ are some LF. Then, in the equation (\[4.7\]), we pass from the variables $q^{\alpha \left[ k\right] },\;q^{g\left[ l\right] },$ $\;k,l=0,1,...$ to ones $q^{\alpha \left[ k_{\alpha }\right] },\;f_{\alpha }^{\left[ l\right] },\;q^{g\left[ l\right] },$ $k_{\alpha }=0,1,...,l_{\alpha
}-1,\;$ $l=0,1,...$ . Such a variable change is not singular. In terms of the new variables, the equation (\[4.7\]) reads$$\sum_{k=0}^{K}\xi ^{\alpha k}f_{\alpha }^{\left[ k\right] }=0\;,\;K<\infty
\;.$$Its general solution is well known $$\xi ^{\alpha k}=\sum_{l}f_{\alpha ^{\prime }}^{\left[ l\right] }u^{l\alpha
^{\prime }|k\alpha }\,,\;u^{l\alpha ^{\prime }|k\alpha }=-u^{k\alpha
|l\alpha ^{\prime }}\;.$$Now we can write the general solution of the equation (\[4.6\]) as: $$\hat{\xi}^{a}=\hat{\xi}^{g}\delta _{g}^{a}+\hat{\xi}_{\mathrm{triv}}^{a},\;\;\hat{\xi}_{\mathrm{triv}}^{a}=\sum_{k,l}\left( \frac{d^{l}}{dt^{l}}\bar{F}_{b}\right) u^{lb|ka}\frac{d^{k}}{dt^{k}},\;\;u^{lb|ka}=-u^{ka|lb}.
\label{4.12}$$Let $b=\left( \alpha ^{\prime },g^{\prime }\right) ,\;$ $a=\left( \alpha
,g\right) $ in (\[4.12\]) . Then $u^{lg^{\prime }|k\alpha }$, $u^{l\alpha
^{\prime }|kg}=-u^{kg|l\alpha ^{\prime }}$ and $u^{lg^{\prime }|kg}$ are arbitrary LF (e.g., they can be selected to be zero). Indeed, the functions $u^{lg^{\prime }|k\alpha }$ and $u^{lg^{\prime }|kg}$ do not enter the expressions for the generators $\hat{\xi}^{a}$. Besides, terms with $u^{l\alpha ^{\prime }|kg}$ affect only the generators $\hat{\xi}^{g}$, which are arbitrary by the construction. Respectively, the general solution of the equation (\[4.1\]) reads: $$\mathbf{\hat{R}}=\hat{\xi}^{g}\mathbf{\hat{R}}_{g}+\mathbf{\hat{R}}_{\mathrm{triv}}\;,\;\;\mathbf{\hat{R}}_{g}=(\hat{R}_{g}^{a}=\delta _{g}^{b}\hat{V}_{b}^{a}=\hat{V}_{g}^{a})\,, \label{4.13}$$and $$\hat{R}_{\mathrm{triv}}^{a}=\hat{\xi}_{\mathrm{triv}}^{b}\hat{V}_{b}^{a}=\sum_{k,l}\left[ \frac{d^{l}}{dt^{l}}\left( \hat{V}_{b}^{c}F_{c}\right) \right] u^{lb|kd}\frac{d^{k}}{dt^{k}}\hat{V}_{d}^{a}=\sum_{k,l}\left( \frac{d^{l}}{dt^{l}}F_{b}\right) T^{lb|ka}\frac{d^{k}}{dt^{k}}\,,$$where $T^{lb|ka}=-T^{ka|lb}\,$ are some LF. The set of the generators $\mathbf{\hat{R}}_{g}=(\hat{R}_{g}^{a}=\hat{V}_{g}^{a})$ is complete and it is presented by LO. Moreover, these generators are independent. Indeed, multiplying the equation $\hat{n}^{g}\hat{R}_{g}^{a}=\hat{O}\left( F\right) $ from the right by $\hat{U}_{a}^{b},$ we get: $\hat{n}^{g}\delta _{g}^{b}=\hat{O}\left( F\right) \Longrightarrow \hat{n}^{g}=\hat{O}\left( F\right) $.
Thus, there exist the following nontrivial identities between the ELE:$$\hat{R}_{g}^{a}\frac{\delta S}{\delta q^{a}}\equiv 0\,,\;g=1,...,r,
\label{4.14}$$with generators $\mathbf{\hat{R}}_{g}$ that are LO. These identities are called the* *gauge identities. As is well known (see for example, [GitTy90,HenTe92]{}), the existence of the gauge identities (\[4.14\]) implies the existence of infinitesimal gauge transformations of the form $$q^{a}\rightarrow q^{a}+\delta q^{a}\,,\;\delta q^{a}=\left( \hat{R}^{T}\right) _{g}^{a}\epsilon ^{g}\;, \label{4.15}$$where $r$ parameters $\epsilon ^{g}=\epsilon ^{g}(t)$ are arbitrary functions of time $t$. Remark that $\mathbf{\hat{R}}^{T}$ are LO as well.
Thus, it was demonstrated that for theories that obey appropriate suppositions of the ranks there exists a constructive procedure of revealing the gauge generators. For such theories all the generators are LO. The number of the independent generators and, therefore, the number of the independent gauge transformations is equal to the number of the gauge coordinates in the ELE.
As a simple mechanical example, consider the action of the form[^10]
$$S=\int Ldt\,,\;\;L=\frac{1}{2}\left( \dot{x}-y\right) ^{2}+\frac{a}{2}\left(
y^{2}-x^{2}\right) \,. \label{e.7}$$
The corresponding ELE are: $$F_{1}=\ddot{x}-\dot{y}+ax=0,\;\;F_{2}=\dot{x}-\left( 1+a\right) y=0\,,
\label{e.8}$$where $F_{2}=0$ is a constraint. The generalized Hessian reads:$$M=\left|
\begin{array}{cc}
\frac{\partial ^{2}L}{\partial \dot{x}^{2}}=1 & \frac{\partial ^{2}L}{\partial \dot{x}\partial y}=-1 \\
\frac{\partial ^{2}L}{\partial y\partial \dot{x}}=-1 & \frac{\partial ^{2}L}{\partial y^{2}}=a+1\end{array}\right| =a\,. \label{e.10}$$
Let $a\neq 0\,,\;M\neq 0.$ In such a nonsingular case the reduction procedure looks as follows: By the help of the consistency condition $\dot{F}_{2}=0\Longrightarrow \ddot{x}=\left( 1+a\right) \dot{y}\,,$ we eliminate $\ddot{x}$ from the first ELE. Thus, we get an equivalent set, which has the canonical form,$$\dot{y}=-x\,,\;\;\,\dot{x}=\left( 1+a\right) y\;. \label{e.11}$$Another canonical form$$\ddot{x}=-\left( 1+a\right) x\,,\;\;y=\left( 1+a\right) ^{-1}\,\dot{x}\,,
\label{e.12}$$we obtain eliminating $\dot{y}$ from the equation $F_{1}=0$ by the help of the consistency condition $\dot{F}_{2}=0\Longrightarrow \dot{y}=\ddot{x}/\left( 1+a\right) $ $.$
Let $a=0\,.\;$The case is singular, $M=0\,,$ and rank of the Hessian matrix is equal to$\;1$ . One can easily see that the equivalence$$\left(
\begin{array}{c}
F_{1} \\
F_{2}\end{array}\right) =\hat{U}\left(
\begin{array}{c}
\dot{x}-y \\
0\end{array}\right) ,\;\hat{U}=\left(
\begin{array}{cc}
d/dt & 1 \\
1 & 0\end{array}\right) ,\;\hat{U}^{-1}=\left(
\begin{array}{cc}
0 & 1 \\
1 & -d/dt\end{array}\right)$$holds true. Then the canonical form of the ELE reads $\dot{x}=y$ and there is a gauge identity$$\hat{R}^{a}F_{a}\,\equiv 0\,,\;\;\hat{R}^{1}=1\,,\;\hat{R}^{2}=-d/dt\;.$$The operators transposed to $\hat{R}^{a}$ are $\left( \hat{R}^{T}\right)
^{a}=\left( \left( \hat{R}^{T}\right) ^{1}=1,\left( \hat{R}^{T}\right) ^{2}=\frac{d}{dt}\right) .$ Thus, at $a=0,$ the action (\[e.7\]) is invariant under the gauge transformation $x\rightarrow x+\epsilon \,,\;y\rightarrow y+\dot{\epsilon}\,.$ In the case under consideration, the ELE have two canonical forms: $\dot{x}=y\;$and $y=\dot{x}$ .
Concluding remarks
==================
We have formulated the reduction procedure which allows one to transform the ELE to the canonical form as well as to establish possible gauge identities between the equations. The latter part of the procedure can be considered as a constructive way of finding all the gauge generators within the Lagrangian formulation. At the same time, it is proven that, for local theories, all the gauge generators are local in time operators. The canonical form of the ELE reveals their hidden structure, in particular, it presents the spectrum of possible initial data, and it allows one to separate coordinates into nongauge and gauge ones. One ought also to remark that the reduction procedure can be, in particular, treated as a procedure of finding constraints in the Lagrangian formulation.
In that respect one can compare the reduction procedure with the well-known Dirac procedure in the Hamiltonian formulation of constrained systems [Dirac64,GitTy90,HenTe92]{}. Recall that the Dirac procedure is applicable to the Hamilton equations with primary constraints, namely to equations of the form$$F\left( \eta ,\dot{\eta}\right) =\dot{\eta}-\left\{ \eta \,,H^{\left(
1\right) }\right\} =0\,,\,\,\Phi ^{\left( 1\right) }\left( \eta \right)
=0\,,\;H^{\left( 1\right) }=H\left( \eta \right) +\lambda \Phi ^{\left(
1\right) }\left( \eta \right) \,. \label{r.1}$$Here $\eta =\left( q^{a},p_{a}\right) $ are phase-space variables; $\Phi
^{\left( 1\right) }\left( \eta \right) =0\;$are primary constraints, $\lambda $’s are Lagrange multipliers to the primary constraints, and $H^{\left( 1\right) }$ is the total Hamiltonian . Via $\{\cdot ,\cdot \}$ the Poisson bracket is denoted. The aim of the procedure is to eliminate as many as possible $\lambda $’s from the equations, to find all the constraints in the theory. The procedure is based on the consistency conditions $\dot{\Phi}^{\left( 1\right) }=0$. Using the equations $F\left( \eta ,\dot{\eta}\right)
=0$, we may transform any consistency condition to the following form:$$\dot{\Phi}^{\left( 1\right) }=\left\{ \Phi ^{\left( 1\right) },H^{\left(
1\right) }\right\} =0\,.$$From these equations one can define some $\lambda $’s as functions of $\eta $ and reveal some new constraints. Then the procedure has to be applied to the latter constraints and so on.
The equations (\[r.1\]) present a particular case of differential equations considered in the present article (indeed, these equations are ELE for a Hamiltonian action). Thus, our reduction procedure may be applied to these equations. Namely, first one has to consider the equations $F_{A}=0,\;\dot{\Phi}^{\left( 1\right) }=0$ and select independent w.r.t. $\dot{\eta}$ equations. Since equations of the primary constraints are independent by construction, we pass to the next step and solve the constraint equations $\Phi ^{\left( 1\right) }=0$ with respect to a part of the variables $\eta ,$ as $\Phi ^{\left( 1\right) }=0\;\rightarrow \;\eta _{1}-\varphi _{1}\left(
\eta _{2}\right) =0$. Then we exclude $\eta _{1}$ and $\dot{\eta}_{1}$ from the equations $F=0$. Thus, we get $F=0\rightarrow \bar{F}_{A}\left( \eta
_{2},\dot{\eta}_{2}\right) =0$. Then one has to select independent w.r.t. $\dot{\eta}_{2}$ functions $\bar{F}_{A_{/1}}$ . At the same time one finds new constraints $\bar{F}_{A_{/2}}\left( \eta _{2}\right) =0$ and so on (see the Subsect. ”First order equations”).
We see that the Dirac procedure differs from our reduction procedure. Indeed, as was mentioned above, in the Dirac procedure one excludes all the derivatives $\dot{\eta}$ by the help of the equations $F=0$ from the consistency conditions $\dot{\Phi}^{\left( 1\right) }=0.$ Thus, one gets equations for the Lagrange multipliers and new constraints. Besides, one of the aim of the Dirac procedure is to maintain the canonical Hamiltonian structure of the equations $F=0.$ The possibility of the Dirac reduction is due to the specific structure of the equations (\[r.1\]). Namely, here the consistency conditions never involve $\dot{\lambda}\,$ and $\partial F/\partial \dot{\eta}=\left[ F\right] =\left[ \eta \right] $.
Besides, one ought to mention the work [@FadJa88] where it was proposed an alternative (to the Dirac procedure) way of reducing the equations of motion for theories with actions of the form $S=\int \left[ \varphi
_{A}\left( \eta \right) \dot{\eta}^{A}-V\left( \eta \right) \right] dt$ . One can verify that, in fact, the procedure of that work, in a part (the procedure does not reveal the gauge identities), is similar to our reduction procedure in the case of the first order equations (see Sect. IV).
However, the reduction procedure proposed in the present article is formulated for a wider class of Lagrangian systems (differential equations). It does not need the introduction of new variables such as momenta and Lagrange multipliers, and it is defined in the framework of the initial Lagrangian formulation. Moreover, its aim is twofold: to reduce ELE to their canonical form and to reveal the gauge identities between the ELE equations.
The consideration in the present article is restricted by finite-dimensional systems. Its application to field theories (theories with infinite number degrees of freedom) demands additional study. We hope to present the corresponding formulation in futures publications. However, in simple cases, one can apply the present reduction procedure with some natural modifications in the infinite-dimensional case. Consider the Maxwell action $S=-\left( 1/4\right) \int \mathcal{F}_{\mu \nu }\mathcal{F}^{\mu \nu }dx\,,$ $\;\mathcal{F}_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ as a common example of a singular field theory. The ELE read:$$\begin{aligned}
F^{i} &=&\frac{\partial S}{\partial A_{i}}=\partial _{\nu }\mathcal{F}^{i\nu
}=\ddot{A}^{i}+\partial _{i}\dot{A}^{0}-\bigtriangleup A^{i}+\partial
_{i}\varphi =0\,, \label{r.2} \\
F^{0} &=&-\frac{\partial S}{\partial A^{0}}=\partial _{\nu }\mathcal{F}^{\nu
0}=\dot{\varphi}+\bigtriangleup A^{0}\,=0\,,\;\varphi =\partial _{k}A^{k}\,.
\label{r.3}\end{aligned}$$The equation $F_{0}=0$ is a constraint. Following the reduction procedure, we have to consider the set $F_{i}=0\,,\;\dot{F}_{0}=0\,.$ The Jacobi matrix $\partial F^{\mu }/\partial \ddot{A}_{\nu }$ has the constant rank $3.$ We can, for example, select the equations (\[r.2\]) as independent with respect to the derivatives $\ddot{A}^{i}$. The equation $\dot{F}_{0}=0$is their consequence. No more constraints appear. Now we exclude $A^{0}$ and $\dot{A}^{0}$ from (\[r.2\]) by the help of (\[r.3\]). That creates the equivalence $$\left(
\begin{array}{c}
F^{i} \\
F^{0}\end{array}\right) =\left(
\begin{array}{cc}
\delta _{k}^{i} & -\frac{\partial _{i}\partial _{0}}{\bigtriangleup } \\
0 & 1\end{array}\right) \left(
\begin{array}{c}
\bar{F}^{k} \\
F^{0}\end{array}\right) \,,\;\;\;\bar{F}^{k}=\left. F^{k}\right| _{F^{0}=0}=\square \left(
A^{k}+\partial _{k}\varphi \right) \,. \label{r.4}$$Now we discover that the functions $\bar{F}^{k}$ are dependent, $\partial
_{k}\bar{F}^{k}\equiv 0.$ In our terms that reads, for example, as the following equivalence$$\left(
\begin{array}{c}
\bar{F}^{1} \\
\bar{F}^{2} \\
\bar{F}^{3}\end{array}\right) =\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
-\partial _{3}^{-1}\partial _{1} & -\partial _{3}^{-1}\partial _{2} & 1\end{array}\right) \left(
\begin{array}{c}
\bar{F}^{1} \\
\bar{F}^{2} \\
0\end{array}\right) \,. \label{r.5}$$The equations $\bar{F}^{1}=0,\;\bar{F}^{2}=0,\;F^{0}=0$ present one of the canonical forms of the Maxwell equations. The identity that follows from the presence of the zero in the right column of (\[r.5\]) reads as $\partial
_{\mu }F^{\mu }=0$ in terms of the initial functions $F^{\mu }$ and implies the invariance of the Maxwell action under gradient gauge transformations.
B. Geyer thanks the foundations FAPESP and DAAD for support and Institute of Physics USP for hospitality; D. Gitman is grateful to the foundations FAPESP, CNPq, DAAD for support as well as to the Lebedev Physics Institute (Moscow) and to the Institute of Theoretical Physics (University of Leipzig) for hospitality; I. Tyutin thanks INTAS 00-00262, RFBR 02-02-16944, 00-15-96566 for partial support.
Appendix
========
Here we present three Lemmas which are used in the reduction procedure to justify equivalence of equations and LF. In this respect it is useful to recall here the relevant definitions from the Sect. II . Namely:
Two sets of equations, $F_{A}\left( q^{\left[ l\right] }\right) =0$ and $f_{\alpha }\left( q^{\left[ l\right] }\right) =0$ are equivalent $F=0\Longleftrightarrow f=0$ whenever they have the same set of solutions. If two sets of LF $F_{A}\left( q^{\left[ l\right] }\right) $ and $\chi
_{A}\left( q^{\left[ l\right] }\right) $ , $\ \left[ F\right] =\left[ \chi \right] \,,$ are related by some LO $\hat{U}$ and $\hat{V}$ as $F=\hat{U}\chi \,$, $\chi =\hat{V}F\,$, $\hat{U}\hat{V}=1\,,$ then we call such LF equivalent and denote this fact as: $F\sim \chi $ . In this case the corresponding equations are strongly equivalent.
$${\Large Lemma\ 1}$$
Let a set of equations $$\Phi _{\mu }\left( x,y^{\left[ l\right] }\right) =0\,,\;F_{a}\left(
x,y\right) =0\,,\;x=\left( x^{\mu }\right) ,\;y=\left( y^{a}\right) \,,
\label{L.3}$$be given, where $\Phi $ are some LF. And let $\det \left. \partial
F_{a}/\partial y^{b}\right| _{x_{0},y_{0}}\neq 0,\;$where the consideration point $\left( x_{0},y_{0}\right) $ is on shell. Then:
a\) The equations $F_{a}\left( x,y\right) =0$ can be solved w.r.t. $y$ as: $y^{a}=\varphi ^{a}(x)\,,$ where $\varphi ^{a}(x)$ are some single-valued functions of $x$ in the vicinity of the point $x_{0}$. In other words, there is the equivalence $$F_{a}\left( x,y\right) \sim y^{a}-\varphi ^{a}(x)\,, \label{L.2}$$which implies the strong equivalence between the equations $F_{a}\left(
x,y\right) =0$ and $y^{a}=\varphi ^{a}(x)\,$.
b\) The following equivalence between the LF holds true:$$\left(
\begin{array}{c}
\Phi _{\mu }\left( x,y^{\left[ l\right] }\right) \\
F_{a}\left( x,y\right)\end{array}\right) \sim \left(
\begin{array}{c}
\bar{\Phi}_{\mu }\left( x^{\left[ l\right] }\right) \\
y^{a}-\varphi ^{a}(x)\end{array}\right) \,,\;\;\bar{\Phi}_{\mu }=\left. \Phi _{\mu }\right| _{y^{\left[ l\right] }=\varphi ^{\left[ l\right] }}\;\,. \label{l.5}$$
The first statement is, in fact, the well-known implicit function theorem [@Dieud68]. Taking into account (\[L.2\]), we have: $F_{a}\left(
x,y\right) =u_{ab}\left( y^{a}-\varphi ^{a}(x)\right) \,,\;\left. \det
\,u\right| _{x_{0},y_{0}}\neq 0.$ On the other side one can write $\Phi
_{\mu }=\bar{\Phi}_{\mu }+\hat{V}_{\mu a}\left[ y^{a}-\varphi ^{a}(x)\right]
\,,$ where $\hat{V}_{Aa}$ is a LO. Thus, $$\left(
\begin{array}{c}
\Phi \\
F\end{array}\right) =\hat{U}\left(
\begin{array}{c}
\bar{\Phi} \\
y-\varphi\end{array}\right) ,\;\;\hat{U}=\left(
\begin{array}{cc}
1 & \hat{V} \\
0 & u\end{array}\right) ,\;\hat{U}^{-1}=\left(
\begin{array}{cc}
1 & -\hat{V} \\
0 & u^{-1}\end{array}\right) , \label{L.1a}$$and the equivalence (\[l.5\]) is justified.$${\Large Lemma\ 2}$$Let a set of equations $$F_{A}\left( q,z\right) =0\,,\;q=\left( q^{a}\right) ,\;z=\left( z^{i}\right)
,\;\;A=1,...,m,\;a=1,...,n\,,\;i=1,...,l\,,$$be given. And let the Jacobi matrix $\partial F_{A}/\partial q^{a}$ have a constant rank in a vicinity $D_{0}$ of the consideration point $\left(
q_{0}\,,z_{0}\right) $, which is on shell ($F_{A}\left( q_{0}\,,z_{0}\right)
=0$), $$\mathrm{rank\,}\left. \frac{\partial F_{A}}{\partial q^{a}}\right| _{q,z\in
D_{0}}=r\,. \label{l.2}$$Then there exists an equivalence $$F_{A}\sim \bar{F}_{A}=\left(
\begin{array}{c}
y^{\mu }-\varphi ^{\mu }(x,z) \\
\Omega _{G}\left( z\right)\end{array}\right) \,,\;\;q^{a}=\left( x^{g},y^{\mu }\right) ,\;\;A=\left( \mu
,G\right) ,\;\left[ \mu \right] =r\,. \label{l.2a}$$
We begin the proof with the remark that due to (\[l.2\]), there exists a division of the indices $A=\left( \mu ,G\right) $, $a=\left( \mu
\,,g\right) \,$, $\left[ \mu \right] =r\,$, $q^{a}=\left( x^{g},y^{\mu
}\right) \,$, such that $$\det \left. \frac{\partial F_{\mu }}{\partial y^{\nu }}\right|
_{q_{0},z_{0}}\neq 0\,\,. \label{l.3}$$Then by virtue of the Lemma 1 we can write $$F_{\mu }=u_{\mu \nu }f^{\nu },\;\;f^{\nu }=y^{\nu }-\varphi ^{\nu
}(x,z),\;\det \left. u\right| _{q_{0},z_{0}}\neq 0\,. \label{l.4}$$Let us present the functions $F_{G}$ in the form $F_{G}\left( x,y,z\right)
=\Omega _{G}\left( x,z\right) +\Pi _{G\mu }f^{\mu }\left( x,y,z\right) ,$ where $\;\Omega _{G}\left( x,z\right) =\left. F_{G}\right| _{y=\varphi
\left( z,x\right) }\,,$ such that $\Omega _{G}\left( x_{0},z_{0}\right)
=0\,. $ Then $$F_{A}=\left(
\begin{array}{c}
F_{\mu } \\
F_{G}\end{array}\right) =U_{AB}\chi _{B}\,,\;\;\chi _{B}=\left(
\begin{array}{c}
f^{\mu } \\
\Omega _{G}\end{array}\right) \,,\;\;U=\left(
\begin{array}{cc}
u & 0 \\
\Pi & 1\end{array}\right) \,,\;\det \left. U\right| _{q_{0},z_{0}}\neq 0\,. \label{l.6}$$In virtue of (\[l.2\]) and (\[l.6\]) $$\mathrm{rank\,}\left. \frac{\partial \chi _{A}}{\partial q^{a}}\right|
_{q,z\in D_{0}}=r\,. \label{l.7}$$The Jacobi matrix $\partial \chi _{A}/\partial q^{a}$has the following structure: $$\frac{\partial \chi _{A}}{\partial q^{a}}=\frac{\partial \left( f^{\mu
},\Omega _{G}\right) }{\partial \left( y^{\nu },x^{g}\right) }=\left(
\begin{array}{cc}
\delta _{\nu }^{\mu } & -\partial \varphi ^{\mu }/\partial x^{g} \\
0 & \partial \Omega _{G}/\partial x^{g}\end{array}\right) \,.$$Therefore,$$\mathrm{rank}\,\left. \frac{\partial \Omega _{G}}{\partial x^{g}}\right|
_{x\in D_{0}}=0\Longrightarrow \left. \frac{\partial \Omega _{G}}{\partial
x^{g}}\right| _{x,z\in D_{0}}=0\,. \label{l.7a}$$Eq. (\[l.7a\]), together with the relation $\Omega _{G}\left(
x_{0},z_{0}\right) =0$, implies $$\left. \Omega _{G}\right| _{x,z\in D_{0}}\,=\Omega _{G}\left( z\right)
\,,\;\;\Omega _{G}\left( z_{0}\right) =0\,.$$Finally, we may write $$F_{A}=U_{AB}\chi _{B}\,,\;\;\chi _{B}=\left(
\begin{array}{c}
f^{\mu }\left( x,y,z\right) \\
\Omega _{G}\left( z\right)\end{array}\right) \,,\;\;\det \left. U\right| _{q_{0},z_{0}}\neq 0\,, \label{l.11a}$$Thus, the equivalence (\[l.2a\]) is justified.$${\Large Lemma\;3}$$As a consequence of the Lemma 2 the following Lemma holds true:
Let a set of equations$$F_{A}\left( q^{a}\right) =0\,,\;A=1,...,m,\;a=1,...,n\,,$$be given. And let the Jacobi matrix $\partial F_{A}/\partial q^{a}$ have a constant rank in a vicinity $D_{0}$ of the consideration point $q_{0}$ which is on shell ($F\left( q_{0}\right) =0$), $$\mathrm{rank\,}\left. \frac{\partial F_{A}}{\partial q^{a}}\right| _{q\in
D_{0}}=r\,.$$Then there exists an equivalence$$F_{A}\sim \bar{F}_{A}=\left(
\begin{array}{c}
y^{\mu }-\varphi ^{\mu }(x) \\
0_{G}\end{array}\right) \,,\;\;A=\left( \mu ,G\right) ,\;0_{G}\equiv 0\,\;\forall G\,,\;\left[ \mu \right] =r\,. \label{l.12}$$
The proof of this Lemma follows the one of the Lemma 2 if one selects there $z=z_{0}.$
[99]{} P.A.M. Dirac, *Lectures on Quantum Mechanics,* (Belfer Graduate School of Science, Yeshiva University, New York 1964)
D.M. Gitman and I.V. Tyutin, *Quantization of Fields with Constraints*, (Springer-Verlag, Berlin 1990)
M. Henneaux and C. Teitelboim, *Quantization of Gauge Systems* (Princeton University Press, Princeton 1992)
D.M. Gitman, and I.V. Tyutin, *Constraint reorganization consistent with Dirac procedure,* hep-th/0112103; Michael Marinov Memorial Volume: *Multiple Facets of Quantization and Supersymmetry*, (World Publishing, Singapore 2002)
V.A. Borochov and I.V. Tyutin, Yadernaya Fizika, **61** (1998) 1715 (Physics of Atomic Nuclei, **61** (1998) 1603); ibid **62** (1999) 1137 (Physics of Atomic Nuclei, **62** (1999) 1070)
I.A. Batalin and G.A. Vilkovisky, Phys. Lett. **B102** (1981) 27; Phys. Rev. **D28** (1983) 2567
D.M. Gitman and I.V. Tyutin, *Canonical quantization of singular theories*, Izw. VUZov Fizika **5** (1983) 3 (Sov. Phys. Journ. **5** (1983) 423)
D.M. Gitman and I.V. Tyutin, *The structure of gauge theories in the Lagrangian and Hamiltonian formalisms*, In *Quantum field theory and quantum statistics* v. I, pp. 143–164, Ed. by Batalin, Isham and Vilkovisky (Adam Hilger, Bristol 1987)
P. Olver, *Applications of Lie groups to differential equations*, Graduated Texts in Mathematics, **107** (Springer-Verlag, Berlin 1986); B.A. Kupershmidt, *Geometry of jet bundles and structure of Lagrangian and Hamiltonian formalism,* 162-218, Lecture Notes in Mathematics **775** (Springer-Verlag, Berlin 1980); I.S. Krasilshchik, V.V. Lychagin, and A.M. Vinogradov, *Geometry of jet spaces and nonlinear partial differential equations*, Adv. Studies Contemp. Math., (Gordon and Breach, New York 1986)
D.M. Gitman, and I.V. Tyutin, Nucl. Phys. **B630** (2002) 509
F.R. Gantmacher, *The theory of matrices*, Vol. 1 (Chelsea Publ. Co., New York 1959); *Matrizentheorie*, (Dt. Verlag Wiss., Berlin 1986)
D.M. Gitman, S.L. Lyakhovich, and I.V. Tyutin, Izw. VUZov Fizika **8** (1983) 61 (Sov. Phys. Journ. **8** (1983) 730)
J. Dieudonné, *Foundations of Modern Analysis,* (Academic Press, NY, London 1968); G.M. Fichtenholz, *Differential- und Integralrechnung*, Vol. I (Dt. Verlag Wiss., Berlin 1989)
L. Faddeev and R. Jackiw, Phys. Rev. Lett. **60** (1988) 1692
[^1]: Naturwissenschaftlich-Theoretisches Zentrum und Institut für Theoretische Physik, Universität Leipzig, Germany; e-mail: [email protected]
[^2]: Institute of Physics, University of Sao Paulo, Brazil; e-mail: [email protected]
[^3]: Lebedev Physics Institute, Moscow, Russia; e-mail: [email protected]
[^4]: The functions $F$ may depend on time explicitly, however, we do not include $t$ in the arguments of the functions.
[^5]: A retangular matrix with elements $\partial A_{\alpha }/\partial x^{i}$ is often denoted as $\partial A/\partial x\,$ and called the Jacobi matrix.
[^6]: We do not indicate here possible external coordinates.
[^7]: Here, and in what follows, we use Lemma 1 to justify the equivalence.
[^8]: By $\hat{O}(F)$ we denote LO of the form (\[2.10\]) with all the LF $u_{Aa}^{k}=O(F)\,,$ where$$\;\;\;\left. O(F)\right| _{F=0}=0\,.$$
[^9]: Here, we do not distinguish possible different proper orders of the coordinates.
[^10]: At $a\neq 0$ we have a finite-dimensional analog of the Proca action, and at $a=0$ we have the analog of the Maxwell action.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi *et al.* (2014, *SIAM J. Matrix Anal. Appl.*, 35, 699-724). This work addresses the setting when the matrix-valued function involved is very large. We describe subspace procedures that convert the original problem into a small-scale one by means of orthogonal projections and restrictions to certain subspaces, and that gradually expand these subspaces based on the optimal solutions of small-scale problems. Global convergence and superlinear rate-of-convergence results with respect to the dimensions of the subspaces are presented in the infinite dimensional setting, where the matrix-valued function is replaced by a compact operator depending on parameters. In practice, it suffices to solve eigenvalue optimization problems involving matrices with sizes on the scale of tens, instead of the original problem involving matrices with sizes on the scale of thousands.\
**Key words.** Eigenvalue optimization, large-scale, orthogonal projection, eigenvalue perturbation theory, parameter dependent compact operator, matrix-valued function\
**AMS subject classifications.** 65F15, 90C26, 47B37, 47B07
author:
- Fatih Kangal
- Karl Meerbergen
- Emre Mengi
- Wim Michiels
bibliography:
- 'largescale\_eigopt.bib'
title: ' A Subspace Method for Large-Scale Eigenvalue Optimization '
---
Introduction
============
We are concerned with the global optimization problems $$\textbf{(MN)} \;\;
{\rm minimize} \:
\{ \lambda_J(\omega) \; | \; {\omega \in \Omega} \}
\;\;\;\;\;\;\; {\rm and} \;\;\;\;\;\;\;
\textbf{(MX)} \;\;
{\rm maximize} \:
\{ \lambda_J(\omega) \; | \; {\omega \in \Omega} \}.$$ The feasible region $\Omega$ of these optimization problems is a compact subset of ${\mathbb R}^d$. Furthermore, letting $\ell^2({\mathbb N})$ denote the sequence space consisting of square summable infinite sequences of complex numbers equipped with the inner product $\langle w, v \rangle = \sum_{k = 0}^\infty \overline{w_k} \: v_k$ as well as the norm $\| v \|_2 = \sqrt{ \sum_{k = 0}^\infty | v_k |^2 }$, the objective function $\lambda_J(\omega)$ is the $J$th largest eigenvalue of a compact self-adjoint operator $$\label{eq:mat_func}
{\mathbf A}(\omega) : \ell^2({\mathbb N}) \rightarrow \ell^2({\mathbb N}), \quad
{\mathbf A}(\omega) := \sum_{\ell = 1}^\kappa f_\ell(\omega) {\mathbf A}_\ell$$ for every $\omega \in \overline{\Omega}$, an open subset of ${\mathbb R}^d$ containing the feasible region $\Omega$. Above ${\mathbf A}_\ell : \ell^2({\mathbb N}) \rightarrow \ell^2({\mathbb N})$ and $f_\ell : \overline{\Omega} \rightarrow {\mathbb R}$ for $\ell = 1,\dots,\kappa$ represent given compact self-adjoint operators and real-analytic functions, respectively. Throughout the text, ${\mathbf A}(\omega)$ for each $\omega$ and ${\mathbf A}_1, \dots, {\mathbf A}_\kappa$ as in (\[eq:mat\_func\]) could intuitively be considered as infinite dimensional Hermitian matrices.
Our interest in the infinite dimensional eigenvalue optimization problems **(MN)** and **(MX)** rise from their finite dimensional counterparts, which for given Hermitian matrices $A_\ell \in {\mathbb C}^{n\times n}$ for $\ell = 1, \dots, \kappa$ involve the matrix-valued function $$\label{eq:mat_func2}
A(\omega) := \sum_{\ell = 1}^\kappa f_\ell(\omega) A_\ell$$ instead of the parameter dependent operator ${\mathbf A}(\omega)$. These problems come with standard challenges due to their nonsmoothness and nonconvexity. But we would like to tackle a different challenge, when the matrices in them are very large, that is $n$ is very large. Thus, the primary purpose of this paper is to deal with large-dimensionality, it does not address the inherent difficulties due to nonconvexity and nonsmoothness. We introduce the ideas in the idealized infinite dimensional setting, only because this makes a rigorous convergence analysis possible.
To deal with large dimensionality, we propose restricting the domain and projecting the range of the map $v \mapsto {\mathbf A}(\omega) v$ to small subspaces. This gives rise to eigenvalue optimization problems involving small matrices, which we call reduced problems. Two greedy procedures are presented here to construct small subspaces so that the optimal solution of the reduced problem is close to the optimal solution of the original problem. For both procedures, we observe a superlinear rate of decay in the error with respect to the subspace dimension. The first procedure is more straightforward and constructs smaller subspaces, shown to converge at a superlinear rate when $d=1$, but lacks a complete formal argument justifying its quick convergence when $d \geq 2$. The second constructs larger subspaces, but comes with a formal proof of superlinear convergence for all $d$.
While the proposed procedures operate on **(MN)** and **(MX)** similarly, there are remarkable differences in their convergence behaviors in these two contexts. The proposed subspace restrictions and projections on the map $v \mapsto {\mathbf A}(\omega) v$ lead to global lower envelopes for $\lambda_J(\omega)$. These lower envelopes in turn make the convergence to the globally smallest value of $\lambda_J(\omega)$ possible as the dimensions of the subspaces by the procedures grow to infinity, which we prove formally. Such a global convergence behavior does not hold for the maximization problem: if the subspace dimension is let grow to infinity, the globally maximal values of the reduced problems converge to a locally maximal value of $\lambda_J(\omega)$, that is not necessarily globally maximal. But the maximization problem possesses a remarkable low-rank property: there exists a $J$ dimensional subspace such that when the map $v \mapsto {\mathbf A}(\omega) v$ is restricted and projected to this subspace, the resulting reduced problem has the same globally largest value as $\lambda_J(\omega)$. The minimization problem does not enjoy an analogous low-rank property.
Motivation
----------
Large eigenvalue optimization problems arise from various applications. For instance, the distance to instability from a large stable matrix with respect to the matrix 2-norm yields large singular value optimization problems [@VanLoan1985], that can be converted into large eigenvalue optimization problems. The computation of the H-infinity norm of the transfer function of a linear time-invariant (LTI) control system can be considered as a generalization of the computation of the distance to instability. This is a norm for the operator that maps inputs of the LTI system into outputs, and plays a major role in robust control. The singular value optimization characterization for the H-infinity norm involves large matrices if the input, output or, in particular, the intermediate state space have large dimension. The state-of-the-art algorithms for H-infinity norm computation [@Boyd1990; @Bruinsma1990] cannot cope with such large-scale control systems. In engineering applications, the largest eigenvalue of a matrix-valued function is often sought to be minimized. A particular application is the numerical scheme for the design of the strongest column subject to volume constraints [@Cox1992], where the sizes of the matrices depend on the fineness of a discretization imposed on a differential operator. These matrices can be very large if a fine grid is employed. The standard semidefinite program (SDP) formulations received a lot of attention by the convex optimization community since the 1990s [@Vandenberghe1996]. They concern the optimization over the cone of symmetric positive semidefinite matrices of a linear objective function subject to linear constraints. The dual problem of an SDP under mild assumptions can be cast as an eigenvalue optimization problem [@Helmberg2000]. If the size of the matrix variable of an SDP is large, the associated eigenvalue optimization problem involves large matrices. The current SDP solvers are usually not suitable to deal with such large-scale problems.
Literature
----------
Subspace projections and restrictions have been applied to particular eigenvalue optimization problems in the past. But general procedures such as the ones in this paper have not been proposed and studied thoroughly. A subspace restriction idea has been employed specifically for the computation of the pseudospectral abscissa of a matrix in [@Kressner2014]. This computational problem involves the optimization of a linear objective function subject to a constraint on the smallest singular value of an affine matrix-valued function. Fast convergence is observed in that work, and confirmed with a superlinear rate-of-convergence result. In the context of standard semidefinite programs, subspace methods have been used for quite a while, in particular the spectral bundle method [@Helmberg2000] is based on subspace ideas. The small-scale optimization problems resulting from subspace restrictions and projections are solved by standard SDP solvers [@Vandenberghe1996]. A thorough convergence analysis for them has not been performed, also their efficiency is not fully realized in practice. Extensions for convex quadratic SDPs [@Lin2012] and linear matrix inequalities [@Miller2000] have been considered. All these large-scale problems connected to SDPs are convex. We present unified procedures and their convergence analyses that are applicable regardless of whether the problem is convex or nonconvex.
Spectral Properties and Operator Norm of ${\mathbf A}(\omega)$
--------------------------------------------------------------
The spectrum of ${\mathbf A}(\omega)$ is defined by $$\sigma \left( {\mathbf A}(\omega) \right) \; := \; \{ z \in {\mathbb R} \; | \; zI - {\mathbf A}(\omega) \; \text{ is not invertible} \}.$$ This set contains countably many real eigenvalues each with a finite multiplicity, also the only accumulation point of these eigenvalues is 0 (see [@Kato1995 page 185, Theorem 6.26]). We assume, throughout the text, that ${\mathbf A}(\omega)$ has at least $J$ positive eigenvalues for all $\omega \in \Omega$. This ensures that $\lambda_J(\omega)$ is well-defined over $\Omega$.
The eigenspace associated with each eigenvalue of ${\mathbf A}(\omega)$ is also finite dimensional [@Kubrusly2001 Corollary 6.44]. Furthermore, the set of eigenvectors of ${\mathbf A}(\omega)$ can be chosen in a way so that they are orthonormal and complete in $\ell^2({\mathbb N})$ [@Davies2007 Theorem 4.2.23], i.e., there exist an orthonormal sequence $\{ v^{(j)} \}$ in $\ell^2({\mathbb N})$ and a sequence $\{ \mu^{(j)} \}$ of real numbers such that ${\mathbf A}(\omega) v^{(j)} = \mu^{(j)} v^{(j)}$ for all $j = 1,2,\dots$, $\mu^{(j)} \rightarrow 0$ as $j \rightarrow \infty$ and any vector $v \in \ell^2({\mathbb N})$ can be expressed as $v = \sum_{j = 1}^\infty \alpha_j v^{(j)}$ for some scalars $\alpha_j$.
Since ${\mathbf A}_1, \dots, {\mathbf A}_\kappa$ and ${\mathbf A}(\omega)$ are compact, they are bounded. Hence, their operator norms $$\begin{aligned}
\| {\mathbf A}_\ell \|_2 \; := \; {\rm sup} \left\{ \| {\mathbf A}_\ell v \|_2 \; | \; v \in \ell^2({\mathbb N}) \text{ such that } \| v \|_2 = 1 \right\}
\text{ for } \ell = 1, \dots, \kappa & \;\;\; \text{ and } \\
\| {\mathbf A}(\omega) \|_2 \; := \; {\rm sup} \left\{ \| {\mathbf A}(\omega) v \|_2 \; | \; v \in \ell^2({\mathbb N}) \text{ such that } \| v \|_2 = 1 \right\}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; &\end{aligned}$$ are also well-defined.
Outline
-------
We formally define the reduced problems, then analyze the relation between the original and the reduced problems in the next section. Some of this analysis, in particular the low-rank property discussed above, apply only to the maximization problem. Last two subsections of the next section are devoted to the introduction of the two subspace procedures for eigenvalue optimization. Interpolation properties between the original problem and the reduced problems formed by these two subspace procedures are also investigated there. Section \[sec:convergence\] concerns the convergence analysis of the two subspace procedures. In particular, Section \[sec:global\_convergence\] establishes the convergence of the subspace procedures globally to the smallest value of $\lambda_J(\omega)$ for the minimization problem as the subspace dimensions grow to infinity. In practice, we observe at least a superlinear rate-of-convergence with respect to the dimension of the subspaces for both of the procedures. We prove this formally in Section \[sec:rate\_of\_convergence\] fully for one of the procedures and partly for the other. Section \[sec:var\_ext\] focuses on variations and extensions of the two subspace procedures for eigenvalue optimization. Specifically, in Section \[sec:subspace\_nopast\] we argue that a variant that disregards the subspaces formed in the past iterations works effectively for the maximization problem but not for the minimization problem, in Section \[sec:singular\_vals\] and \[sec:small\_singular\_vals\] we extend the procedures to optimize a specified singular value of a compact operator depending on several parameters analytically, and in Section \[sec:cutting\_plane\] we provide a comparison of one of the subspace procedures for eigenvalue optimization with the cutting plane method [@Kelley1960], which also constructs global lower envelopes repeatedly. Section \[sec:applications\] describes the MATLAB software accompanying this work, and the efficiency of the proposed subspace procedures on particular applications, namely the numerical radius, the distance to instability from a matrix, and the minimization of the largest eigenvalue of an affine matrix-valued function. The text concludes with a summary as well as research directions for future in Section \[sec:conclusion\].
The Subspace Procedures {#sec:eig_subspace}
=======================
Let ${\mathcal V}$ be a finite dimensional subspace of $\ell^2({\mathbb N})$, and $V := \{v_1, \dots, v_m \}$ be an orthonormal basis for ${\mathcal V}$. The linear operator $$\label{eq:change_coor}
{\mathbf V} : {\mathbb C}^m \rightarrow \ell^2({\mathbb N}), \quad\quad
{\mathbf V} \alpha := \sum_{j=1}^m \alpha_j v_j$$ maps the coordinates of a vector $v \in {\mathcal V}$ relative to $V$ to itself. On the other hand, its adjoint ${\mathbf V}^\ast : \ell^2({\mathbb N}) \rightarrow {\mathbb C}^m$ projects a vector in $\ell^2({\mathbb N})$ orthogonally onto ${\mathcal V}$ and represents the orthogonal projection in coordinates relative to $V$. The procedures that will be introduced in this section are based on operators of the form $$\label{eq:reduced_mat_fun}
{\mathbf A}^{{\mathcal V}}(\omega) := {\mathbf V}^\ast {\mathbf A}(\omega) {\mathbf V},$$ which is the restriction of ${\mathbf A}(\omega)$ that acts only on ${\mathcal V}$, with its input and output represented in coordinates relative to $V$. This operator can be expressed of the form $${\mathbf A}^{{\mathcal V}}(\omega) = \sum_{\ell = 1}^\kappa f_\ell(\omega) {\mathbf A}^{{\mathcal V}}_\ell,
\;\;\;\;\; {\rm where} \;\;\;
{\mathbf A}^{{\mathcal V}}_\ell := {\mathbf V}^\ast {\mathbf A}_\ell {\mathbf V},$$ which is beneficial from computational point of view, because it is possible to form the $m\times m$ matrix representations of the operators ${\mathbf A}^{{\mathcal V}}_\ell$ in advance. The reduced eigenvalue optimization problems are defined in terms of the $J$th largest eigenvalue $\lambda_J^{\mathcal V}(\omega)$ of ${\mathbf A}^{\mathcal V}(\omega)$ as $$\label{eq:reduced_problems}
{\rm minimize} \:
\{ \lambda_J^{\mathcal V}(\omega) \; | \; {\omega \in \Omega} \}
\;\;\;\;\;\;\; {\rm and} \;\;\;\;\;\;\;
{\rm maximize} \:
\{ \lambda_J^{\mathcal V}(\omega) \; | \; {\omega \in \Omega} \}.$$ The discussions above generalize when ${\mathcal V}$ is infinite dimensional. In the infinite dimensional setting, the operators ${\mathbf A}^{\mathcal V}(\omega)$ and ${\mathbf A}^{{\mathcal V}}_\ell$ are defined similarly in terms of an infinite countable orthonormal basis $V = \{ v_j \; | \; j \in {\mathbb Z}^+ \} $ for ${\mathcal V}$ and the associated operator $$\label{eq:change_coor2}
{\mathbf V} \alpha := \sum_{j=1}^\infty \alpha_j v_j.$$
The eigenvalue function $\lambda^{\mathcal V}_J(\omega)$ is Lipschitz continuous, indeed there exists a uniform Lipschitz constant $\gamma$ for all subspaces ${\mathcal V}$ of $\ell^2({\mathbb N})$ as formally stated and proven in the following lemma.
\[lemma:Lipschitz\_continuity\] There exists a real scalar $\gamma > 0$ such that for all subspaces ${\mathcal V}$ of $\ell^2({\mathbb N})$, we have $$\left| \lambda^{\mathcal V}_J(\widetilde{\omega}) - \lambda^{\mathcal V}_J(\omega) \right|
\; \leq \;
\gamma \cdot \| \widetilde{\omega} - \omega \|_2
\quad
\forall \widetilde{\omega}, \: \omega \in \overline{\Omega}.$$
By Weyl’s theorem (see [@Horn1985 Theorem 4.3.1] for the finite dimensional case; its extension to the infinite dimensional setting is straightforward by exploiting the maximin characterization (\[eq:maximin\]) of $\lambda^{\mathcal V}_J(\omega)$ given below) $$\begin{aligned}
\left| \lambda^{\mathcal V}_J(\widetilde{\omega}) - \lambda^{\mathcal V}_J(\omega) \right|
\; & \leq & \;
\| {\mathbf A}^{\mathcal V}( \widetilde{\omega} ) - {\mathbf A}^{\mathcal V}( \omega ) \|_2 \\
\; & \leq & \;
\sum_{\ell=1}^\kappa \left| f_\ell( \widetilde{\omega}) - f_\ell(\omega) \right| \| {\mathbf A}^{\mathcal V}_\ell \|_2
\quad
\forall \widetilde{\omega}, \: \omega \in \overline{\Omega}.\end{aligned}$$ In the last summation $\| {\mathbf A}^{{\mathcal V}}_\ell \|_2 \leq \| {\mathbf A}_\ell \|_2$ and the real analyticity of $f_\ell(\omega)$ implies its Lipschitz continuity, hence the existence of a constant $\gamma_\ell$ satisfying $$\left| f_\ell(\widetilde{\omega}) - f_\ell(\omega) \right|
\; \leq \;
\gamma_\ell \|\widetilde{\omega} - \omega \|_2
\quad
\forall \widetilde{\omega}, \: \omega \in \overline{\Omega}.$$ Combining these observations we obtain $$\left| \lambda^{\mathcal V}_J(\widetilde{\omega}) - \lambda^{\mathcal V}_J(\omega) \right|
\; \leq \;
\left( \sum_{\ell = 1}^\kappa \gamma_\ell \| {\mathbf A}_\ell \|_2 \right) \cdot \| \widetilde{\omega} - \omega \|_2
\quad
\forall \widetilde{\omega}, \: \omega \in \overline{\Omega}$$ as desired.
Throughout the rest of this section, we first investigate the relation between $\lambda_J(\omega)$ and $\lambda_J^{\mathcal V}(\omega)$ as well as their globally maximal values. Some of these theoretical results will be frequently used in the subsequent sections. The second and third parts of this section introduce two subspace procedures for the generation of small dimensional subspaces ${\mathcal V}$, leading to reduced eigenvalue optimization problems that approximate the original eigenvalue optimization problems accurately.
Relations between $\lambda_J^{\mathcal V}(\omega)$ and $\lambda_J(\omega)$
--------------------------------------------------------------------------
We start with a result about the monotonicity of $\lambda_J^{\mathcal V}(\omega)$ with respect to ${\mathcal V}$. This result is an immediate consequence of the following maximin characterization [@Dunford1998 pages 1543-1544] (see also [@Horn1985 Theorem 4.2.11] for the finite dimensional case) of $\lambda^{\mathcal V}_J (\omega)$ for all subspaces ${\mathcal V}$ of $\ell^2({\mathbb N})$: $$\label{eq:maximin}
\lambda^{\mathcal V}_J (\omega)
\;\; = \;\;
\sup_{{\mathcal S} \subseteq {\mathcal V}, \; \dim {\mathcal S} = J} \;\;
\min_{v \in {\mathcal S}, \; \| v \|_2 = 1} \; \langle {\mathbf A}(\omega) v, v \rangle,$$ where $\langle \cdot, \cdot \rangle$ stands for the standard inner product $\langle w, v \rangle := \sum_{k=0}^\infty \overline{w_k} \: v_k$ and the outer supremum is over all $J$ dimensional subspaces of ${\mathcal V}$. Furthermore, if ${\mathbf A}^{\mathcal V}(\omega)$ has at least $J$ positive eigenvalues, that is if $\lambda_J^{\mathcal V}(\omega) > 0$, or if ${\mathcal V}$ is finite dimensional, then the outer supremum in (\[eq:maximin\]) is attained, hence can be replaced by maximum. The monotonicity result presented next will play a central role later when we analyze the convergence of the subspace procedures.
\[lemma:monotonicity\] Let ${\mathcal V}_1$, ${\mathcal V}_2$ be subspaces of $\ell^2 ({\mathbb N})$ of dimension larger than or equal to $J$ such that ${\mathcal V}_1 \;\; \subseteq \;\; {\mathcal V}_2$. The following holds: $$\label{eq:monotone}
\lambda^{{\mathcal V}_1}_J (\omega) \;\; \leq \;\; \lambda^{{\mathcal V}_2}_J (\omega) \;\; \leq \;\; \lambda_J(\omega).$$
-2ex A consequence of this monotonicity result is the following interpolatory property.
\[lemma:interpolatary\] Let ${\mathcal V}$ be a subspace of $\ell^2 ({\mathbb N})$ of dimension larger than or equal to $J$, and ${\mathbf V}$ be the operator defined as in (\[eq:change\_coor\]) or (\[eq:change\_coor2\]) in terms of an orthonormal basis $V$ for ${\mathcal V}$. If $S := {\rm span}\{ s_1, \dots, s_J \} \; \subseteq \; {\mathcal V}$ where $s_1, \dots, s_J$ are eigenvectors corresponding to the $J$ largest eigenvalues of ${\mathbf A}(\omega)$, then the following hold:
1. $\lambda_J(\omega) = \lambda^{\mathcal V}_J(\omega)$;
2. ${\mathbf V}^\ast s_J$ is an eigenvector of ${\mathbf V}^\ast {\mathbf A}(\omega) {\mathbf V}$ corresponding to its eigenvalue $\lambda^{\mathcal V}_J(\omega)$.
**(i)** We assume each $s_\ell \in {\mathcal V}$ for $\ell = 1, \dots, J$ is of unit length without loss of generality. It follows that there exist $\alpha_1, \dots, \alpha_J$ of unit length such that $s_\ell = {\mathbf V} \alpha_\ell$ for $\ell = 1,\dots,J$. Now define ${\mathcal S}_\alpha := {\rm span} \{ \alpha_1, \dots, \alpha_J \} \subseteq {\mathbb C}^m$, and observe $$\begin{aligned}
\lambda_J \left( \omega \right) \; = \;
\langle {\mathbf A} \left( \omega \right) s_J, s_J \rangle & = &
\langle {\mathbf A} \left( \omega \right) {\mathbf V} \alpha_J , {\mathbf V}\alpha_J \rangle \\
& = & \langle {\mathbf V}^\ast {\mathbf A} \left( \omega \right) {\mathbf V} \alpha_J , \alpha_J \rangle \\
& = & \min_{\alpha \in {\mathcal S}_\alpha, \| \alpha \|_2 = 1} \; \langle {\mathbf V}^\ast {\mathbf A} \left( \omega \right) {\mathbf V} \alpha, \alpha \rangle
\;\; \leq \;\; \lambda^{\mathcal V}_J \left( \omega \right). \end{aligned}$$ The opposite inequality is immediate from Lemma \[lemma:monotonicity\], so $
\lambda_J \left( \omega \right)
=
\lambda^{\mathcal V}_J \left( \omega \right)
$ as claimed.\
**(ii)** The equalities $$\langle {\mathbf V}^\ast {\mathbf A} \left( \omega \right) {\mathbf V} \alpha_J, \alpha_J \rangle
\;\; = \;\; \min_{\alpha \in \mathcal S_{\alpha}, \| \alpha \|_2 = 1} \; \langle {\mathbf V}^\ast {\mathbf A} \left( \omega \right) {\mathbf V} \alpha, \alpha \rangle
\;\; = \;\; \lambda^{\mathcal V}_J \left( \omega \right).$$ imply that $\alpha_J = {\mathbf V}^\ast s_J$ is an eigenvector of ${\mathbf V}^\ast {\mathbf A} \left( \omega \right) {\mathbf V}$ corresponding to the eigenvalue $\lambda^{\mathcal V}_J \left( \omega \right)$.
Maximization of the $J$th largest eigenvalue over a low-dimensional subspace is motivated by the next result. According to the result, it suffices to perform the optimization on a proper $J$ dimensional subspace, which is hard to determine in advance. Here and elsewhere, $\arg \max_{\omega \in \Omega}\; \lambda_J \left( \omega \right)$ and $\arg \min \in_{\omega \in \Omega}\; \lambda_J \left( \omega \right)$ denote the set of global maximizers and global minimizers of $\lambda_J(\omega)$ over $\omega \in \Omega$, respectively.
\[thm:low\_rank\] For a given subspace ${\mathcal V} \subset \ell^2({\mathbb N})$ with dimension larger than or equal to $J$, consider the following assertions:
1. $ \max_{\omega \in \Omega} \; \lambda^{\mathcal V}_J \left( \omega \right) \;\; = \;\;
\max_{\omega \in \Omega} \; \lambda_J \left( \omega \right) $;
2. ${\mathcal S}_\ast := {\rm span} \left\{ s_1, \dots, s_J \right\} \; \subseteq \; {\mathcal V}$, where $s_1, \dots, s_J$ are eigenvectors corresponding to the $J$ largest eigenvalues of ${\mathbf A}(\omega_\ast)$ at some $\omega_\ast \in \arg \max_{\omega \in \Omega}\; \lambda_J \left( \omega \right) $.
Assertion **(ii)** implies assertion **(i)**. Furthermore, when $J = 1$, assertions **(i)** and **(ii)** are equivalent.
Suppose $s_1, \dots, s_J \in {\mathcal V}$ satisfies assertion (ii). By Lemma \[lemma:interpolatary\], we have $$\begin{aligned}
\max_{\omega \in \Omega} \; \lambda_J \left( \omega \right) \;\; = \;\;
\lambda_J \left( \omega_\ast \right) \;\; = \;\; \lambda^{\mathcal V}_J \left( \omega_\ast \right)
\;\;\; \leq \;\;\; \max_{\omega \in \Omega} \; \lambda^{\mathcal V}_J \left( \omega \right).\end{aligned}$$ Lemma \[lemma:monotonicity\] implies the opposite inequality, proving assertion (i).
To prove that the assertions are equivalent when $J = 1$, assume that assertion (i) holds. Letting $\omega_\ast$ be any point in $
\arg \max_{\omega \in \Omega} \; \lambda^{\mathcal V}_1 \left( \omega \right),
$ denoting by ${\mathbf V}$ an operator defined as in (\[eq:change\_coor\]) or (\[eq:change\_coor2\]) in terms of an orthonormal basis $V$ for ${\mathcal V}$, and denoting by $\alpha$ a unit eigenvector corresponding to the largest eigenvalue of ${\mathbf V}^\ast {\mathbf A}(\omega_\ast) {\mathbf V}$ (note that $\lambda_1^{\mathcal V}(\omega_\ast) = \lambda_1(\omega_\ast) > 0$, so the unit eigenvector $\alpha$ is well-defined), we have $$\label{eq:eig_inequality}
\max_{\omega \in {\mathbb R}^d} \; \lambda_1 \left( \omega \right)
\; = \;
\lambda^{\mathcal V}_1 \left( \omega_\ast \right)
\; = \;
\langle {\mathbf V}^\ast {\mathbf A}(\omega_\ast) {\mathbf V} \alpha, \alpha \rangle
\; = \;
\langle {\mathbf A}(\omega_\ast) {\mathbf V} \alpha, {\mathbf V} \alpha \rangle
\; \leq \;
\lambda_1 \left( \omega_\ast \right).$$ Thus we deduce $$\lambda_1 \left( \omega_\ast \right)
\;\; = \;\;
\max_{\omega \in \Omega} \; \lambda_1 \left( \omega \right)
\;\; = \;\;
\langle {\mathbf A}(\omega_\ast) {\mathbf V} \alpha, {\mathbf V} \alpha \rangle.$$ Consequently, ${\mathbf V}\alpha$ is a unit eigenvector of ${\mathbf A}(\omega_\ast)$ corresponding to its largest eigenvalue, where $
\omega_\ast
$ belongs to $
\arg \max_{\omega \in \Omega} \; \lambda_1 \left( \omega \right).
$ Furthermore, ${\rm span} \left\{ {\mathbf V} \alpha \right\} \; \subseteq \; {\mathcal V}$. This proves assertion (ii).
When ${\mathcal V}$ does not contain the optimal subspace ${\mathcal S}_\ast$ (as in part (ii) of Lemma \[thm:low\_rank\]), the next result quantifies the gap between the eigenvalues of the original and the reduced operators in terms of the distance from ${\mathcal S}_\ast$ to the $J$ dimensional subspaces of ${\mathcal V}$. For this result, we define the distance between two finite dimensional subspaces $\widetilde{\mathcal S}, {\mathcal S}$ of $\ell^2({\mathbb N})$ of same dimension by $$d\left( \widetilde{\mathcal S}, {\mathcal S} \right)
\;\; := \;\;
\max_{\widetilde{v} \in \widetilde{\mathcal S}, \; \| \widetilde{v} \|_2 = 1} \;\; \min_{v \in {\mathcal S}} \;\;
\| \widetilde{v} - v \|_2.$$ This distance corresponds to the sine of the largest angle between the subspaces $\widetilde{\mathcal S}$ and ${\mathcal S}$. Results of similar nature can be found in the literature, see for instance [@Parlett1998 Theorem 11.7.1] and [@Saad2011 Proposition 4.5] where the bounds are in terms of distances between one dimensional subspaces.
\[thm:accuracy\_rproblems\] Let ${\mathcal V}$ be a subspace of $\ell^2({\mathbb N})$ with dimension $J$ or larger.
1. For each $\omega$ in $\overline{\Omega}$, we have $$\label{eq:rate_of_conv}
\lambda_J \left( \omega \right) \; - \; \lambda^{\mathcal V}_J \left( \omega \right) \;\; = \;\; O \left( \varepsilon^2 \right),$$ where $$\varepsilon \; := \;
\min
\left\{
d\left( \widetilde{\mathcal S}, {\mathcal S} \right)
\;\; | \;\;
\widetilde{\mathcal S} \text{ is a J dimensional subspace of } {\mathcal V}
\right\}$$ and ${\mathcal S}$ is the subspace spanned by the eigenvectors corresponding to the $J$ largest eigenvalues of ${\mathbf A}(\omega)$.
2. The equality $$\label{eq:rate_of_conv2}
\max_{\omega \in \Omega} \; \lambda_J \left( \omega \right) \; - \;
\max_{\omega \in \Omega} \; \lambda^{\mathcal V}_J \left( \omega \right) \;\; = \;\; O \left( \varepsilon_\ast^2 \right)$$ holds. Here, $\varepsilon_\ast$ is given by $$\varepsilon_\ast \; := \;
\min
\left\{
d\left( \widetilde{\mathcal S}, {\mathcal S}_\ast \right)
\;\; | \;\;
\widetilde{\mathcal S} \text{ is a J dimensional subspace of } {\mathcal V}
\right\}$$ for some subspace ${\mathcal S}_\ast$ spanned by the eigenvectors corresponding to the $J$ largest eigenvalues of ${\mathbf A}(\omega_\ast)$ at some $\omega_\ast \in \arg \max_{\omega \in \Omega} \; \lambda_J \left( \omega \right) $.
**(i)** Let $\widehat{\mathcal S}$ be a $J$ dimensional subspace of ${\mathcal V}$ such that $\varepsilon = d\left( \widehat{\mathcal S}, {\mathcal S} \right)$. Furthermore, for a given unit vector $\widehat{v} \in \widehat{\mathcal S}$, let us use the notations $$v\left( \widehat{v} \right) \; := \;
\arg \min \;
\left\{ \| \widehat{v} - v \|_2 \;\; | \;\; v \in {\mathcal S} \right\}
\;\;\;\; {\rm and} \;\;\;\;
\delta\left( \widehat{v} \right) \; := \;
\widehat{v} - v\left( \widehat{v} \right),$$ where $\| \delta\left( \widehat{v} \right) \|_2 = O(\varepsilon)$. Observe that the inner minimization problem in the definition of $d\left( \widehat{\mathcal S}, {\mathcal S} \right)$ is a least-squares problem, so the minimizer $v\left( \widehat{v} \right)$ of $\| \widehat{v} - v \|_2$ over $v \in {\mathcal S}$ defined above is unique and $\delta\left( \widehat{v} \right) \bot \; {\mathcal S}$. Additionally, $\| v(\widehat{v}) \|_2^2 = 1 - O\left( \varepsilon^2 \right)$ due to the properties $\| \widehat{v} \|_2 = \| v(\widehat{v}) + \delta(\widehat{v}) \|_2 = 1$ and $v(\widehat{v}) \; \bot \; \delta(\widehat{v})$.
Now the maximin characterization (\[eq:maximin\]) for $\lambda_J^{\mathcal V}(\omega)$ implies $$\begin{aligned}
\lambda^{\mathcal V}_J(\omega) \;\; \geq \;\; \min_{\widehat{v} \in \widehat{\mathcal S}, \; \| \widehat{v} \|_2 = 1} \;\; \langle {\mathbf A}(\omega) \widehat{v}, \widehat{v} \rangle
&=&
\min_{\widehat{v} \in \widehat{\mathcal S}, \; \| \widehat{v} \|_2 = 1} \;\;
\langle {\mathbf A}(\omega) \left[ v\left( \widehat{v} \right) + \delta\left( \widehat{v} \right) \right], \left[ v\left( \widehat{v} \right) + \delta\left( \widehat{v} \right) \right] \rangle \\
&\geq&
\min_{\widehat{v} \in \widehat{\mathcal S}, \; \| \widehat{v} \|_2 = 1} \;\; \langle {\mathbf A}(\omega) v(\widehat{v}), v(\widehat{v}) \rangle \\
& & \hskip 4ex
+ \min_{\widehat{v} \in \widehat{\mathcal S}, \; \| \widehat{v} \|_2 = 1} \;\; 2\Re \left( \langle {\mathbf A}(\omega) v\left( \widehat{v} \right), \delta\left( \widehat{v} \right) \rangle \right) \\
& & \hskip 4ex
+ \min_{\widehat{v} \in \widehat{\mathcal S}, \; \| \widehat{v} \|_2 = 1} \;\; \langle {\mathbf A}(\omega) \delta\left( \widehat{v} \right), \delta\left( \widehat{v} \right) \rangle \\
& \geq & \lambda_J \left( \omega \right) \min_{\widehat{v} \in \widehat{\mathcal S}, \; \| \widehat{v} \|_2 = 1} \| v(\widehat{v}) \|_2^2
\; - \; O(\varepsilon^2) \\
&=& \lambda_J \left( \omega \right) \; - \; O(\varepsilon^2).\end{aligned}$$ Above, on the third line, we note that $\langle {\mathbf A}(\omega) v\left( \widehat{v} \right), \delta\left( \widehat{v} \right) \rangle = 0$ due to the fact ${\mathbf A}(\omega) v\left( \widehat{v} \right) \in {\mathcal S}$ and $\delta\left( \widehat{v} \right) \bot \; {\mathcal S}$, and on the second to the last line, we employ $\| v(\widehat{v}) \|_2^2 = 1 - O\left( \varepsilon^2 \right)$. The desired equality follows from $\lambda^{\mathcal V}_J \left( \omega \right) \leq \lambda_J \left( \omega \right)$ due to Lemma \[lemma:monotonicity\].
**(ii)** This is an immediate corollary of (i). In particular, $$O(\varepsilon^2_\ast) \;\; = \;\; \lambda_J\left( \omega_\ast \right) \; - \; \lambda^{\mathcal V}_J \left( \omega_\ast \right)
\;\; \geq \;\;
\max_{\omega \in \Omega} \; \lambda_J \left( \omega \right) \; - \; \max_{\omega \in \Omega} \; \lambda^{\mathcal V}_J (\omega)$$ combined with $
\max_{\omega \in \Omega} \; \lambda^{\mathcal V}_J (\omega) \; \leq \;
\max_{\omega \in \Omega} \; \lambda_J (\omega)
$ (due to Lemma \[lemma:monotonicity\]) yield (\[eq:rate\_of\_conv2\]).
The Greedy Procedure
--------------------
The basic greedy procedure solves the reduced eigenvalue optimization problem (\[eq:reduced\_problems\]) for a given subspace ${\mathcal V}$. Denoting a global optimizer of the reduced problem with $\omega_\ast$, the subspace is expanded with the addition of the eigenvectors corresponding to $\lambda_1(\omega_\ast), \dots, \lambda_J(\omega_\ast)$, then this is repeated with the expanded subspace. A formal description is given in Algorithm \[alg\], where ${\mathcal S}_k$ denotes the subspace at the $k$th step of the procedure, and $\lambda^{(k)}_J(\omega) := \lambda^{{\mathcal S}_k}_J(\omega)$. The reduced eigenvalue optimization problems on line 5 are nonsmooth and nonconvex. The description assumes that these problems can be solved globally. The algorithm in [@Mengi2014] works well in practice for this purpose when the number of parameters, $d$, is small. These reduced problems are computationally cheap to solve, the main computational burden comes from line 6 which requires the computation of eigenvectors of the full problem. In the finite dimensional case, these large eigenvalue problems are typically solved by means of an iterative method, for instance by Lanczos’ method.
$\omega^{(1)} \gets$ a random point in $\Omega$. $s^{(1)}_1, \dots, s^{(1)}_J \gets $ eigenvectors corresponding to $\lambda_1(\omega^{(1)}), \dots, \lambda_J(\omega^{(1)})$. ${\mathcal S}_1 \; \gets \;
{\rm span} \left\{ s^{(1)}_1, \dots, s^{(1)}_J \right\}.
$
$\omega^{(k)}$ -0.4ex $\gets$ -0.4ex any $\omega_\ast \in \arg\min_{\omega \in \Omega} \lambda^{(k-1)}_J(\omega) \; $ for the minimization problem **(MN)**, or\
$\omega^{(k)}$ -0.4ex $\gets$ -0.4ex any $\omega_\ast \in \arg\max_{\omega \in \Omega} \lambda^{(k-1)}_J(\omega) \;$ for the maximization problem **(MX)**. $s^{(k)}_1, \dots, s^{(k)}_J \gets $ eigenvectors corresponding to $\lambda_1(\omega^{(k)}), \dots, \lambda_J(\omega^{(k)})$. ${\mathcal S}_k \; \gets \; {\mathcal S}_{k-1} \oplus {\rm span} \left\{ s^{(k)}_1, \dots, s^{(k)}_J \right\}$.
As numerical experiments demonstrate (see Section 5), the power of this greedy subspace procedure is that high accuracy is often reached after a small number of steps, for reduced problems of small size. This can mostly be attributed to the following interpolatory properties between $\lambda^{(k)}_J(\omega)$ and $\lambda_J(\omega)$.
\[thm:first\_der\] The following hold regarding Algorithm \[alg\]:
1. $\lambda_J\left( \omega^{(\ell)} \right) \; = \; \lambda^{(k)}_J\left( \omega^{(\ell)} \right)$ for $\ell = 1,\dots,k$;
2. If $J > 1$, then $\lambda_{J-1}\left( \omega^{(\ell)} \right) \; = \; \lambda^{(k)}_{J-1}\left( \omega^{(\ell)} \right)$ for $\ell = 1,\dots,k$;
3. If $\lambda_J \left( \omega^{(\ell)} \right)$ is simple, then $\lambda^{(k)}_J \left( \omega^{(\ell)} \right)$ is also simple for $\ell = 1,\; 2,\; \dots, k$;
4. If $\lambda_J \left( \omega^{(\ell)} \right)$ is simple, then $\; \nabla \lambda_J \left( \omega^{(\ell)} \right) \; = \; \nabla \lambda^{(k)}_J \left( \omega^{(\ell)} \right) \;\;$ for $\ell = 1,\; 2,\; \dots,k$.
**(i-ii)** Lines 2, 3, 6 and 7 of Algorithm \[alg\] imply that $s^{(\ell)}_1, \dots, s^{(\ell)}_J \in {\mathcal S}_k$ for $\ell = 1, \dots, k$. An application of part (i) of Lemma \[lemma:interpolatary\] with ${\mathcal V} = {\mathcal S}_k$ yields $\lambda_j\left( \omega^{(\ell)} \right) = \lambda^{{\mathcal S}_k}_j \left( \omega^{(\ell)} \right) = \lambda^{(k)}_j \left( \omega^{(\ell)} \right)$ for $\ell = 1, \dots, k$ and $j = J$, as well as $j = J-1$ if $J > 1$, as desired.
**(iii)** Suppose $\lambda^{(k)}_J \left( \omega^{(\ell)} \right)$ is not simple for some $\ell \in \{ 1, 2, \dots, k \}$. In this case there must exist two mutually orthogonal unit eigenvectors $\hat{\alpha}, \tilde{\alpha} \in {\mathbb C}^m$ corresponding to it. Let us denote by ${\mathbf S}_k$ the operator as in (\[eq:change\_coor\]) in terms of a basis $S_k$ for ${\mathcal S}_k$, and ${\mathbf A}^{{\mathcal S}_k} \left( \omega^{(\ell)} \right) = {\mathbf S}_k^\ast {\mathbf A} \left( \omega^{(\ell)} \right) {\mathbf S}_k$. It follows from part (i) that $$\begin{aligned}
\lambda_J \left( \omega^{(\ell)} \right) \; = \;
\lambda^{(k)}_J \left( \omega^{(\ell)} \right) =
\left\langle {\mathbf A} \left( \omega^{(\ell)} \right) {\mathbf S}_k \hat{\alpha}, {\mathbf S}_k \hat{\alpha} \right\rangle =
\min_{\alpha \in \widehat{\mathcal S}, \| \alpha \|_2 = 1} \; \left\langle {\mathbf A} \left( \omega^{(\ell)} \right) {\mathbf S}_k \alpha, {\mathbf S}_k \alpha \right\rangle \\
= \left\langle {\mathbf A} \left( \omega^{(\ell)} \right) {\mathbf S}_k \tilde{\alpha}, {\mathbf S}_k \tilde{\alpha} \right\rangle =
\min_{\alpha \in \widetilde{\mathcal S}, \| \alpha \|_2 = 1} \; \left\langle {\mathbf A} \left( \omega^{(\ell)} \right) {\mathbf S}_k \alpha, {\mathbf S}_k \alpha \right\rangle\end{aligned}$$ for some $J$ dimensional subspaces $\widehat{\mathcal S}, \widetilde{\mathcal S}$ of $\ell^2({\mathbb N})$ such that $\hat{\alpha} \in \widehat{\mathcal S}$, $\tilde{\alpha} \in \widetilde{\mathcal S}$. This shows that ${\mathbf S}_k \hat{\alpha}, \: {\mathbf S}_k \tilde{\alpha} \in \ell^2({\mathbb N})$ are mutually orthogonal eigenvectors corresponding to $\lambda_J \left( \omega^{(\ell)} \right)$, so $\lambda_J\left( \omega^{(\ell)} \right)$ is not simple either.
**(iv)** It follows from part (iii) that $\lambda^{(k)}_J\left( \omega^{(\ell)} \right)$ is also simple, so both $\lambda_J\left( \omega \right)$ and $\lambda^{(k)}_J \left( \omega \right)$ are differentiable at $\omega^{(\ell)}$, furthermore the associated unit eigenvectors can be chosen in a way so that they are also differentiable at $\omega^{(\ell)}$ (see [@Rellich1969 pages 57-58, Theorem 1] for the differentiability of $\lambda_J(\omega)$ and the associated unit eigenvector, and [@Rellich1969 pages 33-34, Theorem 1] for the differentiability of $\lambda_J^{(k)}(\omega)$ and the associated unit eigenvector). By Lemma \[lemma:interpolatary\] part (ii), the eigenvector $\alpha_J$ of $\; {\mathbf A}^{{\mathcal S}_k} \left( \omega^{(\ell)} \right) = {\mathbf S}_k^\ast {\mathbf A} \left( \omega^{(\ell)} \right) {\mathbf S}_k \;$ corresponding to the eigenvalue $\lambda^{(k)}_J \left( \omega^{(\ell)} \right) = \lambda^{{\mathcal S}_k}_J \left( \omega^{(\ell)} \right)$ satisfies $\alpha_J = {\mathbf S}_k^\ast s^{(\ell)}_J$. Equivalently, we have $s^{(\ell)}_J = {\mathbf S}_k \alpha_J$ (since $s^{(\ell)}_J \in {\mathcal S}_k$). By employing the analytical formulas for the derivatives of eigenvalue functions (see [@Lancaster1964] for the finite dimensional matrix-valued case whose derivation exploits the Hermiticity of the matrix-valued function; the generalization to the infinite dimensional case, leading to the formula $\partial \lambda_J(\omega)/\partial \omega_q = \langle \partial {\mathbf A} (\omega) / \partial \omega_q s_J, s_J \rangle$ where $s_J$ is a unit eigenvector corresponding to $\lambda_J(\omega)$, is straightforward by making use of the self-adjointness of ${\mathbf A}(\omega)$), for $q = 1, \dots, d$ we obtain $$\begin{aligned}
\frac{\partial \lambda^{(k)}_J \left( \omega^{(\ell)} \right) } {\partial \omega_q}
\;\; & = & \;\;
\left\langle \frac{\partial {\mathbf A}^{{\mathcal S}_k} \left( \omega^{(\ell)} \right) }{\partial \omega_q} \alpha_J, \alpha_J \right\rangle \\
\;\; & = & \;\;
\left\langle \frac{\partial {\mathbf A} \left( \omega^{(\ell)} \right) }{\partial \omega_q} {\mathbf S}_k \alpha_J, {\mathbf S}_k \alpha_j \right\rangle \\
\;\; & = & \;\;
\left\langle \frac{\partial {\mathbf A} \left( \omega^{(\ell)} \right) }{\partial \omega_q} s^{(\ell)}_J, s^{(\ell)}_J \right\rangle
\;\; = \;\;
\frac{\partial \lambda_J \left( \omega^{(\ell)} \right) } {\partial \omega_q}.\end{aligned}$$ This completes the proof.
The Extended Greedy Procedure
-----------------------------
To better exploit the Hermite interpolation properties of Lemma \[thm:first\_der\], we extend the basic greedy procedure of the previous subsection with the inclusion of additional eigenvectors in the subspaces at points close to the optimizers of the reduced problems. The purpose here is to achieve $$\lim_{k\rightarrow \infty} \:
\left\| \nabla^2 \lambda_J ( \omega^{(k)} ) - \nabla^2 \lambda_J^{(k)} ( \omega^{(k)} ) \right\|_2 \; = \; 0$$ in addition to $\lambda_J(\omega^{(k)}) = \lambda_J^{(k)} (\omega^{(k)})$ and $\nabla \lambda_J(\omega^{(k)}) = \nabla \lambda_J^{(k)} (\omega^{(k)})$. These properties enable us to make an analogy with a quasi-Newton method for unconstrained smooth optimization, and come up with a theoretical superlinear rate-of-convergence result in the next section.
In the extended procedure also the reduced eigenvalue optimization problem (\[eq:reduced\_problems\]) is solved for a subspace ${\mathcal V}$ already constructed. Denoting the optimizer of the reduced problem with $\omega_\ast$, in addition to the eigenvectors corresponding to $\lambda_1(\omega_\ast), \dots, \lambda_{J}(\omega_\ast)$, the eigenvectors corresponding to $\lambda_1(\omega_\ast + h e_{pq}), \dots, \lambda_J(\omega_\ast + h e_{pq})$ are added into the subspace ${\mathcal V}$ for $h$ decaying to zero if convergence occurs, for $p = 1, \dots, d$ and $q = p, \dots, d$. Here $e_{p q} := (1/\sqrt{2}) ( e_p + e_q )$ for $p \neq q$ as well as $e_{pp} := e_p$. We provide a formal description of this extended subspace procedure in Algorithm \[alge\] below.
$\omega^{(1)} \gets$ a random point in $\Omega$. $s^{(1)}_1, \dots, s^{(1)}_{J} \gets $ eigenvectors corresponding to $\lambda_1(\omega^{(1)}), \dots, \lambda_{J}(\omega^{(1)})$. ${\mathcal S}_1 \; \gets \;
{\rm span} \left\{ s^{(1)}_1, \dots, s^{(1)}_{J} \right\}.
$
$\omega^{(k)}$ -0.4ex $\gets$ -0.4ex any $\omega_\ast \in \arg\min_{\omega \in \Omega} \lambda^{(k-1)}_J(\omega) \;$ for the minimization problem **(MN)**, or\
$\omega^{(k)}$ -0.4ex $\gets$ -0.4ex any $\omega_\ast \in \arg\max_{\omega \in \Omega} \lambda^{(k-1)}_J(\omega) \;$ for the maximization problem **(MX)**. $s^{(k)}_1, \dots, s^{(k)}_{J} \gets $ eigenvectors corresponding to $\lambda_1(\omega^{(k)}), \dots, \lambda_{J}(\omega^{(k)})$. $h^{(k)} \gets \| \omega^{(k)} - \omega^{(k-1)} \|_2$ $s^{(k)}_{1,p q}, \dots, s^{(k)}_{J, p q} \gets $ eigenvectors corresponding to\
18ex $\lambda_1(\omega^{(k)} + h^{(k)} e_{pq}), \dots, \lambda_J(\omega^{(k)} + h^{(k)} e_{pq})$. ${\mathcal S}_k \; \gets \; {\mathcal S}_{k-1} \oplus {\rm span} \left\{ s^{(k)}_1, \dots, s^{(k)}_{J} \right\}
\oplus \left\{ \bigoplus_{p = 1, q = p}^d {\rm span} \left\{ s^{(k)}_{1,p q}, \dots, s^{(k)}_{J, p q} \right\} \right\}$.
The reduced eigenvalue functions of the extended procedure possesses additional interpolatory properties stated in the lemmas below. Their proofs are similar to the proofs for Lemma \[thm:first\_der\], so we omit them.
\[thm:first\_der\_extended\] The assertions of Lemma \[thm:first\_der\] hold for Algorithm \[alge\]. Additionally, we have
1. $\lambda_J\left( \omega^{(\ell)} + h^{(\ell)} e_{pq} \right) \; = \; \lambda^{(k)}_J\left( \omega^{(\ell)} + h^{(\ell)} e_{pq} \right) \;\;$,
2. If $\lambda_J \left( \omega^{(\ell)} + h^{(\ell)} e_{pq} \right)$ is simple, then $\lambda^{(k)}_J \left( \omega^{(\ell)} + h^{(\ell)} e_{pq} \right)$ is also simple,
3. If $\lambda_J \left( \omega^{(\ell)} + h^{(\ell)} e_{pq} \right)$ is simple, $\; \nabla \lambda_J \left( \omega^{(\ell)} + h^{(\ell)} e_{pq} \right) = \nabla \lambda^{(k)}_J \left( \omega^{(\ell)} + h^{(\ell)} e_{pq} \right) \;\;$
for every $\ell = 1,\dots,k$, $\: p = 1,\dots,d$ and $q = p,\dots,d$.
The main motivation for the inclusion of additional eigenvectors in the subspaces is the deduction of theoretical bounds on the proximity of the second derivatives, which we present next. This result is initially established under the assumption that the third derivatives of the reduced eigenvalue functions $\omega\mapsto \lambda_J^{(k)}(\omega)$ are bounded uniformly with respect to $k$ provided $k$ is large enough. Subsequently, we show in Proposition \[prop:uniformb\_3rdd\] that this assumption is always satisfied, hence can be dropped.
\[thm:sec\_der\_ext\] Suppose that the sequence $\{ \omega^{(k)} \}$ by Algorithm \[alge\] (or Algorithm \[alg\] when $d=1$ by defining $h^{(k)} := | \omega^{(k)} - \omega^{(k-1)} |$) is convergent, that its limit $\omega_\ast := \lim_{k\rightarrow \infty} \omega^{(k)}$ lies strictly in the interior of $\Omega$, and that $\lambda_J(\omega_\ast)$ is simple. Assume furthermore that in an open ball containing $\omega_*$, all third derivatives of functions $\omega\mapsto \lambda_J^{(k)}(\omega)$ are bounded uniformly with respect to $k\geq k_0$, with $k_0$ sufficiently large. Then the following assertions hold:
1. There exists a constant $C > 0$ such that $$\label{eq:accuracy_sec_der}
\left|
\frac{\partial^2 \lambda_J \left( \omega^{(k)} \right)}{\partial \omega_p \partial \omega_q} - \frac{\partial^2 \lambda^{(k)}_J \left( \omega^{(k)} \right)}
{\partial \omega_p \partial\omega_q}
\right|_2
\; \leq \;
C h^{(k)} \quad {\rm for} \; p,q = 1,\dots,d,$$ in particular $$\left\|
\nabla^2 \lambda_J \left( \omega^{(k)} \right)
-
\nabla^2 \lambda^{(k)}_J \left( \omega^{(k)} \right)
\right\|_2
\; \leq \;
d C h^{(k)},$$ for all $k$ large enough;
2. Additionally, if $\nabla^2 \lambda_J(\omega_\ast)$ is invertible, then $$\label{eq:inv_Hessian}
\left\| \left[ \nabla^2 \lambda_J ( \omega^{(k)} ) \right]^{-1} -
\left[ \nabla^2 \lambda^{(k)}_J ( \omega^{(k)} ) \right]^{-1} \right\|_2
\; = \;
O( h^{(k)} )$$ for all $k$ large enough.
**(i)** First we specify a ball centered at $\omega_\ast$ in which $\lambda_J(\omega)$ and $\lambda_J^{(k)}(\omega)$ for all large $k$ are simple. The argument initially assumes $J > 1$. Letting $\varepsilon := \min \{ \lambda_{J-1}(\omega_\ast) - \lambda_J(\omega_\ast), \lambda_J(\omega_\ast) - \lambda_{J+1}(\omega_\ast) \}$, consider the ball ${\mathcal B}(\omega_\ast, \varepsilon/(8\gamma))$, where $\gamma$ is the Lipschitz constant as in Lemma \[lemma:Lipschitz\_continuity\]. Without loss of generality, let us assume ${\mathcal B}(\omega_\ast, \varepsilon/(8\gamma)) \subseteq \Omega$ (i.e., otherwise choose $\varepsilon$ even smaller so that ${\mathcal B}(\omega_\ast, \varepsilon/(8\gamma)) \subseteq \Omega$). Now, due to $\lambda_{J-1}(\omega_\ast) - \lambda_J(\omega_\ast) \geq \varepsilon$ as well as $\lambda_J(\omega_\ast) - \lambda_{J+1}(\omega_\ast) \geq \varepsilon$, and by Lemma \[lemma:Lipschitz\_continuity\], we have $$\label{eq:gap}
\lambda_{J-1}(\omega) - \lambda_{J}(\omega) \geq 3 \varepsilon / 4
\;\; {\rm and} \;\;
\lambda_J(\omega) - \lambda_{J+1}(\omega) \geq 3 \varepsilon / 4
\quad \forall \omega \in {\mathcal B}(\omega_\ast, \varepsilon/(8\gamma)).$$
Next choose $k$ large enough so that ${\mathcal B}(\omega^{(k)}, h^{(k)}) \subset {\mathcal B}(\omega_\ast, \varepsilon/(8\gamma))$. We will show that $\lambda^{(k)}_J(\omega)$ is also simple in ${\mathcal B}(\omega_\ast, \varepsilon/(8\gamma))$. In this respect we note that $\lambda_j(\omega^{(k)}) = \lambda_j^{(k)}(\omega^{(k)})$ for $j = J-1, J$ due to parts (i) and (ii) of Lemma \[thm:first\_der\], so from (\[eq:gap\]) we have $$\begin{split}
\lambda_{J-1}^{(k)}(\omega^{(k)}) - \lambda_{J}^{(k)}(\omega^{(k)}) \;\; \geq \;\; 3 \varepsilon / 4
\quad {\rm and} \hskip 24ex \\
\lambda_J^{(k)}(\omega^{(k)}) - \lambda_{J+1}^{(k)}(\omega^{(k)}) \;\; \geq \;\; \lambda_J^{(k)}(\omega^{(k)}) - \lambda_{J+1}(\omega^{(k)})
\;\; \geq \;\; 3 \varepsilon / 4,
\end{split}$$ where in the last line we also used $\lambda_{J+1}^{(k)}(\omega^{(k)}) \leq \lambda_{J+1}(\omega^{(k)})$ which holds due to monotonicity (Lemma \[lemma:monotonicity\]). Another application of Lemma \[lemma:Lipschitz\_continuity\] yields $$\lambda_{J-1}^{(k)}(\omega) - \lambda_{J}^{(k)}(\omega) \geq \varepsilon / 4
\quad {\rm and} \quad
\lambda_J^{(k)}(\omega) - \lambda_{J+1}^{(k)}(\omega) \geq \varepsilon / 4
\quad \forall \omega \in {\mathcal B}(\omega_\ast, \varepsilon/(8\gamma)).$$ We remark that the argument above applies to the case $J = 1$ trivially by considering only the gaps $\lambda_J(\omega) - \lambda_{J+1}(\omega)$ as well as $\lambda_J^{(k)}(\omega) - \lambda_{J+1}^{(k)}(\omega)$ and showing they remain bounded away from zero for all $\omega$ inside ${\mathcal B}(\omega_\ast, \varepsilon/(8\gamma))$.
We prove the desired bounds (\[eq:accuracy\_sec\_der\]) first for the sequence $\{ \omega^{(k)} \}$ generated by Algorithm \[alge\]. The simplicity of the eigenvalue functions $\lambda_J(\omega)$ and $\lambda_J^{(k)}(\omega)$ on ${\mathcal B}(\omega_\ast, \varepsilon/(8\gamma))$ imply that, for each $p, q$, the functions $$\label{eq:linesearch_fun}
\ell_{pq}(t) := \lambda_J(\omega^{(k)} + t h^{(k)} e_{pq})
\quad {\rm and} \quad
\ell_{pq,k}(t) := \lambda^{(k)}_J(\omega^{(k)} + t h^{(k)} e_{pq})$$ are analytic on $(0,1)$. By applying Taylor’s theorem to $\ell_{pq}(t), \ell_{pq,k}(t)$ on the interval $(0,1)$, we obtain $$\begin{aligned}
\ell_{pq}(1)
\; & = \;
\ell_{pq}(0)
+
\ell'_{pq}(0)
+
\ell^{''}_{pq}(0) / 2
+
\ell^{'''}_{pq}(\eta) / 6 \\
\ell_{pq,k}(1)
\; & = \;
\ell_{pq,k}(0)
+
\ell'_{pq,k}(0)
+
\ell''_{pq,k}(0) / 2
+
\ell'''_{pq,k}(\eta^{(k)}) / 6\end{aligned}$$ for some $\eta, \eta^{(k)} \in (0,1)$. Now by employing $\ell_{pq}(0) = \ell_{pq,k}(0)$, $\ell'_{pq}(0) = \ell'_{pq,k}(0)$ due to parts (i), (iv) of Lemma \[thm:first\_der\], and $\ell_{pq}(1) = \ell^{(k)}_{pq}(1)$ due to part (i) of Lemma \[thm:first\_der\_extended\], we deduce $$\frac{ \ell''_{pq}(0) - \ell''_{pq,k}(0) }{2}
\; = \;
\frac{ \ell'''_{pq,k} (\eta^{(k)}) - \ell^{'''}_{pq}(\eta) }{6}.$$ In the last expression, $$\ell''_{pq}(0) = \left[ h^{(k)} \right]^2 e_{pq}^T \nabla^2 \lambda_J(\omega^{(k)}) e_{pq}
\quad
{\rm and}
\quad
\ell''_{pq,k}(0) = \left[ h^{(k)} \right]^2 e_{pq}^T \nabla^2 \lambda_J^{(k)}(\omega^{(k)}) e_{pq}$$ as well as $\; \ell'''_{pq,k} (\eta^{(k)}) = O \left( \left[ h^{(k)} \right]^3 \right)$ and $\ell^{'''}_{pq}(\eta) = O\left( \left[ h^{(k)} \right]^3 \right)$, so we have $$\label{eq:Hessian_identity}
\frac
{
\left[ h^{(k)} \right]^2
\left|
e_{pq}^T
\left[
\nabla^2 \lambda_J \left( \omega^{(k)} \right)
-
\nabla^2 \lambda^{(k)}_J \left( \omega^{(k)} \right)
\right]
e_{pq}
\right|
}{2}
\leq
\frac{D}{6} \left[ h^{(k)} \right]^3$$ for some constant $D$ independent of $k$.
Choosing $q = p$ in (\[eq:Hessian\_identity\]) yields $$\left|
\frac{\partial^2 \lambda_J \left( \omega^{(k)} \right)}{\partial \omega_p^2} - \frac{\partial^2 \lambda^{(k)}_J \left( \omega^{(k)} \right)}{\partial \omega_p^2}
\right|
\leq
\frac{D}{3} h^{(k)}.$$ Additionally, for $q \neq p$, rewriting (\[eq:Hessian\_identity\]) as $$\left|
\frac{1}{2}
\left(
\sum_{\ell = p, q}
\frac{\partial^2 \lambda_J \left( \omega^{(k)} \right)}{\partial \omega_\ell^2} - \frac{\partial^2 \lambda^{(k)}_J \left( \omega^{(k)} \right)}{\partial \omega_\ell^2}
\right)
+
\left(
\frac{\partial^2 \lambda_J \left( \omega^{(k)} \right)}{\partial \omega_p \partial \omega_q} - \frac{\partial^2 \lambda^{(k)}_J \left( \omega^{(k)} \right)}{\partial \omega_p \partial \omega_q}
\right)
\right|
\leq
\frac{D}{3} h^{(k)},$$ we obtain $$\left|
\frac{\partial^2 \lambda_J \left( \omega^{(k)} \right)}{\partial \omega_p \partial \omega_q} - \frac{\partial^2 \lambda^{(k)}_J \left( \omega^{(k)} \right)}{\partial \omega_p \partial\omega_q}
\right|
\leq
\frac{2D}{3} h^{(k)}$$ leading us to (\[eq:accuracy\_sec\_der\]). The proof above establishing the bounds (\[eq:accuracy\_sec\_der\]) also applies to the sequence $\{ \omega^{(k)} \}$ generated by Algorithm \[alg\] when $d = 1$ by defining $\widetilde{h}^{(k)} := \omega^{(k-1)} - \omega^{(k)}$ (so that $h^{(k)} = | \widetilde{h}^{(k)} |$) and letting $$\ell_{pq}(t) := \lambda_J(\omega^{(k)} + t \widetilde{h}^{(k)} e_{pq})
\quad {\rm and} \quad
\ell_{pq,k}(t) := \lambda^{(k)}_J(\omega^{(k)} + t \widetilde{h}^{(k)} e_{pq})$$ instead of (\[eq:linesearch\_fun\]).
**(ii)** From part (i), as well as by the existence of $\nabla^2 \lambda_J(\omega^{(k)})$, $\nabla^2 \lambda_J^{(k)}(\omega^{(k)})$ for all $k$ large enough so that ${\mathcal B}(\omega^{(k)}, h^{(k)}) \subset {\mathcal B}(\omega_\ast, \varepsilon/(8\gamma))$ and by the continuity of $\nabla^2 \lambda_J(\omega)$ at $\omega = \omega_\ast$, we have $$\lim_{k\rightarrow \infty} \nabla^2 \lambda_J(\omega^{(k)}) \;\; = \;\; \lim_{k\rightarrow \infty} \nabla^2 \lambda_J^{(k)}(\omega^{(k)}) \;\; = \;\; \nabla^2 \lambda_J(\omega_\ast).$$ Exploiting this and the invertibility of $\nabla^2 \lambda_J(\omega_\ast)$, we deduce that $\nabla^2 \lambda_J(\omega^{(k)})$ and $\nabla^2 \lambda_J^{(k)}(\omega^{(k)})$ are invertible for all $k$ large enough.
For such a $k$, letting $A = \nabla^2 \lambda_J ( \omega^{(k)} ), \widetilde{A} = \nabla^2 \lambda^{(k)}_J ( \omega^{(k)} )$ as well as $X = [ \nabla^2 \lambda_J ( \omega^{(k)} ) ]^{-1}$, $\widetilde{X} = [ \nabla^2 \lambda^{(k)}_J ( \omega^{(k)} ) ]^{-1}$, the classical adjugate formula (i.e., $[B^{-1}]_{ij} = \frac{(-1)^{i+j} {\rm det} (B_{ji})}{ {\rm det} (B)}$ for an invertible $B$, where $B_{ji}$ denotes the matrix obtained from $B$ by removing its $j$th row and the $i$th column) yields $$\widetilde{x}_{ij}
=
\frac{(-1)^{(i+j)}\det(\widetilde{A}_{ji})}{\det(\widetilde{A})}
\quad {\rm for} \; i,j = 1,\dots,d$$ By part (i), in particular equation (\[eq:accuracy\_sec\_der\]), we have $\det(\widetilde{A}_{ji}) = \det(A_{ji}) + O(h^{(k)})$ and $\det(\widetilde{A}) = \det(A) + O(h^{(k)})$, so we deduce $$\widetilde{x}_{ij}
=
\frac{(-1)^{i+j} \det(A_{ji}) + O(h^{(k)})}{\det(A) + O(h^{(k)})}
=
\frac{(-1)^{i+j} \det(A_{ji})}{\det(A)} + O(h^{(k)})
=
x_{ij} + O(h^{(k)})$$ leading to (\[eq:inv\_Hessian\]).
\[prop:uniformb\_3rdd\] Suppose that the sequence $\{ \omega^{(k)} \}$ by Algorithm \[alge\] is convergent, its limit $\omega_\ast := \lim_{k\rightarrow \infty} \omega^{(k)}$ lies strictly in the interior of $\Omega$ and that $\lambda_J(\omega_\ast)$ is simple. Then for $k$ sufficiently large all third derivatives of functions $\omega\mapsto \lambda_J^{(k)}(\omega)$ exist in an open interval containing $\omega_*$, and they can be bounded uniformly with respect to $k$.
For the sake of clarity we first consider the case when $d=1$. Following from the assumptions of the proposition and from the elements spelled out in the proof of Lemma \[thm:sec\_der\_ext\], there exists an open interval $\mathcal{I}:=(\omega_*-r_1,\ \omega_*+r_1)$, and numbers $k_1\in\mathbb{N}$ and $\ell \in\mathbb{R}, \ell>0$, such that for every $k\geq k_1$ and all $\omega\in\mathcal{I}$, the eigenvalue $\lambda_J^{(k)}(\omega)$ is simple, as well as $$\label{fr1}
\min\left\{ \left|\lambda_J^{(k)}(\omega)-\lambda_{J-1}^{(k)}(\omega)\right|,\
\left|\lambda_J^{(k)}(\omega)-\lambda_{J+1}^{(k)}(\omega)\right|\right\}
\geq \ell.$$
Since functions $f_{\ell}$ are assumed real analytic, there exist analytic extensions on the complex plane at $\omega_*$, which we refer to by functions $\hat f_{\ell},\ \ell=1,\ldots \kappa$. We denote by $\mathbf{\hat A}^{\mathcal{S}_k}(\omega)$ the corresponding extension of $\mathbf{A}^{\mathcal{S}_k}(\omega)$. Following the ideas spelled out in the proof of Lemma 3 of [@Kressner2017], we can express $$\label{fr2}
\begin{array}{ll}
\left\|\mathbf{\hat A}^{\mathcal{S}_k}(\omega)-\mathbf{\hat A}^{\mathcal{S}_k}(\omega_*)\right\|_2
&\leq \sum_{\ell=1}^{\kappa} \left\|\mathbf{S}_k^*\mathbf{A}_{\ell} \mathbf{S}_k\right\|_2 \left| \hat f_\ell({\omega})- \hat{f}_\ell(\omega_*)\right|
\\
&\leq \sum_{\ell=1}^{\kappa} \left\|\mathbf{A}_{\ell} \right\|_2 \left| \hat f_\ell({\omega})- \hat{f}_\ell(\omega_*)\right|.
\end{array}$$ Note that if $\omega$ is non-real, the difference $\mathbf{\hat A}^{\mathcal{S}_k}(\omega)-\mathbf{\hat A}^{\mathcal{S}_k}(\omega_*)$ might be non-Hermitian. However, an important conclusion from the above inequalities is that $\| \mathbf{\hat A}^{\mathcal{S}_k}(\omega)-\mathbf{\hat A}^{\mathcal{S}_k}(\omega_*) \|_2$ can be bounded from above uniformly independent of $k$, that is independent of the dimension of the subspace ${\mathcal S}_k$, for all $\omega$ inside a disk centered at $\omega_\ast$. Furthermore, this upper bound can be chosen arbitrarily close to 0 by reducing the radius of the disk.
From (\[fr1\]), (\[fr2\]) and Theorem 5.1 of [@Stewart1990], we conclude that there exists a number $r_2\in (0,\ r_1)$ independent of $k\geq k_1$ (i.e., the dimension of the subspace), such that the eigenvalue function $\hat \lambda_J^{(k)}$, obtained by replacing $\mathbf{ A}^{\mathcal{S}_k}$ with its analytic exension $\mathbf{\hat A}^{\mathcal{S}_k}$, remains simple on and inside the disk $\left\{\omega\in\mathbb{C}:\ |\omega-\omega_*|\leq r_2 \right\}$ for every $k\geq k_1$. Hence $\hat \lambda_J^{(k)}$ is well-defined and analytic inside this disk for every $k\geq k_1$. For $\tilde\omega\in\mathbb{R}$ satisfying $|\tilde\omega-\omega_*|< r_2/2$, we have $$\frac{d^3\hat \lambda_J^{(k)}}{d\omega^3}(\tilde\omega)=\frac{3!}{2\pi {\rm i}} \oint_{|\omega-\tilde\omega|=r_2/2}
\frac{\hat\lambda_J^{(k)}(\omega)}{(\omega-\tilde\omega)^4}d\omega,$$ with ${\rm i}$ the imaginary unit. We can bound $$\label{fr3}
\begin{array}{lll}
\left|\frac{ d^3\hat\lambda_J^{(k)} } {d\omega^3}(\tilde\omega)\right|
&\leq &\frac{96}{r_2^3} \ \max_{\omega\in\mathbb{C},|\omega-\tilde\omega|=r/2} \left|\hat\lambda_J^{(k)}(\omega)\right|
\\
&\leq & \frac{96}{r_2^3}\
\sum_{\ell = 1}^\kappa \left\|{\mathbf A}^{{\mathcal S}_k}_\ell\right\|_2 \left(\max_{\omega\in\mathbb{C},\ |\omega-\tilde\omega|=r_2/2} \left|\hat f_\ell(\omega)\right|\right) \\
&\leq & \frac{96}{r_2^3}\
\sum_{\ell = 1}^\kappa \left\|{\mathbf A}_\ell\right\|_2 \left(\max_{\omega\in\mathbb{C},\ |\omega-\tilde\omega|=r_2/2} \left|\hat f_\ell(\omega)\right|\right) ,
\end{array}$$ hence, an upper bound is established that does not depend on $k\geq k_1$.
For the case $d>1$ only a slight adaptation is needed for the mixed derivatives. We sketch the main principles by means of the case when $d=2$. From the ideas behind (\[fr3\]) a bound on $\hat\lambda_J^{(k)}$ induces a bound, uniform in $k$, on its partial derivative $\frac{\partial^2 \hat\lambda_J^{(k)}}{\partial \omega_1^2}$ in an open set containing $\omega_*$. Likewise the bound on the latter induces a bound on $\frac{\partial^3 \hat\lambda_J^{(k)}}{\partial \omega_1^2\partial\omega_2}$.
Convergence Analysis {#sec:convergence}
====================
In this section we analyze the convergence properties of the two subspace procedures introduced. The first part concerns global convergence: it elaborates on whether the iterates of Algorithm \[alg\] and Algorithm \[alge\] necessarily converge as the subspace dimensions grow to infinity, and if they do converge, where they converge to. The second part establishes a superlinear rate-of-convergence result for the iterates of Algorithm \[alge\] (and Algorithm \[alg\] when $d=1$).
Global Convergence {#sec:global_convergence}
------------------
The subspace procedures, when applied to the minimization problem **(MN)**, converge globally. A formal statement of this global convergence property together with its proof are given in what follows. Note that the sequence $\{\omega^{(k)} \}$ generated by Algorithm \[alg\] or Algorithm \[alge\] belongs to the bounded set $\Omega$, so it must have convergent subsequences.
\[thm:global\_conv\] The following hold for both Algorithm \[alg\] and Algorithm \[alge\].
1. The limit of every convergent subsequence of the sequence $\left\{ \omega^{(k)} \right\}$ for the minimization problem **(MN)** is a global minimizer of $\lambda_J(\omega)$ over $\Omega$.
2. $
\lim_{k\rightarrow \infty} \lambda_J^{(k)}(\omega^{(k+1)}) \;\; = \;\; \lim_{k\rightarrow \infty} \: \min_{\omega \in \Omega} \lambda_J^{(k)}(\omega)
\;\; = \;\; \min_{\omega \in \Omega} \lambda_J(\omega).
$
**(i)** Let $\{ \omega^{(\ell_k)} \}$ be a convergent subsequence of $\{ \omega^{(k)} \}$. By the monotonicity property, that is by Lemma \[lemma:monotonicity\], we have $$\label{eq:ubound}
\begin{split}
\min_{\omega \in \Omega} \: \lambda_J(\omega) \;\; & \geq \;\; \min_{\omega \in \Omega} \: \lambda^{(\ell_{k+1}-1)}_J (\omega) \\
\;\; & = \;\; \lambda^{(\ell_{k+1}-1)}_J ( \omega^{(\ell_{k+1})} )
\;\; \geq \;\; \lambda^{(\ell_k)}_J ( \omega^{(\ell_{k+1})} ),
\end{split}$$ and, by the interpolatory property, that is part (i) of Lemma \[thm:first\_der\], $$\label{eq:lbound}
\min_{\omega \in \Omega} \lambda_J(\omega) \;\; \leq \;\; \lambda_J(\omega^{(\ell_k)}) \;\; = \;\; \lambda^{(\ell_k)}_J (\omega^{(\ell_k)}). \hskip 12ex$$ But observe the Lipschitz continuity of $\lambda_J^{(\ell_k)}(\omega)$ (see Lemma \[lemma:Lipschitz\_continuity\]) implies $$\lim_{k \rightarrow \infty} \;
\left| \lambda^{(\ell_k)}_J ( \omega^{(\ell_{k+1})} ) - \lambda^{(\ell_k)}_J ( \omega^{(\ell_k)} ) \right|
\; = \;
\gamma
\lim_{k \rightarrow \infty} \;
\| \omega^{(\ell_{k+1})} - \omega^{(\ell_k)} \|_2 \; = \; 0.$$ Consequently, taking the limits in (\[eq:ubound\]) and (\[eq:lbound\]) as $k \rightarrow \infty$ and employing the interpolatory property (part (i) of Lemma \[thm:first\_der\]), we deduce $$\label{eq:all_converge}
\lim_{k\rightarrow \infty} \lambda_J (\omega^{(\ell_k)})
\;\; = \;\;
\lim_{k\rightarrow \infty} \lambda^{(\ell_k)}_J (\omega^{(\ell_k)})
\;\; = \;\;
\lim_{k\rightarrow \infty} \lambda^{(\ell_k)}_J (\omega^{(\ell_{k+1})})
\;\; = \;\;
\min_{\omega \in \Omega} \lambda_J(\omega).$$ Finally, by the continuity of $\lambda_J(\omega)$, the sequence $\{ \omega^{(\ell_k)} \}$ must converge to a global minimizer of $\lambda_J(\omega)$.
**(ii)** Let $\lambda_\ast := \min_{\omega \in \Omega} \lambda_J(\omega)$. We first show that the sequence $\{ \lambda_J^{(k)}(\omega^{(k+1)}) \}$ is convergent. In this respect observe that for every pair of positive integers $p, k$ such that $p > k$, due to monotonicity (Lemma \[lemma:monotonicity\]) we have $$\begin{split}
\lambda_\ast \;\; & \geq \;\; \min_{\omega \in \Omega} \lambda_J^{(p)} (\omega) \;\; = \;\; \lambda_J^{(p)}(\omega^{(p+1)}) \\
\;\; & \geq \;\; \lambda_J^{(k)}(\omega^{(p+1)}) \;\; \geq \;\; \min_{\omega \in \Omega} \lambda_J^{(k)} (\omega) \;\; = \;\; \lambda_J^{(k)}(\omega^{(k+1)}).
\end{split}$$ Hence the sequence $\{ \lambda_J^{(k)}(\omega^{(k+1)}) \}$ is monotonically increasing bounded above by $\lambda_\ast$ proving its convergence.
Now let $\{ \omega^{(\ell_k)} \}$ be a convergent subsequence of $\{ \omega^{(k)} \}$ as in part (i), and let us consider the sequence $\{ \lambda_J^{(\ell_{k+1}-1)}(\omega^{(\ell_{k+1})}) \}$, which is a subsequence of $\{ \lambda_J^{(k)}(\omega^{(k+1)}) \}$. We complete the proof by establishing the convergence of this subsequence to $\lambda_\ast$. The monotonicity and the boundedness of $\{ \lambda_J^{(k)}(\omega^{(k+1)}) \}$ from above by $\lambda_\ast$ imply $$\label{eq:order_seq}
\lambda_J^{(\ell_k)}(\omega^{(\ell_{k+1})}) \;\; \leq \;\; \lambda_J^{(\ell_{k+1}-1)}(\omega^{(\ell_{k+1})}) \;\; \leq \;\; \lambda_\ast.$$ Above, as shown in part (i) (in particular see (\[eq:all\_converge\])), $\lim_{k\rightarrow \infty} \lambda_J^{(\ell_k)}(\omega^{(\ell_{k+1})}) = \lambda_\ast$, so we must also have $\lim_{k\rightarrow \infty} \lambda_J^{(\ell_{k+1}-1)}(\omega^{(\ell_{k+1})}) = \lambda_\ast$ as desired.
As for the maximization problem **(MX)**, it does not seem possible to conclude with such a convergence result to a global maximizer. This is because only a lower bound (but not an upper bound) is available in terms of the reduced eigenvalue functions for the maximum value of $\lambda_J(\omega)$ over $\omega \in \Omega$. However, the sequence $\{ \lambda_J^{(k)}(\omega^{(k+1)}) \}$ is still convergent as shown next.
\[thm:global\_conv\_max\] The sequence $\{ \lambda_J^{(k)}(\omega^{(k+1)}) \}$ generated by Algorithm \[alg\] and Algorithm \[alge\] for the maximization problem **(MX)** converges.
Letting $\lambda^\ast := \max_{\omega \in \Omega} \lambda_J(\omega)$, the sequence by Algorithm \[alg\] and Algorithm \[alge\] satisfies $$\label{eq:global_conv_max}
\begin{split}
\lambda_J^{(k-1)} (\omega^{(k)}) \;\; & \leq \;\; \lambda_J (\omega^{(k)}) \;\; = \;\; \lambda_J^{(k)}(\omega^{(k)}) \\
\;\; & \leq \;\; \max_{\omega \in \Omega} \: \lambda_J^{(k)} (\omega) \;\; = \;\; \lambda_J^{(k)}(\omega^{(k+1)}) \;\; \leq \;\; \lambda^\ast
\end{split}$$ where the first and the last inequality follow from the monotonicity, and the equality in the first line is a consequence of the interpolatory property. These inequalities lead to the conclusion that the sequence $\{ \lambda_J^{(k)} (\omega^{(k+1)}) \}$ is monotone increasing and bounded above by $\lambda^\ast$, hence convergent.
We observe in practice that the sequence $\{ \lambda_J^{(k)} (\omega^{(k+1)}) \}$ converges to a $\widetilde{\lambda}$ such that $\lambda_J(\widetilde{\omega}) = \widetilde{\lambda}$ for some $\widetilde{\omega}$ that is a local maximizer of $\lambda_J(\omega)$, that is not necessarily a global maximizer.
We illustrate these convergence results for the minimization as well as for the maximization of the largest eigenvalue $\lambda_1(\omega)$ of $$\label{eq:numrad}
A(\omega)
\;\; = \;\;
\frac{Ae^{i\omega} + A^\ast e^{-i\omega}}{2}
\;\; = \;\;
{\rm cos}(\omega)
\left(
\frac{A + A^\ast}{2}
\right)
+
{\rm sin}(\omega)
\left(
\frac{i A - i A^\ast}{2}
\right)$$ over $\omega \in [0, 2\pi]$ and for a particular matrix $A$. The maximum of the largest eigenvalue of $A(\omega)$ over $\omega \in [0, 2\pi]$ corresponds to the numerical radius of $A$ [@Horn1991; @He1997], see Section \[sec:numrad\] for more on the numerical radius. Here we particularly choose $A$ as the $400\times 400$ matrix whose real part comes from a five-point finite difference discretization of a Poisson equation, and whose complex part has random entries selected independently from a normal distribution with zero mean and standard deviation equal to 20. An application of the basic subspace procedure (Algorithm \[alg\]) for the maximization of $\lambda_1(\omega)$ starting with $\omega^{(1)} = 2$ results in convergence to a local maximizer $\widetilde{\omega} \approx 2.353$, whereas initiating the procedure with $\omega^{(1)} = 5.5$ leads to convergence to $\omega^\ast \approx 6.145$, the unique global maximizer of $\lambda_1(\omega)$ over $[0, 2\pi]$. This is depicted on the top row in Figure \[fig:convergence\] on the left and on the right, respectively. The situation is quite different when the subspace procedure is applied to minimize $\lambda_1(\omega)$. It converges to the global minimizer $\omega_\ast \approx 3.959$ of $\lambda_1(\omega)$ regardless whether the procedure is initiated with $\omega^{(1)} = 2$ or $\omega^{(1)} = 5.5$ as depicted at the bottom row of Figure \[fig:convergence\]. Notice that the subspace procedure for the maximization problem constructs reduced eigenvalue functions that capture $\lambda_1(\omega)$ well only locally around the maximizers. In contrast for the minimization problem the reduced eigenvalue functions capture $\lambda_1(\omega)$ globally, but their accuracy is higher around the minimizers.
-2ex
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ Top and bottom rows illustrate Algorithm \[alg\] to maximize and minimize, respectively, the largest eigenvalue $\lambda_1(\omega)$ (solid curves) of $A(\omega)$ as in (\[eq:numrad\]) over $\omega \in [0,2\pi]$ (the horizontal axis). For both the maximization and the minimization, Algorithm \[alg\] is initiated with two different initial points $\omega^{(1)} = 2$ on the left column and $\omega^{(1)} = 5.5$ on the right column. The dotted curves on the top and at the bottom represent the reduced eigenvalue functions with one and eight dimensional subspaces, respectively. The dashed curves correspond to the reduced eigenvalue functions right before convergence, with five dimensional subspaces on the top and fifteen dimensional subspaces at the bottom. The dots represent the converged points, that is denoting the converged $\omega$ value with $\widehat{\omega}$ and letting $\widehat{\lambda} := \lambda_1(\widehat{\omega})$ it marks $(\widehat{\omega}, \widehat{\lambda})$. []{data-label="fig:convergence"}](conv_max1smm.pdf "fig:"){width=".49\textwidth"} ![ Top and bottom rows illustrate Algorithm \[alg\] to maximize and minimize, respectively, the largest eigenvalue $\lambda_1(\omega)$ (solid curves) of $A(\omega)$ as in (\[eq:numrad\]) over $\omega \in [0,2\pi]$ (the horizontal axis). For both the maximization and the minimization, Algorithm \[alg\] is initiated with two different initial points $\omega^{(1)} = 2$ on the left column and $\omega^{(1)} = 5.5$ on the right column. The dotted curves on the top and at the bottom represent the reduced eigenvalue functions with one and eight dimensional subspaces, respectively. The dashed curves correspond to the reduced eigenvalue functions right before convergence, with five dimensional subspaces on the top and fifteen dimensional subspaces at the bottom. The dots represent the converged points, that is denoting the converged $\omega$ value with $\widehat{\omega}$ and letting $\widehat{\lambda} := \lambda_1(\widehat{\omega})$ it marks $(\widehat{\omega}, \widehat{\lambda})$. []{data-label="fig:convergence"}](conv_max2smm.pdf "fig:"){width=".49\textwidth"}
![ Top and bottom rows illustrate Algorithm \[alg\] to maximize and minimize, respectively, the largest eigenvalue $\lambda_1(\omega)$ (solid curves) of $A(\omega)$ as in (\[eq:numrad\]) over $\omega \in [0,2\pi]$ (the horizontal axis). For both the maximization and the minimization, Algorithm \[alg\] is initiated with two different initial points $\omega^{(1)} = 2$ on the left column and $\omega^{(1)} = 5.5$ on the right column. The dotted curves on the top and at the bottom represent the reduced eigenvalue functions with one and eight dimensional subspaces, respectively. The dashed curves correspond to the reduced eigenvalue functions right before convergence, with five dimensional subspaces on the top and fifteen dimensional subspaces at the bottom. The dots represent the converged points, that is denoting the converged $\omega$ value with $\widehat{\omega}$ and letting $\widehat{\lambda} := \lambda_1(\widehat{\omega})$ it marks $(\widehat{\omega}, \widehat{\lambda})$. []{data-label="fig:convergence"}](conv_min1smm.pdf "fig:"){width=".49\textwidth"} ![ Top and bottom rows illustrate Algorithm \[alg\] to maximize and minimize, respectively, the largest eigenvalue $\lambda_1(\omega)$ (solid curves) of $A(\omega)$ as in (\[eq:numrad\]) over $\omega \in [0,2\pi]$ (the horizontal axis). For both the maximization and the minimization, Algorithm \[alg\] is initiated with two different initial points $\omega^{(1)} = 2$ on the left column and $\omega^{(1)} = 5.5$ on the right column. The dotted curves on the top and at the bottom represent the reduced eigenvalue functions with one and eight dimensional subspaces, respectively. The dashed curves correspond to the reduced eigenvalue functions right before convergence, with five dimensional subspaces on the top and fifteen dimensional subspaces at the bottom. The dots represent the converged points, that is denoting the converged $\omega$ value with $\widehat{\omega}$ and letting $\widehat{\lambda} := \lambda_1(\widehat{\omega})$ it marks $(\widehat{\omega}, \widehat{\lambda})$. []{data-label="fig:convergence"}](conv_min2smm.pdf "fig:"){width=".49\textwidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The Rate-of-Convergence {#sec:rate_of_convergence}
-----------------------
Next we are concerned with how quickly the iterates of the subspace procedures converge when they do converge to a smooth stationary point of $\lambda_J(\omega)$. Note that by Theorem \[thm:global\_conv\], for the minimization problem, if the sequence $\{ \omega^{(k)} \}$ by Algorithm \[alge\] (or Algorithm \[alg\]) converges to a point $\omega_\ast$ where $\lambda_J(\omega)$ is simple, then $\omega_\ast$ must be a smooth stationary point of $\lambda_J(\omega)$. The analysis below applies to Algorithm \[alge\] (and Algorithm \[alg\] when $d = 1$) in a unified way, both for the minimization problem and for the maximization problem.
\[thm:super\_convergence\] Suppose that the sequence $\{ \omega^{(k)} \}$ by Algorithm \[alge\] (or Algorithm \[alg\] when $d = 1$) converges to a point $\omega_\ast$ in the interior of $\Omega$ such that
**(i)** $\lambda_J(\omega_\ast)$ is simple, **(ii)** $\nabla \lambda_J(\omega_\ast) = 0$, and **(iii)** $\nabla^2 \lambda_J(\omega_\ast)$ is invertible.
Then there exists a constant $\alpha > 0$ such that $$\label{eq:quad_conv}
\frac{ \left\| \omega^{(k+1)} - \omega_\ast \right\|_2 }
{ \left\| \omega^{(k)} - \omega_\ast \right\|_2 \: \max \left\{ \left\| \omega^{(k)} - \omega_\ast \right\|_2, \left\| \omega^{(k-1)} - \omega_\ast \right\|_2 \right\} } \;\; \leq \;\; \alpha
\;\;\;\; \forall k.$$
The arguments in the first two paragraphs of the proof of Lemma \[eq:accuracy\_sec\_der\] establish the existence of a ball ${\mathcal B}(\omega_\ast,\eta)$ containing ${\mathcal B}(\omega^{(k)}, h^{(k)})$ for all $k$ large enough, say for $k \geq k'$, such that $\lambda_J(\omega)$ as well as $\lambda_J^{(k)}(\omega)$ for $k \geq k'$ are simple for all $\omega \in {\mathcal B}(\omega_\ast,\eta)$. The Hessians $\nabla^2 \lambda_J(\omega)$ and $\nabla^2 \lambda_J^{(k)}(\omega)$ for $k \geq k'$ are Lipschitz continuous inside this ball.
Additionally, following the arguments in the proof of part (ii) of Lemma \[eq:accuracy\_sec\_der\], the invertibility of $\nabla^2 \lambda_J(\omega_\ast)$ ensures that $\nabla^2 \lambda_J(\omega^{(k)})$ and $\nabla^2 \lambda_J^{(k)}(\omega^{(k)})$ are invertible (indeed $\| [ \nabla^2 \lambda_J(\omega^{(k)}) ]^{-1} \|_2$ and $\| [ \nabla^2 \lambda_J^{(k)}(\omega^{(k)}) ]^{-1} \|_2$ are bounded from above by some constant) for all $k$ large enough, say for $k \geq k'' \geq k'$.
Now for $k \geq k''$ by the Taylor’s theorem with integral remainder $$0 \;\; = \;\;
\nabla \lambda_J\left( \omega_\ast \right)
\;\; = \;\;
\nabla \lambda_J( \omega^{(k)} )
\;\; + \;\;
\int_0^1 \nabla^2 \lambda_J ( \omega^{(k)} + \alpha ( \omega_\ast - \omega^{(k)} ) ) \: ( \omega_{\ast} - \omega^{(k)} ) \; {\mathrm d}\alpha.$$ Employing $\nabla \lambda_J \left( \omega^{(k)} \right) = \nabla \lambda^{(k)}_J \left( \omega^{(k)} \right)$ (Lemma \[thm:first\_der\], part (iii)) in this equation and left-multiplying the both sides of the equation by the inverse of $\nabla^2 \lambda_J \left( \omega^{(k)} \right)$ yield $$\label{eq:cor_Taylor}
\begin{split}
0 \;\; = \;\;
\left[ \nabla^2 \lambda_J ( \omega^{(k)} ) \right]^{-1} \cdot \nabla \lambda^{(k)}_J ( \omega^{(k)} )
\;\; + \;\; (\omega_\ast - \omega^{(k)}) \;\; + \;\; \hskip 21ex \\
\hskip -1ex \left[ \nabla^2 \lambda_J ( \omega^{(k)} ) \right]^{-1}
\int_0^1
\left[
\nabla^2 \lambda_J ( \omega^{(k)} + \alpha (\omega_\ast - \omega^{(k)}) )
-
\nabla^2 \lambda_J ( \omega^{(k)} )
\right]
( \omega_{\ast} - \omega^{(k)} ) \; {\mathrm d}\alpha.
\end{split}$$ A second order Taylor expansion of $\nabla \lambda^{(k)}_J \left( \omega \right) $ about $\omega^{(k)}$, noting $\nabla \lambda^{(k)}_J ( \omega^{(k + 1)} ) = 0$, implies $$\left[ \nabla^2 \lambda^{(k)}_J ( \omega^{(k)} ) \right]^{-1} \nabla \lambda^{(k)} ( \omega^{(k)} )
\;\; = \;\;
- ( \omega^{(k + 1)} - \omega^{(k)} ) \; + \;
O([h^{(k+1)}]^2).$$ Using this equality in (\[eq:cor\_Taylor\]) leads us to $$\label{eq:cor_Taylor2}
\begin{split}
0
\;\; = \;\;
\omega_\ast - \omega^{(k+1)} \;\; + \;\;
\left\{ \left[ \nabla^2 \lambda_J ( \omega^{(k)} ) \right]^{-1} -
\left[ \nabla^2 \lambda^{(k)}_J ( \omega^{(k)} ) \right]^{-1} \right\}
\nabla \lambda_J ( \omega^{(k)} ) \hskip 3ex \\
\;\; + \;\; O([h^{(k+1)}]^2) \;\; + \;\; \hskip 35ex \\
\hskip -1.5ex
\left[ \nabla^2 \lambda_J ( \omega^{(k)} ) \right]^{-1}
\int_0^1
\left[
\nabla^2 \lambda_J ( \omega^{(k)} + \alpha (\omega_\ast - \omega^{(k)}) )
-
\nabla^2 \lambda_J ( \omega^{(k)} )
\right]
( \omega_{\ast} - \omega^{(k)} ) \: {\mathrm d}\alpha,
\end{split}$$ which, by taking the 2-norms and employing the triangle inequality, yields $$\label{eq:final_norms}
\begin{split}
\| \omega^{(k+1)} - \omega_\ast \|_2 \;\; \leq \;\;
\left\| \left[ \nabla^2 \lambda_J ( \omega^{(k)} ) \right]^{-1} -
\left[ \nabla^2 \lambda^{(k)}_J ( \omega^{(k)} ) \right]^{-1} \right\|_2
\| \nabla \lambda_J ( \omega^{(k)} ) \|_2 \hskip 6ex \\
+ \;\; O([h^{(k+1)}]^2) \;\; + \hskip 41ex \\
\left\| \left[ \nabla^2 \lambda_J ( \omega^{(k)} ) \right]^{-1} \right\|_2
\int_0^1
\left\|
\nabla^2 \lambda_J ( \omega^{(k)} + \alpha (\omega_\ast - \omega^{(k)}) )
-
\nabla^2 \lambda_J ( \omega^{(k)} )
\right\|_2
\| \omega^{(k)} - \omega_{\ast} \|_2 \: {\mathrm d}\alpha.
\end{split}$$ To conclude with the desired superlinear convergence result, we bound the terms on the right-hand side of the inequality in (\[eq:final\_norms\]) from above in terms of $\| \omega^{(k+1)} - \omega_\ast \|_2$, $\| \omega^{(k)} - \omega_\ast \|_2$ and $\| \omega^{(k-1)} - \omega_\ast \|_2$. To this end, first note that the terms in the third line of (\[eq:final\_norms\]) is $O(\| \omega^{(k)} - \omega_{\ast} \|_2^2)$, this is because of the boundedness of $\| [ \nabla^2 \lambda_J ( \omega^{(k)} ) ]^{-1} \|_2$ and the Lipschitz continuity of $\nabla^2 \lambda_J ( \omega )$ inside ${\mathcal B}(\omega_\ast,\eta)$. Secondly $\| \nabla \lambda_J ( \omega^{(k)} ) \|_2 = O( \| \omega^{(k)} - \omega_\ast \| )$, as can be seen from a Taylor expansion of $\nabla \lambda_J(\omega)$ about $\omega^{(k)}$ and by exploiting $\nabla \lambda_J(\omega_\ast) = 0$. Finally, due to part (ii) of Lemma \[thm:sec\_der\_ext\], we have $
\| [ \nabla^2 \lambda_J ( \omega^{(k)} ) ]^{-1} - [ \nabla^2 \lambda^{(k)}_J ( \omega^{(k)} ) ]^{-1} \|_2
=
O( h^{(k)} ).
$ Applying all these bounds to (\[eq:final\_norms\]) gives rise to $$\| \omega^{(k+1)} - \omega_\ast \|_2 \;\; \leq \;\; O(h^{(k)} \| \omega^{(k)} - \omega_\ast \|_2) + O([h^{(k+1)}]^2) + O(\| \omega^{(k)} - \omega_{\ast} \|_2^2).$$ The desired result (\[eq:quad\_conv\]) follows noting $h^{(k)} \leq 2 \max \{ \| \omega^{(k)} - \omega_\ast \|_2, \| \omega^{(k-1)} - \omega_\ast \|_2 \}$ and $[h^{(k+1)}]^2 \leq 2 ( \| \omega^{(k)} - \omega_{\ast} \|_2^2 + \| \omega^{(k+1)} - \omega_{\ast} \|_2^2 )$.
Regarding, specifically, the minimization of the largest eigenvalue when $d=1$, the order of the superlinear rate-of-convergence for Algorithm \[alg\] is shown to be at least $1 + \sqrt{2}$ in [@Kressner2017 Theorem 1].
It does not seem straightforward to extend the rate-of-convergence result above to Algorithm \[alg\] when $d \geq 2$, because relations such as the ones given by Lemma \[thm:sec\_der\_ext\] between the second derivatives of $\lambda_J(\omega)$ and $\lambda_J^{(k)}(\omega)$ are not evident. We observe a superlinear rate-of-convergence for Algorithm \[alg\] in practice as well. Algorithm \[alge\] requires a slightly fewer iterations to reach a prescribed accuracy, but Algorithm \[alg\] attains the prescribed accuracy with subspaces of smaller dimension. These observations are illustrated in Table \[table:rate\_conv\] on the problem of minimizing the largest eigenvalue $\lambda_1(\omega)$ of $$\label{eq:affine}
A(\omega) \;\; = \;\; A_0 + \omega_1 A_1 + \dots + \omega_d A_d$$ for given Hermitian matrices $A_0, A_1, \dots, A_d \in {\mathbb C}^{n\times n}$ for $d = 2,3$, where we restrict $\omega_1, \dots, \omega_d$ to the interval $[-60,60]$. Such eigenvalue optimization problems are convex [@Overton1988], and arise from a classical structural design problem [@Cox1992; @Lewis1996]. Additionally, as discussed in Section \[sec:min\_large\_eig\], they are closely related to semidefinite programs. In Table \[table:rate\_conv\] the minimal values $\lambda_1^{(k)}(\omega^{(k+1)})$ of the reduced eigenvalue functions $\lambda_1^{(k)}(\omega)$, as well as the error $\| \omega^{(k+1)} - \omega_\ast \|_2$ converge superlinearly with respect to $k$ both for $d = 2$ and for $d = 3$. In both cases the extended subspace procedure on the right column achieves 10 decimal digits accuracy (in the sense that $\lambda_1(\omega_\ast) - \lambda_1^{(k)}(\omega^{(k+1)}) \leq 10^{-10}$) after 8 iterations but with subspaces of dimension 32 and 56 for $d = 2$ and $d = 3$, respectively. On the other hand, the basic subspace procedure on the left column requires 11 and 15 iterations, which are also the dimensions of the subspaces constructed, for $d=2$ and $d = 3$, respectively, to achieve the same accuracy. Observe also that the minimal values $\lambda_1^{(k)}(\omega^{(k+1)})$ seem to converge at a rate even faster than the rate-of-decay of the errors $\| \omega^{(k+1)} - \omega_\ast \|_2$.
-2ex
[cccc|cccc]{}\
20ex & 20ex\
$k$ & $p$ & $\lambda_1^{(k)}(\omega^{(k+1)})$ & $\| \omega^{(k+1)} - \omega_\ast \|_2$ & $k$ & $p$ & $\lambda_1^{(k)}(\omega^{(k+1)})$ & $\| \omega^{(k+1)} - \omega_\ast \|_2$\
6 & 6 & 27.8402784689 & 0.3245035160 & 3 & 12 & 7.0566934870 & 0.3112458616\
7 & 7 & 27.9933586806 & 0.1134918678 & 4 & 16 & 24.9336629638 & 0.5250223403\
8 & 8 & 28.4522934270 & 0.0247284535 & 5 & 20 & 27.6531948787 & 0.1830729388\
9 & 9 & 28.5008552358 & 0.0007732659 & 6 & 24 & 28.4440816653 & 0.0142076774\
10 & 10 & 28.5010522075 & 0.0000243843 & 7 & 28 & 28.5010327361 & 0.0001556439\
11 & 11 & 28.5010523924 & 0.0000000694 & 8 & 32 & 28.5010523924 & 0.0000000344\
2ex
-2ex
[cccc|cccc]{}\
20ex & 20ex\
$k$ & $p$ & $\lambda_1^{(k)}(\omega^{(k+1)})$ & $\| \omega^{(k+1)} - \omega_\ast \|_2$ & $k$ & $p$ & $\lambda_1^{(k)}(\omega^{(k+1)})$ & $\| \omega^{(k+1)} - \omega_\ast \|_2$\
10 & 10 & 28.1244577720 & 0.1137180915 & 3 & 21 & 22.9802867532 & 0.8996337159\
11 & 11 & 28.1897386962 & 0.0482696055 & 4 & 28 & 26.1225009081 & 0.6602340340\
12 & 12 & 28.2359716412 & 0.0051064698 & 5 & 35 & 27.0151485822 & 0.1737449902\
13 & 13 & 28.2388852363 & 0.0005705897 & 6 & 42 & 28.0850898434 & 0.0374522435\
14 & 14 & 28.2389043164 & 0.0000124871 & 7 & 49 & 28.2387293186 & 0.0002142688\
15 & 15 & 28.2389043663 & 0.0000001102 & 8 & 56 & 28.2389043663 & 0.0000001112\
-5ex
-3ex
In the one parameter case (i.e., $d=1$) Theorem \[thm:super\_convergence\] applies to Algorithm \[alg\] as well to establish its superlinear convergence. The convergence of Algorithm \[alg\] on the example of Figure \[fig:convergence\] (concerning the minimization or maximization of the largest eigenvalue of the matrix-valued function in (\[eq:numrad\])) starting with $\omega^{(1)} = 5.5$ is depicted in Table \[table:rate\_conv2\]. For the maximization and minimization the iterates $\{ \omega^{(k)} \}$ converge to the global maximizer ($\approx 6.145$) and global minimizer ($\approx 3.959$) of $\lambda_1(\omega)$ at a superlinear rate, which is realized earlier for the maximization problem. The optimal values of the reduced eigenvalue function $\lambda_1^{(k)}(\omega^{(k+1)})$ converge to the globally maximal value and minimal value of $\lambda_1(\omega)$ even at a faster rate.
15ex
[cccc]{}\
$k$ & $\lambda_1^{(k)}(\omega^{(k+1)})$ & $| \omega^{(k+1)} - \omega_\ast |$ & $\omega^{(k+1)}$\
1 & 0.9213417656 & 0.6098999764 & 5.5354507094\
2 & 0.9272678333 & 0.4809562556 & 5.6643944303\
3 & 0.9452307484 & 0.2569647709 & 5.8883859150\
4 & 0.9596256053 & 0.0659648186 & 6.0793858672\
5 & 0.9617115893 & 0.0019013105 & 6.1434493753\
6 & 0.9617265293 & 0.0000002797 & 6.1453504061\
15ex
[cccc]{}\
$k$ & $\lambda_1^{(k)}(\omega^{(k+1)})$ & $| \omega^{(k+1)} - \omega_\ast |$ & $\omega^{(k+1)}$\
10 & 0.8905833092 & 1.2006093051 & 5.1592214939\
11 & 0.8953986993 & 2.7480373651 & 1.2105748237\
12 & 0.9039755162 & 0.2734960246 & 3.6851161642\
13 & 0.9112584133 & 0.0063764350 & 3.9522357538\
14 & 0.9112669429 & 0.0000006079 & 3.9586115809\
15 & 0.9112669442 & 0.0000000000 & 3.9586121888\
-5ex
Variations and Extensions {#sec:var_ext}
=========================
A Greedy Subspace Procedure without Past {#sec:subspace_nopast}
----------------------------------------
The rate-of-convergence analysis of the previous section and, in particular, the proof of Theorem \[thm:super\_convergence\], for Algorithm \[alge\] (for Algorithm \[alg\] when $d=1$) makes use of the eigenvectors added into the subspace in the last iteration (in the last two iterations) only. Hence Theorem \[thm:super\_convergence\] and its superlinear convergence assertion still hold for Algorithm \[alge\] even if its line 11 is changed as $${\mathcal S}_k \; \gets \; {\rm span} \left\{ s^{(k)}_1, \dots, s^{(k)}_{J} \right\}
\; \bigoplus \; \left\{ \bigoplus_{p = 1, q = p}^d {\rm span} \left\{ s^{(k)}_{1,p q}, \dots, s^{(k)}_{J, p q} \right\} \right\},$$ that is even if the previous subspace is completely discarded. We refer to this variant of Algorithm \[alge\] as *Algorithm \[alge\] without past*. Similarly for Algorithm \[alg\] when $d = 1$ the superlinear convergence assertion of Theorem \[thm:super\_convergence\] still holds if only the eigenvectors from the last two iterations are kept inside the subspace, that is if line 7 of Algorithm \[alg\] is replaced by $${\mathcal S}_k \; \gets \; {\rm span} \left\{ s^{(k-1)}_1, \dots, s^{(k-1)}_J \right\} \oplus {\rm span} \left\{ s^{(k)}_1, \dots, s^{(k)}_J \right\}.$$ We refer to this variant of Algorithm \[alg\] for the case $d=1$ as *Algorithm \[alg\] without past*.
The main issue with these subspace procedures without past is the convergence. In Section \[sec:global\_convergence\] the convergence of the sequence $\{ \lambda_J^{(k)}(\omega^{(k+1)}) \}$ is established both for the minimization problem and for the maximization problem (by Theorem \[thm:global\_conv\] and Theorem \[thm:global\_conv\_max\], respectively). When the eigenvectors from the past iterations are discarded, the convergence of $\{ \lambda_J^{(k)}(\omega^{(k+1)}) \}$ is not guaranteed anymore for the minimization problem, because the monotonicity of $\lambda_J^{(k)}(\omega)$ with respect to $k$ is no longer true. On the other hand, all the equalities and inequalities in (\[eq:global\_conv\_max\]) concerning the monotonicity and boundedness of $\{ \lambda_J^{(k)}(\omega^{(k+1)}) \}$ for the maximization problem can be verified to hold, so this sequence is still guaranteed to converge.
The remarks of the previous paragraph are illustrated in Table \[table:nopast\] which concerns the application of Algorithm \[alg\] without past to the example of Table \[table:rate\_conv2\]. For the maximization problem, the maximizers $\lambda_1^{(k)}(\omega^{(k+1)})$ of the reduced eigenvalue functions $\lambda_1^{(k)}(\omega)$ converge to the globally largest value of $\lambda_1(\omega)$ at a superlinear rate with respect to $k$, even though for every $k$ the subspace dimension is two. For the minimization problem, the values in the table depict that the sequence $\{ \lambda_1^{(k)}(\omega^{(k+1)}) \}$ does not converge.
The subspace procedures without past work effectively in practice for the maximization problem. On the other hand, for the minimization problem, it may fail to converge. In [@Kressner2017] a convergent subspace framework making use of three dimensional subspaces is devised for the Crawford number computation, which involves the maximization of the smallest eigenvalue (equivalently the minimization of the largest eigenvalue) of a matrix-valued function depending on one variable. The three dimensional subspaces are formed of eigenvectors from the past iterations, but not necessarily the eigenvectors from the last three iterations. The approach is built on the concavity of the smallest eigenvalue function involved and the knowledge of an interval containing the global maximizer. It does not seem easy to extend the ideas in [@Kressner2017] to the general nonconvex setting that involves the minimization of $\lambda_J(\omega)$.
23ex
----- ----------------------------------- ----- -----------------------------------
$k$ $\lambda_1^{(k)}(\omega^{(k+1)})$ $k$ $\lambda_1^{(k)}(\omega^{(k+1)})$
1 0.9213417656 45 -0.3266668608
2 0.9322945953 46 -0.5147292664
3 0.9595070069 47 -0.3348522694
4 0.9615567360 48 -0.4859510415
5 0.9617263748 49 -0.3259949226
6 0.9617265294 50 -0.5140279537
----- ----------------------------------- ----- -----------------------------------
: The optimal values $\lambda_1^{(k)}(\omega^{(k+1)})$ of the reduced eigenvalue functions by the variant of Algorithm \[alg\] that keeps only the subspaces from the last two iterations are listed with respect to $k$ for the example of the right column of Figure \[fig:convergence\] (which involves the maximization and minimization of the largest eigenvalue of a matrix-valued function of the form (\[eq:numrad\]) starting with $\omega^{(1)} = 5.5$).[]{data-label="table:nopast"}
-3ex
Subspace Procedures for Singular Value Optimization {#sec:singular_vals}
---------------------------------------------------
The ideas of the previous sections can be extended to maximize or minimize the $J$th largest singular value $\sigma_J(\omega)$ of a compact operator $$\label{eq:oper_param}
{\mathbf B}(\omega) := \sum_{\ell = 1}^\kappa f_\ell(\omega) {\mathbf B}_\ell$$ over a compact subset $\Omega$ of ${\mathbf R}^d$. As before ${\mathbf B}(\omega)$ is defined over $\omega$ that belongs to an open subset $\overline{\Omega}$ containing the feasible region $\Omega$. In representation (\[eq:oper\_param\]) of ${\mathbf B}(\omega)$ above $f_\ell : \overline{\Omega} \rightarrow {\mathbb R}$ is a real-analytic function and ${\mathbf B}_\ell : \ell^2({\mathbb N}) \rightarrow \ell^2 ({\mathbb N})$ is a compact operator for $\ell = 1,\dots,\kappa$. Once again, the infinite-dimensional problem is motivated by the finite dimensional case where the matrix-valued function $$\label{eq:mat_func2d}
B(\omega) := \sum_{\ell = 1}^\kappa f_\ell(\omega) B_\ell,$$ for given $B_\ell \in {\mathbb C}^{p\times q} \; \ell = 1,\dots,\kappa$ with large dimensions, takes the role of ${\mathbf B}(\omega)$.
Let ${\mathcal U}, {\mathcal V}$ be two subspaces of $\ell^2({\mathbb N})$ with equal dimension, and let $U, \: V$ be orthonormal bases for these subspaces. The subspace procedures to optimize $\sigma_J(\omega)$ are based on the optimization of the $J$th largest singular value $\sigma_J^{{\mathcal U}, {\mathcal V}}(\omega)$ of an operator of the form $$\label{eq:oper_param_red}
{\mathbf B}^{\mathcal U, \mathcal V}(\omega) := {\mathbf U}^\ast {\mathbf B}(\omega) {\mathbf V}$$ where ${\mathbf V}$ (${\mathbf U}$) represents the operator as in (\[eq:change\_coor\]) or (\[eq:change\_coor2\]) that maps the coordinates of a vector in ${\mathcal V}$ (${\mathcal U}$) relative to the basis $V$ ($U$) to itself. This operator can be written as $${\mathbf B}^{{\mathcal U}, {\mathcal V}}(\omega) =
\sum_{\ell = 1}^\kappa f_\ell(\omega) {\mathbf B}^{{\mathcal U}, {\mathcal V}}_\ell
\;\;\; {\rm where} \;\;\;
{\mathbf B}^{{\mathcal U}, {\mathcal V}}_\ell := {\mathbf U}^\ast {\mathbf B}_\ell {\mathbf V},$$ which helps to reduce computational costs.
The subspace projections and restrictions here for singular value optimization are closely related to the ones employed for eigenvalue optimization. Indeed $\sigma_J(\omega)$ and $\sigma_J^{{\mathcal U}, {\mathcal V}}(\omega)$ correspond to the $J$th largest eigenvalues of $$\begin{split}
\left[
\begin{array}{cc}
0 & {\mathbf B}(\omega) \\
{\mathbf B}^\ast (\omega) & 0 \\
\end{array}
\right]
\quad
{\rm and} \quad & \hskip 2.5ex
\left[
\begin{array}{cc}
0 & {\mathbf U}^\ast {\mathbf B}(\omega) {\mathbf V} \\
{\mathbf V}^\ast {\mathbf B}^\ast(\omega) {\mathbf U} & 0 \\
\end{array}
\right] \\
& =
\left[
\begin{array}{cc}
{\mathbf U}^\ast & {\mathbf V}^\ast
\end{array}
\right]
\left[
\begin{array}{cc}
0 & {\mathbf B}(\omega) \\
{\mathbf B}^\ast (\omega) & 0 \\
\end{array}
\right]
\left[
\begin{array}{c}
{\mathbf U} \\
{\mathbf V} \\
\end{array}
\right],
\end{split}$$ respectively. Hence the results in Section \[sec:eig\_subspace\], specifically Lemma \[lemma:Lipschitz\_continuity\] - \[thm:low\_rank\] and Theorem \[thm:accuracy\_rproblems\], extend to relate the singular values $\sigma_J(\omega)$ and $\sigma_J^{{\mathcal U}, {\mathcal V}}(\omega)$. For instance, monotonicity amounts to the following: for four subspaces ${\mathcal U}_1, {\mathcal U}_2, {\mathcal V}_1, {\mathcal V}_2$ of $\ell^2({\mathbb N})$ such that $\; {\mathcal U}_1 \subseteq {\mathcal U}_2$ and ${\mathcal V}_1 \subseteq {\mathcal V}_2$, we have $$\sigma_J^{{\mathcal U}_1, {\mathcal V}_1}(\omega) \quad \leq \quad \sigma_J^{{\mathcal U}_2, {\mathcal V}_2}(\omega) \quad \leq \quad \sigma_J(\omega).$$
The extended greedy subspace procedure for singular value optimization forms ${\mathcal U}$ and ${\mathcal V}$ from the left singular vectors and right singular vectors of ${\mathbf B}(\omega)$ at the optimizers of the reduced problems and at nearby points. A precise description is given in Algorithm \[algse\] below, where ${\mathcal U}_k$ and ${\mathcal V}_k$ denote the left and the right subspace at step $k$ and $\sigma_J^{(k)}(\omega) := \sigma_J^{{\mathcal U}_k, {\mathcal V}_k}(\omega)$. The basic greedy procedure is defined similarly by modifying lines 11 and 12 as $$\label{eq:algsvdb}
{\mathcal U}_k \; \gets \; {\mathcal U}_{k-1} \oplus {\rm span} \left\{ u^{(k)}_1, \dots, u^{(k)}_{J} \right\}
\;\;
{\rm and}
\;\;
{\mathcal V}_k \; \gets \; {\mathcal V}_{k-1} \oplus {\rm span} \left\{ v^{(k)}_1, \dots, v^{(k)}_{J} \right\}.$$ In the description, by a consistent pair of left and right singular vectors $u$ and $v$ corresponding to a singular value $\sigma$ of ${\mathbf B}(\omega)$, we mean the vectors satisfying ${\mathbf B}(\omega) v = \sigma u$ and ${\mathbf B}^\ast (\omega) u = \sigma v$ simultaneously.
$\omega^{(1)} \gets$ a random point in $\Omega$. $u^{(1)}_1, \dots, u^{(1)}_{J}$ and $v^{(1)}_1, \dots, v^{(1)}_{J}$ $\gets $ consistent left and right singular vectors corresponding to $\sigma_1(\omega^{(1)}), \dots, \sigma_{J}(\omega^{(1)})$. ${\mathcal U}_1 \; \gets \;
{\rm span} \left\{ u^{(1)}_1, \dots, u^{(1)}_{J} \right\}.
$ and ${\mathcal V}_1 \; \gets \;
{\rm span} \left\{ v^{(1)}_1, \dots, v^{(1)}_{J} \right\}.
$
$\omega^{(k)}$ -0.4ex $\gets$ -0.4ex any $\omega_\ast \in \arg\min_{\omega \in \Omega} \sigma^{(k-1)}_J(\omega) \;\;$ for the minimization problem, or\
$\omega^{(k)}$ -0.4ex $\gets$ -0.4ex any $\omega_\ast \in \arg\max_{\omega \in \Omega} \sigma^{(k-1)}_J(\omega) \;\;$ for the maximization problem. $u^{(k)}_1, \dots, u^{(k)}_{J}$ and $v^{(1)}_1, \dots, v^{(1)}_{J}$ $ \gets $ consistent left and right singular vectors corresponding to $\sigma_1(\omega^{(k)}), \dots, \sigma_{J}(\omega^{(k)})$. $h^{(k)} \gets \| \omega^{(k)} - \omega^{(k-1)} \|_2$ $u^{(k)}_{1,p q}, \dots, u^{(k)}_{J, p q}$ and $v^{(k)}_{1,p q}, \dots, v^{(k)}_{J, p q}$ $\gets $ consistent left and right singular\
vectors corresponding to $\sigma_1(\omega^{(k)} + h^{(k)} e_{pq}), \dots, \sigma_J(\omega^{(k)} + h^{(k)} e_{pq})$. ${\mathcal U}_k \; \gets \; {\mathcal U}_{k-1} \oplus {\rm span} \left\{ u^{(k)}_1, \dots, u^{(k)}_{J} \right\}
\oplus \left\{ \bigoplus_{p = 1, q = p}^d {\rm span} \left\{ u^{(k)}_{1,p q}, \dots, u^{(k)}_{J, p q} \right\} \right\}$. ${\mathcal V}_k \; \gets \; {\mathcal V}_{k-1} \oplus {\rm span} \left\{ v^{(k)}_1, \dots, v^{(k)}_{J} \right\}
\oplus \left\{ \bigoplus_{p = 1, q = p}^d {\rm span} \left\{ v^{(k)}_{1,p q}, \dots, v^{(k)}_{J, p q} \right\} \right\}$.
Observe that the reduced singular value function $\sigma_J^{(k)}(\omega)$ is the same as $\lambda_J^{(k)}(\omega)$ formed by Algorithm \[alge\] when it is applied to $$\label{eq:sval_matfun}
{\mathbf A}(\omega)
:=
\left[
\begin{array}{cc}
0 & {\mathbf B}(\omega) \\
{\mathbf B}^\ast(\omega) & 0
\end{array}
\right],$$ so Algorithm \[algse\] applied to ${\mathbf B}(\omega)$ and Algorithm \[alge\] applied to ${\mathbf A}(\omega)$ lead the same sequence $\{ \omega^{(k)} \}$. Similarly, the basic greedy subspace procedure for singular value optimization is equivalent to Algorithm \[alg\] operating on ${\mathbf A}(\omega)$. All of the convergence results deduced in Section \[sec:convergence\] for eigenvalue optimization, in particular
1. global convergence for the minimization problem (Theorem \[thm:global\_conv\]),
2. convergence of the sequence of maximal values of the reduced problems for the maximization problem (Theorem \[thm:global\_conv\_max\]),
3. superlinear rate-of-convergence for smooth optimizers (Theorem \[thm:super\_convergence\]),
carry over to this singular value optimization setting. Some of these results in Section \[sec:convergence\] are proven assuming the simplicity of $\lambda_J(\omega_\ast)$ at a particular $\omega_\ast$, which translates into a simplicity and a positivity assumption on $\sigma_J(\omega_\ast)$ in the singular value setting. Regarding issue 2. above, we observe in practice the convergence of $\{ \sigma_J^{(k)} (\omega^{(k+1)}) \}$ for the maximization problem to $\sigma_\ast$ such that $\sigma_J(\widetilde{\omega}) = \sigma_\ast$ for some local maximizer $\widetilde{\omega}$, that is not necessarily a global maximizer, similar to what we observe in the eigenvalue optimization setting.
Optimization of the $J$th Smallest Singular Value {#sec:small_singular_vals}
-------------------------------------------------
The minimum (maximum) of the $J$th smallest eigenvalue $\lambda_{-J}(\omega)$ of ${\mathbf A}(\omega)$ is equal to the negative of the maximum (minimum) of the $J$th largest eigenvalue of $-{\mathbf A}(\omega)$. Hence Algorithm \[alg\] and Algorithm \[alge\] can be adapted to optimize $\lambda_{-J}(\omega)$.
The optimization of the $J$th smallest singular value has a different nature. In particular it cannot be converted into an optimization problem involving the $J$th largest singular value. This is partly seen by the observation that the $J$th smallest singular value $\sigma_{-J}(\omega)$ of an operator ${\mathbf B}(\omega)$ of the form (\[eq:oper\_param\]) corresponds to an eigenvalue of ${\mathbf A}(\omega)$ as in (\[eq:sval\_matfun\]), right in the middle of its spectrum. For the restricted operator ${\mathbf B}^{{\mathcal U},{\mathcal V}}(\omega)$ defined as in (\[eq:oper\_param\_red\]) and its $J$th smallest singular value $\sigma_{-J}^{{\mathcal U}, {\mathcal V}}(\omega)$, monotonicity is lost as the subspaces ${\mathcal U}, {\mathcal V}$ expand. In particular $\sigma_{-J}^{{\mathcal U},{\mathcal V}}(\omega) \geq \sigma_{-J}(\omega)$ does not necessarily hold; the restriction of the domain of ${\mathbf B}(\omega)$ to ${\mathcal V}$ causes an increase in the $J$th smallest singular value, whereas the projection of the range onto ${\mathcal U}$ causes a decrease in the singular value. As a consequence of the loss of monotonicity, the interpolation properties do not hold anymore either.
A neater and theoretically more-sound approach is to employ only the restrictions from the right-hand side, that is a subspace procedure that operates on ${\mathbf B}^{\mathcal V}(\omega) := {\mathbf B}(\omega) {\mathbf V}$ and its $J$th smallest singular value. The resulting subspace procedures are equivalent to those for eigenvalue optimization applied to ${\mathbf B}^\ast(\omega) {\mathbf B}(\omega)$, so monotonicity and interpolation properties as well as all the theoretical convergence properties are regained.
A Comparison with the Cutting-Plane Methods {#sec:cutting_plane}
-------------------------------------------
The cutting plane method was introduced by Kelley to solve convex minimization problems [@Kelley1960]. For unconstrained minimization problems, the approach under-estimates convex functions with piece-wise linear functions globally, solves the resulting linear program, and refines the under-estimator with the addition of the new linear approximation about the optimizer of the linear program [@Bonnans2006 Section 9.3.2].
Let us consider again the minimization of the largest eigenvalue $\lambda_1(\omega)$ of an affine and Hermitian matrix-valued function $$A(\omega) \; = \; A_0 + \omega_1 A_j + \dots + \omega_d A_d$$ for given Hermitian matrices $A_0, \dots, A_d \in {\mathbb C}^{n\times n}$. Recall that the largest eigenvalue $\lambda_1(\omega)$ is convex [@Overton1988], so the cutting-plane method is applicable for its minimization as outlined in Algorithm \[algcut\]. In this description $\widehat{\lambda}^{(k)}_1(\omega)$ represents the piece-wise linear under-estimator for $\lambda_1(\omega)$ and is defined by $$\begin{aligned}
\widehat{\lambda}^{(k)}_1(\omega) & := & \max_{\ell = 1,\dots,k} \;\;
\lambda_1\left( \widehat{\omega}^{(\ell)} \right) +
\nabla \lambda_1 \left( \widehat{\omega}^{(\ell)} \right)^T \left( \omega - \widehat{\omega}^{(\ell)} \right) \\
& = &
\max_{\ell = 1,\dots,k} \;\; \left( \widehat{v}^{(\ell)} \right)^\ast A(\omega) \: \widehat{v}^{(\ell)}, \end{aligned}$$ where $\widehat{v}^{(\ell)}$ denotes a unit eigenvector corresponding to the largest eigenvalue of $A\left( \widehat{\omega}^{(\ell)} \right)$. Furthermore, in line 4 of the outline, $\widehat{\omega}^{(k)}$ denotes the unique minimizer of the convex function $\widehat{\lambda}^{(k-1)}_1(\omega)$.
$\widehat{\omega}^{(1)} \gets$ a random vector in ${\mathbb R}^d$. $\widehat{v}^{(1)} \; \gets \;$ a unit eigenvector corresponding to largest eigenvalue of $A\left( \widehat{\omega}^{(1)} \right)$. $\widehat{\omega}^{(k)} \gets \arg\min_{\omega \in \Omega} \widehat{\lambda}^{(k-1)}_1(\omega) \;\;$. $\widehat{v}^{(k)} \; \gets \;$ a unit eigenvector corresponding to largest eigenvalue of $A\left( \widehat{\omega}^{(k)} \right)$.
The basic greedy subspace procedure Algorithm \[alg\] for the minimization of the largest eigenvalue $\lambda_1(\omega)$ in this finite dimensional matrix-valued setting minimizes $\lambda^{(k)}_1(\omega)$, which, by the monotonicity property, under-estimates $\lambda_1(\omega)$. The sequences $\{ \omega^{(k)} \}$ and $\{ \widehat{\omega}^{(k)} \}$ generated by Algorithm \[alg\] and Algorithm \[algcut\] are not the same in general, but, for simplicity, let us suppose $\omega^{(\ell)} = \widehat{\omega}^{(\ell)}$ for $\ell = 1,\dots,k$ and for some $k$. In this case, for each $\omega \in \Omega$, we have $$\begin{aligned}
\lambda_1(\omega) \;\; \geq \;\; \lambda^{(k)}_1(\omega) & \;\; = & \max_{\alpha \in {\mathbb C}^m, \; \| \alpha \|_2 = 1} \;\; \alpha^\ast V_k^\ast A(\omega) V_k \alpha \\
& \;\;\geq & \;\; \max_{\ell = 1,\dots,k} \;\; \left( \widehat{v}^{(\ell)} \right)^\ast A(\omega) \: \widehat{v}^{(\ell)}
\;\; = \;\; \widehat{\lambda}^{(k)}_1(\omega)
\end{aligned}$$ where the columns of the matrix $V_k$ form an orthonormal basis for the subspace ${\rm span} \left\{ \widehat{v}^{(1)}, \dots, \widehat{v}^{(k)} \right\}$. This illustrates that, for this special case concerning the minimization of the largest eigenvalue of an affine matrix-valued function, the under-estimators used by the basic greedy subspace procedure are more accurate than those used by the cutting-plane method.
The better accuracy of the subspace procedure is apparent in Figure \[fig:support\] for the matrix-valued function $A(\omega) = A_0 + \omega A_1$, where $A_0, A_1$ are $10\times 10$ random symmetric matrices. In this figure, the function $\widehat{\lambda}_1^{(2)}(\omega)$ used by the cutting plane method is the maximum of two linear approximations about $\widehat{\omega}^{(1)} = 0$ and $\widehat{\omega}^{(2)} = 0.2$, while a two dimensional subspace is used by the subspace procedure with $\omega^{(1)} = 0$ and $\omega^{(2)} = 0.2$.
4ex
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ A comparison of the approximations by the cutting plane method and the subspace procedure: (Solid Curve) The largest eigenvalue of $A(\omega) = A_0 + \omega A_1$ for randomly selected $10\times 10$ symmetric $A_0, A_1$ with respect to $\omega \in [-0.2,0.4]$; (Dashed Curve) The function $\widehat{\lambda}_1^{(2)}(\omega)$ used by the cutting plane method with $\widehat{\omega}^{(1)} = 0$ and $\widehat{\omega}^{(2)} = 0.2$; (Dashed-Dotted Curve) The function $\lambda_1^{(2)}(\omega)$ used by the subspace procedure with $\omega^{(1)} = 0$ and $\omega^{(2)} = 0.2$; Additionally, the black dots represent $(0, \lambda_1(0))$ and $(0.2, \lambda_1(0.2))$. []{data-label="fig:support"}](support_functions3.pdf "fig:"){width=".85\textwidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Numerical Experiments {#sec:applications}
=====================
The next section specifies the Matlab software accompanying this work, and some important implementation details of this software. The rest of the section is devoted to three applications of large-scale eigenvalue and singular value optimization, namely, the numerical radius, the distance to instability from a matrix or a pencil, and the minimization of the largest eigenvalue of an affine matrix-valued function, which is closely related to semi-definite programming. On these examples we illustrate the power of the subspace procedures, introduced and analyzed in the previous sections, in practice.
Software and Implementation Details
-----------------------------------
MATLAB implementations accompanying this work are made publicly available on the internet[^1]. The current version includes generic routines for the following purposes (for a prescribed positive integer $J$):
- minimization of the $J$th largest eigenvalue (based on Algorithm \[alg\]);
- maximization of the $J$th largest eigenvalue (based on Algorithm \[alg\] without past if $d = 1$, or Algorithm \[alge\] without past if $d > 1$);
- minimization of the $J$th smallest singular value (based on an adaptation of Algorithm \[alg\] without past if $d = 1$, or Algorithm \[alge\] without past if $d > 1$, for singular value optimization).
The Matlab routines above terminate if one of the conditions $$\begin{split}
\textbf{(1)} \; \left| \lambda_J^{(k)}(\omega^{(k+1)}) - \lambda_J^{(k-1)}(\omega^{(k)}) \right| \; \leq \texttt{tol}
\;\;\: \left( {\rm or} \;\; \left| \sigma_J^{(k)}(\omega^{(k+1)}) - \sigma_J^{(k-1)}(\omega^{(k)}) \right| \; \leq \texttt{tol} \right),
\\
\textbf{(2)} \; \ \# \text{ subspace iterations } \; > \; \sqrt{n}, \hskip 50.8ex
\end{split}$$ hold for a prescribed tolerance `tol`, where $n$ is the size of the matrices $A_\ell$ in (\[eq:mat\_func2\]) for eigenvalue optimization, or the maximum of the dimensions $p$ and $q$ of the matrices $B_\ell$ in (\[eq:mat\_func2d\]) for singular value optimization. For all of the experiments in this section $\texttt{tol} = 10^{-12}$ is used unless otherwise specified. The second condition is never used in practice for the examples in this section, because the other condition is fulfilled after a few subspace iterations.
The reduced eigenvalue optimization and singular value optimization problems are solved by means of the the MATLAB package `eigopt` [@Mengi2014 Section 10]. These routines keep a lower bound and an upper bound for the optimal value of the reduced problem, and terminate when they differ by less than a prescribed tolerance. We set this tolerance equal to $0.1 \: \texttt{tol}$. Additionally, `eigopt` requires a global lower (upper) bound $\gamma$ for the minimization (maximization) problem on the second derivatives of the eigenvalue functions or the singular value functions, particular choices for the three applications in this section are specified below. Finally, `eigopt` performs the optimization on a box, which must be supplied by the user. Once again, particular box choices for the applications in this section are specified below.
Large-scale eigenvalue and singular value problems are solved iteratively by means of `eigs` and `svds` in Matlab. If the optimal values of the reduced eigenvalue or singular value functions at two consecutive iterations are close enough (i.e., if they differ by an amount less than $10^{-2}$), then we provide the shift $\lambda_{\rm cur} + 5 | \lambda_{\rm cur} - \lambda_{\rm pre} |$ ($\sigma_{\rm cur} - 5 | \sigma_{\rm cur} - \sigma_{\rm pre} |$ ) - with $\lambda_{\rm cur}, \lambda_{\rm pre}$ ($\sigma_{\rm cur}, \sigma_{\rm pre}$) denoting the optimal values of the current, previous reduced $J$th largest eigenvalue functions ($J$th smallest singular value functions) - to `eigs` (`svds`) and require it to compute the $J$ eigenvalues (singular values) closest to this shift. Otherwise, `eigs` and `svds` are called without shifts.
Numerical Radius {#sec:numrad}
----------------
The numerical radius $r(A)$ of a matrix $A \in {\mathbb C}^{n\times n}$, as also indicated at the end of Section \[sec:global\_convergence\], is the modulus of the outermost point in the field of values [@Horn1991; @He1997] formally defined by $$r(A) \;\; := \;\; \{ | z^\ast A z | \; | \; z \in {\mathbb C}^n, \; \| z \|_2 = 1 \}.$$ This quantity is used, for instance, to analyze the convergence of the classical iterative schemes for linear systems [@Axelsson1994; @Eiermann1993]. Recall from Section \[sec:global\_convergence\] that it has an eigenvalue optimization characterization, specifically $$r(A) \;\; = \;\; \max_{\omega \in [0, 2\pi]} \lambda_1(\omega) \quad\quad {\rm where} \;\;
A(\omega)
\; = \;
\frac{Ae^{i\omega} + A^\ast e^{-i\omega}}{2}.$$
We apply Algorithm \[alg\] without past as discussed in Section \[sec:subspace\_nopast\] for the computation of the numerical radius of two family of matrices. The box and $\gamma$ supplied to `eigopt` are $[0, 2\pi]$ and $2 \| A \|_2$, respectively. The latter is not guaranteed to be an upper bound on the second derivatives of the eigenvalue function, but it works well in practice in our experience.
[**Example 1:**]{} The Grcar matrix is a Toeplitz matrix with 1s on the main, first, second, third superdiagonals, -1s on the first subdiagonal, which exhibits ill-conditioned eigenvalues. It is used as a test matrix in the previous works [@He1997; @Uhlig2009] concerning the estimation of the numerical radius. Table \[table:numrad\_Grcar\] lists the computed values of the numerical radius by the subspace procedure and run-times in seconds for the Grcar matrices of sizes varying between $320 - 20480$. As the matrix sizes increase, the solution of large-scale eigenvalue problems take nearly all of the computation time. In contrast to this, the time spent to solve reduced eigenvalue optimization problems is very small and it is more or less constant as the sizes of the matrices increase. The reported values of the numerical radius for the Grcar matrices of sizes 320 and 640 match with the results reported in [@Uhlig2009] up to 12 decimal digits.
[**Example 2:**]{} This example concerns the gear matrix which is another Toeplitz matrix with 1s on the superdiagonal, subdiagonal, 1 at position $(1,n)$ and -1 at $(n,1)$. This matrix also has ill-conditioned, real eigenvalues lying in the interval $(-2,2)$. The computed values of the numerical radius by the subspace procedure for the gear matrix of size $n$ with $n$ varying in $320 - 20480$, as well as the run-times, are given in Table \[table:numrad\_Gearmat\]. Once again, the computation time for increasing $n$ is dominated by large-scale eigenvalue computations. The results reported for the gear matrices of sizes 320 and 640 match again with those reported in [@Uhlig2009] up to prescribed accuracy.
2.5ex
$n$ $\#$ iter $r(A)$ total time eigval comp reduced prob
------- ----------- ---------------- ------------ ------------- --------------
320 11 3.240793870067 8.3 0.8 7.4
640 12 3.241243679341 9.1 1.4 7.7
1280 13 3.241357030535 9.7 2.5 7.2
2560 15 3.241385481170 10.9 4.6 6.3
5120 16 3.241392607964 14.4 8.8 5.4
10240 18 3.241394391431 30.8 26.6 3.9
20480 19 3.241394837519 87.4 82.8 4.0
: The computed value of the numerical radius by Algorithm \[alg\] without past (i.e, the subspaces ${\mathcal S}_k$ are spanned by the eigenvectors at the last two iterations, hence two dimensional), the number of subspace iterations and the run-times in seconds are listed for the Grcar matrix of size $n$ for various $n$. Specifically, the number of subspace iterations (2nd column) to reach the prescribed accuracy, the numerical radius (3rd column), the total run-time (4th column), the time spent for large-scale eigenvalue computations (5th column) and the time spent for the solution of the reduced eigenvalue optimization problems (6th column) are reported w.r.t. the sizes of the matrices. []{data-label="table:numrad_Grcar"}
-3ex
2.5ex
$n$ $\#$ iter $r(A)$ total time eigval comp reduced prob
------- ----------- ---------------- ------------ ------------- --------------
320 5 1.999904217490 5.2 0.4 4.8
640 5 1.999975979457 5.5 0.8 4.6
1280 6 1.999993985476 6.2 1.2 5.0
2560 5 1.999998495194 5.8 1.9 3.9
5120 5 1.999999623651 6.3 3.0 3.3
10240 5 1.999999905895 7.7 5.6 2.1
20480 5 1.999999976471 16.4 14.6 1.6
: This table lists the computed value of the numerical radius by Algorithm \[alg\] without past (i.e, the subspaces ${\mathcal S}_k$ are spanned by the eigenvectors at the last two iterations, hence two dimensional) for the gear matrix of size $n$ for various $n$, as well as the number of subspace iterations, and run-times in seconds. For the meaning of each column see the caption of Table \[table:numrad\_Grcar\]. []{data-label="table:numrad_Gearmat"}
-5ex
Distance to Instability {#sec:distinstab}
-----------------------
-4ex The distance to instability from a square matrix $A \in {\mathbb C}^{n\times n}$ with all eigenvalues on the open left half of the complex plane, defined by $${\mathcal D}(A) \;\; := \;\;
\min
\{
\| \Delta \|_2 \; | \;
\exists \omega \in {\mathbb R}, \;\; \det (A + \Delta - \omega {\rm i} I) = 0
\},$$ is suggested in [@VanLoan1985] as a measure of robust stability for the autonomous system $x'(t) = Ax(t)$. An application of the Eckart-Young theorem [@Golub1996 Theorem 2.5.3] yields the characterization $${\mathcal D}(A) \;\; = \;\; \min_{\omega \in {\mathbb R}} \; \sigma_{-1} ( \omega )$$ where $\sigma_{-1}(\omega)$ denotes the smallest singular value of $B(\omega) = A - \omega {\rm i} I$.
Here also, we adopt Algorithm \[alg\] without past (see Section \[sec:subspace\_nopast\]), noting that at step $k+1$ the two dimensional right subspace is the span of the right singular vectors corresponding to $\sigma_{-1}(\omega^{(k)})$ and $\sigma_{-1}(\omega^{(k-1)})$. We illustrate numerical results on two family of sparse matrices. We set $\gamma$, the global lower bound for the second derivatives of the singular value function for `eigopt`, equal to $-2 \| A \|_2$, which is a heuristic that works well in practice. The boxes supplied to `eigopt` are $[-60,60]$ and $[150, 160]$ for the first and second family, respectively. These boxes indeed contain the global minimizers.
**Example 3:** Tolosa matrices arise from the stability analysis of an airplane. They are used as test examples in previous works [@He1998; @Freitag2011] concerning the computation of the distance to instability. These are stable matrices with all eigenvalues lying in the left half of the complex plane, but they are nearly unstable (see Figure 4 of [@Freitag2011] for the spectrum of the $340\times 340$ Tolosa matrix), indeed their perturbations at a distance 0.002 have eigenvalues on the right half plane. We run the subspace procedure on the Tolosa matrices of size 340, 1090, 2000, 4000, which are all available through the matrix market [@Boisvert]. According to Table \[table:distinstab\_Tolosa\] only four iterations suffice to reach prescribed accuracy. The time required for large-scale singular value computations increases with respect to the size of the matrices. However the majority of the time is consumed for the solution of the reduced problems, but this is only because the matrices are relatively small. The computed value of ${\mathcal D}(A)$ is the same in each case and match with the result reported in [@Freitag2011] up to prescribed accuracy.
2.5ex
$n$ $\#$ iter ${\mathcal D}(A)$ total time singval comp reduced prob
------ ----------- ------------------- ------------ -------------- --------------
340 4 0.001999796888 9.1 1.0 8.0
1090 4 0.001999796888 9.7 1.4 8.2
2000 4 0.001999796888 9.9 1.8 8.1
4000 4 0.001999796888 10.3 2.7 7.5
: The run-times in seconds, number of iterations (2nd column) and computed values of ${\mathcal D}(A)$ (3rd column) are listed for Algorithm \[alg\] without past applied to compute the distance to instability from the Tolosa matrices of size 340, 1090, 2000, 4000. The subspaces ${\mathcal S}_k$ are two dimensional and spanned by the right singular vectors computed at the last two iterations. The total run-times, the time for large-scale singular value computations and the time for the reduced optimization problems in seconds are provided in the 4th, 5th and 6th column, respectively. []{data-label="table:distinstab_Tolosa"}
-5ex
**Example 4:** This example is taken from [@He1998]. A finite difference discretization of an Orr-Sommerfeld operator with step-size $h$ for planar Poiseuille flow leads to an $n\times n$ generalized eigenvalue problem $B_n v = \lambda L_n v$ or an $n\times n$ standard eigenvalue problem $L_n^{-1} B_n v = \lambda v$ where $n = 2/h - 1$, $$\begin{split}
L_n \; & = \; \frac{1}{h^2} {\rm tridiag} (1, -(2 + h^2), 1) \\
B_n \; & = \; \frac{1}{100} L_n^2 - {\rm i} (U_n L_n + 2I),
\end{split}$$ and $U_n = {\rm diag} (1 - x_1^2, \dots, 1 - x_n^2)$ with $x_k = 1 + k\cdot h$ for $k = 1,\dots, n$. Matrix $A_n = L_n^{-1} B_n$ is stable with eigenvalues on the left half plane, yet nearly unstable. We apply the subspace procedure for the computation of the distance to instability for the Orr-Sommerfeld matrix $A_n$ of size $n = 400, 1000, 2000, 4000, 8000, 16000$. Note that $A_n$ is not sparse, yet applications of Arnoldi’s method for large-scale smallest singular value computations on $A_n - \omega {\rm i} I$ require the solutions of the linear systems of the form $(A_n - \omega {\rm i} I) v_{k+1} = v_k$ for $v_{k+1}$ for a given $v_k$. We equivalently solve the sparse linear system $$\label{eq:OS_linsys}
(B_n -\omega {\rm i} L_n) v_{k+1} = L_n v_k$$ in practice. In Table \[table:distinstab\_OrrSommerfeld\] we again observe that the total computation time is dominated by large-scale singular value computations as $n$ increases, whereas the contribution of the time for the solution of the reduced problems to the total running time is very little for large $n$. The computed values of ${\mathcal D}(A)$ in the table are listed only to six decimal digits, because for large $n$ `eigs` could not compute singular values beyond 7-8 decimal digits in a reliable fashion. We attribute this to the fact that the norm of $B_n$ and $L_n$ increase considerably as $n$ increases, so it is not possible to solve the linear system (\[eq:OS\_linsys\]) with high accuracy, for instance $\| B_{16000} \|_2 \approx 6.467 \cdot 10^{13}$. For smaller $n$ the computed solutions are accurate up to 12 decimal digits, for instance the computed value of the distance to instability for $A_{400}$ by the subspace procedure is 0.001978172281 which differ from the result reported in [@Freitag2011] by an amount less than $10^{-12}$.
5ex
$n$ $\#$ iter ${\mathcal D}(A)$ total time singval comp reduced prob
------- ----------- ------------------- ------------ -------------- --------------
400 9 0.001978 5.0 0.9 4.1
1000 8 0.001978 5.3 1.7 3.4
2000 7 0.001978 5.9 3.0 2.8
4000 8 0.001978 7.9 5.5 2.3
8000 8 0.001979 15.2 11.7 3.4
16000 7 0.001938 30.8 27.7 2.9
: The number of iterations, computed values of the distance to instability and run-times are given for Algorithm \[alg\] without past applied to estimate the distance to instability from the Orr-Sommerfeld matrices. Once again the subspaces are always two dimensional and spanned by the right singular vectors computed at the last two iterations. For the meaning of each column we refer to the caption of Table \[table:distinstab\_Tolosa\]. []{data-label="table:distinstab_OrrSommerfeld"}
-5ex
Minimization of the Largest Eigenvalue {#sec:min_large_eig}
--------------------------------------
-4ex A problem that drew substantial interest late 1980s and early 1990s [@Overton1988; @Fan1995] concerns the minimization of the largest eigenvalue of $\lambda_1(\omega)$ of $$\label{eq:affine_matrix}
A(\omega)
:=
A_0 + \omega_1 A_1 + \dots + \omega_d A_d$$ for given symmetric matrices $A_0, \dots, A_d \in {\mathbb R}^{n\times n}$. This problem is already discussed in Section \[sec:rate\_of\_convergence\] in the more general case, when $A_0, \dots, A_d$ are complex and Hermitian. Numerical results over there on random matrices indicate that the sequences generated by both the basic subspace procedure and the extended one converge at least superlinearly. An important application is in the context of semidefinite programming: under mild assumptions the dual of a semidefinite program can be expressed as an unconstrained minimization problem with the objective function $\lambda_1(\omega) + b^T \omega$ for some $b \in {\mathbb R}^d$ and with $\lambda_1(\omega)$ denoting the largest eigenvalue of a matrix-valued function $A(\omega)$ of the form (\[eq:affine\_matrix\]).
We apply the subspace procedures Algorithm \[alg\] and Algorithm \[alge\] for a particular notoriously difficult family of matrix-valued functions depending on two parameters. In this example the subspaces from the previous iterations are kept fully. Note that the Matlab software is based on Algorithm \[alg\], additionally we apply Algorithm \[alge\] for comparison purposes. As for the parameters for `eigopt`, since the largest eigenvalue function is convex, $\gamma$ (the global lower bound on the second derivatives of the eigenvalue function) in theory can be chosen zero, instead we set $\gamma = -10^{-6}$ for numerical reliability. We have specified the box containing the minimizer as $[-10,10]\times [-10, 10]$.
**Example 5:** In [@Overton1988], for $A_0 \in {\mathbb R}^{n\times n}$ with its $(k,j)$ entry equal to $$\begin{cases}
\min \{ k , j\} & \text{if $|k - j | > 1$} \\
\min \{ k , j\} + 0.1 & \text{if $|k-j| = 1$} \\
0 & k = j
\end{cases} \;\; ,$$ the spectral radius (i.e., the absolute value of the eigenvalue furthest away from the origin) of $C_n(\omega) = A_0 - \sum_{j=1}^d \omega_j e_j e_j^T$ is minimized. It is observed in that paper on the $n= 10$ case that at the optimal $\omega$, the eigenvalue with the largest modulus has multiplicity three. This problem would correspond to a semidefinite program relaxation of a max-cut problem, if $A_0$ had been a Laplace matrix of a graph [@Helmberg2000 Section 7].
Here we minimize the spectral radius of $$\label{eq:minlargesteig}
\widetilde{C}_n(\omega) = \frac{1}{100 n} A_0 - \omega_1 I_u - \omega_2 I_l$$ for $n = 250, 500, 1000, 2000$, where $I_u = {\rm diag} (I_{n/2}, 0_{n/2})$ and $I_l = {\rm diag} (0_{n/2}, I_{n/2})$. The scaling in front of $A_0$ is to make sure that the unique minimizer of the problem is inside $[-10, 10]\times [-10,10]$. This problem can equivalently be posed as the minimization of the largest eigenvalue of $A(\omega) := {\rm diag} (\widetilde{C}_n(\omega), -\widetilde{C}_n(\omega))$, so fits within the problem class described by (\[eq:affine\_matrix\]). Table \[table:specrad\] lists the minimal spectral radius values computed by the basic and extended subspace procedures along with number of subspace iterations and computation time. The total computation times are again dominated by the solutions of large eigenvalue problems. Furthermore, even though the basic subspace procedure usually requires more iterations to reach the prescribed accuracy, overall it solves fewer large eigenvalue problems and takes less computation time as compared to the extended subspace procedure.
In all cases the computed minimizer $\omega_\ast$ is such that the largest eigenvalue of $A(\omega_\ast)$ has algebraic multiplicity three, so $\lambda_1(\omega)$ is not differentiable at $\omega_\ast$. For instance for $n = 500$, that is when the matrix-valued function $A(\omega)$ is of size 1000, the five largest eigenvalues of $A(\omega_\ast)$ are listed in Table \[table:specrad\_extra\] on the left. Even the gaps between the remaining 497 positive eigenvalues are very small, as indeed among 500 positive eigenvalues 495 of them lie in an interval of length 0.017. The fact that most of the eigenvalues belong to a small interval causes poor convergence properties for `eigs`. On the other hand, it appears that the nonsmoothness does not affect the superlinear convergence of the iterates $\{ \lambda_1^{(k)}(\omega^{(k+1)}) \}$, as depicted in Table \[table:specrad\_extra\] on the right. This quick convergence is also apparent from the number of subspace iterations in Table \[table:specrad\]. Theorem \[thm:super\_convergence\] does not apply to this nonsmooth case. It is an open problem to come up with a formal argument explaining the quick convergence in this nonsmooth setting.
1ex
$n$ $\#$ iter $p$ $\rho_\ast$ total time eigval comp reduced prob
------ ----------- ----------- ---------------- ------------ ------------- --------------
250 7 $\:$7$\:$ 0.509646245274 3.8 1.5 2.2
500 7 7 1.016261471669 4.8 2.8 1.6
1000 8 8 3.584040976076 13.9 11.8 1.2
2000 7 7 4.055903987776 68.7 65.8 0.7
: The table lists the number of subspace iterations, the dimension of the subspace ${\mathcal S}_k$ at termination ($p$), computational times in seconds and the computed minimal value ($\rho_\ast$) of the spectral radius when Algorithm \[alg\] (on the top) and Algorithm \[alge\] (at the bottom) are applied to minimize the largest eigenvalue of $A(\omega) = {\rm diag} (\widetilde{C}_n(\omega), -\widetilde{C}_n(\omega))$ for the matrix-valued function $\widetilde{C}_n(\omega)$ defined as in (\[eq:minlargesteig\]). Note that $A(\omega)$ whose largest eigenvalue is minimized is of size $2n$. Also note that the subspaces from all of the previous iterations are kept in these example. []{data-label="table:specrad"}
1ex
$n$ $\#$ iter $p$ $\rho_\ast$ total time eigval comp reduced prob
------ ----------- ----- ---------------- ------------ ------------- --------------
250 6 24 0.509646245274 4.7 2.8 1.6
500 6 24 1.016261471669 7.6 5.4 1.1
1000 7 28 3.584040976076 22.1 17.8 1.0
2000 7 28 4.055903987776 115.9 106.7 0.8
: The table lists the number of subspace iterations, the dimension of the subspace ${\mathcal S}_k$ at termination ($p$), computational times in seconds and the computed minimal value ($\rho_\ast$) of the spectral radius when Algorithm \[alg\] (on the top) and Algorithm \[alge\] (at the bottom) are applied to minimize the largest eigenvalue of $A(\omega) = {\rm diag} (\widetilde{C}_n(\omega), -\widetilde{C}_n(\omega))$ for the matrix-valued function $\widetilde{C}_n(\omega)$ defined as in (\[eq:minlargesteig\]). Note that $A(\omega)$ whose largest eigenvalue is minimized is of size $2n$. Also note that the subspaces from all of the previous iterations are kept in these example. []{data-label="table:specrad"}
-3ex
[cc]{}
14ex
Eigenvalues of $A(\omega_\ast)$
---------------------------------
1.016261471669
1.016261471669
1.016261471669
1.016234975093
1.016234803766
: (Left) The five largest eigenvalues of $A(\omega) = {\rm diag} (\widetilde{C}_{500}(\omega), -\widetilde{C}_{500}(\omega))$ at $\omega_\ast$ where $\widetilde{C}_n(\omega)$ defined as in (\[eq:minlargesteig\]) and $\omega_\ast$ is the minimizer of the largest eigenvalue of $A(\omega)$ for $n = 500$. (Right) The last four iterates $\{ \lambda_1^{(k)}(\omega^{(k+1)}) \}$ of Algorithm \[alg\] (which keeps all of the subspaces from the previous iterations) when the large eigenvalue computations are performed directly by calling `eig` in MATLAB. []{data-label="table:specrad_extra"}
4ex & 4ex
[c]{}\
$k$ $\lambda_1^{(k)}(\omega^{(k+1)})$
----- -----------------------------------
3 1.016260414001
4 1.016261417096
5 1.016261471669
6 1.016261471669
: (Left) The five largest eigenvalues of $A(\omega) = {\rm diag} (\widetilde{C}_{500}(\omega), -\widetilde{C}_{500}(\omega))$ at $\omega_\ast$ where $\widetilde{C}_n(\omega)$ defined as in (\[eq:minlargesteig\]) and $\omega_\ast$ is the minimizer of the largest eigenvalue of $A(\omega)$ for $n = 500$. (Right) The last four iterates $\{ \lambda_1^{(k)}(\omega^{(k+1)}) \}$ of Algorithm \[alg\] (which keeps all of the subspaces from the previous iterations) when the large eigenvalue computations are performed directly by calling `eig` in MATLAB. []{data-label="table:specrad_extra"}
-5ex
Concluding Remarks {#sec:conclusion}
==================
-3ex We have proposed subspace procedures to cope with large-scale eigenvalue and singular value optimization problems. To optimize the $J$th largest eigenvalue of a Hermitian and analytic matrix-valued function $A(\omega)$ over $\omega$ for a prescribed integer $J$, the subspace procedures operate on a small matrix-valued function that acts like $A(\omega)$ in a small subspace. The subspace is expanded with the addition of the eigenvectors of $A(\omega)$ at the optimizer of the eigenvalue function of the small matrix-valued function and, possibly, at nearby points. A similar strategy is adopted to optimize the $J$th largest singular value of an analytic matrix-valued function $B(\omega)$. In that context, it is advantageous to use two different subspace restrictions on the input and the output to $B(\omega)$ so that the resulting small matrix-valued function acts like the original one only in these small input and output spaces. The subspaces are expanded with the inclusion of the left and right singular vectors of $B(\omega)$ at the optimizers of the small problems and at nearby points. The optimization of the $J$th smallest singular value involves some subtlety, here it seems suitable to apply restrictions only on the input to $B(\omega)$, which is equivalent to the frameworks for eigenvalue optimization applied to $B(\omega)^\ast B(\omega)$. The preferred subspace procedures for particular cases are summarized in the table below.
[|p[2.7cm]{}||p[9.5cm]{}|]{}
**Problem**
&
**Preferred Subspace Procedure**
\
**(1)** $\; \min_{\omega \in \Omega} \lambda_J(\omega)$
&
Algorithm \[alg\]
\
**(2)** $\; \max_{\omega \in \Omega} \lambda_J(\omega)$
&
Algorithm \[alge\] (Algorithm \[alg\]) without past if $d > 1$ (if $d=1$)
\
**(3)** $\;\: \min_{\omega \in \Omega} \sigma_J(\omega)$
&
Algorithm \[algse\](b)
\
**(4)** $\;\: \max_{\omega \in \Omega} \sigma_J(\omega)$
&
Algorithm \[algse\] (Algorithm \[algse\](b)) without past if $d > 1$ (if $d=1$)
\
**(5)** $ \min_{\omega \in \Omega} \sigma_{-J}(\omega)$
&
Algorithm \[alge\](s) (Algorithm \[alg\](s)) without past if $d > 1$ (if $d = 1$)
\
**(6)** $ \max_{\omega \in \Omega} \sigma_{-J}(\omega)$
&
Algorithm \[alg\](s)
\
In the table, Algorithm \[algse\](b) refers to the basic greedy procedure for singular value optimization that uses two-sided projections, i.e., Algorithm \[algse\] but with lines 11 and 12 replaced by (\[eq:algsvdb\]). Additionally, Algorithm \[alg\](s) and Algorithm \[alge\](s) refer to the adaptations of Algorithm \[alg\] and Algorithm \[alge\] for singular value optimization, which form the subspace from the right singular vectors rather than the eigenvectors. Note that the minimization and maximization of a $J$th smallest eigenvalue are not listed in the table, since they can be posed as the maximization and minimization of a $J$th largest eigenvalue, respectively.
We have performed convergence and rate-of-convergence analyses for these subspace procedures by extending $A(\omega)$ and $B(\omega)$ to infinite dimension, so by replacing them with compact operators ${\mathbf A}(\omega) : \ell^2({\mathbb N}) \rightarrow \ell^2({\mathbb N})$ and ${\mathbf B}(\omega) : \ell^2({\mathbb N}) \rightarrow \ell^2({\mathbb N})$, former of which is also self-adjoint. Most remarkably, Theorem \[thm:global\_conv\] establishes global convergence for Algorithms \[alg\]-\[alge\] when the $J$th largest eigenvalue is minimized, and Theorem \[thm:super\_convergence\] establishes a superlinear rate-of-convergence for Algorithm \[alg\] when $d = 1$ and Algorithm \[alge\] for minimizing and maximizing the $J$th largest eigenvalue. The superlinear convergence result is established under the simplicity assumption on the $J$th largest eigenvalue at the optimizer, even though we observe superlinear convergence in numerical experiments (e.g., see Example 5 in Section \[sec:min\_large\_eig\]) where this simplicity assumption is violated. The convergence results do also extend to Algorithm \[algse\] for the optimization of the $J$th largest singular value. The convergence properties of the proposed subspace frameworks for the six problems in the table above are as follows: **(1)$\&$(3)$\&$(6)** Proven global convergence at a proven (observed) superlinear rate if $d = 1$ ($d > 1$); **(2)$\&$(4)$\&$(5)** Proven convergence but necessarily to a global solution, observed local convergence at a proven superlinear rate.
Two additional problems where the subspace procedures and their convergence analyses developed here may be applicable are large-scale sparse estimation problems and semidefinite programs. For instance, sparse estimation problems that can be cast as nuclear norm minimization problems [@Recht2010] seem worth exploring because of their connection with singular value optimization. As noted in the text, the dual of a standard semidefinite program can often be cast as an eigenvalue optimization problem involving the minimization of the largest eigenvalue. A systematic integration of the subspace frameworks proposed here for eigenvalue optimization into large-scale semidefinite programs, motivated by their theoretical convergence properties, is a direction that is worth investigating.
**Acknowledgement.** We are grateful to the two anonymous referees, Daniel Szyld and Daniel Kressner for valuable comments on this manuscript.
[^1]: `http://home.ku.edu.tr/`$\sim$`emengi/software/leigopt`
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Integrating artificial intelligence (AI) into wireless networks has drawn significant interest in both industry and academia. A common solution is to replace partial or even all modules in the conventional systems, which is often lack of efficiency and robustness due to their ignoring of expert knowledge. In this paper, we take deep reinforcement learning (DRL) based scheduling as an example to investigate how expert knowledge can help with AI module in cellular networks. A simulation platform, which has considered link adaption, feedback and other practical mechanisms, is developed to facilitate the investigation. Besides the traditional way, which is learning directly from the environment, for training DRL agent, we propose two novel methods, i.e., learning from a dual AI module and learning from the expert solution. The results show that, for the considering scheduling problem, DRL training procedure can be improved on both performance and convergence speed by involving the expert knowledge. Hence, instead of replacing conventional scheduling module in the system, adding a newly introduced AI module, which is capable to interact with the conventional module and provide more flexibility, is a more feasible solution.'
author:
- 'Emails: {wangjian23, xuchen14, huangfuyourui, lirongone.li, yiqun.ge, justin.wangjun}@huawei.com'
bibliography:
- 'rlscheduling.bib'
title: Deep Reinforcement Learning for Scheduling in Cellular Networks
---
artificial intelligence, cellular networks, deep reinforcement learning, scheduling, proportional fair
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Let $k$ be a number field, with algebraic closure $\bar{k}$, and let $\mathcal{A}$ be an abelian variety over $k$ of dimension $n=2^h$, where $h\geq 0$. Let $p$ be a prime number and let ${{\mathcal{A}}}[p]$ denote the $p$-torsion subgroup of ${{\mathcal{A}}}$. We prove that for every $h$, there exists a prime $p_h$, depending only on $h$, such that if ${{\mathcal{A}}}[p]$ is either an irreducible or a decomposable ${{\rm Gal}}(\bar{k}/k)$-module, then for all primes $p>p_h$ the local-global divisibility by $p$ holds in ${{\mathcal{A}}}(k)$ and $\Sha^1 (k,{{\mathcal{A}}}[p])$ is trivial. In particular, when ${{\mathcal{A}}}$ has dimension 2 or 4, we show $p_h=3$. This result generalizes some previous ones proved for elliptic curves. In the case when ${{\mathcal{A}}}$ is principally polarized, the vanishing of $\Sha^1 (k,{{\mathcal{A}}}[p])$ implies that the elements of the Tate-Shafarevich group $\Sha(k,{{\mathcal{A}}})$ are divisible by $p$ in the Weil-Châtelet group $H^1(k,{{\mathcal{A}}})$ and the local-global principle for divisibility by $p$ holds in $H^r(k,{{\mathcal{A}}})$, for all $r\geq 0$.'
author:
- 'Laura Paladino[^1]'
date:
title: Divisibility questions in abelian varieties
---
startsection [section]{}[1]{}[@]{}[-5.5ex plus -.5ex minus -.2ex]{}[1ex plus .2ex]{}[****]{}
============================================================================================
\[section\] \[thm\][Main Theorem]{}
\[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{}
\[thm\][Property]{}
\[thm\][Remark]{} \[thm\][Definition]{}
0.5cm
1.5cm
Introduction
============
We consider two local-global problems, strongly related, that recently arose as generalizations of some classical questions. The setting is the one of an abelian variety ${{\mathcal{A}}}$ of dimension $g$ defined over number field $k$. Let $\bar{k}$ be the algebraic closure of $k$ and let $M_k$ be the set of places $v$ of $k$. For every positive integer $q$, we denote by ${{\mathcal{A}}}[q]$ be the $q$-torsion subgroup of ${{\mathcal{A}}}$ and by $k({{\mathcal{A}}}[q])$ the number field obtained by adding to $k$ the coordinates of the $q$-torsion points of ${{\mathcal{A}}}$. It is well-known that ${{\mathcal{A}}}[q]\simeq ({{\mathbb Z}}/q{{\mathbb Z}})^{2g}$ and that the Galois group ${{\rm Gal}}(k({{\mathcal{A}}}[q])/k)$ is isomorphic to the image of the representation of the absolute Galois group ${{\rm Gal}}(\bar{k}/k)$ in the general linear group ${{\rm GL}}_{2g}({{\mathbb Z}}/q{{\mathbb Z}})$. The behaviour of ${{\rm Gal}}(k({{\mathcal{A}}}[q])/k)$ is related to the answer of the following question, known as *Local-global divisibility problem*.
\[prob1\] Let $P\in {\mathcal{A}}(k)$ and let $q$ be a positive integer. Assume that for all but finitely many valuations $v\in k$, there exists $D_v\in {\mathcal{A}}(k_v)$ such that $P=qD_v$. Is it possible to conclude that there exists $D\in {\mathcal{A}}(k)$ such that $P=qD$?
This problem was stated in 2001 by Dvornicich and Zannier in the more general case when ${{\mathcal{A}}}$ is a commutative algebraic group and its formulation was motivated by a particular case of the famous Hasse Principle on quadratic forms and by the Grunwald-Wang Theorem (see [@DZ] and [@DZ2]). The vanishing of the first cohomology group $H^1({{\rm Gal}}(k({{\mathcal{A}}}[q])/k),{{\mathcal{A}}}[q])$ assures a positive answer (see for instance [@DZ], [@Won]). Clearly a solution to Problem \[prob1\] for all powers $p^l$ of prime numbers $p$ is sufficient to get an answer for all integers $q$, by the unique factorization in ${{\mathbb Z}}$ and Bézout’s identity.
In the case of elliptic curves the problem has been widely studied since 2001 and recently a complete answer has been proved when $k={{\mathbb Q}}$. The answer is affirmative when $q$ is a prime $p$ (see [@DZ], [@Won]) and for powers $p^l$, where $p\geq 5$ and $l\geq 2$ (see [@PRV2]). On the contrary, the answer is negative for $q=p^l$, with $p\in \{2,3\}$ and $l\geq 2$ (see [@Cre], [@DZ2], [@Pal], [@Pal3]). For a general number field $k$, the answer is still positive when $q$ is a prime $p$ (see [@DZ], [@Won]). With a mild hypothesis on $k$, the proof of [@DZ3 Theorem 1] implies the following statement (see also [@PRV]).
\[DZ1\] Let $p$ be a prime. Let ${{\mathcal{E}}}$ be an elliptic curve defined over a number field $k$ which does not contain the field ${{\mathbb Q}}({{\zeta}}_p+{{\zeta}}_p^{-1})$, where ${{\zeta}}_p$ is a primitive $p$th root of the unity. If ${{\mathcal{E}}}$ does not admit any $k$-rational isogeny of degree $ p $, then the local-global principle holds for divisibility by $p^l$ in ${{\mathcal{E}}}$ over $k$, for every positive integer $l$.
The hypothesis that $k$ does not contain ${{\mathbb Q}}({{\zeta}}_p+{{\zeta}}_p^{-1})$ is necessary (see [@PRV2 Sec. 6]). Stronger criteria for the local-global divisibility in elliptic curves have been given in [@PRV] and [@PRV2]. In particular there exists a prime $p_k$, depending only on $k$, such that if $p>p_k$ then the answer is positive for divisibility by $p^l$, for all $l\geq 1$. Here we prove the following statements that assures the local-global divisibility by $p$ on some abelian varieties of higher dimension satisfying certain conditions.
\[P17\_gal\] Let $p$ be a prime number. Let $k$ be a number field that does not contain ${{\mathbb Q}}({{\zeta}}_p+{{\zeta}}_p^{-1})$. Let ${{\mathcal{A}}}$ be an abelian variety defined over $k$, of dimension $n=2^h$, where $h\geq 0$. For every $h$, there exists a prime $p_h$, depending only on $h$, such that if ${{\mathcal{A}}}[p]$ is either an irreducible or a decomposable ${{\rm Gal}}(\bar{k}/k)$-module, then the local-global divisibility by $p$ holds in ${{\mathcal{A}}}(k)$, for all $p\geq p_h$. In particular, for abelian varieties of dimension 2 and 4, we have $p_1=p_2=3$.
Evidently, Theorem \[P17\_gal\] implies the following result that reminds of Theorem \[DZ1\] for divisibility by $p$ in higher dimension.
\[P17\] Let $p$ be a prime number. Let $k$ be a number field that does not contain ${{\mathbb Q}}({{\zeta}}_p+{{\zeta}}_p^{-1})$. Let ${{\mathcal{A}}}$ be an abelian variety defined over $k$, of dimension $n=2^h$, where $h\geq 0$. For every $h$, there exists a prime $p_h$, depending only on $h$, such that if ${{\mathcal{A}}}$ does not admit a $k$-rational isogeny of degree $p^{\alpha}$, with $1\leq \alpha\leq 2n-1$, then the local-global divisibility by $p$ holds in ${{\mathcal{A}}}(k)$, for all $p\geq p_h$. In particular, for abelian varieties of dimension 2 and 4, we have $p_1=p_2=3$.
Both Theorem \[P17\_gal\] and its direct consequence Corollary \[P17\] follow immediately by the proof of the next statement.
\[P1\_bis\] Let $p$ be a prime number and let $l,m$ be positive integers. Let $k$ be a number field that does not contain ${{\mathbb Q}}({{\zeta}}_p+{{\zeta}}_p^{-1})$. Let ${{\mathcal{A}}}$ be an abelian variety defined over $k$, of dimension $2^h$, where $h\geq 0$. Let $n=2^{h+1}$ and assume that ${{\rm Gal}}(k({{\mathcal{A}}}[p^l])/k)$ is isomorphic to a subgroup of ${{\rm GL}}_{n}(p^m)$, for some positive integer $m$. For every $h$, there exists a prime $p_h$, depending only on $h$, such that if $p>p_h$ and the local-global divisibility by $p$ fails in ${{\mathcal{A}}}(k)$, then ${{\rm Gal}}(k({{\mathcal{A}}}[p^l])/k)$ acts reducibly but not decomposably over ${{\mathcal{A}}}[p^l]$. In particular, for abelian varieties of dimension 2 and 4, we have $p_1=p_2=3$.
Our proof of Theorem \[P1\_bis\] shows that the Tate-Shafarevich group $\Sha^1(k,{{\mathcal{A}}}[p])$ is trivial when ${{\mathcal{A}}}[p]$ is either an irreducible or a decomposable ${{\rm Gal}}(\bar{k}/k)$-module and $p>p_h$. If ${{\mathcal{A}}}$ is principally polarized, then the triviality of $\Sha^1(k,{{\mathcal{A}}})$ implies $\Sha(k,{{\mathcal{A}}})\subseteq p H^r(k,{{\mathcal{A}}})$, for all $r\geq 0$, by [@Cre2 Theorem 2.1]. In that case we have an affirmative answer to the following second and more general problem, for all $r$.
\[prob2\] Let $q$ be a positive integer and let $\sigma\in H^r(k,{{\mathcal{A}}})$. Assume that for all $v\in M_k$ there exists $\tau_v\in H^r(k_v,{{\mathcal{A}}})$ such that $q\tau_v=\sigma$. Can we conclude that there exists $\tau \in H^r(k,{{\mathcal{A}}})$, such that $q\tau=\sigma$?
Problem \[prob2\] was firstly considered by Cassel for $r=1$ in the case when ${{\mathcal{A}}}$ is an elliptic curve ${{\mathcal{E}}}$ (see [@Cas Problem 1.3]). In particular Cassels questioned if the elements of the Tate-Shafarevich group $\Sha(k,{{\mathcal{E}}})$ were divisible by $p^l$ in the Weil-Châtelet group $H^1(k,{{\mathcal{E}}})$, for all $l$. Tate produced an affirmative answer for divisibility by $p$, but the question for powers $p^l$, with $l\geq 2$ remained open (see [@Cas2]). The mentioned results to Problem \[prob1\] imply an answer to Problem \[prob2\] too, since the proofs show the triviality or the non triviality of the corresponding Tate-Shafarevich group. So Cassel’s question has an affirmative answer for $p\geq 5$ and a negative one for $p\in \{2,3\}$ in elliptic curves. The problem was afterwards considered for abelian varieties by Bašmakov (see [@Bas]) and lately by Çiperiani and Stix, who gave some sufficient conditions for a positive answer (see [@CS]). In [@Cre] Creutz proved that for every prime $p$, there exist infinitely many non-isomorphic abelian varieties $A$ defined over ${{\mathbb Q}}$ such that $\Sha(k,A) \not\subseteq pH^1(k,A)$. In abelian varieties of dimension strictly greater than 1, even the local-global divisibility by $p$ may fail for both Problem 1 and Problem 2 (see also [@DZ §3]). Here we prove the following statement.
\[P\_Sha\] Let $p$ be a prime number. Let $k$ be a number field that does not contain ${{\mathbb Q}}({{\zeta}}_p+{{\zeta}}_p^{-1})$. Let ${{\mathcal{A}}}$ be a principally polarized abelian variety defined over $k$, of dimension $2^h$, where $h\geq 0$. There exists a prime $p_h$, depending only on $h$, such that if the $p$-torsion subgroup ${{\mathcal{A}}}[p]$ of ${{\mathcal{A}}}$ is either an irreducible or a decomposable ${{\rm Gal}}(\bar{k}/k)$-module, then the elements of $\Sha(k,{{\mathcal{A}}})$ are divisible by $p$ in the Weil-Châtelet group $H^1(k,{{\mathcal{A}}})$, i. e. $\Sha(k,{{\mathcal{A}}})\subseteq p H^1(k,{{\mathcal{A}}})$, for all $p>p_h$. In particular for abelian varieties of dimension 2 and 4 we have $p_h=3$.
As mentioned above, the conclusion of Theorem \[P\_Sha\] assures an affirmative answer to Problem 2, for all $r\geq 0$, in the case when ${{\mathcal{A}}}$ is abelian variety satisfying the hyphoteses of the statement and $p>p_h$. Then, for such abelian varieties and $p>p_h$, we have a local-global principle for divisibility by $p$ in $H^r(K,{{\mathcal{A}}})$, for all $r$. The result is particularly interesting for abelian varieties of dimension 2 or 4, since we have an explicit $p_h=3$.
A few preliminary results in the theory of groups and in local-global divisibility are stated in next section. In Section 2 we treat the special case in which ${{\rm Gal}}(k({{\mathcal{A}}}[p^l])/k)$ acts decomposably over $k({{\mathcal{A}}}[p^l])$, in particular when ${{\mathcal{A}}}$ is a product of elliptic curves. In the last and main part of the paper, we proceed with the proof of Theorem \[P1\_bis\].
Preliminary results
===================
We recall some known results about local-global divisibility and about group theory, that will be useful for the proof of Theorem \[P1\_bis\].
As above, let $k$ be a number field and let ${{\mathcal{A}}}$ be an abelian variety of dimension $g$, defined over $k$. Let $q:=p^l$, where $p$ is a prime number and $l$ is a positive integer. As introduced before, the $p^l$-torsion subgroup of ${{\mathcal{A}}}$ will be denoted by ${{\mathcal{A}}}[p^l]$ and the number field obtained by adding to $k$ the coordinates of the points in ${{\mathcal{A}}}[p^l]$ will be denoted by $F:=k({{\mathcal{A}}}[p^l])$. The $p$-torsion subgroup ${{\mathcal{A}}}[p^l]$ of ${{\mathcal{A}}}$ is a $G_k$-module, where $G_k$ denotes the absolute Galois group ${{\rm Gal}}(\bar{k}/k)$. Since ${{\mathcal{A}}}[p^l]\simeq ({{\mathbb Z}}/p^l{{\mathbb Z}})^{n}$, with $n=2g$, then $G_k$ acts over ${{\mathcal{A}}}[p^l]$ as a subgroup of ${{\rm GL}}_{n}({{\mathbb Z}}/p^l{{\mathbb Z}})$ isomorphic to $G:=\textrm{Gal}(k({{\mathcal{A}}}[p^l])/k)$. We still denote by $G$ the representation of $G_k$ in ${{\rm GL}}_{n}({{\mathbb Z}}/p^l{{\mathbb Z}})$. If $l=1$, in particular $G\leq {{\rm GL}}_{{n}}(p)$.
Let $\Sigma$ be a subset of $M_k$ containing all but finitely many places $v$, such that $v\notin \Sigma$, for all $v$ ramified in $F$. For every $v\in \Sigma$, we denote by $G_v$ the Galois group ${{\rm Gal}}(F_w/k_v)$, where $w$ is a place of $F$ extending $v$. In [@DZ] Dvornicich and Zannier proved that the answer to the local-global question for divisibility by $q$ of points in ${{\mathcal{A}}}(k)$ is linked to the behaviour of the following subgroup of $H^1(G,{{\mathcal{A}}}[q])$
$$\label{h1loc}
H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[q]):=\bigcap_{v\in \Sigma} \ker H^1(G,{{\mathcal{A}}}[q])\xrightarrow{\makebox[1cm]{{\small $res_v$}}} H^1(G_v,{{\mathcal{A}}}[q]),$$
where $res_v$, as usual, denotes the restriction map. By substituting $M_k$ to $\Sigma$ in , i. e. by letting $v$ vary over all the valuations of $k$, we get the classical definition of the Tate-Shafarevich group $\Sha^1(k,{{\mathcal{A}}}[q])$ (up to isomorphism)
$$\Sha^1(k,{{\mathcal{A}}}[q]):=\bigcap_{v\in M_k} \ker H^1(k,{{\mathcal{A}}}[q])\xrightarrow{\makebox[1cm]{{\small $res_v$}}} H^1(k_v,{{\mathcal{A}}}[q]).$$
In particular, the vanishing of $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[q])$ assures the triviality of $\Sha^1(k,{{\mathcal{A}}}[q])$, that is a sufficient condition to get an affirmative answer to Problem \[prob2\], for all $r\geq 0$, in the case when ${{\mathcal{A}}}$ is principally polarized (see [@Cre2 Theorem 2.1]). Furthermore the vanishing of $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[q])$ is a sufficient condition for an affirmative answer to Problem \[prob1\] (see [@DZ Proposition 2.1]).
Because of Čebotarev’s Density Theorem, the group $G_v$ varies over all cyclic subgroups of $G$ as $v$ varies in $\Sigma$, then in [@DZ] Dvornicich and Zannier gave the following equivalent definition of $H_{\textrm{loc}}^1(G,A[q])$.
\[loc\_cond\] A cocycle $\{Z_{\sigma}\}_{\sigma\in G}\in H^1(G,{{\mathcal{A}}}[q])$ satisfies the local conditions if, for every $\sigma\in G$, there exists $A_{\sigma}\in {{\mathcal{A}}}[q]$ such that $Z_{\sigma}=(\sigma-1)A_{\sigma}$. The subgroup of $H^1(G,{{\mathcal{A}}}[q])$ formed by all the cocycles satisfying the local conditions is called *first local cohomological group* of $G$ with values in ${{\mathcal{A}}}[q]$ and it is denoted by $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[q])$.
The description of $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[q])$ given in Definition \[loc\_cond\] is useful in proving its triviality and even in producing counterexamples to the local-global divisibility. We keep the notation $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[q])$ used in almost all previous papers about the topic, but it is worth to mention that in [@San] Sansuc already treated similar modified Tate-Shafarevich groups as in and introduced the notation $\Sha^1_{\Sigma}(k,{{\mathcal{A}}})$.
The vanishing of $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])$ is strongly related to the behaviour of $H^1_{\textrm{loc}}(G_p,{{\mathcal{A}}}[p^l])$, where $G_p$ is the $p$-Sylow subgroup of $G$ (see [@DZ]).
\[Sylow\] Let $G_p$ be the $p$-Sylow subgroup of $A$. An element of $H^1_{\textrm{loc}}({{\mathcal{A}}}, {{\mathcal{A}}}[p^l])$ is zero if and only if its restriction to $H^1_{\textrm{loc}}(G_p, {{\mathcal{A}}}[p^l])$ is zero.
In some cases, a quick way to show that both $H^1_{\textrm{loc}}(G, {{\mathcal{A}}}[p^l])$ and $H^1_{\textrm{loc}}(G_p, {{\mathcal{A}}}[p^l])$ are trivial is the use of Sah’s Theorem (see [@Lan Theorem 5.1]).
\[Sah\] Let $G$ be a group and let $M$ be a $G$-module. Let $\alpha$ be in the center of $G$. Then $H^1 (G, M )$ is annihilated by the map $x \rightarrow \alpha x - x$ on $M$. In particular, if this map is an automorphism of $M$, then $H^1 (G, M ) = 0$.
By Lemma \[Sah\], if $G$ is a subgroup of ${{\rm GL}}_n(p^l)$ that contains a non-trivial scalar matrix, then $H^1(G,{{\mathbb Z}}/q{{\mathbb Z}})=0$. Thus, in particular, the same holds for $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[q])=0$.
\[scalar\] Let $G\leq {{\rm GL}}_n(q)$, for some positive integers $n$ and $q$. If $\lambda\cdot I_n\in G$ is a nontrivial scalar matrix, then $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[q])=0$.
0.1cm
In our proof of Theorem \[P1\_bis\], a crucial tool is the use of Aschbacher’s Theorem on the classification of maximal subgroups of ${{\rm GL}}_n(q)$ (see [@Asc]). Aschbacher proved that the maximal subgroups of ${{\rm GL}}_n(q)$ could be divided into 9 specific classes ${{\mathcal{C}}}_i$, $1\leq i\leq 9$. For a big $n$, it is a very hard open problem to find the maximal subgroups of ${{\rm GL}}_n(q)$ of type ${{\mathcal{C}}}_9$. We have an explicit list of such groups only for $n\leq 12$ (see [@BHR]). On the contrary, the maximal subgroups of ${{\rm GL}}_n(q)$ of geometric type (i. e. of class ${{\mathcal{C}}}_i$, with $1\leq i\leq 8$) have been described for every $n$ (see [@KL]). We recall some notations in group theory and then we resume the description of the maximal subgroups of geometric type in the following Table 1 (see [@KL Table 1.2.A and $\S$ 3.5]). 0.2cm
Let $n,q$ be positive integers and let ${{\mathbb F}}_q$ be the finite field with $q$ elements. Let $\omega_q$ be a primitive element of ${{\mathbb F}}_q^*$. We use the standard notations for the special linear group ${{\rm SL}}_n(q)$, the projective special linear group ${{\rm PSL}}_n(q)$, the special orthogonal group $\textrm{SO}_n(q)$, the unitary group ${{\rm U}}_n(q)$, the symplectic group ${{\rm Sp}}_n(q)$, the simmetric group $S_n$ and the alternating group $A_n$. By $C_n$ we denote a cyclic group of order $n$, by $E_n$ an elementary abelian group of order $n$ and by $p^{1+2n}$ an extraspecial group of order $p^{1+2n}$. Furthermore, if $n$ is even and $q$ is odd we denote by (see [@BHR])
: ${{\rm GO}}_n^+(q)$ the stabilizer of the non-degenerate symmetric bilinear antidiagonal form (1,...,1);
: ${{\rm SO}}_n^+(q)$ the subgroup of ${{\rm GO}}_n^+(q)$ formed by the matrices with determinant 1;
: ${{\rm GO}}_n^-(q)$ the stabilizer of non-degenerate symmetric bilinear form $I_n$, when $n\equiv 2 (\textrm{mod 4})$ and $q\equiv 3 (\textrm{mod 4})$ and the stabilizer of non-degenerate symmetric bilinear diagonal form $(\omega_q,1,...,1)$, when $n\not\equiv 2 (\textrm{mod 4})$ and $q\not\equiv 3 (\textrm{mod 4})$;
: ${{\rm SO}}_n^-(q)$ the subgroup of ${{\rm GO}}_n^-(q)$ formed by the matrices with determinant 1.
0.2cm
Let $A, B$ be two groups. We denote by
: $A\rtimes B$, the semidirect product of $A$ with $B$ (where $A\trianglelefteq A\rtimes B$);
: $A\circ B$, the central product of $A$ and $B$;
: $A\wr B$, the wreath product of $A$ and $B$;
: $A.B$, a group $\Gamma$ that is an extension of its normal subgroup $A$ with the group $B$ (then $B\simeq \Gamma/A$), in the case when we do not know if it is a split extension or not;
: $A^{.}B$, a group $\Gamma$ that is a non-split extension of its normal subgroup $A$ with the group $B$ (then $B\simeq \Gamma/A$);
: $A:B$, a group $\Gamma$ that is a split extension of its normal subgroup $A$ with the group $B$ (then $B\simeq \Gamma/A$ and $\Gamma\simeq A\rtimes B$).
[|c|c|c|]{} type & description & structure\
${{\mathcal{C}}}_1$ &
stabilizers of a totally singular or non$–$singular subspace
& maximal parabolic group\
${{\mathcal{C}}}_2$ &
stabilizers of a direct sum decomposition
$V =\bigoplus_{i=1}^{r} V_i$, with each $V_i$ of dimension $t$
&
${{\rm GL}}_t(q)\wr S_r, n=rt$
\
${{\mathcal{C}}}_3$ &
stabilizers of an extension field of ${{\mathbb F}}_q$ of prime index $r$
& ${{\rm GL}}_t(q^r).C_r, n=rt, r$ prime\
${{\mathcal{C}}}_4$ &
stabilizers of tensor product decomposition $V=V_1\otimes V_2$
& ${{\rm GL}}_t(q)\circ {{\rm GL}}_r(q), n=rt$\
${{\mathcal{C}}}_5$ &
stabilizers of subfields of ${{\mathbb F}}_q$ of prime index $r$
& ${{\rm GL}}_n(q_0)$, $q=q_0^r$, $r$ prime\
0.6cm
${{\mathcal{C}}}_6$
&
normalizers of symplectic-type $r$-groups
($r$ prime) in absolutely irreducible
representations
&
$E_{r^{2t}}.{{\rm Sp}}_{2t}(r)$, $n=r^{t}, r$ prime
0.1cm
$2^{1+2t}.\textrm{O}^{-}_{2t}(r)$, $n=2^t$
0.1cm
$2_{+}^{1+2t}.\textrm{O}^{+}_{2t}(r)$, $n=2^{t}$
0.1cm
\
${{\mathcal{C}}}_7$ &
stabilizers of decompositions
$V=\bigotimes_{i=1}^t V_i, \textrm{dim}(V_i)=r$
& $ \underbrace{ ({{\rm GL}}_r(q)\circ ... \circ {{\rm GL}}_r(q)) }_{t} .S_r, n=r^t $\
0.2cm
${{\mathcal{C}}}_8$
&
0.2cm
classical subgroups
&
${{\rm Sp}}_n(q)$, $n$ even
$\textrm{O}_n^{\epsilon}(q)$, $q$ odd
${{\rm U}}_n(q^{\frac{1}{2}}),$ $q$ a square
\
\
Although we generally do not know explicitly the maximal subgroups of type ${{\mathcal{C}}}_9$, by Aschbacher’s Theorem, we have such a characterization of them:
“if $\Gamma$ is a maximal subgroup of ${{\rm GL}}_n(q)$ of class ${{\mathcal{C}}}_9$ and $Z$ denotes its center, then for some nonabelian simple group $T$, the group $\Gamma/(\Gamma \cap Z)$ is almost simple with socle $T$; in this case the normal subgroup $(\Gamma ∩ Z) .T$ acts absolutely irreducibly, preserves no nondegenerate classical form, is not a subfield group, and does not contain ${{\rm SL}}_n(q)$.”
For very small integers $n$ there are a few subsequent and more explicit versions of Aschbacher’s Theorem, that describe explicitly the subgroups of class ${{\mathcal{C}}}_9$. To prove Theorem \[P17\] we will use the classification of the maximal subgroups of ${{\rm SL}}_n(q)$ appearing in [@BHR], for $n\in \{4,8\}$.
Decomposable actions and products of elliptic curves
====================================================
First of all we investigate what happens when the group $G=\textrm{Gal}(k({{\mathcal{A}}}[q])/k,{{\mathcal{A}}}[q])$ acts decomposably on ${{\mathcal{A}}}[q]$, i. e., the representation of $G_k$ in ${{\rm GL}}_n(q)$ is a group of matrices with diagonal blocks. For instance, this is the case when ${{\mathcal{A}}}$ is a direct product of elliptic curves.
\[reducible\] Let $q$ be a positive integer. Suppose that $G$ acts decomposably on ${{\mathcal{A}}}[q]$, i. e. the representation of $G$ in ${{\rm GL}}_n({{\mathbb Z}}/q{{\mathbb Z}})$ is of the form
$$\label{diag_blocks} \left(
\begin{array}{ccccc}
B_1 & 0 & ... & & 0\\
0 & B_2 & 0 & ... & 0 \\
\vdots & & \ddots & &\vdots\\
& & & \ddots & 0\\
0 & ... & & 0 & B_s
\end{array}
\right)$$
where $B_i \in {{\rm GL}}_{n_i}$, for $i\in \{1, 2, ... ,s\}$ and $\sum_{i=1}^{s}n_i=n$. Let $G_i$ denote the subgroup of ${{\rm GL}}_{n_i}$ formed by the matrices $B_i$, for all $1\leq i\leq s$. Then $H^1_{{{\rm loc}}}(G,{{\mathbb Z}}/q{{\mathbb Z}}^n)=0$ if and only if $H^1_{{{\rm loc}}}(G_i,{{\mathbb Z}}/q{{\mathbb Z}}^{n_i})=0$, for all $1\leq i\leq s$.
The conclusion is a direct consequence of $H^1(G,-)$ being an additive functor and $H_{\textrm{loc}}^1(G,-)$ being a subfunctor of his (see for instance [@HS] and [@JL]); anyway we show a proof involving local cocycles. We prove the statement when $s=2$. When $s> 2$, the conclusion follows by induction. Assume that the representation of $G=\textrm{Gal}(k({{\mathcal{A}}}[q])/k,{{\mathcal{A}}}[q])$ in ${{\rm GL}}_n({{\mathbb Z}}/q{{\mathbb Z}})$ is of the form
$$\left(
\begin{array}{cc}
B_1 & 0 \\
0 & B_2 \\
\end{array}
\right),$$
where $B_1 \in {{\rm GL}}_{n_1}({{\mathbb Z}}/q{{\mathbb Z}})$, $B_2\in {{\rm GL}}_{n-n_1}({{\mathbb Z}}/q{{\mathbb Z}})$. Of course $G_1$ and $G_2$ can be identified with subgroups of $G$. If a cocycle of $G$ satisfies the local conditions, in particular its restriction to any subgroup of $G$ satisfies the local conditions too. Thus $H^1_{{{\rm loc}}}(G,{{\mathbb Z}}/q{{\mathbb Z}}^n)=0$ implies $H^1_{{{\rm loc}}}(G_i,{{\mathbb Z}}/q{{\mathbb Z}}^{n_i})=0$, for $i\in \{1,2\}$. Let $\{a_{i,j}\}_{1\leq i,j\leq n}$ denote a matrix in $G$; consequently $\{a_{i,j}\}_{1\leq i, j\leq n_1}$ and $\{a_{i,j}\}_{n_1+1\leq i, j\leq n}$ are matrices in $G_1$ and $G_2$, respectively. Suppose that there exists a cocycle $\{Z_{\sigma}\}_{\sigma\in G}$ of $G$ with values in $({{\mathbb Z}}/q{{\mathbb Z}})^{n}$ satisfying the local conditions, with $Z_{\sigma}=(z_{\sigma,1}, ..., z_{\sigma,n})$. We define two new cocycles, one of $G_1$ with values in $({{\mathbb Z}}/q{{\mathbb Z}})^{n_1}$ and the other of $G_2$ with values in $({{\mathbb Z}}/q{{\mathbb Z}})^{n-n_1}$, respectively by $Z_{\sigma,B_1}:=(z_{\sigma,1}, ..., z_{\sigma,n_1})\in ({{\mathbb Z}}/q{{\mathbb Z}})^{n_1}$ and $Z_{\sigma,B_2}:=(z_{\sigma,n_1+1}, ..., z_{\sigma,n})\in ({{\mathbb Z}}/q{{\mathbb Z}})^{n-n_1}$. Since $\{Z_{\sigma}\}_{\sigma\in G}$ satisfies the local conditions, then $\{Z_{\sigma,B_1}\}_{B_1\in G_1}$ and $\{Z_{\sigma,B_2}\}_{B_2\in G_2}$ satisfy the local conditions too. Because of our hypothesis that $H^1_{{{\rm loc}}}(G_1,({{\mathbb Z}}/q{{\mathbb Z}})^{n_1})=0$, there exists $W_1=(w_{1}, ..., w_{n_1})\in ({{\mathbb Z}}/q{{\mathbb Z}})^{n_1}$, such that $(B_1-I_{n_1})W_1=Z_{\sigma,B_1}$, for all $B_1 \in G_1$. In the same way, since $H^1_{{{\rm loc}}}(G_2,({{\mathbb Z}}/q{{\mathbb Z}})^{n-n_1})=0$, then there exists $W_2=(w_{n_1+1}, ..., w_n)\in ({{\mathbb Z}}/q{{\mathbb Z}})^{n-n_1}$, such that $(B_2-I_{n-n_{1}})W_2=Z_{\sigma,B_2}$, for all $B_2 \in G_2$. Let $W=(w_{1}, ..., w_{n_1}, w_{n_1+1}, ..., w_n)\in ({{\mathbb Z}}/q{{\mathbb Z}})^{n}$. Therefore $(G-I_{n})W=Z_{\sigma}$, for all $\sigma \in G$. We have proved that every cocycle of $G$ with values in $({{\mathbb Z}}/q{{\mathbb Z}})^{n}$ and satisfying the local conditions is a coboundary; thus $H^1_{{{\rm loc}}}(G,({{\mathbb Z}}/q{{\mathbb Z}})^{n})=0$.
\[rem\_red\] Observe that the conclusion of Lemma \[reducible\] holds even if we suppose that the image of the representation of $\textrm{Gal}(k({{\mathcal{A}}}[q])/k)$ in ${{\rm GL}}_n({{\mathbb Z}}/q{{\mathbb Z}})$ is isomorphic to a subgroup of ${{\rm GL}}_n(p^m)$ (for some prime $p$ and some positive integer $m$). We have such a technical assumption in the statement of Theorem \[P1\_bis\].
With Lemma \[reducible\] and a known answer to the problem for elliptic curves, the case of products of elliptic curves is quite obvious to solve. Anyway it is worth to be mentioned here for completeness.
\[product\_ell\_curv\] Let $k$ be a number field and let ${{\mathcal{E}}}_1, {{\mathcal{E}}}_2$ be elliptic curves with Weierstrass form respectively $y^2=x^3+b_ix+c_i$, for $i\in \{1,2\}$, where $b_i, c_i\in k$. Let $p$ be a prime number and $l$ be a positive integer. The local-global divisibility by $p^l$ holds in the product ${{\mathcal{E}}}_1\times {{\mathcal{E}}}_2$ over $k$ if and only if it holds in both ${{\mathcal{E}}}_1$ over $k$ and ${{\mathcal{E}}}_2$ over $k$.
To ease notation let ${{\mathcal{A}}}={{\mathcal{E}}}_1\times {{\mathcal{E}}}_2$. The fundamental observation is that the representation of the Galois group $\textrm{Gal}(k({{\mathcal{A}}}[p^l])/k)$ in ${{\rm GL}}_4({{\mathbb Z}}/p^l{{\mathbb Z}})$ is a group of matrices with two diagonal blocks (each of them with 2 rows and 2 columns). It is not true in general that the whole automorphism group of a product of elliptic curves is formed by matrices with diagonal blocks. Anyway, this is the exact situation when we restrict to automorphisms corresponding to actions of $\textrm{Gal}(k({{\mathcal{A}}}[p^l])/k)$. In fact, every automorphism of ${{\mathcal{A}}}$ in $\textrm{Gal}(k({{\mathcal{A}}}[p^l])/k)$ corresponds to a Galois homomorphism of the extension $k({{\mathcal{A}}}[p^l])/k$, whose action on the points of ${{\mathcal{A}}}$ can be viewed as two separate actions on the points of ${{\mathcal{E}}}_1$ and ${{\mathcal{E}}}_2$ (even when ${{\mathcal{E}}}_1={{\mathcal{E}}}_2$). We can apply Lemma \[reducible\] to get the conclusion.
The argument in the previous proof works also if we have an abelian variety of dimension $g$, that is the product of elliptic curves ${{\mathcal{E}}}_1, ..., {{\mathcal{E}}}_g$ satisfying the hypotheses of Theorem \[product\_ell\_curv\]. Then, more generally, we have the following statement.
\[product\_ell\_curv\_gen\] Let $k$ be a number field, let $g$ be a positive integer and let ${{\mathcal{E}}}_1, {{\mathcal{E}}}_2, ..., {{\mathcal{E}}}_g$ be elliptic curves with Weierstrass form respectively $y^2=x^3+b_ix+c_i$, for $i\in \{1,2, ..., g\}$, where $b_i, c_i\in k$. Let $p$ be a prime number and $l$ be a positive integer. The local-global divisibility by $p^l$ holds in the product ${{\mathcal{E}}}_1\times {{\mathcal{E}}}_2 ... \times {{\mathcal{E}}}_g$ over $k$ if and only if holds in every curve ${{\mathcal{E}}}_i$, over $k$, for all $1\leq i \leq g$.
By using Theorem \[product\_ell\_curv\_gen\] and [@PRV Corollary 2], we get the next result.
\[product\_cor1\] Let $k$ be a number field, let $g$ be a positive integer and let ${{\mathcal{E}}}_1, {{\mathcal{E}}}_2, ..., {{\mathcal{E}}}_g$ be elliptic curves with Weierstrass form respectively $y^2=x^3+b_ix+c_i$, for $i\in \{1,2, ..., g\}$, where $b_i, c_i\in k$. Let $p$ be a prime number. Then there exists a number $C([k:{{\mathbb Q}}])$ depending only on the degree $[k:{{\mathbb Q}}]$, such that, if $p>C([k:{{\mathbb Q}}])$, then the local-global divisibility by $p^l$ holds in the product ${{\mathcal{E}}}_1\times {{\mathcal{E}}}_2 ... \times {{\mathcal{E}}}_g$ over $k$, for every positive integer $l$.
Furthermore, if $k={{\mathbb Q}}$, we can combine Theorem \[product\_ell\_curv\] with the results appearing in [@DZ], [@PRV2], [@Pal3] and [@Cre2], to get a complete answer to the local-global divisibility in products of elliptic curves defined over the rationals.
\[product\_cor2\] Let $g$ be a positive integer and let ${{\mathcal{E}}}_1, {{\mathcal{E}}}_2, ..., {{\mathcal{E}}}_g$ be elliptic curves defined over ${{\mathbb Q}}$ with Weierstrass form respectively $y^2=x^3+b_ix+c_i$, for $i\in \{1,2, ..., g\}$, where $b_i, c_i\in {{\mathbb Q}}$. Let $p$ be a prime number. If $p\geq 5$, then the local-global divisibility by $p^l$ holds in the product ${{\mathcal{E}}}_1\times {{\mathcal{E}}}_2 ... \times {{\mathcal{E}}}_g$ over ${{\mathbb Q}}$, for every positive integer $l$. If $p\in \{2,3\}$, then the local-global divisibility by $p^l$ holds in the product ${{\mathcal{E}}}_1\times {{\mathcal{E}}}_2 ... \times {{\mathcal{E}}}_g $ over ${{\mathbb Q}}$ only when $l=1$; on the contrary, when $l\geq 2$, there are counterexamples.
**Counterexamples.** For powers of 2 (resp. powers of 3), the explicit counterexamples to the local-global divisibility appearing in [@Pal2] and [@Pal3] give also explicit counterexamples to the local-global divisibility by $2^l$ (resp. $3^l$), for every $l\geq 2$, in products of elliptic curves defined over ${{\mathbb Q}}$ (resp. over the cyclotomic field ${{\mathbb Q}}({{\zeta}}_3)$), for all $g$. It suffices to take the product of elliptic curves ${{\mathcal{E}}}_1, ..., {{\mathcal{E}}}_g$ with at least one of the ${{\mathcal{E}}}_i's$ being a curve giving a counterexample.
Proof of Theorem \[P1\_bis\]
============================
First note that if $\dim(A)=2^h$, then ${{\rm Gal}}(k({{\mathcal{A}}}[p])/k)$ is isomorphic to a subgroup of ${{\rm GL}}_{2^{h+1}}(p)$. As above, to ease notation we set $n=2^{h+1}$, so that we can simply refer to ${{\rm GL}}_n(p)$ and, more generally, to ${{\rm GL}}_n(p^m)$, $m\geq 1$.
Our assumption that $G$ is isomorphic to a subgroup of ${{\rm GL}}_{2^{h+1}}(p^m)$ (instead of simply ${{\rm GL}}_{2^{h+1}}(p)$) is just a technical one, since dealing with powers of $p$ in lieu of $p$ will be useful when $G$ is of type ${{\mathcal{C}}}_3$ (and it is isomorphic to a subgroup of ${{\rm GL}}_t(p^r).C_r$, with $n=tr$ and $r$ prime) or $G$ is of type ${{\mathcal{C}}}_5$ (and it is isomorphic to a subgroup of ${{\rm GL}}_n(p^r)$, with $r$ a prime dividing $m$).
For $h=0$ and $G< {{\rm GL}}_2(p^m)$, the following statement can be deduced from the proof of [@DZ3 Theorem 1] and from Remark \[rem\_red\].
\[caso\_n=2bis\] Let $k$ be a number field that does not contain ${{\mathbb Q}}({{\zeta}}_p+{{\zeta}}_p^{-1})$. Let ${{\mathcal{A}}}$ be an algebraic group defined over $k$, such that ${{\rm Gal}}(k({{\mathcal{A}}}[p^l])/k)\lesssim {{\rm GL}}_2(p^m)$, where $p>3$ is a prime number and $l,m$ are positive integers. If ${{\mathcal{A}}}[p^l])$ is either an irreducible or a decomposable $G_k$-module, then the local-global principle holds for divisibility by $p^l$ in ${{\mathcal{A}}}$ over $k$.
In particular that result holds when $l=m=1$. From now on, we may assume that $h\geq 1$ (i. e. $\dim(A)\geq 2$ and $n\geq 4$). Let $G$ be a subgroup of ${{\rm GL}}_{n}(p^m)$ and let $\widetilde{G}:=G\cap {{\rm SL}}_n(p^m)$. Since $|{{\rm GL}}_n(p^m)|=(p^m-1)|{{\rm SL}}_n(p^m)|$, then the $p$-Sylow subgroup of ${{\rm GL}}_n(p^m)$ coincides with the $p$-Sylow subgroup of ${{\rm SL}}_n(p^m)$. By Lemma \[Sylow\], we have $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^m])=0$ if and only if $H^1_{\textrm{loc}}(\widetilde{G},{{\mathcal{A}}}[p^m])=0$. Therefore we may assume $G\leq {{\rm SL}}_n(p^m)$. If $G ={{\rm SL}}_n(p^m)$, then, being $n=2^{h+1}$ and $p\neq 2$, the nontrivial scalar matrix $-I$ belongs to ${{\mathcal{A}}}$. By Corollary \[scalar\] we get the triviality of $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^m])$. From now on we will assume, without loss of generality, that $G$ is a proper subgroup of ${{\rm SL}}_n(p^m)$.
For $h=1$ we give a proof of Theorem \[P1\_bis\] based on a case by case analysis of the possible maximal subgroups of ${{\rm SL}}_4(p^m)$. Then we prove the statement for a general $h$, using the classification of the possible maximal subgroups of ${{\rm SL}}_n(q)$ resumed in Table 1, combined with induction for some of the classes ${{\mathcal{C}}}_i$ of groups. By the proof it will be clear that for subgroups of geometric type ${{\mathcal{C}}}_i$, with $i\neq 6$, everything works for every $p>3$ too. The class ${{\mathcal{C}}}_6$ is the hardest to be described explicitly between the ones of geometric type. Because of the groups $G$ in that class we possibly have to choose $p_h\neq 3$, for $h\geq 3$. Furthermore, for $h\geq 3$ a complete classification of the subgroups of type ${{\mathcal{C}}}_9$ is unknown, then for such integers we cannot show an explicit $p_h$, even if we can prove its existence. In the very last part of the proof, looking at the maximal subgroups of ${{\rm SL}}_8(p^m)$, we establish an explicit $p_2$.
The case of abelian varieties of dimension 2 {#sec4}
--------------------------------------------
In this subsection we prove Theorem \[P1\_bis\] for abelian varieties of dimension 2. Assume that ${{\rm Gal}}(k({{\mathcal{A}}}[p^l])/k)$ is isomorphic to a proper subgroup of $SL_4(p^m)$, for any positive integer $m$. First of all, we recall some notation and some group isomorphisms.
Let $s$ be a positive integer. The extraspecial 2-group of *minus type* $2_{-}^{1+2s}$ is a central product of a quaternion group of order 8 with one or more dihedral groups of order 8 (see for instance [@KL]). The extraspecial 2-group of *plus type* $2_{+}^{1+2s}$ is a central product of dihedral groups of order 8. The *symplectic type* is given by a central product of either type of extraspecial $2$-groups with a cyclic group of order $4$.
\[isom\] The following isomorphisms hold
1)
: ${{\rm SO}}_4^+(q)\cong {{\rm SL}}_2(q)\times {{\rm SL}}_2(q)$;
2)
: ${{\rm SO}}_4^-(q)\cong {{\rm SL}}_2(q^2)$.
0.5cm We also recall the classification of the maximal subgroups of ${{\rm SL}}_4(p^m)$ and the classification of the maximal subgroups of the sympleptic group $\textrm{Sp}_4(q)$ (when $q$ is odd) appearing in [@BHR].
\[M+B+L-lem\] Let $q=p^m$, where $p$ is an odd prime and $m$ is a positive integer. Let $d:=\gcd(q-1,4)$. The maximal subgroups of ${{\rm SL}}_4(q)$ are
(a)
: a group of type ${{\mathcal{C}}}_1$, the stabilizer of a projective point, i. e. the group $C_q^3:{{\rm GL}}_3(q)$, having order $q^6 (q^3-1)(q^2-1)(q-1)$;
(b)
: a group of type ${{\mathcal{C}}}_1$, the stabilizer of a projective line, having order $q^4|{{\rm SL}}_4(q)|^2(q-1)=q^6(q^2-1)^2(q-1)$;
(c)
: a group of type ${{\mathcal{C}}}_1$, the stabilizer of two distinct projective points and a projective line, having order $q^5|{{\rm GL}}_2(q)|(q-1)=q^6(q^2-1)(q-1)^3$;
(d)
: a group of type ${{\mathcal{C}}}_1$, a group isomorphic to ${{\rm GL}}_3(q)$, that stabilizes both a projective point and a projective plane, whose direct sum is ${{\mathbb F}}_q^4$, having order $q^3(q^3-1)(q^2-1)(q-1)$;
(e)
: a group of type ${{\mathcal{C}}}_2$, the stabilizer of a decomposition of four subspaces of dimension 1 whose direct sum is ${{\mathbb F}}_q^4$, i. e. a group of order $(q-1)^34!$;
(f)
: a group of type ${{\mathcal{C}}}_2$, the stabilizer of a decomposition of two subspaces of dimension 2 whose direct sum is ${{\mathbb F}}_q^4$, i. e. a group of order $2|{{\rm SL}}_2(q^2)|(q-1)=2q^2(q^2-1)^2(q-1)$;
(g)
: a group of type ${{\mathcal{C}}}_3$, a group of order $2q^2(q^4-1)(q+1)$, which has ${{\rm SL}}_2(q^2)$ as a normal subgroup;
(h)
: a group of type ${{\mathcal{C}}}_6$, the group $C_4\circ 2^{1+4}\hspace{0.1cm} ^{\cdot} S_6$;
(i)
: a group of type ${{\mathcal{C}}}_6$, the group $C_4\circ 2^{1+4}\hspace{0.1cm} _{\cdot} A_6$;
(j)
: a group of type ${{\mathcal{C}}}_8$, a group of order $d|{{\rm SO}}_4^+(q)|$, which has ${{\rm SO}}_4^+(q)$ as a normal subgroup;
(k)
: a group of type ${{\mathcal{C}}}_8$, a group of order $d|{{\rm SO}}_4^-(q)|$, which has ${{\rm SO}}_4^-(q)$ as a normal subgroup;
(l)
: a group of type ${{\mathcal{C}}}_8$, the group ${{\rm Sp}}_4(q).C_2$ of order $2q^4(q-1)(q^2-1)(q^4-1)$;
(m)
: a group of type ${{\mathcal{C}}}_9$, the group $A_7$ (only if $p=2$);
(n)
: a group of type ${{\mathcal{C}}}_9$, the group $C_d\circ C_2\hspace{0.1cm}^{\cdot}{{\rm SL}}_2(7)$;
(o)
: a group of type ${{\mathcal{C}}}_9$, the group $C_d\circ C_2\hspace{0.1cm}^{\cdot}A_7$;
(p)
: a group of type ${{\mathcal{C}}}_9$, the group $C_d\circ C_2\hspace{0.1cm}^{\cdot}\textrm{U}_4(2)$.
0.3cm
\[Sp4\_sub\] Let $q=p^m$, where $p$ is an odd prime and $m$ is a positive integer. The maximal subgroups of $\textrm{Sp}_4(q)$ are
(l.1)
: a group of type ${{\mathcal{C}}}_1$, the group $E_{q}\hspace{0.1cm}. E_q^2:((q-1)\times {{\rm Sp}}_2(q))$, of order $q^4(q-1)^2$;
(l.2)
: a group of type ${{\mathcal{C}}}_1$, the group $E_q^3:{{\rm GL}}_3(q)$, of order $q^(q^3-1)(q^2-1)(q-1)$;
(l.3)
: a group of type ${{\mathcal{C}}}_2$, the stabilizer of a decomposition of two subspace of dimension 2 whose direct sum is ${{\mathbb F}}_q^4$, i. e. $\textrm{Sp}_2(q)^2\rtimes C_2$;
(l.4)
: a group of type ${{\mathcal{C}}}_2$, the group ${{\rm GL}}_2(q).C_2$;
(l.5)
: a group of type ${{\mathcal{C}}}_3$, the group $\textrm{Sp}_2(q^2)\rtimes C_2$;
(l.6)
: a group of type ${{\mathcal{C}}}_6$, the group $2_{-}^{1+4} \hspace{0.1cm}_{\cdot}S_5$;
(l.7)
: a group of type ${{\mathcal{C}}}_6$, the group $2_{-}^{1+4} \hspace{0.1cm}_{\cdot}A_5$;
(l.8)
: a group of type ${{\mathcal{C}}}_9$, the group $C_2\hspace{0.1cm}^{\cdot}A_6$;
(l.9)
: a group of type ${{\mathcal{C}}}_9$, the group $C_2\hspace{0.1cm}^{\cdot}S_6$;
(l.10)
: a group of type ${{\mathcal{C}}}_9$, the group $C_2\hspace{0.1cm}^{\cdot}A_7$ (only for $p=7$);
(l.11)
: a group of type ${{\mathcal{C}}}_9$, the group ${{\rm SL}}_2(q)$.
**Proof of Theorem \[P1\_bis\] for ***h***=1.** Let $p>3$. Without loss of generality we assume that $G$ is contained in a proper subgroup of ${{\rm SL}}_4(q)$, where $q=p^l$, and we use Lemma \[M+B+L-lem\]. We furthermore assume that ${{\mathcal{A}}}[p^l]$ is either an irreducible or a decomposable $G$-module. In particular we are not in one of the cases **(a), (b), (c)**, unless we can apply Remark \[rem\_red\] and, because of Lemma \[caso\_n=2bis\], to get the vanishing of $H^1_{{{\rm loc}}}(G_p,{{\mathcal{A}}}[p^l])$.
If we are in case **(d)**, then $G$ is of type ${{\mathcal{C}}}_1$, but acts decomposably on ${{\mathcal{A}}}[p^l]$. Thus $H^1_{{{\rm loc}}}(G_p,{{\mathcal{A}}}[p^l])=0$, by Remark \[rem\_red\] and Lemma \[caso\_n=2bis\] again.
If we are in cases **(e)**, then $p\nmid |G|$, the $p$-Sylow subgroup of $G$ is trivial and $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$.
In case **(f)**, the $p$-Sylow subgroup $G_p$ of $G$ has shape
$$\left(
\begin{array}{cccc}
1 & a & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & b \\
0 & 0 & 0 & 1 \\
\end{array}
\right)$$
where $a,b\in {{\mathbb Z}}/p{{\mathbb Z}}$. By Lemma \[caso\_n=2bis\] and Remark \[rem\_red\], the first local cohomology group $H^1_{{{\rm loc}}}(G_p,{{\mathcal{A}}}[p^l])$ is trivial. Thus $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$.
In case **(g)**, we have that $G$ is contained in a group that has a normal subgroup isomorphic to ${{\rm SL}}_2(p^2)$, with index not divisible by $p$. Observe that the $p$-Sylow subgroup $G_p$ of $G$ is contained in $G':={{\rm SL}}_2(p^2)\cap G$ . We use Lemma \[caso\_n=2bis\] to get $H^1_{{{\rm loc}}}(G_p,{{\mathcal{A}}}[p^l])=0$, that is equivalent to $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$.
If we are in case **(h)** (resp. case **(i)**) and $p>5$, then the $p$-Sylow subgroup of $G$ is trivial too and $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$. If we are in case **(h)** (resp. case **(i)**) and $p=5$, then the $5$-Sylow subgroup of $G$ is cyclic and $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$.
In cases **(j)** and **(k)** we use the group isomorphisms listed in Lemma \[isom\]. Then both cases are covered by Lemma \[caso\_n=2bis\].
Consider case **(l)**, i. e. $G$ is isomorphic to a subgroup of ${{\rm Sp}}_4(p^m).C_2$. Since $p\neq 2$, then $G_p$ is contained in ${{\rm Sp}}_4(p^m)$. To ease notation, without loss of generality, we may assume $G\lesssim {{\rm Sp}}_4(p^m)$. If $G={{\rm Sp}}_4(p^m)$, then $-I_4\in G$. Since $p\neq 2$, then $G$ contains a nontrivial scalar matrix and, by Corollary \[scalar\], we have $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$. If $G=\langle I_4\rangle$, then $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])$ is trivial too. Therefore, we assume that $G$ is a proper subgroup of ${{\rm Sp}}_4(p^m)$ and we use Lemma \[Sp4\_sub\].
In cases **(l.1)** and **(l.2)**, the group $G$ acts reducibly over ${{\mathcal{A}}}[p^l]$.
In cases **(l.3)** and **(l.4)**, the group $G_p$ acts decomposably over ${{\mathcal{A}}}[p^l]$ and $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$, by Remark \[rem\_red\] and Lemma \[Sylow\].
Consider case **(l.5)**. Then $G_p$ is isomorphic to a subgroup of the $p$-Sylow subgroup of $\textrm{Sp}_2(p^{2m})$. In particular $G_p$ is isomorphic to a subgroup of ${{\rm SL}}_2(p^{2m})$. By Lemma \[caso\_n=2bis\], we get $H^1_{{{\rm loc}}}(G_p,{{\mathcal{A}}}[p^l])=H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$.
In cases **(l.6)** and **(l.7)**, if $p>5$, then $p\nmid |G|$, implying $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$. If $p=5$, we have that the $5$-Sylow subgroup of $G$ is cyclic and $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$ too.
In cases **(l.8)**, **(l.9)** and **(l.10)**, the $p$-Sylow subgroup of $G$ is either trivial or cyclic, for all $p>3$.
Case **(l.11)** is covered by Lemma \[caso\_n=2bis\] again.
We are left with the cases when $G$ is of type ${{\mathcal{C}}}_9$ and it is not contained in ${{\rm Sp}}_4(p)$. In all those cases **(m)**, **(n)**, **(o)** and **(p)** the $p$-Sylow subgroup $G_p$ of $G$ is either trivial or cyclic, for all $p>3$. Then $H^1_{{{\rm loc}}}(G,{{\mathcal{A}}}[p^l])=0$. [${\Box}$ ]{}
0.5cm
From the proof, it is clear that the conclusion holds not only for abelian varieties of dimension 2, but even for all algebraic groups ${{\mathcal{A}}}$ such that ${{\mathcal{A}}}[p]\simeq ({{\mathbb Z}}/p^l{{\mathbb Z}})^4$ and ${{\rm Gal}}(k({{\mathcal{A}}}[p^l])/k)\lesssim {{\rm GL}}_4(p^m)$, with $m\geq 1$.
General Case {#gen_case}
------------
We are going to prove Theorem \[P1\_bis\], for the general case of an abelian variety of dimension $2^h$, that corresponds to the proof of the following proposition. We will prove $p_2=3$ in next subsection.
\[P1\_prop1\] Let $p$ be a prime number and let $l,h$ be positive integers. Let $k$ be a number field that does not contain ${{\mathbb Q}}({{\zeta}}_p+{{\zeta}}_p^{-1})$. Let ${{\mathcal{A}}}$ be an abelian variety defined over $k$, of dimension $2^h$, where $h\geq 0$. Let $n=2^{h+1}$ and assume that ${{\rm Gal}}(k({{\mathcal{A}}}[p^l])/k)$ is isomorphic to a subgroup of ${{\rm GL}}_{n}(p^m)$, for some positive integer $m$. For every $h$, there exists a prime $p_h$, depending only on $h$, such that if $p>p_h$ and the local-global divisibility by $p$ fails in ${{\mathcal{A}}}(k)$, then ${{\rm Gal}}(k({{\mathcal{A}}}[p^l])/k)$ acts reducibly but not decomposably over ${{\mathcal{A}}}[p^l]$.
Let $p>3$ and, as above, let $n=2^h$. Suppose that $G$ acts either irreducibly or decomposably on ${{\mathcal{A}}}[q]$, where $q=p^l$. We use the description of the subgroups of ${{\rm GL}}_n(q)$ of geometric type given in Table 1. For some classes of groups we proceed by induction, having already proved the statement for $ h\in \{0,1\}$. Thus, assume that the proposition holds for all integers $h' <h$. We will prove it for $h$.
Because of our assumptions, the group $G$ is not of class ${{\mathcal{C}}}_1$, unless its action on ${{\mathcal{A}}}[q]$ is decomposable. For all $p>p_{h-1}$, the triviality of $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[q])$ follows from Remark \[rem\_red\] and induction.
Suppose that $G$ is of type ${{\mathcal{C}}}_2$. Then $G$ is the wreath product of a group $G'$ of matrices with $2^{\alpha}$ diagonal blocks by a symmetric group $S_{2^{\alpha}}$, where $\alpha\leq h$. Since $p> 2$, then the $p$-Sylow subgroup $G_p$ of $G$ is contained in $G'$. Thus, by Remark \[rem\_red\] and by induction, we get $H^1_{\textrm{loc}}(G_p,{{\mathcal{A}}}[p^l])=0$, for all $p>p_{h-1}$. Consequently $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$, for all $p>p_{h-1}$, because of Lemma \[Sylow\].
Suppose now that $G$ is of type ${{\mathcal{C}}}_3$. Then $G$ is isomorphic to a subgroup of ${{\rm GL}}_t(p^{mr}).C_r$, where $r$ is a prime and $n=tr$. Since $n=2^{h+1}$, then $r=2$ and $t=2^{h}$. Furthermore $p$ does not divide $r$ and we may assume without loss of generality that $G$ is isomorphic to a subgroup of ${{\rm GL}}_t(p^{mr})$. Since $t|n$, $t\neq n$, we use induction to get $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$, for every $p>p_{h-1}$.
Suppose that $G$ is of type ${{\mathcal{C}}}_4$. Then $G$ is isomorphic to a subgroup of a central product ${{\rm GL}}_t(p^m)\circ {{\rm GL}}_r(p^m)$ acting on a tensor product $V_1\otimes V_2={{\mathcal{A}}}[p^l]$, where $rt=n=2^h$ and $V_1$, $V_2$ are vectorial spaces over ${{\mathbb F}}_{p^m}$, with dimension respectively $t$ and $r$. A central product $\Gamma$ of two groups is a quotient of their direct product by a subgroup of its center. Then every subgroup of $\Gamma$ is a central product too. So let $G=G_t\circ G_r$, with $G_t$ acting on $V_1$ and $G_r$ acting on $V_2$. Consider $Z_{\sigma\otimes \tau}$, with $\sigma\otimes \tau\in G_t\circ G_r$, representing a cocycle of $G$ with values in ${{\mathcal{A}}}[p^l] =V_1\otimes V_2$. If $Z_{\sigma\otimes \tau}$ satisfies the local conditions, then there exists $A_{\sigma\otimes\tau}\in V_1\otimes V_2$ such that $Z_{\sigma\otimes \tau}=(\sigma\otimes \tau - 1\otimes 1)A_{\sigma\otimes\tau}$, for all $\sigma\otimes \tau\in G_t\circ G_r$. Observe that $A_{\sigma\otimes\tau}=A_{\sigma\otimes\tau,1}\otimes A_{\sigma\otimes\tau,2}$, for some $A_{\sigma\otimes\tau,1}\in V_1$ and $A_{\sigma\otimes\tau,2}\in V_2$. We have two separated actions of $G_t$ on $V_1$ and $G_r$ on $V_2$. Then we can construct a cocycle $Z_{\sigma}:=(\sigma-1)A_{\sigma}$, with $\sigma\in G_t$, by choosing $A_{\sigma}$ among the possible $A_{\sigma\otimes\tau,1}\in V_1$. In the same way we can construct a cocycle $Z_{\tau}:=(\tau-1)A_{\tau}$, with $\tau\in G_r$, by choosing $A_{\tau}$ among the possible $A_{\sigma\otimes\tau,2}\in V_2$. For the tensor product construction, a priori we could have more than one choice of $A_{\sigma}$ (respectively $A_{\tau}$) for each $\sigma$ (resp. $\tau$). Anyway, we choose just one $A_{\sigma}$ (resp. $A_{\tau}$). We will have no problems about this choice, because of the two separated actions of $G_t$ and $G_r$ respectively on $V_1$ and $V_2$. Observe that even in the general case of Definition \[loc\_cond\], when a cocycle satisfies the local conditions, there could exist various $A_{\sigma}$ giving the equality $Z_{\sigma}=(\sigma-1)A_{\sigma}$. Anyway we make just one choice for $A_{\sigma}\in {{\mathcal{A}}}[q]$, for each $\sigma\in G$. Since $r=2^{\alpha}$, with $\alpha< h$, by induction $H^{1}_{\textrm{loc}}(G_r,V_1)=0$, for every $p>p_{\alpha}$, unless $G_r$ acts reducibly but not decomposably over $V_1$. Observe that if $G_r$ acts reducibly over $V_1$, then also $G_r\otimes G_t$ is a parabolic group and $G$ acts reducibly over ${{\mathcal{A}}}[p^l]$ too. That is a contradiction with our assumptions. Then $H^{1}_{\textrm{loc}}(G_r,V_1)=0$ and there exists $A\in V_1$, such that $Z_{\sigma}=(\sigma-1)A$, for all $\sigma\in G_r$. In the same way, by induction, since $t=2^{\beta}$, with $\beta<h$ (and $\beta+\alpha=h$), by induction, for all $p>p_{\beta}$, we have $H^{1}_{\textrm{loc}}(G_t,V_2)=0$, unless $G_t$ acts reducibly but not decomposably over $V_2$. As above, if $G_t$ acts reducibly over $V_2$, then also $G_r\otimes G_t$ is a parabolic group and $G$ acts reducibly over ${{\mathcal{A}}}[p^l]$ too. Since this contradicts our assumptions, then there exists $B\in V_2$, such that $Z_{\tau}=(\tau-1)B$, for all $\tau\in G_t$. Therefore $Z_{\sigma\otimes \tau}=(\sigma\otimes \tau - 1\otimes 1)A\otimes B$, for all $\sigma\otimes \tau\in G_t\circ G_r$ and $H^{1}_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$. Since $2^h$ is the greatest proper divisor of $n$, then for every $p>p_{h-1}$, we have $H^{1}_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$.
If $G$ is of class ${{\mathcal{C}}}_5$, then $G$ is isomorphic to a subgroup of ${{\rm GL}}_n(p^t)$, where $m=tr$, with $t$ a positive integer and $r$ a prime. If $G$ is the whole group ${{\rm GL}}_n(p^t)$, then $G$ contains $-I$ and $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$. If $G$ is trivial, then $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])$ is trivial too. Suppose that $G$ is a proper subgroup of ${{\rm GL}}_n(p^t)$. If $G$ is still of class ${{\mathcal{C}}}_5$, then $G$ is isomorphic to a subgroup of ${{\rm GL}}_n(p^{t_2})$, for some integer $t_2$, such that $t=r_2t_2$, with $r_2$ prime. Again, if $G={{\rm GL}}_n(p^{t_2})$, then $-I\in G$ and $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$ and we may assume that $G$ is a proper subgroup of ${{\rm GL}}_n(p^{t_2})$. And so on. Since $m$ is finite and we are assuming that $G$ is not trivial, then $G$ is isomorphic to a subgroup of ${{\rm GL}}_n(p^{t_j})$ (for some positive integer $t_j$ dividing $m$) of class ${{\mathcal{C}}}_i$, with $i\neq 5$. We may then repeat the arguments used for other classes ${{\mathcal{C}}}_i$, with $i\notin \{1,5\}$, to get $H^{1}_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$.
Suppose that $G$ is of class ${{\mathcal{C}}}_6$, i. e. $G$ lies in the normalizer of an extraspecial group. This is possible only when $m=1$, which is the case of main interest for us. When $n=2^h$, we have the following possible types of maximal subgroups of class ${{\mathcal{C}}}_6$ (see [@KL §3.5]): $E_{2^{2h}}.{{\rm Sp}}_{2h}(2)$; $2^{1+2h}.{{\cal O}}^{-}_{2h}(2)$; $2_{+}^{1+2h}.{{\cal O}}^{+}_{2h}(2)$. If $p$ does not divide $\prod_{i=1}^{h}(2^{2i}-1)$, then it does not divide neither $|{{\rm Sp}}_{2h}(2)|$, nor $|{{\cal O}}^{\epsilon}_{2h}(2)|$, for every $\epsilon \in \{+,-\}$. Let $p_{\bar{h}}$ be the greatest prime dividing $\prod_{i=1}^{h}(2^{2i}-1)$. If $p> p_{\bar{h}}$, then $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$.
Assume that $G$ is of class ${{\mathcal{C}}}_7$. Thus $G$ is the stabilizer of a tensor product decomposition $\bigotimes_{i=1}^{t}V_r$, with $n=r^t$ and $\textrm{dim}(V_i)=r$, for every $1\leq i\leq t$. By using induction on $t$ and the argument given in the case when $G$ is of class ${{\mathcal{C}}}_4$ as the base of the induction, we get $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$, for every $p>p_{h-1}$.
Suppose that $G$ is of class ${{\mathcal{C}}}_8$. Since $p^m$ is odd and $n=2^h$ is even, then $G$ is contained either in the group ${{\rm Sp}}_n(p^m)$, or in a group $\textrm{O}_n^{\epsilon}(p^m)$, for any $\epsilon\in\{+,-\}$, or in the group $U_n(p^{\frac{m}{2}})$, with $m$ even too. If $G$ is one of the whole groups ${{\rm Sp}}_n(p^m)$ or $\textrm{O}_n^{\epsilon}(p^m)$ or $U_n(p^{\frac{m}{2}})$, then it contains a scalar multiple of the identity and $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$. Suppose that $G$ is a proper subgroup of one of those three groups. From the classification of the maximal subgroups of ${{\rm Sp}}_n(p^m)$ and $\textrm{O}_n^{\epsilon}(p^m)$ and $U_n(p^{\frac{m}{2}})$ (see [@KL], in particular Table 3.5B, Table 3.5C, Table 3.5D and Table 3.5E), we have that $\textrm{O}_n^{\epsilon}(p^m)$ and $U_n(p^{\frac{m}{2}})$ do not contain groups of class ${{\mathcal{C}}}_8$ and that the subgroups of ${{\rm Sp}}_n(p^m)$ of class ${{\mathcal{C}}}_8$ are $\textrm{O}_n^{\epsilon}(p^m)$ themselves, where $\epsilon \in \{+,-\}$ (and $n\geq 4$). Since we are assuming that $G$ is strictly contained in one of those three groups, then it is a subgroup of class ${{\mathcal{C}}}_i$, for some $i\neq 8$. We get the conclusion by the same arguments used for those classes of groups.
For every $n$, there is a finite number of subgroups of ${{\rm GL}}_n(p^m)$ of type ${{\mathcal{C}}}_9$. Unfortunately, as said above, for $n\>12$ an explicit classification of those groups is not known. Anyway, there exists a prime $p_{h'}$, that is the greatest prime dividing the order of at least one of those subgroups of type ${{\mathcal{C}}}_9$. If $G$ is of class ${{\mathcal{C}}}_9$, then $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[q])=0$, for all $p>p_{h'}$.
Let $p_{h}:=\textrm{max}\{p_{\bar{h}},p_{h-1},p_{h'}\}$. Then $H^1_{\textrm{loc}}(G,{{\mathcal{A}}}[p^l])=0$, for all $p>p_h$.
The case of abelian varieties of dimension 4
--------------------------------------------
To complete the proof of Theorem \[P1\_bis\] (and consequently of Theorem \[P17\_gal\], Corollary \[P17\] and Theorem \[P\_Sha\]), we have to show $p_2=3$. By the proof of Proposition \[P1\_prop1\], it is clear that $p_h$ depends only on $p_{h-1}$ and on the subgroups of ${{\rm SL}}_{2^{h+1}}(q)$ of class ${{\mathcal{C}}}_6$ and of class ${{\mathcal{C}}}_9$. In the next lemma we recall the classification of the maximal subgroups of ${{\rm SL}}_8(q)$ of those two classes (see [@BHR]).
\[M+B+L-lem\] Let $q=p^m$, where $p$ is an odd prime and $m$ is a positive integer. Let $d:=\gcd(q-1,4)$. Let $d:=\gcd(q-1,8)$. The only maximal subgroup of ${{\rm SL}}_8(q)$ of type ${{\mathcal{C}}}_6$ is $(C_d\circ 2^{1+6})^{\cdot} {{\rm Sp}}_6(2)$. The maximal subgroups of ${{\rm SL}}_8(q)$ of type ${{\mathcal{C}}}_9$ are isomorphic to the following groups
(a)
: $C_4\hspace{0.1cm}^{\cdot}{{\rm PSL}}_3(4)$, for $q=p=5$;
(b)
: $C_d\circ C_4\hspace{0.1cm}^{\cdot}{{\rm PSL}}_3(4)$, for $q=p\equiv 9,21,29,41,61,69 {{\textrm{ (mod) }}}80$;
(c)
: $C_8\circ C_4\hspace{0.1cm}^{\cdot}{{\rm PSL}}_3(4).C_2$, for $q=p\equiv 1,49 {{\textrm{ (mod) }}}80$;
(d)
: $C_8\circ C_4\hspace{0.1cm}^{\cdot}{{\rm PSL}}_3(4)$, for $q=p^2$, $p\equiv \pm 3,\pm 13,\pm 27,\pm 37 {{\textrm{ (mod) }}}80$;
(e)
: $C_8\circ C_4\hspace{0.1cm}^{\cdot}{{\rm PSL}}_3(4).C_2$, for $q=p^2$, $p\equiv \pm 7,\pm 17,\pm 23,\pm 33 {{\textrm{ (mod) }}}80$.
**Proof of Theorem \[P17\] for $\mathbf{h=2}$.** As noted above, by Theorem \[P1\_bis\], if the group $G$ lies in one of the classes ${{\mathcal{C}}}_i$, for $i\notin \{1,6,9\}$, then $H^1_{\textrm{loc}}(G, {{\mathcal{A}}}[p^l])=0$, for all $p>3$. Assume that $G$ is of class ${{\mathcal{C}}}_6$. By Lemma \[M+B+L-lem\], we have that $G$ is a subgroup of $(C_d\circ 2^{1+6})^{\cdot} {{\rm Sp}}_6(2)$, where $d=\gcd(p^l-1,8)$. Therefore the cardinality of $|G|$ divides $2^{19}\cdot 3\cdot 7$. For every prime $p>3$ the $p$-Sylow subgroup of $G$ is either trivial or cyclic (this last case occurs only if $p=7$). In both cases $H^1_{\textrm{loc}}(G, {{\mathcal{A}}}[q])=0$. If $G$ is a group of class ${{\mathcal{C}}}_9$, then, by Lemma \[M+B+L-lem\], the cardinality of $G$ divides $2^6|{{\rm PSL}}_3(4)|$. Since ${{\rm SL}}_3(4)$ has a trivial center, then ${{\rm PSL}}_3(4)={{\rm SL}}_3(4)$ and the cardinality of $G$ divides $ 2^6 \cdot 3\cdot 5 \cdot 7$. Again, for all $p>3$ the $p$-Sylow subgroup of $G$ is either trivial or cyclic (this last case occurs only if $p=5$ or $p=7$). We have $H^1_{\textrm{loc}}(G, {{\mathcal{A}}}[q])=0$. Thus $p_2=3$. $\Box$
*Acknowledgments*. I am grateful to John van Bon, Roberto Dvornicich and Gabriele Ranieri for useful discussions. I wrote the last part of this paper at the Max Planck Institute for Mathematics in Bonn. I would like to thank all people there for their kind hospitality.
[Pal]{}
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0.5cm Laura Paladino
Max Planck Institute for Mathematics
Vivatgasse, 7
53111 Bonn
Germany
e-mail address: [email protected]
[^1]: Partially supported by Istituto Nazionale di Alta Matematica F. Saveri with grant “Assegno di ricerca Ing. Giorgio Schirillo”
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Kes 69 is a mixed-morphology (MM) supernova remnant (SNR) that is known to be interacting with molecular clouds based on 1720 MHz hydroxyl (OH) maser emission observations in the northeastern and southeastern regions. We present an investigation of Kes 69 using $\sim$67 ks [*Suzaku*]{} observation. The X-ray spectrum of the whole SNR is well fitted by a non-equilibrium ionization model with an electron temperature of $kT_{\rm e}$ $\sim$ 2.5 keV, ionization time-scale of $\tau$ $\sim$ 4.1$\times10^{10}$ cm$^{-3}$ s and absorbing column density of $N_{\rm H}$ $\sim$ 3.1$\times10^{22}$ cm$^{-2}$. We clearly detected the Fe-K$\alpha$ line at $\sim$6.5 keV in the spectra. The plasma shows slightly enhanced abundances of Mg, Si, S and Fe indicating that the plasma is likely to be of ejecta origin. We find no significant feature of a recombining plasma in this SNR. In order to characterize radial variations in the X-ray spectral parameters, we also analyze annular regions in the remnant. We investigate the explosive origin of Kes 69 and favor the core-collapse origin. Additionally, we report a lack of significant gamma-ray emission from Kes 69, after analyzing the GeV gamma-ray data taken for about 9 years by the Large Area Telescope on board [*Fermi*]{}. Finally, we discuss the properties of Kes 69 in the context of other interacting MM SNRs.'
author:
- |
A. Sezer,$^{1}$[^1] T. Ergin$^{2}$[^2], R. Yamazaki$^{3}$, Y. Ohira$^{4}$ and N. Cesur$^{5}$\
$^{1}$Department of Electrical-Electronics Engineering, Avrasya University, 61250, Trabzon, Turkey\
$^{2}$TUBITAK Space Technologies Research Institute, ODTU Campus, 06800, Ankara, Turkey\
$^{3}$Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara 252-5258, Japan\
$^{4}$Department of Earth and Planetary Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\
$^{5}$Department of Physics, Y[i]{}ld[i]{}z Technical University, 34220, Istanbul, Turkey
title: 'A [*Suzaku*]{} X-ray study of the mixed-morphology supernova remnant Kes 69 and searching for its gamma-ray counterpart'
---
\[firstpage\]
ISM: individual objects: Kesteven 69 (G21.8$-$0.6) $-$ ISM: supernova remnants $-$ X-rays: ISM $-$ gamma-rays: ISM.
Introduction
============
Galactic supernova remnant (SNR) Kesteven 69 (also known as G21.8$-$0.6, hereafter Kes 69) has an extended incomplete radio shell with an angular diameter of $\sim$20 arcmin (e.g. @Sh70). [*ROSAT*]{} and [*Einstein*]{} observations showed that this SNR has an irregular X-ray morphology, which shows correlation with an incomplete radio shell [@Se90; @Yu03]. Based on the bright X-ray emission from the interior of the radio shell and a possible thermal X-ray spectrum, this remnant was classified as a mixed-morphology (MM) SNR [@Yu03]. @Gr97 detected a compact OH maser at velocity of $\sim$69.3 km s$^{-1}$ in the northeastern part of the remnant using the Very Large Array (VLA) and Australia Telescope Compact Array (ATCA) observations. At $\sim$85 km s$^{-1}$ velocity, @He08 detected extended OH maser emission toward the southern bright radio shell in the Green Bank Telescope OH maser survey. Millimeter band observations of CO and HCO$^{+}$ lines toward Kes 69 provided strong evidence of an association between SNR and the $\sim$$+$85 km s$^{-1}$ component of molecular gas [@Zh09]. [*Spitzer*]{} observations of the remnant revealed the bright molecular emission lines of OH, CO and H$_{2}$, excited by a shock [@Re06; @He09a].
X-ray observations of Kes 69 were carried out by [*Einstein*]{}, [*ROSAT*]{}, [*ASCA*]{} and [*XMM-Newton*]{} [@Se90; @Yu03; @Bo12; @Se13]. @Yu03 used [*ROSAT*]{} Position Sensitive Proportional Counter data and found that the thermal X-ray spectrum is well fitted by VMEKAL model with an absorbing column density of $N_{\rm H}$ $\sim$ 2.4 $\times10^{22}$ cm$^{-2}$ and an electron temperature of $kT_{\rm e}$ $\sim$ 1.6 keV. @Bo12 searched for compact hard X-ray sources in the field of Kes 69 using [*XMM-Newton*]{} data and reported on the detection of 18 hard X-ray sources in the 3.0$-$10.0 keV. @Se13 analyzed [*XMM-Newton*]{} data and showed that the plasma of Kes 69 has a collisional ionization equilibrium (CIE) condition and yielded a plasma temperature of $kT_{\rm e}$ $\sim$ 0.62 keV and a column absorption of $N_{\rm H}$ $\sim$ 2.85 $\times10^{22}$ cm$^{-2}$.
The kinematic distance to Kes 69 is estimated as $\sim$11.2 kpc [@Gr97] from the OH maser velocity, $\sim$5.5 kpc from H[i]{} and $^{13}$CO spectra [@Ti08], and $\sim$5.2 kpc from the SNR/molecular cloud (MC) association [@Zh09]. Assuming the Sedov stage solution, @Bo12 estimated an age of the SNR about 10, 000 yr from [*XMM-Newton*]{} data. @Se13 derived the age of the remnant to be $\sim$$2.5\times10^{4}$ yr assuming a distance of 5.5 kpc.
Kes 69 was searched in TeV and GeV gamma-ray bands by H.E.S.S. and the Large Area Telescope detector on board [*Fermi*]{} Gamma-Ray Space Telescope ([*Fermi*]{}-LAT), respectively. In the 1st [*Fermi*]{}-LAT Supernova Remnant Catalog [@Ac16], the upper limits on the flux of this SNR were reported on Table 3, i.e. among the not-detected SNRs, and H.E.S.S. missed Kes 69 in TeV gamma rays [@Bo11]. Recently, although a preliminary analysis of [*Fermi*]{}-LAT data showed the detection of GeV gamma rays from Kes 69, @Er16 cautioned that there are many bright gamma-ray sources in the close neighborhood of Kes 69 that could be contributing to this result.
Several SNRs are known as MM SNRs, which are identified by shell emission in the radio and centrally brightened thermal emission in the X-ray band with little or no shell brightening [@Rh98; @La06; @Vi12]. However, the nature of the centrally brightened X-ray emission is poorly understood. The overionized/recombining plasma (RP) was observed in many MM SNRs (e.g. @Ya09 [@Oz09; @Er17; @Su18]). In these SNRs, the ionization temperature ($kT_{\rm z}$) has been found to be significantly higher than the electron temperature ($kT_{\rm e}$). Most MM SNRs exhibit the GeV/TeV gamma-ray emission (e.g., @Ab09 [@Ab10a; @Ca10]). They are associated with MCs as indicated by CO line emission and/or OH (1720 MHz) masers. There is a strong association between maser-emitting (ME) and MM SNRs, which implies that MM SNRs require dense molecular gas in their environment to show the unusual centrally filled X-ray morphology (e.g., @Sl17 [@Yu03; @Wh91]). Kes 69 has a bright X-ray emission from the interior of the radio shell and interacting with MC based on 1720 MHz OH maser emission observations; thus, Kes 69 is considered to be a member of ME-MM SNRs [@Yu03].
In this paper, we study the high spectral resolution [*Suzaku*]{} observation of Kes 69 to investigate the nature of the X-ray emission and address the observed spectral characteristics. Additionally, we search for the gamma-ray counterpart of Kes 69 using archival [*Fermi*]{}-LAT data. The rest of the paper is organized as follows: We first summarize observations and data reduction in Section 2. In Section 3, we describe our analysis. Then, in Section 4, we discuss the nature of the thermal X-ray emission with a comparison to previous X-ray studies (Section 4.1), gamma-ray results (Section 4.2), its properties in the context of other interacting MM SNRs (Section 4.3) and explosion origin of Kes 69 (Section 4.4). The conclusions are summarized in Section 5.
Observations and Data Reduction
===============================
X-ray Data
----------
Kes 69 was observed with the [*Suzaku*]{} X-ray Imaging Spectrometer (XIS: @Ko07a) on 2014 September 27-29 (ObsID: 509037010). The net exposure of the cleaned event data was $\sim$67.2 ks. The XIS consists of four charge-coupled device (CCD) cameras placed at the focus of the X-Ray Telescope (XRT: @Se07), covering the 0.2$-$12 keV energy range. Each XIS sensor has 1024$\times$1024 pixels and covers a 17.8$\times$17.8 arcmin$^2$ field of view (FoV). The XIS 0, 2, and 3 are front side illuminated CCDs, whereas XIS1 is a back-side illuminated CCD. Since 2006 November 9, the XIS2 has not been available for observations. We retrieved [*Suzaku*]{} archival data of Kes 69 from the Data Archives and Transmission System (DARTS)[^3]. Data reduction and analysis were made with [Headas]{} software version 6.20[^4] and [xspec]{} version 12.9.1 [@Ar96] with AtomDB v3.0.9[^5] [@Sm01; @Fo12]. We used a cleaned event file created by the [*Suzaku*]{} team, combined the 5$\times$5 and 3$\times$3 editing mode event files using [xselect]{} v2.4d and generated the Redistribution Matrix Files and Ancillary Response Files using [xisrmfgen]{} and [xissimarfgen]{} tools [@Is07]. All spectra are binned to a minimum of 25 counts per bin to allow use of the $\chi^{2}$ statistic using the ftool [grppha]{}.
Gamma-ray Data
--------------
[*Fermi*]{}-LAT data obtained between 2008-08-04 and 2017-03-07 were used in this analysis. We selected the [*Fermi*]{}-LAT Pass 8 ‘Source’ class and front$+$back type events coming from zenith angles smaller than 90$^{\circ}$ and from a circular region of interest (ROI) with a radius of 30$^{\circ}$ centred at the SNR’s radio position using `gtselect` of Fermi Science Tools (FST).
The exposure map produced by the likelihood calculations gives the total exposure (in terms of effective area multiplied by time) for a given position on the sky producing counts inside the ROI. The time that [*Fermi*]{}-LAT observed at a given position on the sky at a given off-axis angle is called the livetime and it is used to calculate the exposure map. The FST `gtltcube` tool produces an array of livetimes at all points on the sky and saves it into a livetime cube. Then the `gtexpcube2` tool of FST can be used to apply the livetime cube to the ROI to generate a binned exposure map. In our analysis, the livetime calculated over the whole sky ranged between 26 mins and 388 hours and 14 exposure maps were produced for the analysis region of 15$^{\circ}$ $\times$ 15$^{\circ}$, where each of the maps corresponds to a different logarithmic energy bin assigned between 200 MeV and 300 GeV.
Analysis
========
X-ray Spectral Analysis
-----------------------
In Figure 1, we present the XIS1 image of Kes 69 in the 0.3$-$10.0 keV energy band. The radio data from National Radio Astronomy Observations (NRAO) Very Large Array (VLA) Sky Survey (NVSS: @Co98) are overlaid for comparison. The cross shows the position of maser emission at velocity $\sim$69.3 km s$^{-1}$ [@Gr97]. The spectral extraction regions are shown with the circles, while the eliminated hard point-like sources detected by [*XMM-Newton*]{} [@Bo12] are shown with magenta circles (radius 0.33 arcmin). The calibration sources at the corner of the CCD chips are excluded from the image.
### Background Estimation
For the background analysis of [*XMM-Newton*]{} data, @Bo12 used an empty area in the northeastern part of the FoV as a background region. We extracted the background spectra from the FoV of the XIS, shown by the dashed area in Figure 1, excluding the calibration sources at the corners of the FoV, the point-like sources, and the emission from the remnant.
Recent studies have demonstrated that an accurate estimation of the X-ray background is particularly important to generate the X-ray spectrum of extended sources such as SNRs (e.g., G346.6$-$0.2: @Ya13; G32.8$-$0.1: @Ba16a). For example, the flux of the Galactic ridge X-ray emission (GRXE) strongly depends on the location of the SNRs (e.g., @Uc13 [@YaS16]). We therefore consider the following background components: the non-X-ray background (NXB) as an instrumental background component and the GRXE and cosmic X-ray background (CXB) components as the X-ray background. We generated the NXB using [xisnxbgen]{} [@Ta08] and subtracted it from the extracted spectrum. Then, we fitted the NXB-subtracted background spectra with the following model:
$$\begin{aligned}
{Abs_{\rm CXB} \times (power-law)_{\rm CXB} + Abs_{\rm GRXE} \times (apec + apec)}
\label{eqn:pi0}\end{aligned}$$
where the apec is a CIE plasma model in the [xspec]{}. The second term is the GRXE component. We assumed the CXB spectrum as a power-law of photon index 1.4 and CXB having a surface brightness of 5.4$\times$10$^{-15}$ erg s$^{-1}$ cm$^{-2}$ arcmin$^{-2}$ in the 2$-$10 keV energy band [@Ku02]. The X-ray background spectrum was simulated by using the [fakeit]{} command in [xspec]{} and then it was subtracted from the observed spectrum.
### Spectral Fits
{width="53.00000%"}
In order to characterize the X-ray emitting plasma of the whole SNR, we first extracted the XIS spectra from the circular region with a radius of $\sim$6.1 arcmin, which is shown in Figure 1. To search for spectral variations in the temperature, ionization state and elemental abundances of Kes 69, we also extracted spectra from three annular regions of 0$-$2, 2$-$4 and 4$-$6 arcmin centred around $\rmn{RA}(J2000)=18^{\rmn{h}} 32^{\rmn{m}} 58^{\rmn{s}}$, $\rmn{Decl.}~(J2000)=-10\degr 07\arcmin 51\arcsec$, which are regions 1, 2 and 3, respectively, as shown in Figure 1.
The previous observations suggested that the X-ray spectrum of Kes 69 is characterized by a thermal plasma in CIE [@Yu03; @Bo12; @Se13]. Therefore, we first applied an absorbed (TBABS: @Wi00) single-component variable-abundance CIE model (VMEKAL model in [xspec]{}). The absorbing column density $N_{\rm H}$, electron temperature $kT_{\rm e}$, abundances of Mg, Si, S and Fe, and normalization were allowed to vary freely during the fitting. The other metal abundances were fixed to the solar values [@Wi00]. In this case, the electron temperature $kT_{\rm e}$ was obtained to be $\sim$1.2 keV, but this model was statistically unacceptable (with a reduced-$\chi^{2}$ $>$ 1.4).
As a next step, we employed an absorbed non-equilibrium ionization (NEI) plasma model (VNEI model in [xspec]{}), where the free parameters were the interstellar absorption, $N_{\rm H}$, electron temperature, $kT_{\rm e}$, ionization time-scale, $n_{\rm e}t$, where $n_{\rm e}$ and $t$ represent the electron density and the time after the shock heating, respectively. The abundances of Mg, Si, S and Fe were also free parameters, while the other abundances were fixed to the solar values. We obtained a statistically acceptable fit (with a reduced-$\chi^{2}$ value of 1.19) for the whole region. To confirm the spectral parameters, we fitted the spectra with a single-component plane-parallel shock model with variable abundances ([xspec]{} model VPSHOCK; @Bo01) modified by interstellar absorption. We obtained an equally good fit with similar parameters to VNEI.
In order to check whether a cool component is needed across the remnant, we applied the two-component thermal plasma model (VNEI+VMEKAL) to the spectrum. For VNEI component, abundances of Mg, Si, S and Fe were free parameters. The abundances of all elements were assumed to the solar values for VMEKAL component. This model slightly improved the fit ($\chi^{2}_{\nu}$/dof=1.13/1539), but it did not give well-constrained spectral parameters. We therefore employed one-temperature NEI plasma model to fit the spectra of our spectral regions and concluded that the VNEI model represents the spectra. The background-subtracted 0.9$-$8.0 keV spectra are presented in Figure 2. For the annulus regions, we fixed the absorption $N_{\rm H}$ to that obtained for the whole SNR and then fitted each spectrum to a VNEI model. We obtain statistically acceptable fits (with reduced-$\chi^{2}$ values of 1.04$-$1.13). In Table 1, we present the best-fitting parameters with 90 per cent confidence ranges for all regions.
We also tried an absorbed RP model (VRNEI[^6] in [xspec]{}) to search for the RP. This model is characterized by final electron temperature ($kT_{\rm e}$) and initial electron temperature ($kT_{\rm init}$), elemental abundances and a single ionization time-scale ($\tau$). We find that an initial plasma temperature ($\sim$0.57 keV) much smaller than the current plasma temperature ($\sim$2.59 keV) indicating that the plasma of this SNR is under-ionized. We also obtain $kT_{\rm init}$ $<$ $kT_{\rm e}$ for our annular regions indicating that the plasma is still ionizing.
Component Parameters Whole Region 1 Region 2 Region 3
----------- --------------------------------------------- --------------------- --------------------- --------------------- ---------------------
TBABS $N_{\rm H}$ ($10^{22}$ cm$^{-2})$ $3.1_{-0.2}^{+0.2}$ 3.1 (fixed) 3.1 (fixed) 3.1 (fixed)
VNEI $kT_{\rm e}$ (keV) $2.5_{-0.2}^{+0.4}$ $2.6_{-0.4}^{+0.5}$ $2.1_{-0.2}^{+0.4}$ $2.1_{-0.2}^{+0.2}$
Mg $1.3_{-0.2}^{+0.2}$ $1.3_{-0.1}^{+0.2}$ $1.4_{-0.2}^{+0.1}$ $1.2_{-0.2}^{+0.2}$
Si $1.5_{-0.2}^{+0.2}$ $1.2_{-0.1}^{+0.3}$ $1.6_{-0.2}^{+0.2}$ $1.6_{-0.3}^{+0.2}$
S $1.6_{-0.2}^{+0.2}$ $1.8_{-0.2}^{+0.4}$ $1.9_{-0.2}^{+0.3}$ $1.2_{-0.1}^{+0.2}$
Fe $1.6_{-0.5}^{+0.6}$ $1.5_{-0.4}^{+0.4}$ $1.4_{-0.2}^{+0.3}$ $1.4_{-0.2}^{+0.1}$
$\tau$=$n_{\rm e}t$ ($10^{10}$ cm$^{-3}$ s) $4.1_{-0.5}^{+0.6}$ $3.3_{-0.6}^{+0.4}$ $5.2_{-0.4}^{+0.4}$ $4.3_{-0.6}^{+0.4}$
Norm (10$^{-3}$ cm$^{-5}$) $4.8_{-0.7}^{+0.6}$ $0.7_{-0.1}^{+0.2}$ $2.4_{-0.1}^{+0.1}$ $2.4_{-0.2}^{+0.5}$
reduced-$\chi^{2}$ (dof) 1.19 (1541) 1.04 (309) 1.09 (793) 1.13 (1056)
[**Notes.**]{} The normalization of the VNEI, norm=$10^{-14}$$\int n_{\rm e} n_{\rm H} dV$/($4\pi d^{2}$) (cm$^{-5}$), where $n_{\rm e}$, $n_{\rm H}$, $V$ and $d$ are the electron and hydrogen densities (cm$^{-3}$), emitting volume (cm$^{3}$) and distance to the source (cm), respectively. Abundances are given relative to the solar values of @Wi00.
{width="75.00000%"}
Gamma-ray Analysis
------------------
We used `fermipy`[^7] analysis toolkit to apply the maximum likelihood fitting method [@Ma96] on spatially and spectrally binned data for 0.2 $-$ 300 GeV by taking P8R2$_{-}$SOURCE$_{-}\!\!$V6 as the instrument response function. The analysis yields the test statistics (TS) value, which is the square root of the detection significance. A large TS value shows that the maximum likelihood value for a model without an additional source (the null hypothesis) is not valid.
The background model of the analysis region consists of the diffuse background sources and all the point-like and extended sources from the 3rd [*Fermi*]{}-LAT Source Catalog [@Ac15] located within 15$^{\circ}$ $\times$ 15$^{\circ}$ region centred on the ROI centre. All parameters of the diffuse Galactic emission (*gll$_{-}$iem$_{-}$v6.fits*) and the isotropic component (*iso$_{-}$P8R2$_{-}$SOURCE$_{-}\!\!$V6$_{-}\!$v06.txt*) were freed. The normalization parameters of all sources within 3$^{\circ}$ are set free. In addition, we freed all sources with TS $>$ 10 and fixed all sources with TS $<$ 10.
Results and Discussion
======================
X-ray Properties of Kes 69
--------------------------
The X-ray spectra of the SNR were previously studied by @Yu03 with [*ROSAT*]{} and [*ASCA*]{}, by @Bo12 and by @Se13 with [*XMM-Newton*]{}. Although their spectral fits indicate that the X-ray-emitting gas is characterized by a CIE model, we find that the plasma is well represented by a NEI plasma model. Our estimated absorption is slightly higher than [*ROSAT*]{} and [*XMM-Newton*]{} results. The CIE plasma temperatures found are $\sim$1.6 keV [@Yu03], $\sim$0.8 keV [@Bo12] and $\sim$0.62 keV [@Se13], which are significantly lower than our result. These inconsistencies may be related to the detection of Fe-K line in the [*Suzaku*]{} spectra, which produces significantly higher $kT_{\rm e}$ compared with previous studies. Also, in our analysis we use different background regions, different background estimation method, the different spectral capabilities of the X-ray satellites, the elemental abundances by @Wi00 and the recently updated AtomDB database.
The X-ray spectra of the annular regions surrounding the X-ray centre of Kes 69 show no strong evidence of spectral variability. A characteristic feature of MM SNRs is a flat electron temperature profile [@Rh98]. Kes 69 shows a flat temperature of $\sim$2.3 keV across its centre, which is consistent with MM SNRs. The ionization time-scales for all regions indicate that the X-ray emitting plasma across the whole remnant is in an NEI condition. We find enhanced abundances of Mg, Si, S and Fe in all regions indicating that the X-ray emitting plasma has an ejecta origin. However, this result is not conclusive when the error bars are taken into account (see Table 1).
The density of the X-ray emitting gas was estimated from the normalization, norm=$n_{\rm e}n_{\rm H}
V$/($4\pi d^{2}$$10^{14}$). Assuming that the emitting region to be a sphere of radius 6.1 arcmin, the SNR is at a distance of 5.2 kpc and $n_{\rm e}=1.2n_{\rm H}$, we estimated the emission volume to be $V\sim 9.6\times10^{58}fd_{5.2}^{3}$ ${\rm cm^{3}}$, where $f$ is the volume filling factor and $d_{5.2}$ is the distance scaled to 5.2 kpc. Consequently, we found an ambient gas density of $\sim$0.14$f^{-1/2}d_{5.2}^{-1/2}$ ${\rm cm}^{-3}$ and age of $\sim$$9.2\times10^{3}f^{1/2}d_{5.2}^{1/2}$ yr. Our estimated age of the remnant is comparable to the derived by @Bo12 and lower than that derived by @Se13 in their [*XMM-Newton*]{} analysis. We then derived the total X-ray-emitting mass, $M_{\rm X}$ $\sim 15.8 f^{1/2}d_{5.2}^{5/2}{M\sun}$ using $M_{\rm X}$=1.4$m_{\rm H}n_{\rm e}V$. This result indicated that the X-ray-emitting plasma is dominated by ejecta material of a core-collapse (CC) supernova (SN), because the X-ray-emitting mass is much larger than the Chandrasekhar mass.
Assuming the @Se59 model, we calculated the SNR radius of $R$ $\sim$ 18.6 pc and the swept-up interstellar medium (ISM) mass of $M_{\rm sw}$ $\sim$ $56M_{\sun}$ using the initial ISM density $n_{0}$=0.14 cm$^{-3}$, SNR age $t=9200$ yr and explosion energy $E=10^{51}$ erg. The swept-up ISM mass of Kes 69 is reasonably explained if the X-ray-emitting plasma is dominated by the shocked ambient medium. We note that a large swept-up ISM mass may be expected for MM SNRs, which appear to be interacting with MCs.
We also calculated the total thermal energy of $\sim$$10^{51}$ erg by assuming $M$=56$M_{\sun}$ and ion temperature $T_{\rm i}$=2.5 keV. We therefore cannot rule out the ISM origin if $T_{\rm e}$=$T_{\rm i}$. However, the relaxation time-scale due to the Coulomb interaction between $T_{\rm i}$ and $T_{\rm e}$ is about $10^{13}$ sec ($n$/0.1 cm$^{-3}$)$^{-1}$, which is much larger than expected age of the SNR. So, we may expect $T_{\rm i}$ $>>$ $T_{\rm e}$ and the ISM model requires explosion energies much larger than $10^{51}$ erg.Ê In this case, the X-ray emitting plasma would be more like ejecta in origin.Ê
Gamma-ray Results
-----------------
In the energy range of 0.2 - 300 GeV no excess gamma-ray emission was found from the direction of Kes 69. The upper limit at 95 per cent confidence level (CL) on the flux and energy flux were found to be 3.1 $\times$ 10$^{-6}$ MeV cm$^{-2}$ s$^{-1}$ and 2.1 $\times$ 10$^{-9}$ cm$^{-2}$ s$^{-1}$, respectively. The flux upper limit is comparable to the upper limit given in the 1st [*Fermi*]{}-LAT Supernova Remnant Catalog [@Ac16], which is 1.8 $\times$ 10$^{-9}$ cm$^{-2}$ s$^{-1}$ for a spectral index of 2.5 and 95 per cent upper limit. Assuming the distance of 5.5 kpc to the SNR, the upper limit of the gamma-ray luminosity is 1.8 $\times$ 10$^{34}$ erg s$^{-1}$, which is slightly smaller than the reported gamma-ray luminosities for interacting SNRs [@Ac16].
Kes 69 is an MM SNR in Sedov phase and shows signs of interaction with MC thorough the detection of OH (1720 MHz) masers. However, due to narrow physical conditions needed to excite the OH (1720 MHz) masers, the lack of them would not give any information about hadronic gamma rays being emitted from this SNR or not [@Fr11]. Besides, although the cosmic rays may be responsible for the local enhancement in SNRs, which is sufficient to produce OH abundance in the post shock gas, the ionization rate from the interior X-ray emission is comparable to the one from cosmic rays. The dominant emission mechanism might depend on the gas location with respect to the interior X-ray emitting plasma and cosmic ray acceleration site [@He09b].
@He08 reported extended OH (1720 MHz) maser emission from Kes 69 using GBT data, which was previously not detected by VLA (summarized in Table 3 of @He08). They reported a single compact maser emission at +69 km s$^{-1}$, which was also detected during the VLA observations [@Gr97]. However, faint OH (1720 MHz) emission is present at +85 km s$^{-1}$ that appears across the southern ridge of the SNR, which has broader line widths and a velocity gradient. Since both the extended maser emission and molecular material are found at around +85 km s$^{-1}$, it raises suspicion about the Kes 69 origin of the maser found at +69 km s$^{-1}$. If +69 km s$^{-1}$ maser is not associated with Kes 69, then the OH (1720 MHz) maser emission arising from Kes 69 is only extended in nature. These extended masers are about 20 times lower in brightness than compact masers and they may be produced by the interior X-ray emission of the SNR. @Zh09 showed that a molecular arc is present at 77-86 km s$^{-1}$ at the southeast part of the shell of Kes 69, which overlaps with the 1.4 GHz radio continuum and mid-infrared (IR) observations at the same region of the shell. They also found HCO$^{+}$ emission at 85 km s$^{-1}$ on the shell. Both the molecular arc and the HCO$^{+}$ emission at $\sim$85 km s$^{-1}$ were reported to be consistent with the presence of the extended 1720 MHz OH emission along the southeastern boundary of Kes 69.
According to @Zh09, the multi-wavelength emissions along the southeastern shell of Kes 69 were caused by the impact of the SNR shock on a dense, clumpy patch of molecular gas, which pre-existed and is possibly the cooled and clumpy left-over debris of the interstellar molecular gas swept up by the progenitors stellar wind. Therefore, a possible explanation of the lack of GeV gamma-ray emission from Kes 69 is that the extended maser emission is produced by the interior X-ray emitting plasma rather than by cosmic rays accelerated at regions away from the molecular gas clumps.
Kes 69: A Maser Emitting Mixed-morphology SNR
---------------------------------------------
Kes 69 is a middle-aged SNR, its centre-filling X-ray emission surrounded by a shell-like radio structure suggests that Kes 69 belongs to the category of MM SNRs. There are two main models to explain the centrally filled X-ray emission in the MM SNRs. One is the evaporating cloudlet model (e.g., @Wh91) and the other is the thermal conduction model (e.g., @Co99). But, the true origin of the centre-filled thermal X-ray emission from middle-aged MM SNRs is still unknown. It is unclear what leads to their centre-filled morphology and whether these SNRs are dominated by the ejecta or by the shocked ISM. The origin of emission from ejecta-dominated MM SNRs also remains unknown. Our XIS analysis shows that Kes 69 is likely to be of ejecta origin.
Recently, @Sl17 performed multi-dimensional hydrodynamics simulations to discuss the origin of centrally peaked X-ray profile. They do not include any piston ejecta but put thermal energy in their simulations. Then the central diffuse gas expands due to large thermal pressure, pushing ISM to form the shock. According to @Sl17 the origin of the hot gas is the ISM and not the ejecta. In addition, @Sl17 suggest that the SNR’s centre-filled morphology is due to the expansion into a cloudy medium and ejecta is not a dominant factor in the formation of this morphology. As mentioned in Section 4.1, X-ray properties of Kes 69 show that the detected hot gas in Kes 69 is likely ejecta in origin rather than ISM.
Here, we discuss Kes 69 in comparison with other MM SNRs known to be interacting with MCs. We list the SNRs identified as interacting MM SNRs type and summarize their properties, such as detection of RP and gamma-ray in Table 2. As seen in Table 2, all of the SNRs that have detected RP emission, show also gamma-ray emission except for G346.6$-$0.2. Other interacting MM SNRs, like W51C and G298.6$-$0.0 in Table 2, show gamma-ray emission, but no RP emission. There has been no published analysis results about RP in Sgr A East and Kes 41, but they present gamma-ray emission. However, Kes 69 is the only candidate in the list for which no gamma-rays nor RP emission is detected.
----------------------- ------------ ---------------- ----------- ----------------------- ------------------------ --------------------------------------------- --------------- --------------------
SNR Other ${\gamma}$-ray RP MC $kT_{\rm init}$ $kT_{\rm e}$ Age$^{\rm d}$ References
name detection detection interaction$^{\rm a}$ (keV) (keV) (kyr)
G0.0+0.0 Sgr A East GeV/TeV ? Y - $1.21_{-0.03}^{+0.02}$, $6.0_{-0.5}^{+0.4}$ 1.2$-$10 \[1, 2, 3, 4\]
G6.4$-$0.1 W28 GeV/TeV Y Y 3 (fixed) $0.40_{-0.03}^{+0.02}$ 33$-$36 \[5, 6, 7\]
G21.8$-$0.6$\dagger$ Kes 69 N N Y $0.57_{-0.02}^{+0.03}$ $2.57_{-0.34}^{+0.26}$ 5$-$25 This work
G31.9+0.0 3C 391 GeV Y Y $1.8_{-0.6}^{+1.6}$ 0.495$\pm$0.015 3.7$-$4.4 \[8, 9\]
G34.7$-$0.4$\dagger$ W44 GeV Y Y $1.07_{-0.06}^{+0.08}$ 0.48$\pm$0.02 20$-$28 \[10, 11\]
G49.2$-$0.7 W51C GeV/TeV N Y - $0.69_{-0.05}^{+0.06}$ 18$-$30 \[12, 13, 14\]
G89.0+4.7$\dagger$ HB21 GeV Y Y$^{\rm b}$ $0.58_{-0.07}^{+0.09}$ $0.17_{-0.02}^{+0.01}$ 4.8$-$15 \[15, 16\]
G189.1+3.0$\dagger$ IC 443 GeV/TeV Y Y 10 (fixed) 0.65$\pm$0.04 3$-$30 \[17, 18, 19, 20\]
G298.6$-$0.0 GeV N Y$^{\rm c}$ - $0.78_{-0.08}^{+0.09}$ - \[21\]
G337.8$-$0.1 Kes 41 GeV ? Y - $1.80_{-0.47}^{+0.82}$ 12$-$16 \[22, 23\]
G346.6$-$0.2 N Y Y 5 (fixed) $0.30_{-0.01}^{+0.03}$ 4.2$-$16 \[24, 25\]
G348.5+0.1 CTB 37A GeV Y Y 5 (fixed) $0.49_{-0.06}^{+0.09}$ 10$-$30 \[26, 27\]
G359.1$-$0.5$\dagger$ GeV/TeV Y Y $0.77_{-0.08}^{+0.09}$ 0.29$\pm$0.02 $\geq$10 \[28, 29, 30\]
----------------------- ------------ ---------------- ----------- ----------------------- ------------------------ --------------------------------------------- --------------- --------------------
[**Notes.**]{} $^{\rm a}$OH detection from @Ji10. $^{\rm b}$Interaction shown by CO MA & LB, CO ratio, H$_{2}$, NIR. $^{\rm c}$Possible detection of IR emission [@Re06]. $^{\rm d}$SNR catalog in @Fe12. $\dagger$ Ejecta-dominated.
[**References.**]{} \[1\] @Ah06; \[2\] @Ac16; \[3\] @Aj17; \[4\] @Ko07b; \[5\] @Ah08a; \[6\] @Ab10c; \[7\] @Sa12; \[8\] @Er14; \[9\] @Sa14; \[10\] @Ab10b; \[11\] @Uc12; \[12\] @Ab09; \[13\] @Al12; \[14\] @Ha13; \[15\] @Pi13; \[16\] @Su18; \[17\] @Ab10a; \[18\] @Alb07; \[19\] @Ya09; \[20\] @Oh14; \[21\] @Ba16; \[22\] @Li15; \[23\] @Zh15; \[24\] @Er12; \[25\] @Ya13; \[26\] @Ca10; \[27\] @YaS14; \[28\] @Ah08b; \[29\] @Hu11; \[30\] @Oh11.
------- ------------------------ ------------- --------------- ---------------- ---------------- -------------- -------------- -------------- -------------- -------------- --------------
Ratio Kes 69 W7$\dagger$ WDD2$\dagger$ PDDe$\ddagger$ DDTe$\ddagger$ 11$M_{\sun}$ 12$M_{\sun}$ 15$M_{\sun}$ 20$M_{\sun}$ 25$M_{\sun}$ 30$M_{\sun}$
Mg/Si $0.81_{-0.18}^{+0.20}$ 0.06 0.02 0.0017$^\star$ 0.025$^\star$ 0.57 0.12 0.70 0.16 0.45 1.79
S/Si $1.02_{-0.18}^{+0.19}$ 1.07 1.17 1.5 1.4 0.87 1.53 0.62 1.28 0.96 0.24
Fe/Si $1.06_{-0.35}^{+0.39}$ 1.56 0.85 0.89 0.91 1.37 0.23 0.70 0.88 0.13 0.15
------- ------------------------ ------------- --------------- ---------------- ---------------- -------------- -------------- -------------- -------------- -------------- --------------
[**Notes.**]{} $\dagger$ @No97. $\ddagger$ @Ba03. $^\ast$@Wo95. $^\star$@Ra06.
The explosion origin of Kes 69
------------------------------
As mentioned before, Kes 69 is an ME SNR and [*Spitzer*]{} detection of spectral lines associated with shocked H$_2$, which are strong evidences of CC origin. We consider other methods to estimate the progenitor for this SNR. There are several methods to determine the origin of SNRs; (i) X-ray morphology: Using high-resolution [*Chandra*]{} images of SNRs, @Lo09 [@Lo11] found that the CC SNRs are more spatially asymmetric than the Type Ia SNRs; (ii) Fe-K line: @Ya14 concluded that Fe-K centroid energies for Type Ia SNRs are lower ($<$6.55 keV) than those of CC SNRs; (iii) Abundance of ejecta; (iv) Presence of compact object or pulsar wind nebulae (PWN); (v) Environment.
To determine the Fe-K centroid energy, we fitted the 5.0$-$8.0 keV spectrum with an absorbed power-law plus a Gaussian model. We note that the Gaussian line width parameter was fixed to zero. For comparison with the results of @Ya14, we estimated the centroid energy of Fe-K$\alpha$ as $6.47_{-0.03}^{+0.03}$, $6.50_{-0.06}^{+0.05}$, $6.49_{-0.04}^{+0.04}$ and $6.45_{-0.05}^{+0.05}$ keV for whole, Regions 1, 2 and 3, respectively, which agree on a Type Ia origin.
The GRXE also emits K-shell lines from neutral iron (at 6.4 keV), helium-like iron (at 6.7 keV) and hydrogen-like iron (at 7.0 keV) (e.g. @Uc13). Therefore, we compare the flux of Fe-K line from Kes 69 and GRXE. The Gaussian line at $\sim$6.5 keV of the remnant has a surface brightness of $\sim$0.8$\times$10$^{-7}$ photons cm$^{-2}$ s$^{-1}$ arcmin$^{-2}$, while that of the background is very faint (less than $\sim$1 per cent of the $\sim$6.5 keV line flux detected in the remnant). This result indicates that 6.5 keV Fe-K line is not a contamination of the Fe K-shell line in the GRXE and the detection of the $\sim$6.5 keV line feature is robust.
Table 3 presents the best-fitting abundances of the whole region relative to Si, the abundance ratios expected for CC SN with different progenitor masses CC [@Wo95] and Type Ia models [@No97; @Ba03]. Mg/Si ratios are taken from @Ra06, since they were not given in @Ba03. As seen from Table 3, we find that the CC model with 11$M_{\sun}$ is feasible.
We also plotted the element masses in the X-ray-emitting gas, the Type Ia models of @No97 and @Ba03 and the CC nucleosynthesis models with various progenitor masses of @Wo95 [@No06; @Th96; @Su16] and @Fr18 for comparison in Figure 3 and 4, similar to @Zh16. In some CC models (e.g., @Fr18), the yields are strongly dependent on the SN explosion energy. To see how the abundance depends on the progenitor mass and the explosion energy, we plotted the yields for different SN explosion energies (15, 20 and 25$M_{\sun}$) as given by @Fr18, which is shown in Figure 4. We see that none of the 25$M_{\sun}$ models can explain the observed the elemental mass for this SNR. As shown in Figure 3 and 4, we found that the estimated values of $M_{\rm Mg}$ $\sim$ 0.01, $M_{\rm Si}$ $\sim$ 0.016, $M_{\rm S}$ $\sim$ 0.009 and $M_{\rm Fe}$ $\sim$ 0.0018 $M_{\sun}$ are in good agreement with the progenitor mass of 11$M_{\sun}$ [@Wo95] and between 9 and 12$M_{\sun}$ [@Su16] of the CC models.
The environment of Kes 69 is suggestive of a massive progenitor star: it is interacting with MCs and the detection of IR emission from shocked molecular gas. Also its X-ray morphology has highly asymmetric nature and it is located in the Galactic plane. All these environmental and morphological typing methods support that Kes 69 has a CC origin. We note that there are no features for central compact object or a PWN in the remnant.
In summary, based on the total ejecta mass (as discussed in Section 4.1) and the result of our comparison between the element masses in the X-ray-emitting plasma and the predicted SN yields (as seen in Figure 3), we conclude that the origin of Kes 69 is a CC SN.
The centroid energy of Fe-K$\alpha$ line ($\sim$6.5 keV) corresponds to a charge number of about +18 (see Figure 1, left; @Ya14), which is consistent with $n_{\rm e}t$ $\sim$ $4\times10^{10}$ cm$^{-3}$ s (see Figure 5 of @Ha15). The low $n_{\rm e}t$ value and low centroid energy of Fe-K$\alpha$ suggest that the CC SN exploded in the low density region.
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{width="55.00000%"} {width="55.00000%"} {width="55.00000%"}
Conclusions
===========
In this paper, we analyze the archival [*Suzaku*]{} and [*Fermi*]{}-LAT data of Kes 69 and report on the spectral properties of the X-ray emission of this remnant. Our main conclusions are summarized below:
1. The centre-filled X-ray emission and the radio shell-like morphology suggest that Kes 69 is a member of the MM SNR class, as noted before by @Yu03.
2. We confirmed the thermal origin of the central plasma and detected slightly enhanced abundances of Mg, Si, S and Fe, which suggest the possible presence of SN ejecta in Kes 69. Thus, we concluded that Kes 69 is an ejecta-dominated MM SNR candidate. Future X-ray observations with the superb high angular resolution satellites like [*Chandra*]{} will allow us to study the X-ray structure of the ejecta and the surrounding medium.
3. The spectra can be well described by a single NEI model with a temperature of $\sim$$2.5$ keV. The ionization time-scale is $\sim$4.1$\times10^{10}$ cm$^{-3}$ s, which implies that the plasma is far from ionization equilibrium, still ionizing. We found no spectral variation across the SNR. We detect the Fe-K$\alpha$ emission line at $\sim$6.5 keV in the spectra.
4. We searched for RP in Kes 69 and found that the initial plasma temperature much smaller than the current plasma temperature. Therefore, we concluded that the X-ray emitting plasma of Kes 69 is in the NEI state, not overionized.
5. We investigated the origin of SN type and concluded that the origin of Kes 69 is a CC SN and its progenitor mass is most likely between 9 and 12$M_{\sun}$ (see Figure 3).
6. We found no significant gamma-ray emission detected from Kes 69. The lack of gamma-ray emission from this interacting MM SNR can be explained by the dominant emission mechanism depending on the location of the gas with respect to the X-ray emitting plasma and the cosmic ray acceleration site. Assuming that the only associated OH (1720 MHz) maser emission is the extended maser emission at $\sim$85 km s$^{-1}$, which is lower in brightness in comparison to the one of compact maser emission, it could be a result of the interior X-ray emitting plasma rather than by the cosmic rays accelerated by the SNR shocks.
7. We compared Kes 69 with other interacting MM SNRs and concluded that Kes 69 offers an interesting case because of non-detection both RP and gamma-ray emission.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Dr. Aya Bamba for her valuable comments to improve this paper. We thank all the [*Suzaku*]{} team members for their support of the observation and software development. We also thank the referee for constructive comments and recommendations. AS is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) through the BİDEB-2219 fellowship program. TE thanks to the support of the Science Academy Young Scientists Program (BAGEP-2015). This work is supported in part by grant-in-aid from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, No.15K05088(RY), No.18H01232(RY) and No.16K17702(YO).
$~$
Note added in proof: While this paper was under review, Nobukawa et al. (2018; arXiv:1801.07881), reported a discovery of the neutral iron line emission from Kes 69 and its flux is higher than the average of the GRXE, which agrees with our result.
$~$
[*Facility*]{}: [*Suzaku*]{} and [*Fermi*]{}.
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[^1]: E-mail: [email protected] (AS)
[^2]: [email protected] (TE)
[^3]: http://www.darts.isas.jaxa.jp/astro/suzaku/
[^4]: https://heasarc.nasa.gov/lheasoft/
[^5]: http://www.atomdb.org
[^6]: https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node210.html
[^7]: http://fermipy.readthedocs.io/en/latest/index.html
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We develop the statistical theory of the stimulated Brillouin backscatter (BSBS) instability of a spatially and temporally partially incoherent laser beam for laser fusion relevant plasma. We find a new regime of BSBS which has a much larger threshold than the classical threshold of a coherent beam in long-scale-length laser fusion plasma. Instability is collective because it does not depend on the dynamics of isolated speckles of laser intensity, but rather depends on averaged beam intensity. We identify convective and absolute instability regimes. Well above the incoherent threshold the coherent instability growth rate is recovered. The threshold of convective instability is inside the typical parameter region of National Ignition Facility (NIF) designs although current NIF bandwidth is not large enough to insure dominance of collective instability and suggests lower instability threshold due to speckle contribution. In contrast, we estimate that the bandwidth of KrF-laser-based fusion systems would be large enough.'
author:
- 'Pavel M. Lushnikov$^{1}$ and Harvey A. Rose$^2$'
title: Collective stimulated Brillouin backscatter
---
Inertial confinement fusion (ICF) experiments require propagation of intense laser light through underdense plasma subject to laser-plasma instabilities which can be deleterious for achievement of thermonuclear target ignition because they can cause the loss of target symmetry and hot electron production [@Lindl2004]. Among laser-plasma instabilities, the backward stimulated Brillouin backscatter (BSBS) has been considered for a long time as serious danger because the damping threshold of BSBS of coherent laser beams is typically several order of magnitude lower compared with the required laser intensity $\sim 10^{15}\mbox{W}/\mbox{cm}^2$ for ICF. Recent experiments for a first time achieved conditions of fusion plasma and indeed demonstrated that large levels of BSBS (up to tens percent of reflectivity) are possible[@FroulaPRL2007].
Theory of laser-plasma interaction (LPI) instabilities is well developed for coherent laser beam [@Kruer1990]. However, ICF laser beams are not coherent because temporal and spatial beam smoothing techniques are currently used to produce laser beams with short correlation time, $T_c,$ and lengths to suppress laser-plasma interactions. The laser intensity forms a speckle field - a random in space distribution of intensity with transverse correlation length $l_c\simeq 2F/k_0$ and longitudinal correlation length (speckle length) $L_{speckle}\simeq 7F^2\lambda_0$, where $F$ is the optic $f/\#$ and $\lambda_0=2\pi/k_0$ is the wavelength (see e.g. [@RosePhysPlasm1995; @GarnierPhysPlasm1999]). Beam smoothing is a part of most constructed and suggested ICF facilities. However, instability theory of smoothed laser beam interaction with plasma is not well developed. There are intense experimental and simulation ongoing efforts [@FroulaPRL2007] to determine BSBS threshold for smoothed beams which appears to be in some cases quite low so that it is now under discussion that laser intensity at the National Ignition Facility (NIF) should lowered by a factor of few compared with original NIF designs[@Lindl2004] with intensities $\sim 2\times 10^{15}\mbox{W}/\mbox{cm}^2$.
Here we develop a theory of collective BSBS instability (CBSBS), which is a new BSBS regime, for propagation of laser beam with finite $T_c$ in homogeneous plasma. CBSBS has threshold comparable with NIF intensities. CBSBS requires $T_c$ small enough to suppress contribution from speckles. If we additionally assume that $T_c\gg L_{speckle}/c$ then CBSBS threshold does not depend on $T_c$. Such $T_c$ is accessible to KrF lasers [@Weaver2007], $T_c\simeq 0.7\mbox{ps}$, but not for NIF glass lasers with beam smoothing up to 3[Å]{} at $1\omega$, implying $T_c\simeq 4\mbox{ps}$ at $3\omega$. This is consistent with the numerical simulations which show that BSBS threshold in NIF emulation experiments is dominated by speckles [@FroulaPRL2007; @BergerPrivate2007]. We show below that speckle-dominated threshold is lower by a factor 7 than CBSBS threshold. Since plasma inhomogeneity can only increase instability threshold [@Kruer1990], The CBSBS threshold is a lower bound. Fig. \[fig:fig0\] depicts CBSBS between large $T_c$ speckle regime [@RoseDuBois1994] and random phase approximation (RPA) [@vedenov1964; @DuBoisBezzeridesRose1992; @PesmeBerger1994] regime.
Assume that laser beam propagates in plasma with frequency $\omega_0$ along $z$ with the electric field $\cal E$ given by $$\begin{aligned}
\label{EBdef}
{\cal E}=(1/2)e^{-i\omega_0 t}\Big [E e^{ik_0 z}+Be^{-ik_0
z-i\Delta\omega t}\Big ]+c.c.,\end{aligned}$$ where $E({\bf r}, z,t)$ is the envelope of laser beam and $B({\bf
r}, z,t)$ is the envelope of backscattered wave, ${\bf r}=(x,y)$, and c.c. means complex conjugated terms. Frequency shift $\Delta
\omega=-2k_0c_s$ is determined by coupling of $E$ and $B$ through ion-acoustic wave with phase speed $c_s$ and wavevector $2k_0$ with plasma density fluctuation $\delta n_e$ given by $\frac{\delta n_e}{n_e}=\frac{1}{2}\sigma
e^{2ik_0z+i\Delta\omega t}+c.c.,$ where $\sigma({\bf r}, z,t)$ is the slow envelope and $n_e$ is the average electron density, assumed to be small compared to critical density, $n_c$. The coupling of $E$ and $B$ to plasma density fluctuations gives, ignoring light wave damping, $$\begin{aligned}
\label{EBeq1}
\left [ i\Big (c^{-1}{\partial_t}+{\partial_z}\Big )+(2k_0)^{-1}\nabla^2
\right ]E=\frac{k_0}{4}\frac{n_e}{n_c}\sigma B, \\
\left [ i\Big (c^{-1}{\partial_t}-{\partial_z}\Big )+(2k_0)^{-1}\nabla^2
\right ]B=\frac{k_0}{4}\frac{n_e}{n_c}\sigma^* E, \label{EBeq2}\end{aligned}$$ $\nabla=({\partial_x},{ \partial_y})$, and $\sigma$ is described by the acoustic wave equation coupled to the pondermotive force $\propto {\cal E}^2$ which results in the envelope equation $$\begin{aligned}
\label{sigma1}
[ i ({c_s^{-1}}{\partial_t}+2\nu_{ia} k_0+{\partial_z} )-(4k_0)^{-1}\nabla^2
]\sigma^*=-2k_0 E^*B,\end{aligned}$$ where we neglected terms $\propto |E|^2, \ |B|^2$ in r.h.s. which are responsible for self-focusing effects, $\nu_L$ is the Landau damping of ion-acoustic wave and $\nu_{ia}=\nu_L/2k_0c_s$ is the scaled acoustic damping coefficient. $E$ and $B$ are in thermal units (see e.g. [@LushnikovRosePRL2004]). Assume that laser beam was made partially incoherent through induced spacial incoherence beam smoothing [@LehmbergObenschain1983] which defines stochastic boundary conditions at $z=0$ for the spacial Fourier transform (over ${\bf r}$) components $ \hat E({\bf k})$, of laser beam amplitude [@LushnikovRosePRL2004]: $$\begin{aligned}
\label{phik}
\hat E({\bf k },z=0,t)= |E_{\bf k}|\exp [ i\phi_{\bf
k}(t) ], \nonumber \\
\langle \exp i [\phi_{\bf
k}(t)-\phi_{{\bf k}'}(t') ] \rangle =\delta_{{\bf k
k}'}\exp (-|t-t'|/T_c),
\nonumber \\
|E_{\bf k}|=const, \ k<k_m; \ E_{\bf k}=0, \ k>k_m,\end{aligned}$$ chosen to the idealized “top hat” model of NIF optics [@polarization]. Here $k_m\simeq k_0/(2F)$ and the average intensity, $ \langle I
\rangle \equiv \langle |E|^2 \rangle =I$ determines the constant.
In linear approximation, assuming $|B|\ll |E|$ so that only laser beam is BSBS unstable, we can neglect right hand side (r.h.s.) of Eq. (\[EBeq1\]). The resulting linear equation with top hat boundary condition (\[phik\]) has the exact solution as decomposition of $E$ into Fourier series, $E({\bf r},z,t)=\sum_j E_{{\bf k}_j}$ with $E_{{\bf k}_j} \propto \exp\big [ i(\phi_{{\bf k}_j}(t-z/c)+{\bf k}_j\cdot {\bf r}-{\bf k}_j^2z/2k_0)\big ].$ Eq. (\[EBeq2\]) is linear in $B$ and $E$ which implies that $B$ can be also decomposed into $B=\sum_j B_{{\bf k}_j}.$ We approximate r.h.s. of (\[sigma1\]) as $E^*B\simeq \sum_j
E_{{\bf k}_j}^*
B_{{\bf k}_j}$ so that $$\begin{aligned}
\label{sigma2}
\left [ i (c_s^{-1}{\partial_t}+2\nu_{ia} k_0+{\partial_z})-(4k_0)^{-1}\nabla^2
\right ]\sigma^*\nonumber \\ =-2k_0 \sum_jE_{{\bf k}_j}^*B_{{\bf k}_j},\end{aligned}$$ which means that we neglect off-diagonal terms $E_{{\bf k}_j}^*
B_{{{\bf k}_j}'}, \quad j\neq j'.$ Since speckles of laser field arise from interference of different Fourier modes, $j\neq j',$ we associate the off-diagonal terms with speckle contribution to BSBS (independent hot spot model [@RoseDuBois1993; @RosePhysPlasm1995]). Speckle contribution can can be neglected if [@MounaixPRL2000] $$\begin{aligned}
\label{TcTsaturation}
T_c \ll t_{sat},\end{aligned}$$ where $t_{sat}$ is the characteristic time scale at which BSBS convective gain saturates.
We use the linear part of the theory of Ref. [@MounaixPRL2000] to estimate $T_{sat}$ for speckle contribution to backscatter as $t_{sat}=(L_{speckle}/c)\big[2+(\gamma_0/\nu_L)^2\big ]$, where $\gamma_0^2=k_0^2 c c_s I_{speckle}n_e/2n_c$ and we choose the typical intensity of light in speckle $I_{speckle}=3I$, where $I$ is the spatial average of laser intensity $|E|^2$ [@RoseDuBois1993; @GarnierPhysPlasm1999; @MounaixPRL2000]. In such a case $T_c
/t_{sat}\simeq T_ck_0c_s\nu_{ia}/(4 \tilde I)$, where here and below $\tilde I$ designates the scaled dimensionless laser intensity defined as $\tilde I=\frac{4F^2}{\nu_{ia}}\frac{n_e}{n_c} I$. For typical NIF parameters $\tilde I\sim 1$ [@Lindl2004; @LushnikovRosePlasmPhysContrFusion2006], $\lambda_0=351 \mbox{nm}$ and $c_s=6\times 10^{7}\ \mbox{cm s}^{-1}$ we obtain from (\[TcTsaturation\]) the estimate $T_c \ll 0.4/\nu_{ia}\ \mbox{[ps]}$ which is not satisfied for low plasma ionization number $Z$ plasma in NIF which typically has $\nu_{ia}\sim 0.1$. However, CBSBS can still be relevant for NIF in gold plasma near hohlraum [@Lindl2004] with $\nu_{ia}\sim 0.01$. Similar estimate for KrF lasers ($\lambda_0=248 \mbox{nm}, \ F=8$) gives $T_c \ll 0.3/\nu_{ia}\ \mbox{[ps]}$ which is easier to satisfy because of smaller $T_c$ and suggests that KrF lasers are better suited for applicability of CBSBS.
If we look for solution of Eqs. (\[EBeq2\]) and (\[sigma2\]) in exponential form $B_j, \sigma^* \propto e^{i(\kappa z+{\bf k}\cdot {\bf r}-\omega t)}$, we arrive at the following dispersion relation in dimensionless units $$\begin{aligned}
\label{dispk1}
-i\omega+\mu+i\kappa-(i/4)k^2\nonumber \\
=8iF^4\frac{n_e}{n_c}\sum\limits_{j=1}^N \frac{|E_j|^2}{\omega\frac{c_s}{c}+\kappa-k_j^2-\frac{k^2}{2}-{\bf k}_j\cdot {\bf k}},\end{aligned}$$ where $\mu\equiv 2\nu_{ia} k_0^2/k_m^2,$ $1/k_m$ is the transverse unit of length, $k_0/k_m^2$ is the unit in $z$ direction, $k_0/k_m^2 c_s$ is the time unit and $I=\sum_j|E_{{\bf k}_j}|^2$.
The dispersion relation (\[dispk1\]) is correct provided the temporal growth rate $\omega_i=Im(\omega)$ is small compare to inverse time of light propagation along speckle, $\omega_i\ll c/L_{spekle}$, and if during time $T_c$ light travels much further than a speckle length, $L_{speckle}\ll cT_c$. That second condition ensures that term $\propto\phi'_{{\bf k}_j}(t-z/c)\sim 1/T_c$ could be neglected in Eq. (\[EBeq2\]) allowing the time dependence of $E$ in Eqs. (\[EBeq2\]) and (\[sigma2\]) to be ignored and in such case density fluctuation $\sigma$ evolves without fluctuations. E.g. for typical NIF parameters, $T_ck_0c_s/2F\sim 1$ we obtain that $2c/7c_sF\gg 1$ which is well satisfied for NIF optics [@Lindl2004].
In the continuous limit $N\to \infty$, sum in (\[dispk1\]) is replaced by integral which gives for most unstable mode ${\bf k}=0$: $$\begin{aligned}
\label{dispk0cont}
\Delta(\omega,\kappa)=-i\omega+\mu+i\kappa+i\frac{\mu}{4}\tilde I\ln\frac{1-\kappa-\omega\frac{c_s}{c}}{-\kappa-\omega\frac{c_s}{c}}=0.\end{aligned}$$ Eq. (\[dispk0cont\]) has branch cut in complex $\kappa$ plane determined by points $\kappa_1=1-\omega\frac{c_s}{c}$ and $\kappa_2=-\omega\frac{c_s}{c}$. Standard analysis of convective vs. absolute instabilities (see e.g. [@Briggs1964]) should be modified to include that branch cut. In discrete case with $N \gg 1$ instead of branch cut the discrete dispersion relation (\[dispk1\]) has solutions located near the line $(\kappa_1,\kappa_2)$. These solutions are highly localized around some ${\bf k}_j$ so they cannot be approximated by (\[dispk0cont\]) but they are stable for $N \gg 1$. Generally there are two solutions of (\[dispk0cont\]), however for $Im(\omega)\to \infty$ one solution is absorbed into branch cut. Second solution is stable. Above the convective CBSBS threshold, $$\begin{aligned}
\label{I0convthresh}
\tilde I_{convthresh}=4/\pi,\end{aligned}$$ the first solution crosses real $\kappa$ axis from below as $Im(\omega)\to 0$ so it describes instability of backscattered wave with $Im(\kappa)>0.$
However, above the absolute CBSBS threshold, which can be approximated from solution of Eq. (\[dispk0cont\]) as $$\begin{aligned}
\label{I0absthresh}
\tilde I_{absthresh}\simeq(1/2)\Big (\mu^{-1}+\mu+\sqrt{\mu^2-2}\Big ), \quad \mu\gtrsim 4,\end{aligned}$$ the contour $Im(\omega)=Const$ cannot be moved down to real $\omega$ axis because of pinching of two solutions of (\[dispk0cont\]) which defines growth rate of absolute instability. We conclude that classical analysis of instabilities still holds for incoherent beam if we additionally allow the absorption of one solution branch into branch cut. This effect results from incoherence of pump beam which has infinitely many transverse Fourier modes in approximation of Eq. (\[dispk0cont\]) and there is no counterpart of that effect for coherent beam.
For $\mu\gg 1 $ the absolute threshold (\[I0absthresh\]) reduces to the coherent absolute BSBS instability threshold $$\begin{aligned}
\label{I0absthreshcoherent}
\tilde I_{absthreshcoherent}= \mu.\end{aligned}$$ For NIF parameters, $T_e\simeq 5\mbox{keV}, \ F=8,\ \ n_e/n_c=0.1, \ \lambda_0=351 \mbox{nm}$ with moderate acoustic damping, $\nu_{ia}\simeq 0.1$, we obtain in dimensional units $I_{convthresh}\simeq 2\times 10^{15}\mbox{W}/\mbox{cm}^2$ and $I_{absthresh}\simeq 9\times 10^{16}\mbox{W}/\mbox{cm}^2.$ For high $Z$ plasma (e.g. gold plasma near the wall of NIF hohlraum [@Lindl2004], $\nu_{ia}\simeq 0.01$) we obtain $I_{convthresh}\simeq 2\times 10^{14}\mbox{W}/\mbox{cm}^2$ and $I_{absthresh}\simeq 9\times
10^{14}\mbox{W}/\mbox{cm}^2.$ Typical intensity of NIF laser shots is between $10^{15}\mbox{W}/\mbox{cm}^2$ and $2 \times 10^{15}\mbox{W}/\mbox{cm}^2$ so we conclude that in different parts of NIF plasma both convective and absolute instabilities are possible. Fig. \[fig:fig1\] compares instability gain rate of coherent and incoherent beams for $\mu=51.2$.
In contrast with Eq. (\[I0convthresh\]), the convective instability threshold in coherent case is 0 because we neglect damping of $B$ in Eq. (\[EBeq2\]). Retaining collisional light damping gives finite threshold $\tilde I_{convcoherent}=16F^2 \nu_B/k_0 c\ll 1$, where $\nu_B=\frac{n_e}{n_c}\frac{\nu_{ei}}{2}$ [@Kruer1990] is the collisional damping of backscattered wave $B$ and $\nu_{ei}$ is the electron-ion collision frequency. That threshold is several orders of magnitude smaller compared with (\[I0convthresh\]) and is neglected here. Qualitatively incoherence of laser beam can be considered as effective damping of $B$ with effective damping rate $\nu_{effective}=\frac{\pi k_0 c_s}{16F^2}$.
Depending on laser incoherence we have a hierarchy of thresholds: (a)Spatially incoherent laser beam with large $T_c$ has threshold, $\tilde I_{threshold}=\tilde I_{thresholdspeckle}=4/7\pi$ which is dominated by intense speckles [@RoseDuBois1994]. (b)Spatial and temporary incoherent beam with $T_c$ satisfying (\[TcTsaturation\]) is given by (\[I0convthresh\]) which factor $7$ times higher compared with speckle threshold and does not depends on $T_c.$ It indicates practical limit of how threshold of BSBS instability can be increased by decreasing $T_c.$ (c)For much smaller $T_c$, such that it is smaller than both inverse acoustic damping $T_c\ll 1/\nu_L$ and inverse temporal growth rate $T_c\ll 1/\omega_i$, the classical RPA regime is recovered which has ignorable diffraction ([@vedenov1964; @DuBoisBezzeridesRose1992; @PesmeBerger1994]). This limit (e.g. for $\lambda_0=351\mbox{nm}$ and $\nu_{ia}=0.15$ it requires $T_c\ll 0.3\mbox{ps}$) is not practical for ICF as $T_c$ is too small. Cases (a)-(c) are shown in Fig. \[fig:fig0\].
Current NIF 3[Å]{} beam smoothing design is between regimes (a) and (b). KrF laser with $T_c\simeq 0.7\mbox{ps}$ would be in regime (b). Thus generally we expect that KrF-laser-based ICF allows access to CBSBS regime although CBSBS threshold for NIF can be possibly initiated by self-induced temporal incoherence (see e.g. [@SchmittAfeyan1998]). Another possibility for self-induced temporal incoherence is through collective forward stimulated Brillouin scatter (CFSBS) instability [@LushnikovRosePRL2004; @LushnikovRosePlasmPhysContrFusion2006]. Above CFSBS threshold correlation length decreases with beam propagation length and may decrease $T_c$. For low $Z$ plasma threshold for CFSBS is close to (\[I0convthresh\]) [@LushnikovRosePRL2004]. As $Z$ increases (which can be achieved by adding high $Z$ dopant), CFSBS threshold decreases below (\[I0convthresh\]) and might result in decrease of $T_c.$
To distinguish contribution to BSBS from speckles (regime (a)) and CBSBS (regime (b)) we propose to look at angular divergence $\triangle \theta=\triangle k/k_0$ of BSBS. In general one expects gain narrowing of the scattered light: the modes close to the most unstable mode, with gain rate $(\kappa_i)_{max}$, dominate. Here $\kappa_i \equiv Im(\kappa)$. Fig. \[fig:fig2\] shows $2F\triangle \theta$ from CBSBS as a function of laser intensity above CBSBS threshold at propagation distance $L=10/ (\kappa_i)_{max}$. $L$ is chosen from the physical condition that there is sufficient convective CBSBS gain, to amplify the energy of thermal acoustic fluctuations at wavenumber $2k_0$ to have reflectivity $\sim 1$, and for fusion plasma this is typically $\exp(G)=\exp(20)$ (see e.g. [@FroulaPRL2007]), where $G=2\kappa_i L$ is the power convective gain exponent. Then $\triangle \theta$ is conventionally defined by half width at half maximum: $\exp[G(\triangle \theta)] = 0.5 \exp[G(\theta)]|_{\theta=0}$. Important feature of CBSBS seen in Fig. \[fig:fig2\] is that $\triangle\theta\neq 0$ at threshold with $\kappa_i( k)\simeq\kappa_i(0)(1-\tilde \alpha k^2)$ and $\tilde \alpha\simeq\mu\tilde I/(\mu\tilde I-1)$ near threshold. Fig. \[fig:fig2\] should be compared with $\triangle \theta$ from speckle-dominated backscatter. Previous work [@DivolMounaixPRE1998] suggested that speckles can also cause $\triangle \Theta$ below top-hat width, $1/2F$, for very intense speckle backscatter. We estimate based on Refs. [@RosePhysPlasm1995; @GarnierPhysPlasm1999]) that for nominal ICF plasma ($\sim 10^5$ speckle volumes), most intense speckle is $\sim 15 I$ which gives $G_{intense}= 15 \langle G_0\rangle \simeq 100$ near CBSBS threshold, where $\langle G_0\rangle =2\kappa_i L_{speckle }\simeq 7$ is the the gain over speckle with the average intensity $I.$ We performed direct simulations of backscatter from $G_{intense}= 100$ speckle and found that $\triangle \theta\simeq 1/2F$ which means that asymptotic [@DivolMounaixPRE1998] is still not applicable. In other words, finite size plasma effects dominates over asymptotic theory of infinite plasma. We conclude that regime (a) can be easily distinguished from CBSBS regime (b): near CBSBS threshold with condition (\[TcTsaturation\]) satisfied one should see backscattered light spectrum with essential peak whose width is given by Fig. \[fig:fig2\] and wide weak background determined by speckles.
In summary, we found a novel coherence time regime in which $T_c$ is too large for applicability of well-known statistical theories (RPA) but rather an intermediate regime, $T_c$ is small enough to suppress speckle BSBS. Unlike coherent beam CBSBS has threshold typically much larger than that determined by damping while for laser intensity many times above convective instability threshold for incoherent beam, the coherent theory is recovered.
We acknowledge helpful discussions with B. Afeyan, R. Berger, L. Divol, D. Froula, and N. Meezan. This work was carried out under the auspices of the NNSA of the DOE at LANL under Contract No. DE-AC52-06NA25396
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In psychology and neuroscience it is common to describe cognitive systems as input/output devices where perceptual and motor functions are implemented in a purely feedforward, open-loop fashion. On this view, perception and action are often seen as encapsulated modules with limited interaction between them. While embodied and enactive approaches to cognitive science have challenged the idealisation of the brain as an input/output device, we argue that even the more recent attempts to model systems using closed-loop architectures still heavily rely on a strong separation between motor and perceptual functions. Previously, we have suggested that the mainstream notion of modularity strongly resonates with the separation principle of control theory. In this work we present a minimal model of a sensorimotor loop implementing an architecture based on the separation principle. We link this to popular formulations of perception and action in the cognitive sciences, and show its limitations when, for instance, external forces are not modelled by an agent. These forces can be seen as variables that an agent cannot directly control, i.e., a perturbation from the environment or an interference caused by other agents. As an alternative approach inspired by embodied cognitive science, we then propose a nonmodular architecture based on the active inference framework. We demonstrate the robustness of this architecture to unknown external inputs and show that the mechanism with which this is achieved in linear models is equivalent to integral control.'
author:
-
title: Nonmodular architectures of cognitive systems based on active inference
---
modularity, separation principle, active inference, Bayesian inference, optimal control
Introduction
============
In cognitive science it is often assumed that agents can be described as input/output systems, an idea based on traditional, computational accounts of cognition [@newell1972human; @fodor1983modularity; @hurley2001perception]. In these models, the emphasis is on internal models of the world, central processing and the sense-model-plan-act framework, often neglecting embodiment, situatedness and feedback from the environment [@brooks1991new]. More recent attempts, e.g., [@kawato1999internal; @wolpert2000computational; @todorov2004optimality], have proposed closed-loop descriptions of cognitive system using internal forward/inverse models in an attempt to provide better accounts of behaviour in living organisms. However in both the input/output and the closed-loop architectures advocated by these approaches, the role of perceptual and motor processes is thought to be fundamentally modular [@fodor1983modularity], i.e., these functions can be described as nearly independent, (informationally) encapsulated components with minimal interactions.
In recent years, theories of estimation and control have become increasingly popular accounts of perception [@knill1996perception; @rao1999predictive; @lee2003hierarchical] and action [@kawato1999internal; @wolpert2000computational; @todorov2004optimality] respectively. In this context, the Kalman-Bucy filter is used as a model of perception [@rao1999predictive; @wolpert2011principles] while LQR (linear quadratic regulator) constitutes the basis of various accounts of motor control [@li2004iterative; @stevenson2009bayesian]. In previous work [@baltieri2018modularity] we claimed that the idea of modularity of action and perception can be seen as an analogy of the separation principle in control theory [@wonham1968separation; @anderson1990optimal; @stengel1994optimal]. According to this principle, problems of estimation and control of a system can be solved separately and their solutions can be *optimally* combined under a set of assumptions. Following this, one can sequentially combine a Kalman-Bucy filter and LQR to create the LQG (linear quadratic Gaussian) architecture, used as a general methodology for several models of sensorimotor loops, e.g., [@wolpert2000computational; @todorov2002optimal; @stevenson2009bayesian; @yeo2016optimal]. The “classical sandwich” [@hurley2001perception] of cognitive science thus survives, we claim, even in the forward/inverse models formulation of perception and motor control.
The fields of embodied and enactive cognitive science on the other hand emphasise the deep integration of perception and action, seen as fundamentally intertwined [@clark1998being; @wilson2002six; @di2017sensorimotor]. In [@baltieri2018modularity] we proposed to use a framework based on the formulation of perception and action as estimation and control while not implementing the conditions for the separation principle, i.e., active inference. Active inference is a process theory based on the free energy principle [@Friston2010nature] describing cognitive functions (perception and action, but also learning and attention) as processes of minimisation of sensory surprisal [@Friston2010nature; @buckley2017free]. More precisely, since this quantity is not directly accessible by an agent, it is thought that the variational free energy (an upper bound to sensory surprisal) is minimised in its place. In active inference, perceptual and motor processes are often described as entangled and inseparable [@friston2011optimal; @wiese2016action; @pezzulo2017model] providing thus a new possible methodology combining estimation and control following embodied/enactive theories of the mind. We previously presented a conceptual account of active inference and its role for nonmodular architectures of cognitive systems [@baltieri2018modularity]. Here we introduce a minimal agent model highlighting the different implementations (LQG vs. active inference) especially in presence of unknown external stimuli affecting an agent’s observations.
LQG and the separation principle
================================
The framework provided by LQG control and based on the separation principle linearly combines two processes of 1) estimation or inference of hidden properties of the environment and 2) control or regulations of variables of interest. The estimation of hidden variables is based on the presence of a Kalman (for discrete time systems) or Kalman-Bucy (for continuous time systems) filter, while the control of the desired variables on LQR [@wonham1968separation; @anderson1990optimal; @stengel1994optimal]. In particular, this combination is provably optimal according to a set of assumptions:
1. the estimator is implemented through a state-space model where only linear process dynamics and observation laws describe the environment and its latent states
2. uncertainty or noise in both dynamics and observations are represented by *white*, zero-mean Gaussian variables
3. the properties of these random variables, in particular their (co)variance matrices, are known
4. the performance of the regulator can be evaluated using a quadratic cost function
5. all the inputs/forces applied to the agent are known, e.g., external disturbances and internal signals such as motor actions.
Following the separation principle, the LQG controller produces optimal estimation and optimal control for linear systems, sequentially combining two separate sub-systems, a Kalman-Bucy filter and LQR, in an optimal (i.e., minimum-variance) way [@anderson1990optimal; @stengel1994optimal]. The Kalman-Bucy filter provides the optimal state-estimate of a signal and the LQR controller uses such estimate (i.e., the mean) to implement the optimal deterministic controller: LQG control makes use of the estimated mean and feeds it into an LQR controller.
A general linear system to be regulated in the presence of noise on the observed state is described by: $$\begin{aligned}
d \bm{x} = A \bm{x} \: dt + B \bm{a} \: dt + d \bm{w} \quad \quad \quad
\bm{y} = C \bm{x} + d\bm{z}
\label{eq:LQGSSM}\end{aligned}$$ where all the variables and parameters are the same as previously defined for Kalman-Bucy filters and LQR. Using the separation principle, it can then be shown that minimising the expected value of the cost-to-go is equivalent to minimising the cost-to-go for the expected (estimated) state [@stengel1994optimal] $$\begin{aligned}
c(\bm{x}, \bm{a}) = c(\hat{\bm{x}}, \bm{a}) = \frac{1}{2} \hat{\bm{x}}^T Q \hat{\bm{x}} + \frac{1}{2} \bm{a}^T R \bm{a}
\label{eq:costrateLQG}\end{aligned}$$ where we replaced states $\bm{x}$ with their estimates $\hat{\bm{x}}$, meaning that the optimal control can be computed using only the state estimate (i.e., the mean) $\hat{\bm{x}}$ rather than $\bm{x}$. The combined problem of estimation and control in LQG terms is then implemented by the following system combining Kalman-Bucy filter and LQR equations: $$\begin{aligned}
\dot{\hat{\bm{x}}} = & A \hat{\bm{x}} + B \bm{a} + K (y - C \hat{\bm{x}}) \IEEEnonumber \\
\bm{a} = & - L \hat{\bm{x}} \IEEEnonumber \\
K = & P H^T (\Sigma_z)^{-1} \IEEEnonumber \\
L = & R^{-1} B^T V \IEEEnonumber \\
\dot{P} = & \Sigma_w + A P + P A^T - K (\Sigma_z) K^T \IEEEnonumber \\
- \dot{V} = & Q + A^T V + V A - L^T R L.
\label{eq:LQGEstimationControl}\end{aligned}$$
Active inference {#sec:AI}
================
Active inference is a process theory proposed to explain brain functioning and other functions of living systems based on Bayesian inference and optimal control theory [@Friston2010biocyb; @Friston2010nature; @buckley2017free]. In this section we establish its relations to the LQG architecture, starting by building an active inference version of the regulation of a linear multivariate system, and highlighting differences, limitations and possible extensions proposed for the control problem. As with LQG control, we build an estimator of the hidden states $\bm{x}$. In this case however, we will give a variational account of the estimator in generalised coordinates of motion that generalises the MLE/MAP derivation of Kalman-Bucy filters [@chen2003bayesian] using Variational Bayes with a Laplace approximation [@Friston2008a; @buckley2017free]. We start by defining a generative model for an agent capturing the dynamics of the system to control and how these relate to observations and represented in a *generalised* state-space form [@Friston2008c; @buckley2017free]: $$\begin{aligned}
\bm{x}' = \hat{A} \bm{x}' + \hat{B} \bm{v} + \bm{w} \quad \quad \quad
\bm{y} = \hat{C} \bm{x} + \bm{z}\end{aligned}$$ where the hat over the matrices simply represents the fact that the matrices used in the generative model don’t necessarily mirror their counterparts describing the world dynamics ([@baltieri2017active; @brown2013active]), as shown in our model later. The main difference with respect to LQG however is that LQG explicitly mirrors (by construction in the linear case) the dynamics of the observed system, thus including knowledge of inputs $\bm{a}$. On the other hand, in active inference this vector is not explicitly modelled by an agent, assuming that such information is not available to a system, in accordance with evidence in motor neuroscience suggesting the lack of knowledge of self-produced controls (i.e., efference copy) [@feldman2009new; @friston2011optimal; @feldman2016active]. It is in fact proposed that a deeper duality of estimation and control exists whereby, in the simplest case (i.e., a purely reflexive account), actions are simply responses to the presence of prediction errors at the proprioceptive level, irrespectively of the cause of sensations (self-generated or external forces) [@friston2011optimal; @brown2013active]. The vector $\bm{v}$ in the generative model encodes instead external or exogenous inputs in a state-space models context or, from a Bayesian perspective, priors or “desired” outcomes generated by higher layers in hierarchical (Bayesian) implementations [@lee2003hierarchical; @Friston2008c]. In this light, priors can be used to effectively bias the estimator to “infer” desired rather than observed states, with a controller instantiating actions on the world to fulfil the “observed” (= desired) states of an agent. Variables $\bm{z}, \bm{w}$ model the real noise in the environment making, however, use of the definition of state space models in generalised coordinates of motion [@Friston2008a; @Friston2008c], where $z, w$ are treated as analytical noise with non-zero autocorrelation, generalising the definition of Wiener processes with Markov property.
This state-space model can then be written down in a probabilistic form, mapping the measurements equation to a likelihood $P(\bm{y} | \bm{\hat{x}})$ (no direct influence of inputs on observations), and a the dynamics to a prior $P(\bm{\hat{x}}, \bm{v})$ [@Friston2008c; @buckley2017free; @baltieri2017active]. The two multivariate Gaussian probabilities densities can then be combined and used in the general formulation of Laplace encoded variational free energy defined in [@Friston2008a; @buckley2017free] (without constants): $$\begin{aligned}
F \approx - \ln P(\bm{y}, \bm{x}, \bm{v}) \Bigr \rvert_{\bm{x} = \bm{\mu_x}, \bm{v} = \bm{\mu_v}}
\label{eq:freeEnergyLaplace}\end{aligned}$$ the free energy for a generic linear multivariate system becomes then: $$\begin{aligned}
F & \approx \frac{1}{2} \bigg[ \Big( \bm{y} - \hat{C} \bm{\mu_x} \Big)^T \Pi_z \Big( \bm{y} - \hat{C} \bm{\mu_x} \Big) + \IEEEnonumber \\
& + \Big (\bm{\mu_x'} - \hat{A} \bm{\hat{\mu}_x} - \hat{B} \bm{\mu_v} \Big)^T \Pi_w \Big (\bm{\mu_x'} - \hat{A} \bm{\mu_x} - \hat{B} \bm{\mu_v} \Big) + \IEEEnonumber \\
& - \ln \big| \Pi_z \big| - \ln \big| \Pi_w \big| + (m + n) \ln 2 \pi \bigg]
\label{eq:freeEnergyMultivariate}\end{aligned}$$ where we explicitly replaced $\bm{x}, \bm{v}$ with their expectations $\bm{\mu_x}, \bm{\mu_v}$ since under the Laplace assumption this represents the best estimate of $\bm{x}, \bm{v}$ (i.e., covariances of the approximate, variational density can be recovered analytically [@Friston2008a; @buckley2017free]). Variables $m, n$ represent the length of vectors $\bm{y}$ and $\bm{x}$ respectively. Expectations $\bm{\mu_x}$ play the same role of estimates $\bm{\hat{x}}$ in LQG, we simply decided to use a notation consistent with some of our previous work [@buckley2017free; @baltieri2017active; @baltieri2018propabilistic]. We also defined precision matrices $\Pi_z, \Pi_w$ as the inverse of covariance matrices $\Sigma_z, \Sigma_w$ and used $|\cdot|$ to define the determinant of a matrix. It is important to highlight that, in general, the covariance matrices used in the generative model can be different from the ones used to describe the environment or generative process [@brown2013active; @baltieri2017active]. To simplify the already heavy notation we will however represent them in the same way.
The recognition dynamics, encoding perception and action in a system minimising free energy [@Friston2008a; @buckley2017free] and equivalent to estimation and control functions respectively, are implemented in standard active inference formulations as a gradient descent scheme minimising the free energy with respect to the variables $\bm{\mu_{x}}$ for perception/estimation: $$\begin{aligned}
\bm{\dot{\mu}_x} & = D \bm{\mu_x} - \frac{\partial F}{\partial \bm{\mu_x}} = \bm{\mu_x'} + \hat{C}^T \Pi_z \Big( \bm{y} - \hat{C} \bm{\mu_x} \Big) + \IEEEnonumber \\
& + \hat{A}^T \Pi_w \Big (\bm{\mu_x'} - \hat{A} \bm{\mu_x} - \hat{B} \bm{\mu_v} \Big) \IEEEnonumber \\
\bm{\dot{\mu}_x'} & = D \bm{\mu_x'} - \frac{\partial F}{\partial \bm{\mu_x'}} = \bm{\mu_x''} - \Pi_w \Big (\bm{\mu_x'} - \hat{A} \bm{\mu_x} - \hat{B} \bm{\mu_v} \Big)
\label{eq:perceptionActiveInference}\end{aligned}$$ and actions $\bm{a}$ for action/control, assuming only that actions have an effect on observations $\bm{y}$ [@Friston2010biocyb]: $$\begin{aligned}
\bm{\dot{a}} & = - \frac{\partial F}{\partial \bm{a}} = - \frac{\partial F}{\partial \bm{y}} \frac{\partial \bm{y}}{\partial \bm{a}} = - \frac{\partial \bm{y}}{\partial \bm{a}}^T \Pi_{z} \Big( \bm{y} - \hat{C} \bm{\mu_{x}} \Big).
\label{eq:actionActiveInference}\end{aligned}$$ The estimation expressed in prescribes a generalisation of Kalman-Bucy filters to trajectories with arbitrary embedding orders where random variables are not treated as Markov processes [@Friston2008a]. In , we also include an extra term $D \bm{\mu_{x}}$ that represents the “mode of the motion” (also the mean for Gaussian variables) for the minimisation in generalised coordinates of motion [@Friston2008c; @buckley2017free], with $D$ as a differential operator shifting the order of motion, i.e., $D \bm{\mu_{x}} = \bm{\mu_{x}}'$. More intuitively, since we are now minimising the components of a generalised state representing a trajectory rather than a static variable, variables are in a moving framework of reference where the minimisation is achieved for $\bm{\dot{\mu}_x} = \bm{\mu'_x}$ rather than $\bm{\dot{\mu}_x} = \bm{0}$. Action as expressed in may appear similar to the traditional LQR/LQG form, but is fundamentally different since it depends explicitly on observations $\bm{y}$ rather than estimated hidden states $\bm{\mu_x}$.
The model
=========
The double integrator is a canonical example used in control theory and represents one of the most fundamental problems in optimal control, modelling single degree of freedom motion of different physical systems [@rao2001naive; @aastrom2010feedback]. In the case presented here, this could be thought of as a block on frictionless surface. In motor neuroscience, this is the simplest model of single-joint movement [@gottlieb1993computational] and can, in some cases, be easily generalised to multiple degrees of freedom [@Friston2010biocyb]. The standard double integrator is usually described as a deterministic system. The control policy is thus defined using a feedback law applied directly to the *known* dynamics, as the full state of the system is measured with no uncertainty [@rao2001naive]. For the purposes of this work, where uncertainty and noise are crucial components, we will introduce process and measurement noise into the system, making the estimation of hidden states necessary. This will then allow us to compare LQG and active inference in one of the simplest possible examples in the control theory literature with direct applications to the study of motor systems and behaviour [^1]. The double integrator is described by the following state-space model: $$\begin{aligned}
\dot{\bm{x}} = A \bm{x} + B \bm{a} + \bm{w}
\end{aligned}
\quad
\begin{aligned}
\bm{y} = C \bm{x} + \bm{z}
\end{aligned}
\label{eq:doubleIntegratorSSM}$$ where matrices $A, B, C$ are defined as: $$\begin{aligned}
A =
\begin{bmatrix}
0 & 1 \\
0 & 0
\end{bmatrix}\IEEEnonumber
\quad
B =
\begin{bmatrix}
0 & 0\\
0 & 1
\end{bmatrix}\IEEEnonumber
\quad
C =
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}\IEEEnonumber
\label{eq:doubleIntegratorMatrices}\end{aligned}$$ and covariance matrices $\Sigma_z, \Sigma_w$ as: $$\begin{aligned}
\Sigma_z =
\begin{bmatrix}
\exp(0) & 0 \\
0 & \exp(0)
\end{bmatrix}
\quad
\Sigma_w =
\begin{bmatrix}
0 & 0 \\
0 & \exp(-1)
\end{bmatrix}\IEEEnonumber\end{aligned}$$
![**The generative process, a double integrator.** The double integrator models the motion of a system with a single degree of freedom, corresponding to a block of mass=1kg placed on a surface with no friction. The block is initialised at a random position with a random velocity and needs to stop, $x'=0$, at position $x=0$.[]{data-label="fig:DoubleIntegratorGP"}](./BlockFrictionless){width=".7\linewidth"}
The LQG solution to the double integrator
-----------------------------------------
For LQG we implement using the same matrices $A, B, C, \Sigma_z, \Sigma_w$ specified above and furthermore define: $$\begin{aligned}
Q =
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\quad
R =
\begin{bmatrix}
4 & 0 \\
0 & 4
\end{bmatrix}
\end{aligned}
\label{eq:doubleIntegratorLQRWeights}$$ with no specific optimisation of these parameters since it is beyond the scope of this work. For further analysis see for instance [@rao2001naive].
[.5]{} ![**The double integrator solved using LQG.** (a) Five examples with different initial conditions showing in blue the observed trajectories of different blocks in the phase-space and in red the agent’s estimates of the same trajectories. (b) Actions taken by the five agents.[]{data-label="fig:DoubleIntegratorLQG"}](./DoubleIntegratorLQG "fig:"){width="\linewidth"}
[.5]{} ![**The double integrator solved using LQG.** (a) Five examples with different initial conditions showing in blue the observed trajectories of different blocks in the phase-space and in red the agent’s estimates of the same trajectories. (b) Actions taken by the five agents.[]{data-label="fig:DoubleIntegratorLQG"}](./DoubleIntegratorLQGAction "fig:"){width="\linewidth"}
[.5]{} ![**The double integrator solved using LQG.** (a) Five examples with different initial conditions showing in blue the observed trajectories of different blocks in the phase-space and in red the agent’s estimates of the same trajectories. (b) Actions taken by the five agents after an external force is introduced (black line).[]{data-label="fig:DoubleIntegratorLQGNoExternalForce"}](./DoubleIntegratorLQGNoExternalForce "fig:"){width="\linewidth"}
[.5]{} ![**The double integrator solved using LQG.** (a) Five examples with different initial conditions showing in blue the observed trajectories of different blocks in the phase-space and in red the agent’s estimates of the same trajectories. (b) Actions taken by the five agents after an external force is introduced (black line).[]{data-label="fig:DoubleIntegratorLQGNoExternalForce"}](./DoubleIntegratorLQGNoExternalForceAction "fig:"){width="\linewidth"}
As we can see in [Fig. \[fig:DoubleIntegratorLQGAgents\]]{}, the block is effectively driven to the desired position $x=0$ and velocity $x'=0$ from a set of 5 randomly initialised conditions (position and velocity are sampled from zero-mean Gaussian distributions, sd=300). In [Fig. \[fig:DoubleIntegratorLQGAction\]]{} we then show the actions over time of the same 5 example agents, all converging to zero since the agents effectively reach their desired target. The main feature of LQG, and from which active inference will depart, is the reliability of estimates of both position and velocity (the red line in the phase space), using a Kalman-Bucy filter. In LQG, accurate estimates are necessary to then enact the LQR component implementing a negative feedback mechanism based on estimates $\bm{\hat{x}}$ rather than true hidden states $\bm{x}$. In [Fig. \[fig:DoubleIntegratorLQGNoExternalForce\]]{} we introduced a new external force not modelled by the agents, equivalent to a disturbance from the environment (black line in [Fig. \[fig:DoubleIntegratorLQGNoExternalForceAction\]]{}). [Fig. \[fig:DoubleIntegratorLQGNoExternalForceAgents\]]{} then shows that the agents are incapable of regulating their position/velocity against this unknown input (blue lines), after an initial convergence towards the desired state, they in fact move away from it when the unexpected force is introduced. Furthermore, these agents are incapable of correctly inferring their trajectories, providing inaccurate estimates of their sensed variables (red lines). In [Fig. \[fig:DoubleIntegratorLQGNoExternalForceAction\]]{} we see that all of these agents attempt to counteract the effects of unexpected stimuli (they minimise their velocity after the force is introduced), however the lack of an appropriate mechanism to track their position correctly (e.g., integral action) pushes them away from the target.
The double integrator with active inference
-------------------------------------------
To solve the same control problem, active inference relies on the generation of predictions of proprioceptive sensations (position, velocity as in LQG, and also acceleration in this case), followed by the implementation of actions in the world via (trivial) reflex arcs. The proprioceptive modality is essentially treated as other inputs (vision, audition, etc.) and estimates/predictions are generated using the same generative model taking advantage of incoming proprioceptive sensations. This produces a considerably different control system, with state estimates and actions now created by the same model, making it hard to clearly separate processes of perception and action. The copy of motor control signals (cf. efference copy [@von1950reafferenzprinzip]), necessary in standard LQG settings to meet the observability constraints of Kalman-Bucy filters [@anderson1990optimal; @stengel1994optimal] is not included in this formulation, as explained in [section \[sec:AI\]]{}. Active inference postulates in fact that direct representations of the causes or actions $\bm{a}$ of self-generated sensations need not be discounted during the prediction of new incoming sensory inputs. This could be seen as a limitation of active inference, but in general this speaks to the robustness of this approach in face of unknown inputs (i.e., motor actions produced by an agent or exogenous forces from the environment), see [@baltieri2018propabilistic]. In this framework, inputs can also be estimated using an appropriate generative model of the world dynamics [@Friston2008a], a feature thought to be fundamental in biological systems [@sontag2003adaptation]. Simple and effective approximations are also possible, for example with integral control, thought to be the most basic heuristic dealing with the problem of uncertain inputs in biological systems down to the unicellular level [@yi2000robust; @sontag2003adaptation] and already shown to be consistent with formulations of active inference [@baltieri2018propabilistic].
![**The generative model.** To implement the regulation of position and velocity, the agent implements a model whereby an imaginary spring pulls the block back to the origin ($x=0$) while an imaginary damper slows it down ($x' = 0$).[]{data-label="fig:DoubleIntegratorGM"}](./BlockSpringDamper){width=".9\linewidth"}
To derive an active inference solution to the double integrator, we start by defining a generative model for the agent, i.e., the block: $$\begin{aligned}
\bm{x}' = \hat{A} \bm{x} + \hat{B} \bm{v} + \bm{w}
\end{aligned}
\quad
\begin{aligned}
\bm{y} = \hat{C} \bm{x} + \bm{z}
\end{aligned}
\label{eq:doubleIntegratorSSMGenerativeModel}$$ where matrix $\hat{A}$ is: $$\begin{aligned}
\hat{A} =
\begin{bmatrix}
0 & 1 & 0 \\
-\alpha_1 & -\alpha_2 & 0 \\
0 & 0 & 0
\end{bmatrix}\IEEEnonumber
\label{eq:doubleIntegratorMatricesActiveInference}\end{aligned}$$ while $\hat{B}$ is diagonal with $\exp (1)$ values, $\hat{C}$ is zero everywhere but in $C_{2,2}$ where the motor action is applied (with a value of 1) and covariance matrices $\Sigma_z, \Sigma_w$ are also diagonal with, respectively, $\exp (1)$ and $\exp (8)$ values on the main diagonals. The agent implements beliefs of a world where it is pulled back to the desired state $x=x'=0$ by an imaginary spring and slows down thanks to an imaginary piston-like damper, “designed” (in this case by us, but more in general one could imagine evolutionary processes for biological system [@Friston2010biocyb]) to favour normative behaviour.
Following , the variational free energy for our controller is then described by: $$\begin{aligned}
F & \approx \frac{1}{2} \bigg[ \pi_{z} (y - \mu_x)^2 + \pi_{z'} (y' - \mu_x')^2 + \pi_{z''} (y'' - \mu_x'')^2 + \IEEEnonumber \\
& + \pi_{w'} (\mu_x'' - \mu_v)^2 - \ln (\pi_{z} \pi_{z'} \pi_{z''} \pi_{w'}) + (3+2)\ln 2 \pi \bigg]\end{aligned}$$ where precisions $\pi$ are taken from the diagonals of precision matrices $\Pi_z, \Pi_w$ (inverse covariances matrices $\Sigma_z, \Sigma_w$ defined in the generative model). After explicitly writing out the equations derived from the matrix formulation in , we get the following formulation of perceptual inference: $$\begin{aligned}
\dot{\mu}_x = & \mu_x' + \pi_{z} (y - \mu_x) + \pi_w (\mu_x' + \alpha \mu_x - \beta \mu_v) \IEEEnonumber \\
\dot{\mu}_x' = & \mu_x'' + \pi_{z'} (y' - \mu_x') + \pi_{w'} (\mu_x'' + \alpha \mu_x' - \beta \mu_v') \IEEEnonumber\\
\dot{\mu}_x'' = & \mu_x''' + \pi_{z''} (y'' - \mu_x'') + \pi_{w''} (\mu_x'' + \alpha \mu_x' - \beta \mu_v')
\label{eq:ActiveInferenceEstimation}\end{aligned}$$ and $$\begin{aligned}
\dot{\mu}_x' = & - \pi_w (\mu_x' + \alpha \mu_x - \beta \mu_v) \IEEEnonumber \\
\dot{\mu}_x'' = & - \pi_{w'} (\mu_x'' + \alpha \mu_x' - \beta \mu_v')\end{aligned}$$ showing the lack of the Kalman gain $K$ and an important difference derived from its absence: if $K$ is non-diagonal as in this case (one can simply verify this claim with standard functions solving continuous Riccati equations, as in the provided code), both orders of motion are present in the optimal filter problem in , but only one appears in since the precision matrices are assumed to be diagonal in our formulation. More in general, in active inference the Kalman gain $K$ matrix is replaced by learning rates such as in this work or [@baltieri2017active], or by clever implementations that allow for adaptive update schemes with varying integration steps as in [@Friston2008a].
The action component is, however, the one most significantly different, starting from the assumption that direct knowledge of motor signals is not available and thus not modelled in the generative model (motor commands $\bm{a}$ are replaced by inputs $\bm{v}$ acting as priors). This entails a new approach to the problem, with active inference suggesting that the only information needed comes from observations $\bm{y}$, see . On this account, action reduces to $$\bm{\dot{a}} = - \bigg( \frac{\partial \bm{y}'}{\partial a} \bigg)^T \Pi_{z} (\bm{y} - \hat{C} \bm{\mu}_x)
\label{eq:ActiveInferenceControlMatrixVersion}$$ and with the assumption that $$\frac{\partial \bm{y}}{\partial a} =
\begin{bmatrix}
1 & 1 & 1
\end{bmatrix}^T$$ the explicit, scalar version of action becomes $$\dot{a} = - \pi_{z} (y - \mu_x) - \pi_{z'} (y' - \mu_x') - \pi_{z''} (y'' - \mu_x''),$$ replacing the LQR component in . This type of control is equivalent to a PID controller, and is the “optimal” linear solution when knowledge of inputs $\bm{a}$ is not available in the generative model [@baltieri2018propabilistic]. As in the case of filtering, the feedback gain $L$ is missing in the active inference formulation, once again replaced by learning rates of the gradient descent or by other approximations.
[.5]{} ![**The double integrator solved using active inference ($\bm{\alpha_1 = \exp (2), \alpha_2 = \exp (1)}$).** Same layout as [Fig. \[fig:DoubleIntegratorLQG\]]{}. (a) Five examples with different initial conditions showing in blue the observed trajectories of different blocks in the phase-space and in red the agent’s estimates of the same trajectories. (b) Actions taken by the five agents.[]{data-label="fig:DoubleIntegratorActiveInference"}](./DoubleIntegratorActiveInference "fig:"){width="\linewidth"}
[.5]{} ![**The double integrator solved using active inference ($\bm{\alpha_1 = \exp (2), \alpha_2 = \exp (1)}$).** Same layout as [Fig. \[fig:DoubleIntegratorLQG\]]{}. (a) Five examples with different initial conditions showing in blue the observed trajectories of different blocks in the phase-space and in red the agent’s estimates of the same trajectories. (b) Actions taken by the five agents.[]{data-label="fig:DoubleIntegratorActiveInference"}](./DoubleIntegratorActiveInferenceAction "fig:"){width="\linewidth"}
[.5]{} ![**The double integrator solved using active inference ($\bm{\alpha_1 = \exp (2), \alpha_2 = \exp (1)}$).** Same layout as [Fig. \[fig:DoubleIntegratorLQG\]]{}. (a) Five examples with different initial conditions showing in blue the observed trajectories of different blocks in the phase-space and in red the agent’s estimates of the same trajectories. (b) Actions taken by the five agents after an external force is introduced (black line).[]{data-label="fig:DoubleIntegratorActiveInferenceNoExternalForce"}](./DoubleIntegratorActiveInferenceNoExternalForce "fig:"){width="\linewidth"}
[.5]{} ![**The double integrator solved using active inference ($\bm{\alpha_1 = \exp (2), \alpha_2 = \exp (1)}$).** Same layout as [Fig. \[fig:DoubleIntegratorLQG\]]{}. (a) Five examples with different initial conditions showing in blue the observed trajectories of different blocks in the phase-space and in red the agent’s estimates of the same trajectories. (b) Actions taken by the five agents after an external force is introduced (black line).[]{data-label="fig:DoubleIntegratorActiveInferenceNoExternalForce"}](./DoubleIntegratorActiveInferenceNoExternalForceAction "fig:"){width="\linewidth"}
In [Fig. \[fig:DoubleIntegratorActiveInference\]]{} we can see an example implementation of the double integrator using active inference. Five agents are initialised at random position and velocity (zero-mean Gaussian distributed, sd=300) and converge to the target solution where the output actions are essentially zero (excluding some noise), as expected [Fig. \[fig:DoubleIntegratorActiveInferenceAction\]]{}. The most striking feature is that estimates of both position and velocity of the block are very inaccurate but the agent nonetheless reaches the desired target in the phase space, [Fig. \[fig:DoubleIntegratorActiveInferenceAgents\]]{}. These differences are given by the generative implemented by the agent, encoding an imaginary spring-damper system that pulls it towards its desired state [Fig. \[fig:DoubleIntegratorGM\]]{}. [Fig. \[fig:DoubleIntegratorActiveInferenceNoExternalForce\]]{} shows the robustness of this implementation when an external force is introduced: by implementing integral control [@baltieri2018propabilistic], active inference can in this case counteract the effects of unexpected inputs. The presence of integral action perfectly counteracts the effects of disturbances [Fig. \[fig:DoubleIntegratorActiveInferenceNoExternalForceAction\]]{} (cf. [Fig. \[fig:DoubleIntegratorLQGNoExternalForceAction\]]{}), and more importantly allows for the desired regulation of the agents’ positions, [Fig. \[fig:DoubleIntegratorActiveInferenceNoExternalForceAgents\]]{}, which is impossible in LQG accounts assuming perfect knowledge of the world (cf. [Fig. \[fig:DoubleIntegratorLQGNoExternalForceAgents\]]{}).
Discussion
==========
LQG-based architectures are modular in nature, with perception and action seen as separate problems solved nearly independently. According to this view, a system should initially find accurate estimates of the hidden properties of its observations, and only once such estimates are available should an agent attempt to regulate variables that are of interest to achieve its goals, e.g., temperature, oxygen level, etc.. On the other hand, we can define a framework based on mathematical formulations of control problems where the separation principle is not included or required. According to one such proposal, that we identified in active inference [@Friston2010biocyb; @Friston2010nature], perception and action are combined in an inseparable sensorimotor loop described by the minimisation of variational free energy for an agent. In this set up, action and perception are seen as instances of a fundamentally unique process [@clark1998being], using different labels for our (i.e., the observers’) convenience. In particular, the idea of precise inferences of world variables is called into question [@clark2015radical; @baltieri2017active], to the point that inaccurate perception is not only possible but becomes a pre-requisite to act on the world [@brown2013active; @wiese2016action]. In architectures based on the separation principle, the estimated state of a system is thought of as a relevant account of real observations, e.g., their means and covariances. Conversely, in active inference it becomes clear that estimates of latent variables of the world are deeply connected to the current goal of an agent, e.g., to regulate its observations, cf. [@powers1973behavior]. To do so, its targets are encoded as prior expectations and used to bias inferential processes toward its desires so that prediction errors are created as the mismatch of observations and the estimates of hidden variables. These errors are then minimised by acting on the world [@Friston2010biocyb], taking advantage of proprioceptive prediction errors that enact reflex arcs to make observation better accord with existing predictions [@clark2015surfing; @wiese2016action]. More in general, the active inference formulation allows also for accurate estimates of the latent variables generating observations, see for instance [@Friston2008a], but this modality fundamentally excludes the possibility of acting: if no prediction errors are generated for action to minimise, an agent becomes a simple mirror of its world with no strong desire or even necessity to act [@friston2012dark; @brown2013active; @baltieri2017active]. In other words, depending on different precision weights an agent can accurately estimate its observations without acting or potentially discard its sensations to only pursue its desires, generating all possible cases in between as a balanced mix of weighted prediction errors [@allen2018cognitivism].
Conclusions
===========
In recent years the more traditional understanding of perceptual and motor as nearly independent processes as been put into discussion by different authors, especially in neuroscience [@ahissar2016perception; @busse2017sensation; @buckley2018theory]. It is clear that many experimental set ups are limited [@krakauer2017neuroscience], requiring new and ethologically meaningful paradigms for an appropriate study of different aspects of living systems [@najafi2018perceptual]. In this context, we propose some new ideas that could drive future experiments. These ideas are centred around a critical appraisal of LQG as a model architecture for cognitive systems, focusing in particular on the assumptions made by the use of Kalman-Bucy filters, central to these proposals [@todorov2002optimal; @wolpert2011principles; @franklin2011computational]. One of the key requirements for Kalman-Bucy filters to generate an accurate estimate of the hidden state of a system is to have access to *all* the outputs (the observations) and *all* the inputs (forces that affect the state) of a system. The inputs, in particular, include both motor commands, which in classical forward/inverse models are identified using the idea of efference copy [@von1950reafferenzprinzip] (see for instance [@kawato1999internal; @wolpert2000computational; @todorov2004optimality]), and external forces/signals from the environment that cannot be in principle accounted by an organism, i.e., a sudden change in weather conditions or unexpected interactions with other agents.
In this work we focused on the latter, since the presence of external unaccounted forces is often overlooked in many experimental set-ups with fixed or predictable conditions (e.g., the classic and still dominating two-alternative forced choice paradigm). In more realistic and ethological scenarios, however, one should expect that external and unpredictable stimuli constantly affect the behaviour of an agent [@krakauer2017neuroscience; @najafi2018perceptual; @buckley2018theory]. In this case, introducing noise or varying experimental conditions may help in testing the robustness of LQG-based architectures. In practice, if some inputs are not known, one should expect LQG to perform rather poorly until these inputs can be estimated and adaptation (e.g., learning) to new conditions can take place. However, one should then explain how such forces can be described in LQG since Kalman-Bucy filters cannot estimate inputs [@chen2003bayesian] (cf. DEM [@Friston2008a]). More in general, if a system is well adapted to deal with unpredictable stimuli, simple mechanisms such as integral control could be in place, as shown formally in [@sontag2003adaptation] and in experiments on chemotactic adaptation in E. Coli [@yi2000robust] for instance. More recently, some promising results have been presented in [@ritz2018control], supporting the idea that integral feedback control, unlike Kalman(-Bucy) filters, is a good model for adaptation in environments with varying conditions. Integral control constitutes a linear approximation to problems of control with unknown forces affecting the observations of an agent [@aastrom2010feedback; @baltieri2018propabilistic], providing a robust solution with fast responses to problems that otherwise would require slower learning mechanisms [@yi2000robust], which may be ineffective in fast-paced environments [@ashby1957introduction].
Acknowledgments
===============
This work was supported in part by a BBSRC Grant BB/P022197/1.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[^1]: The code is available at https://github.com/mbaltieri/doubleIntegrator
| {
"pile_set_name": "ArXiv"
} |
\[1994/12/01\]
[**THE INEQUALITIES FOR\
SOME TYPES OF $q$-INTEGRALS**]{}
**Predrag M. Rajković**
Depart. of Mathematics, Faculty of Mechanical Engineering
[*e-mail*]{}: [[email protected]]{}
**Sladjana D. Marinković**
Dep. of Mathematics, Faculty of Electronic Engineering
[*e-mail*]{}: [[email protected]]{}
**Miomir S. Stanković**
Department of Mathematics, Faculty of Occupational Safety
[*e-mail*]{}: [[email protected]]{}
**University of Niš, Serbia and Montenegro**
[**Abstract.**]{} We discuss the inequalities for $q$-integrals because of the fact that the inequalities can be very useful in the future mathematical research. Since $q$-integral of a function over an interval $[a,b]$ is defined by the difference of two infinite sums, there a lot of unexpected troubles in analyzing analogs of well-known integral inequalities. In this paper, we will signify to some directions to exceed this problem.
[**Mathematics Subject Classification:**]{} 33D60, 26D15
[**Key words:**]{} integral inequalities, $q$-integral.
Introduction
============
The integral inequalities can be used for the study of qualitative and quantitative properties of integrals. In order to generalize and spread the existing inequalities, we specify two ways to overcome the problems which ensue from the general definition of $q$-integral. The first one is the restriction of the $q$-integral over $[a,b]$ to a finite sum (see [@Gauchman]). The second one is indicated in [@RSM] and it means introduction the definition of the $q$-integral of the Riemann type. At the start sections, we give all definitions of the $q$-integrals, their correlations and properties. In the other sections, we elaborate the $q$-analogues of the well–known inequalities in the integral calculus, as Chebyshev, Grüss, Hermite-Hadamard for all the types of the $q$-integrals. At last, we give a few new inequalities which are valid only for some types of the $q$-integrals.
In the fundamental books about $q$-calculus (for example, see [@Gasper] and [@Hahn]), the $q$-integral of the function $f$ over the interval $[0,b]$ is defined by $$I_q(f;0,b)=\int_0^b f(x)d_q x =
b(1-q)\sum_{n=0}^{\infty}f(bq^n)q^n\quad (0<q<1).
\label{(1.1)}$$ If $f$ is integrable over $[0,b]$, then $$\lim_{q\nearrow 1} I_q(f;0,b) = \int_0^b f(x)\ d x = I(f;0,b).$$ Generally accepted definition for $q$-integral over an interval $[a,b]$ is $$I_q(f;a,b)=\int_a^b f(x)d_q x= \int_0^b f(x)d_q x - \int_0^a f(x)
d_q x\quad (0<q<1). \label{(1.2)}$$ The values of such defined $q$-integrals of the polynomials have very similar form to those in the standard integral calculus. So, for example, we it is valid $$\int_{a}^{b} x^n d_q x=\frac{b^{n+1}-a^{n+1}}{[n+1]_q},$$ where $$[n]_q=\frac{1-q^n}{1-q}=1+q+\cdots+q^{n-1},\qquad n\in\mathbb N.$$
But, the problems come when the integrand $f$ is defined in $[a,b]$ and it is not defined in $[0,a]$. Obviously this definition cannot be applied on evaluating of the integrals of the form $$\int_2^3 \ln (x-1)d_q x.$$
The $q$-integrals and correlations
===================================
Let $a, b$ and $q$ be some real numbers such that $0<a<b$ and $q\in (0,1)$.
Beside the $q$-integrals defined by (\[(1.1)\]) and (\[(1.2)\]) we will consider two other types of the $q$-integrals.
In the paper [@Gauchman], H. Gauchman has introduced [*the restricted $q$-integral*]{} $$G_q(f;a,b)=\int_{a}^b f(x)\ d_q^{G} x=
b(1-q)\sum_{k=0}^{n-1}f(bq^k)q^k \quad (a=bq^n).
\label{(2.3)}$$ Let us notice that lower bound of integral is $a=bq^n$, i.e. it is tied by chosen $b$, $q$ and positive integer $n$.
In the paper [@RSM], we have introduced [*Riemann-type $q$-integral*]{} by $$R_q(f;a,b)=\int_a^b f(t)d_q^{R}t=
(b-a)(1-q)\sum_{k=0}^{\infty}f\bigl(a+(b-a)q^k\bigr)q^k.
\label{(2.4)}$$ This definition includes only point inside the interval of the integration.
The different types of the $q$-integral defined by (\[(1.1)\])–(\[(2.4)\]) can be denoted in the unique way by $J_q(\,\cdot\,;a_{(J)},b)$, where $J$ can be $G$, $I$ or $R$. Interval of the integration $E_{(J)}=[a_{(J)},b]$ of $q$-integral $J_q(\,\cdot\,;a_{(J)},b)$ depends on its type:
$a_{(G)}=bq^n$, $n\in\mathbb N$, for $G_q(\,\cdot\,;a,b)$;
$a_{(I)}=0$, for $I_q(\,\cdot\,;0,b)$;
$a_{(I)} \in [0,b]$, for $I_q(\,\cdot\,;a,b)$;
$a_{(R)} \in [0,b]$, for $R_q(\,\cdot\,;a,b)$.
We can say that a real function $f$ is [*$q$-integrable*]{} on $[0,b]$ or $[a,b]$ if the series in (1) and (2) converge. In the similar way, we say that $f$ is [*$qR$-integrable*]{} on $[a,b]$ if the series in (4) converges.
From now on, it will be assumed that the function $f$ is $q$-integrable on $[0,b]$ ($qR$-integrable on $[a,b]$) whenever $I_q(f;0,b)$ or $I_q(f;a,b)$ ($R_q(f;a,b)$) appears in the formula.
In this research it is convenient to define the operators $$\aligned
&\widehat {}\ :f\mapsto \widehat f,\qquad \widehat f(x)=f\bigl(a+(b-a)x\bigr),\\
&\widetilde {}\ :f\mapsto \widetilde f,\qquad \widetilde f(x)=bf(bx)-af(ax),\\
&\breve{}\ :f\mapsto\breve f,\qquad \breve f(x)=f(bx)-f(ax),
\endaligned$$ such that associate the functions defined on $[0,1]$ to the function defined on $[a,b]$. Notice that, for $x\in[0,1]$, it is $$\widehat{(fg)}(x)=\widehat f(x)\ \widehat g(x),\quad
\widetilde{(fg)}(x)=\frac 1{b-a}\bigl(\widetilde f(x) \widetilde g(x)-ab\
\breve f(x) \breve g(x)\bigr). \label{(2.10)}$$
The correlations between the $q$-integrals defined by (\[(1.1)\])–(\[(2.4)\]) are given in the following lemma.
\[2.1\] If the real function $f$ is $q$-integrable on $[0,b]$ or $qR$-integrable on $[a,b]$ $(0<a<b)$, then it holds $$\begin{aligned}
I_q(f;0,b)&=&\lim_{n\to\infty}G_q(f;bq^n,b), \label{(2.5)}\\
I_q(f;a,b)&=&I_q(\widetilde f;0,1),\quad \textrm{where}\quad
\widetilde f(x)=bf(bx)-af(ax), \label{(2.6)}\\
R_q(f;a,b)&=&(b-a)I_q(\widehat f;0,1), \quad\textrm{where}\quad
\widehat f(x)=f(a+(b-a)x), \label{(2.7)}\end{aligned}$$
[*Proof.*]{} Since $G_q(f;bq^n,b)$ $(n\in\mathbb N)$ is the partial sum of the series $I_q(f;0,b)$, the relation (\[(2.5)\]) is evident.
The equalities (\[(2.6)\]) and (\[(2.7)\]) are valid because of $$I_q(f;a,b)
=(1-q)\sum_{k=0}^{\infty}\bigl(bf(bq^k)-af(aq^k)\bigr)q^k
=I_q(\widetilde f;0,1)$$ and $$R_q(f;a,b)=(b-a)(1-q)\sum_{k=0}^{\infty}f(a+(b-a)q^k)q^k
=(b-a)I_q(\widehat f;0,1).\ \square$$ The mentioned connections can be used to derive the inequalities for all types of the $q$-integrals. By (\[(2.5)\]), the inequalities for the infinite sum $I_q(f;0,b)$ can be derived in the limit process from this ones for $G_q(f;a,b)$ which are defined by the finite sum. Using (\[(2.6)\]) and (\[(2.7)\]), the integrals $I_q(f;a,b)$ and $R_q(f;a,b)$ can be considered as the $q$-integrals over $[0,1]$. Nevertheless, the results for $I_q(f;a,b)$ are quite rough because the points outside of the interval of integration (i.e. points on $[0,a]$) are included.
According to (\[(2.10)\]) and Lemma \[2.1\], the following integral relations are valid: $$\begin{aligned}
&&R_q(fg;a,b)=(b-a)I_q\bigl(\widehat{(fg)};0,1\bigr)
=(b-a)I_q\bigl(\widehat f\ \widehat g;0,1\bigr), \label{(2.8)} \\
&&I_q(fg;a,b)=I_q\bigl(\widetilde{(fg)};0,1\bigr)
=\frac1{b-a}\Bigl(I_q\bigl(\widetilde f \ \widetilde g;0,1\bigr)
-ab\ I_q\bigl(\breve f\ \breve g;0,1\bigr)\Bigr).\qquad \label{(2.9)}\end{aligned}$$
$q$-Chebyshev inequality
========================
In this section we give the $q$-analogues of Chebyshev inequality for the monotonic functions (see [@Mitrinovic], pp. 239.). The discrete case of this inequality is used in [@Gauchman] for the restricted $q$-integrals. We derive its variants for the rest of the $q$-integrals.
The function $f:[a,b]\to \mathbb R$ is called $q$-[*increasing*]{} ($q$-decreasing) on $[a,b]$ if $f(qx)\le f(x)$ ($f(qx)\ge f(x)$) whenever $x,qx\in[a,b]$. It is easy to see that if the function $f$ is increasing (decreasing), then it is $q$-increasing ($q$-decreasing) too.
\[3.1\] Let $f,g:E_{(J)}\to\mathbb R$ be two real functions, both $q$-decreasing or both $q$-increasing. If $J_q(\,\cdot\,;a_{(J)},b)$ is the $q$-integral defined by $(\ref{(1.1)})$, $(\ref{(2.3)})$ or $(\ref{(2.4)})$, it holds $$J_q(fg;a_{(J)},b) \ge \frac1{b-a_{(J)}}J_q(f;a_{(J)},b)\
J_q(g;a_{(J)},b). \$$
[*Proof.*]{} For $J_q(\,\cdot\,;a_{(J)},b)=G_q(\,\cdot\,;a,b)$, $a=bq^n$, the inequality is proven in [@Gauchman]. So, the inequalities $$G_q(fg;bq^n,b)\ge\frac 1{b-bq^n}G_q(f;bq^n,b) \ G_q(g;bq^n,b)$$ are valid for all $n=1,2,\ldots$ . When $n\to\infty$, using (\[(2.5)\]) we get the desired inequality for $J_q(\,\cdot\,;a_{(J)},b)=I_q(\,\cdot\,;0,b)$. In the case $J_q(\,\cdot\,;a_{(J)},b)=R_q(\,\cdot\,;a,b)$, from the $q$-monotonicity of the functions $f$ and $g$ on $[a,b]$ follows the $q$-monotonicity of the functions $\widehat f$ and $\widehat
g$ on $[0,1]$. Hence, we have $$I_q(\widehat f \ \widehat g;0,1)\ge I_q(\widehat f;0,1) \
I_q(\widehat g;0,1).$$ According to (\[(2.6)\]) and (\[(2.7)\]) we get the required inequality. $\square$
The Chebyshev inequality in the source form is not valid for $I_q(\,\cdot\,;a,b)$, where $0<a<b$.
[**Example 3.1**]{} For $f(x)=x^3$ and $g(x)=x^4$ on the interval $[1,2]$ we have $$I_q(x^3\cdot x^4;1,2) - I_q(x^3;1,2)I_q(x^4;1,2)
=255\frac{1-q}{1-q^8} - 465 \frac{(1-q)^2}{(1-q^4)(1-q^5)},$$ wherefrom we conclude that the inequality holds only for $q>1/2$, but it has opposite sign for $q<1/2$.
\[3.2\] Let the function $f:[0,b]\to\mathbb R$ be increasing and $0<a<b$. If there exist two positive constants $l$ and $L$ such that $a^2/b^2\le l/L$ and for every $x,y\in [0,b]$ the inequality $$l \le \frac{f(x)-f(y)}{x-y} \le L$$ is valid, then the function $\widetilde f:[0,1]\to\mathbb R$ is increasing too.
[*Proof.*]{} Under the conditions of the Lemma, for every $0\le x <y \le b$ we have $$l(y-x)\le f(y)-f(x) \le L(y-x).$$ Then it holds $$\aligned \widetilde f(y)-\widetilde f(x)
&=b\bigl(f(by)-f(bx)\bigr)-a\bigl(f(ay)-f(ax)\bigr)\\
&\ge (b^2 l-a^2 L)(y-x) \ge 0.\ \square
\endaligned$$
Let $f,g:[0,b]\to \mathbb R$ be two real increasing functions. If there exist the constants $l_f$, $L_f$, $l_g$ and $L_g$ such that $a^2/b^2\le l_f/L_f$, $a^2/b^2\le l_g/L_g$ and $$l_f \le \frac{f(x)-f(y)}{x-y}\le L_f,\quad l_g \le
\frac{g(x)-g(y)}{x-y}\le L_g$$ holds, then the inequalities are valid: $$\aligned
(a)\quad I_q(fg;a,b) & \ge \frac 1{b-a}I_q(f;a,b)
I_q(g;a,b)
-\frac{ab(b-a)}{[3]_q} L_f L_g \\
(b) \quad I_q(fg;a,b) & \ge \frac 1{b-a}I_q(f;a,b) I_q(g;a,b)
-\frac{ab}{b-a}\bigl(f(b)-f(0)\bigr)\bigl(g(b)-g(0)\bigr).
\endaligned$$
[*Proof.*]{} Suppose that $f$ and $g$ are both increasing on $[0,b]$. Then, according to Lemma \[3.2\], $\widetilde f$ and $\widetilde g$ are both increasing and hence $q$-increasing on $[0,1]$. With respect to (\[(2.9)\]) we can write $$I_q(fg;a,b)=\frac 1{b-a}\Bigl( I_q(\widetilde f\ \widetilde
g;0,1)- ab\ I_q(\breve f\ \breve g;0,1)\Bigr).$$ Using Theorem \[3.1\], we have $$I_q(\widetilde f \ \widetilde g;0,1)\ge I_q(\widetilde f;0,1) \
I_q(\widetilde g;0,1),$$ wherefrom $$I_q(fg;a,b)\ge \frac 1{b-a}\Bigl(I_q(f;a,b)\
I_q(g;a,b)-abI_q(\breve f\ \breve g;0,1)\Bigr). \label{3.3}$$ (a) Under the conditions satisfied by the functions $f$ and $g$ on $[0,b]$, it holds $$\aligned I_q(\breve f\ \breve g;0,1)
&=(1-q)\sum_{k=0}^\infty \bigl(f(bq^k)-f(aq^k)\bigr)\bigl(g(bq^k)-g(aq^k)\bigr)q^k\\
&\le(1-q)\sum_{k=0}^\infty L_f L_g(bq^k-aq^k)^2 q^k
%\\&=(1-q)L_f L_g (b-a)^2 \sum_{k=0}^\infty q^{3k}
=L_f L_g(b-a)^2\frac{1-q}{1-q^3}
\endaligned$$ Substituting this estimation in (\[3.3\]), we get the first inequality.
\(b) Since the functions $f$ and $g$ are increasing on $[0,b]$, it holds $$I_q(\breve f\ \breve g;0,1)\le
(1-q)\bigl(f(b)-f(0)\bigr)\bigl(g(b)-g(0)\bigr)\sum_{k=0}^\infty
q^k =\bigl(f(b)-f(0)\bigr)\bigl(g(b)-g(0)\bigr),$$ what with (\[3.3\]) gives the second inequality. $\square$
$q$-Grüss inequality
====================
The Grüss inequality (see [@Mitrinovic], pp. 296) can be understood as conversion of Chebyshev one.
\[4.1\] Let $f,g:E_{(J)}\to\mathbb R$ be two real functions, such that $m \le f(x) \le M$, $\varphi \le g(x) \le \Phi$ on $E_{(J)}$, where $m,M,\varphi,\Phi$ are given real constants. If $J_q(\,\cdot\,;a_{(J)},b)$ is the $q$-integral defined by $(\ref{(1.1)})$, $(\ref{(2.3)})$ or $(\ref{(2.4)})$, it holds $$\begin{aligned}
\Biggm| \frac 1{b-a_{(J)}}J_q(fg;a_{(J)},b)-
\frac1{\bigl(b-a_{(J)}\bigr)^2}J_q(f;a_{(J)},b)\ J_q(g;a_{(J)},b)\Biggm| \qquad \qquad \\
\qquad \qquad \le \frac 14(M-m)(\Phi-\varphi).\end{aligned}$$
[*Proof.*]{} For the restricted $q$-integrals $G_q(\,\cdot\,;bq^n,b)$, the inequality is proven in [@Gauchman]. So, for any arbitrary positive integer $n$, the inequality $$\begin{aligned}
\Bigm| \dfrac 1{b-bq^n}G_q(fg;bq^n,b)-\frac
1{(b-bq^n)^2}G_q(f;bq^n,b) \ G_q(g;bq^n,b)\Bigm| \qquad \qquad \\
\hspace{3cm}
\le \frac14(M-m)(\Phi-\varphi)\end{aligned}$$ is valid. When $n\to\infty$, we get the required inequality for $I_q(\,\cdot\,;0,b)$ via (\[(2.5)\]). Finally, providing the conditions of the theorem, the functions $\widehat f$ and $\widehat g$ are bounded on $[0,1]$ by the constants $m,M,\varphi,
\Phi$ respective. Then, $$\Bigm| I_q(\widehat f \ \widehat g;0,1)-I_q(\widehat f;0,1) \
I_q(\widehat g;0,1)\Bigm| \le \frac 14(M-m)(\Phi-\varphi)$$ holds and using the relation (\[(2.7)\]), we get the inequality for $R_q(\,\cdot\,;a,b)$. $\square$
[**Example 4.1**]{} For $f(x)=x$ and $g(x)=x^2$ on the interval $[1,2]$ we have $$I_q(x\cdot x^2;1,2) - I_q(x;1,2)I_q(x^2;1,2)
=(1-2q)\frac{3\ (2-q)}{(1+q)(1+q^2)(1+q+q^2)}.$$ Including the boundaries of the functions $f(x)$ and $g(x)$, we can see that the formula of Grüss inequality will not be hold on for $q\in (0, 1/3)$.
Let $f,g:[0,b]\to \mathbb R$ be two bounded such that $m\le f(x)
\le M$, $\varphi \le g(x) \le \Phi$ on $[0,b]$, where $m,M,\varphi,\Phi$ are given real constants. Then it holds $$\begin{aligned}
\Bigm| \frac 1{b-a}I_q(fg;a,b) -
\frac1{(b-a)^2}I_q(f;a,b) I_q(g;a,b)\Bigm| \hspace{4cm}\\
\hspace{5cm}\le \frac
14(M-m)(\Phi-\varphi)\biggl(1+\frac{4ab}{(b-a)^2}\biggr).\end{aligned}$$
[*Proof.*]{} Having in mind the boundaries of $f$ and $g$ on $[0,b]$, we have $$bm-aM\le \widetilde f(x)\le bM-am,\qquad
b\varphi-a\Phi\le\widetilde g(x)\le b\Phi-a\varphi,$$ where $\widetilde f$ and $\widetilde g$ are the function defined on $[0,1]$. According to Theorem \[4.1\], we have $$\begin{aligned}
\Bigm| I_q(\widetilde f\ \widetilde g;0,1)
-I_q(\widetilde f;0,1) \ I_q(\widetilde g;0,1)\Bigm| \hspace{5cm}\\
\hspace{4cm}\le \frac
14(bM-am-bm+aM)(b\Phi-a\varphi-b\varphi+a\Phi).\end{aligned}$$ By using (\[(2.9)\]), we obtain $$\begin{aligned}
\phantom{\le} &\Bigm|(b-a)I_q(fg;a,b)-I_q(f;a,b) \
I_q(g;a,b)\Bigm| -ab\Bigm|I_q(\breve f\ \breve g;0,1)\Bigm|
\hspace{2truecm}
\\
&\hspace{2truecm}\le \Bigm|(b-a)I_q(fg;a,b)-I_q(f;a,b) \
I_q(g;a,b)+ab\ I_q(\breve f\ \breve g;0,1)\Bigm|
\\
&\hspace{7truecm}\le \frac
14(b-a)^2(M-m)(\Phi-\varphi).\end{aligned}$$
With respect to the boundaries of $f$ and $g$ on $[0,b]$, the estimation $$\Bigm|I_q(\breve f\ \breve g;0,1)\Bigm|\le (M-m)(\Phi-\varphi)$$ holds, what, finally, proves the statement. $\square$
$q$-Hermite–Hadamard inequality
===============================
The Hermite–Hadamard inequality (see [@Mitrinovic], pp. 10) is related to the Jensen inequality for the convex function. In [@Gauchman] there is proved a variant of its analogue for the restricted $q$-integrals. Here we will formulate and prove another variant of the $q$-Hermite–Hadamard inequality for the restricted $q$-integrals and for the other types of $q$-integrals.
\[5.1\] Let $f:[a,b]\to \mathbb{R}$ $(a=bq^n)$ be a convex function. Then it holds $$f\biggl(\frac{a+b}{[2]_q}\biggr) \le \frac1{b-a}G_q(f;a,b) \le
\frac1{[2]_q}\biggl(q\ f\Bigl(\frac aq\Bigr)+f(b)\biggr).$$
[*Proof.*]{} According to the definition of the restricted $q$-integral, we have $$\frac1{b-a}G_q(f;a,b)=\frac{1-q}{1-q^n} \sum_{k=0}^{n-1}f(bq^k)q^k
=\Biggl(\sum\limits_{k=0}^{n-1}q^k\Biggr)^{-1}
\Biggl(\sum\limits_{k=0}^{n-1}f(bq^k)q^k\Biggr)\ .$$ If we assign $$\overline x=\Biggl(\sum\limits_{k=0}^{n-1}\ q^k\Biggr)^{-1}
\Biggl(\sum\limits_{k=0}^{n-1}bq^k\ q^k\Biggr)
=\frac{b(1+q^n)}{1+q}=\frac{a+b}{1+q}$$ and apply Jensen inequality for the convex functions on the last term, we obtain $$\frac1{b-a}G_q(f;a,b)\ge f(\overline
x)=f\biggl(\frac{a+b}{1+q}\biggr).$$ On the other side, using a variant of the reverse Jensen inequality (see [@Mitrinovic], pp. 9.), we get $$\aligned \frac1{b-a}G_q(f;a,b)&\le \frac{b-\overline
x}{b-bq^{n-1}}\ f(bq^{n-1})
+\frac{\overline x-bq^{n-1}}{b-bq^{n-1}}\ f(b)\\
&=\biggl(b-\frac aq\biggr)^{-1}\biggl(\Bigl(b-\frac{a+b}{1+q}\Bigr)f\Bigl(\frac
aq\Bigr)+
\Bigl(\frac{a+b}{1+q}-\frac aq\Bigr)f(b)\biggr)\\
&=\frac 1{1+q}\biggl(q\ f\Bigl(\frac aq\Bigr)+f(b)\biggr).\
\square
\endaligned$$
\[5.2\] Let $f:[0,b]\to \mathbb{R}$ be a continuous convex function. Then, $$f\biggl(\frac{b}{[2]_q}\biggr) \le \frac1{b}I_q(f;0,b) \le
\frac1{[2]_q}\Bigl(q\ f(0)+f(b)\Bigr).$$
[*Proof.*]{} Since the function $f$ satisfies the conditions of Theorem \[5.1\] on the intervals $[bq^n,b]$ for every $n\in\mathbb N$, the inequalities $$f\biggl(\frac{bq^n+b}{[2]_q}\biggr) \le
\frac1{b-bq^n}G_q(f;bq^n,b) \le \frac1{[2]_q}\biggl(q\
f\biggl(\frac {bq^n}q\biggr)+f(b)\biggr)$$ are valid. When $n\to\infty$, we obtain the desired inequality because $f$ is continuous and (\[(2.5)\]) is satisfied. $\square$
\[5.3\] Let $f:[a,b]\to \mathbb{R}$ be a continuous convex function. Then, $$f\biggl(\frac{aq+b}{[2]_q}\biggr) \le \frac1{b-a}R_q(f;a,b)) \le
\frac1{[2]_q}\Bigl(q\ f(a)+f(b)\Bigr).$$
[*Proof.*]{} Under the conditions which are satisfied by the function $f$ on $[a,b]$, the function $\widehat
f(x)=f\bigl(a+(b-a)x\bigr)$ satisfies the conditions of the Theorem \[5.2\] on $[0,1]$. Hence $$\widehat f\biggl(\frac{1}{[2]_q}\biggr) \le I_q(\widehat f;0,1)
\le \frac1{[2]_q}\Bigl(q\ \widehat f(0)+\widehat f(1)\Bigr).$$ According to (\[(2.8)\]) and the continuity of the function $f$, we get the desired inequality. $\square$
Let us remember that the function $f$ is convex on $[0,b]$ if for all $x,y\in[0,b]$ and $p_1+p_2>0$ $$f\biggl(\frac{p_1 x+p_2 y}{p_1+p_2}\biggr) \le \frac{p_1 f(x)+p_2 f(y)}{p_1+p_2}$$ holds. The convexity of the function $\widetilde f$ on $[0,1]$ is due to the existence of the appropriate constants $l$ and $L$ such that the condition $$\label{5.10}
l \le \frac{p_1 f(x)+p_2 f(y)}{p_1+p_2}
-f\biggl(\frac{p_1 x+p_2 y}{p_1+p_2}\biggr) \le L$$ is satisfied.
\[5.4\] Let the function $f:[0,b]\to\mathbb R$ be convex. If there exist two positive constants $l$ and $L$ such that $bl \ge aL$ and for every $x,y\in [0,b]$ and $p_1+p_2>0$ the condition $(\ref{5.10})$ is satisfied, then the function $\widetilde
f:[0,1]\to\mathbb R$ is convex too.
[*Proof.*]{} Under the conditions of the Lemma, for every $0\le x,y \le b$ and $p_1+p_2>0$ we have $$\begin{aligned}
&&\frac{p_1 \widetilde f(x)+p_2 \widetilde f(y)}{p_1+p_2} -
\widetilde f\biggl(\frac{p_1 x+p_2 y}{p_1+p_2}\biggr) \\
&&\hspace{4cm}=b\Biggl(\frac{p_1 f(bx)+p_2 f(by)}{p_1+p_2}
-f\biggl(\frac{p_1 bx+p_2 by}{p_1+p_2}\biggr)
\Biggr)\\
&&\hspace{4cm}-a\Biggl(\frac{p_1 f(ax)+p_2 f(ay)}{p_1+p_2}
-f\biggl(\frac{p_1 ax+p_2 ay}{p_1+p_2}\biggr)
\Biggr)\\
&&\hspace{4cm}\ge bl-aL \ge 0.\ \square\end{aligned}$$
\[5.5\] Let $f:[0,b]\to \mathbb{R}$ be a continuous and convex function. If there exist two positive constants $l$ and $L$ such that $bl \ge a L$ and for every $x,y\in [0,b]$, $p_1+p_2>0$ the condition $(\ref{5.10})$ is satisfied, then it holds $$\label{5.6}
b f\Bigl(\dfrac{b}{[2]_q}\Bigr)-a f\Bigl(\dfrac{a}{[2]_q}\Bigr)\le
I_q(f;a,b) \le \frac{(b-a)qf(0)+ b f(b)-af(a)}{[2]_q}.$$
[*Proof.*]{} According to Lemma \[5.4\], the function $\widetilde f$ is convex on $[0,1]$. Then, using Theorem \[5.2\], we have $$\widetilde f\biggl(\frac{1}{[2]_q}\biggr) \le I_q(\widetilde
f;0,1) \le \frac1{[2]_q}\biggl(q\ \widetilde f(0)+\widetilde
f(1)\biggr).$$ Applying the relation (\[(2.6)\]) we get the statement. $\square$
\[5.5\] Let $f:[0,a+b]\to \mathbb{R}$ be a continuous and convex function. If there exist two positive constants $l$ and $L$ such that $bl \ge a L$ and for every $x,y\in [0,a+b]$, $p_1+p_2>0$ the condition $(\ref{5.10})$ is satisfied, then it holds $$l+f\Bigl(\frac{a+b}{[2]_q}\Bigr)\le\frac1{b-a} I_q(f;a,b) \le
\frac1{[2]_q}\Bigl(qf(0)+ f(a+b)+L\Bigr).$$
[*Proof.*]{} Let $p_1=b/(b-a)$, $p_2=-a/(b-a)$. Applying the condition (\[5.10\]) with $x=b/(1+q)$, $y=a/(1+q)$ on the left term and $x=a$, $y=b$ on the right term in (\[5.6\]), we get the statement. $\square$
The other inequalities
======================
In this section we will formulate some new inequalities for $G_q(\,\cdot\,;a,b)$, $I_q(\,\cdot\,;0,b)$ and $R_q(\,\cdot\,;a,b)$. They will be proven only for $G_q(\,\cdot\,;a,b)$. In the way presented in the previous sections, these inequalities for the other two types follow directly. Furthermore, it seems that the corresponding inequalities for the integral $I_q(\,\cdot\,;a,b)$ defined by (\[(1.2)\]), exist and have different forms because of the previously mentioned difficulties related to estimating of the difference of series.
So, let $J_q(\,\cdot\,)=J_q(\,\cdot\,;a_{(J)},b)$ denotes the $q$–integral defined by (\[(1.1)\]), (\[(2.3)\]) or (\[(2.4)\]). In the formulation and proofs of the theorems we follow the inequalities for the finite sums given in [@S.S.; @Dragomir].
The first class are the inequalities the Cauchy-Buniakowsky-Schwarz type.
Let $f,g:E_{(J)}\to \mathbb R$ be two real functions and $\alpha,
\beta >1$ the numbers satisfying $\dfrac 1\alpha+\dfrac
1\beta=1$. Then the following inequalities hold:
$$\begin{aligned}
(i) \quad &&\frac 1\alpha J_q(|f|^\alpha) + \frac 1\beta
J_q(|g|^\beta) \ge \frac 1{b-a_{(J)}} J_q(|f|) J_q(|g|), \\
\\
(ii) \quad &&\frac 1\alpha J_q(|f|^\alpha) J_q(|g|^\alpha) + \frac
1\beta J_q(|f|^\beta) J_q(|g|^\beta)
\ge \Bigl(J_q(|fg|)\Bigr)^2 ,\\
\\
(iii) \quad &&\frac 1\alpha J_q(|f|^\alpha) J_q(|g|^\beta) +\frac
1\beta J_q(|f|^\beta) J_q(|g|^\alpha)
\ge J_q(|f||g|^{\alpha-1})J_q(|f||g|^{\beta-1}),\\
\\
(iv) \quad &&J_q(|f|^\alpha) J_q(|g|^\beta) \ge J_q(|fg|)
J(|f|^{\alpha-1}|g|^{\beta-1}).\end{aligned}$$
[*Proof.*]{} If in well-known Young inequality (see [@Mitrinovic], pp. 381) $$\frac 1\alpha x^\alpha+\frac 1\beta y^\beta\ge xy
\qquad\qquad
(x,y\ge0,\quad \alpha, \beta >1:\ \frac 1\alpha+\frac
1\beta=1),$$ we put $x=|f(bq^i)|$, $y=|g(bq^j)|$, where $i,j=0,1,\ldots, n-1$, we have $$\frac 1\alpha |f(bq^i)|^\alpha+\frac 1\beta |g(bq^j)|^\beta \ge
|f(bq^i)||g(bq^j)|,\quad i,j=0,1,\ldots, n-1.$$ Multiplying by $q^{i+j}$ and summing over $i$ and $j$, we obtain $$\begin{aligned}
\frac 1\alpha \sum_{j=0}^{n-1}q^j
\sum_{i=0}^{n-1}q^i|f(bq^i)|^\alpha + \frac 1\beta
\sum_{i=0}^{n-1}q^i \sum_{j=0}^{n-1}q^j|g(bq^j)|^\beta
\ge
\sum_{i=0}^{n-1}q^i|f(bq^i)| \sum_{j=0}^{n-1}q^j|g(bq^j)|\end{aligned}$$ and, finally, inequality ([*i*]{}). The rest of inequalities can be proved in the same manner by the next choice of the parameters in Young inequality: $$\begin{aligned}
(ii)\ & x=|f(bq^j)|\ |g(bq^i)|, \ &y=|f(bq^i)|\ |g(bq^j)|,\\
(iii)\ & x=|f(bq^j)|/|g(bq^j)|, \ &y=|f(bq^i)|/|g(bq^i)|,\quad \bigl(g(bq^j) \ g(bq^i)\ne 0\bigr),\\
(iv)\ & x=|f(bq^i)|/|f(bq^j)|, \ &y=|g(bq^i)|/|g(bq^j)|,\quad
\bigl(f(bq^j)\ g(bq^j)\ne 0\bigr),\end{aligned}$$ where additional conditions about not vanishing for $f$ and $g$ do not have influence on final conclusion. $\square$
Let $f,g:E_{(J)}\to \mathbb R$ be two real functions and $\alpha,
\beta >1$ the numbers satisfying $\dfrac 1\alpha+\dfrac
1\beta=1$. Then the following inequalities hold: $$\aligned
(i) \ &\frac 1\alpha J_q(|f|^\alpha) J_q(|g|^2) +
\frac 1\beta J_q(|f|^2) J_q(|g|^\beta)
\ge J_q(|fg|) J_q(|f|^{2/\beta}|g|^{2/\alpha}),\\
(ii) \ &\frac 1\alpha J_q(|f|^2) J_q(|g|^\beta) +
\frac 1\beta J_q(|f|^\alpha) J_q(|g|^2)
\ge J_q(|f|^{2/\alpha}|g|^{2/\beta})
J_q(|f|^{\alpha-1}|g|^{\beta-1}), \\
(iii) \ & \qquad\qquad \ J_q(|f|^2) J_q\Bigl(\frac 1\alpha |g|^\alpha
+\frac 1\beta |g|^\beta\Bigr)\ge J_q(|f|^{2/\alpha}|g|)
J_q(|f|^{2/\beta}|g|).
\endaligned$$
[*Proof.*]{} As previous, the proof is based on Young inequality with appropriate choice of the parameters:
$$\begin{aligned}
(i)\ &x=|f(bq^i)|\ |g(bq^j)|^{2/\alpha}, \
&y=|f(bq^j)|^{2/\beta}\ |g(bq^i)|,\\
(ii)\ &x=|f(bq^i)|^{2/\alpha}/|f(bq^j)|, \
&y=|g(bq^i)|^{2/\beta}/|g(bq^j)|\quad
\ (f(bq^j)g(bq^j)\ne 0),\\
(iii)\ &x=|f(bq^i)|^{2/\alpha}\ |g(bq^j)|, \
&y=|f(bq^j)|^{2/\beta}\ |g(bq^i)|.\ \square\end{aligned}$$
The following few inequalities include the boundaries of the functions.
\[6.3\] If $f,g:E_{(J)}\to \mathbb R$ are two positive functions and $$m=\min_{a\le x\le b}\frac{f(x)}{g(x)},\qquad M=\max_{a\le x\le
b}\frac{f(x)}{g(x)},$$ then the following inequalities hold: $$\aligned
(i) \quad & \qquad \qquad \quad 0\le J_q(f^2)J_q(g^2) \le
\frac{(m+M)^2}{4mM}\Bigl(J_q(fg)\Bigr)^2,\\
(ii)\quad & 0\le \sqrt{(J_q(f^2) J_q(g^2)}-J_q(fg)
\le \frac{(\sqrt{M}-\sqrt{m})^2}{2\sqrt{mM}}J_q(fg),\\
(iii)\quad & 0\le J_q(f^2) J_q(g^2)-\Bigl(J_q(fg)\Bigr)^2
\le \frac{(M-m)^2}{4mM}\Bigl(J_q(fg)\Bigr)^2.
\endaligned$$
[*Proof.*]{} With respect to the definition of $G_q(\,\cdot\,;a,b)$, the inequality ([*i*]{}) is the immediate consequence of the Cassels inequality (see [@S.S.; @Dragomir], pp. 72). The inequalities ([*ii*]{}) and ([*iii*]{}) can be obtained by a few transformations of ([*i*]{}). $\square$
If $f,g:E_{(J)}\to \mathbb R$ are two positive functions such that $$0<c\le f(x)\le C<\infty,\qquad 0<d\le g(x)\le D<\infty,$$ then the following inequalities hold: $$\aligned
(i)\quad \ & \qquad \qquad \qquad 0\le J_q(f^2)J_q(g^2) \le
\frac{(cd+CD)^2}{4cdCD}\Bigl(J_q(fg)\Bigr)^2,\\
(ii)\quad & \quad 0\le \sqrt{J_q(f^2) J_q(g^2)}-J_q(fg)
\le \frac{(\sqrt{CD}-\sqrt{cd})^2}{2\sqrt{cdCD}}J_q(fg),\\
(iii)\quad & 0\le J_q(f^2) J_q(g^2)-\Bigl(J_q(fg)\Bigr)^2
\le \frac{(CD-cd)^2}{4cdCD}\Bigl(J_q(fg)\Bigr)^2.
\endaligned$$
[*Proof.*]{} Under the conditions satisfied by the functions $f$ and $g$, we have $$\frac cD \le \frac{f(x)}{g(x)} \le \frac Cd
%,\quad a\le x\le b
.$$ Applying Theorem \[6.3\] we get the inequality ([*i*]{}) and, using it, ([*ii*]{}) and ([*iii*]{}). $\square$
Let $f:E_{(J)}\to \mathbb R$ be a positive function such that $$0<c\le f(x)\le C<\infty.$$ Then the following inequality holds: $$J_q(f^2) \le \frac{(c+C)^2}{4cC\bigl(b-a_{(J)}\bigr)}\Bigl(J_q(f)\Bigr)^2.$$
The next few inequalities are obtained via Jensen inequality for the convex functions.
Let $f,g:E_{(J)}\to \mathbb R$ be two positive functions and $p\ne
0$ a real number. Then it holds $$\aligned
\Bigl(J_q(fg)\Bigr)^p &\le \Bigl(J_q(f^2)\Bigr)^{p-1}
J_q(f^{2-p}g^p),\quad \textrm{for} \quad p\notin (0,1),\\
\Bigl(J_q(fg)\Bigr)^p &\ge \Bigl(J_q(f^2)\Bigr)^{p-1}
J_q(f^{2-p}g^p),\quad \textrm{for} \quad p\in (0,1).
\endaligned$$
[*Proof.*]{} For $p\notin (0,1)$ the function $t\mapsto
t^p$ is convex. Applying the Jensen inequality for convex functions (see [@Mitrinovic], pp.6.) we have $$\Biggl(\frac{\sum\limits_{k=0}^{n-1} f(bq^k) g(bq^k)q^k}
{\sum\limits_{k=0}^{n-1}\bigl(f(bq^k)\bigr)^2 q^k}\Biggr)^p \le
\frac 1{\sum\limits_{k=0}^{n-1}\bigl(f(bq^k)\bigr)^2 q^k} \
\sum_{k=0}^{n-1} \Bigl(\frac{g(bq^k)}{f(bq^k)}\Bigr)^p
\bigl(f(bq^k)\bigr)^2 q^k,$$ i.e., $$\biggl(\sum_{k=0}^{n-1} f(bq^k) g(bq^k)q^k\biggr)^p \le
\biggl(\sum_{k=0}^{n-1}\bigl(f(bq^k)\bigr)^2 q^k\biggr)^{p-1}
\biggl(\sum_{k=0}^{n-1} \bigl(g(bq^k)\bigr)^p
\bigl(f(bq^k)\bigr)^{2-p} q^k\biggr).$$ According to the definition of $G_q(\,\cdot \,;a,b)$ we get the inequality. The reverse case is obtained for $p\in (0,1)$ because of the concave function $t\mapsto t^p$. $\square$
Let $f:E_{(J)}\to \mathbb R$ be a positive function and $p\ne 0$ a real number. Then it holds $$\Bigl(J_q(f)\Bigr)^p \le \bigl(b-a_{(J)}\bigr)^{p-1}\ J_q(f^p),$$ for $p\notin (0,1)$, or reverse for $p\in(0,1)$.
If $f,g:E_{(J)}\to \mathbb R$ are two positive functions such that $$0<m\le \frac{g(x)}{f(x)}\le M<\infty$$ and $p\ne 0$ a real number, then it holds $$\aligned
J_q(f^{2-p}g^p)+\frac{mM(M^{p-1}-m^{p-1})}{M-m}
J_q(f^p) \le \frac{M^p-m^p}{M-m}J_q(fg),
\endaligned$$ for $p\notin (0,1)$, or reverse for $p\in(0,1)$. Especially, for $p=2$, we have $$J_q(g^2) + mM J_q(f^2)\le (M+m) J_q(fg).$$
[*Proof.*]{} The inequality is based on the Lah-Ribarić inequality (see [@Mitrinovic], pp. 9., [@S.S.; @Dragomir], pp. 123). $\square$
**Acknowledgement**
This research was supported by the Science Foundation of Republic Serbia, Project No. 144023 and Project No. 144013.
[7]{}
, [*“A survey on Cauchy-Bunyakovsky-Schwarz type discrete inequalities”*]{}, J. Inequalities in Pure and Applied Mathematics, [**4**]{}(3) Art. 63, 2003.
, [*Integral Inequalities in $q$-Calculus*]{}, Computers and Mathematics with Applications, vol. 47, (2004), 281–300.
, [*“Basic hypergeometric series”*]{}, Cambridge University Press, London and New York, 1990.
, [*“Lineare Geometrische Differenzengleichungen”*]{}, 169 Berichte der Mathematisch-Statistischen Section im Forschungszentrum Graz, 1981.
, [*“Classical and New Inequalities in Analysis”*]{}, Kluwer Academic Publishers, 1993.
, [*The zeros of polynomials orthogonal with respect to $q$-integral on several intervals in the complex plane*]{}, Proceedings of The Fifth International Conference on Geometry, Integrability and Quantization, 2003, Varna, Bulgaria (ed. I.M. Mladenov, A.C.Hirsshfeld), 178-188.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded, Hölder continuous, and that they satisfy a suitable Harnack inequality. Hence, we provide an extension of celebrated results of M. Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a particular class of fractional Sobolev functions, reminiscent of that considered by E. De Giorgi in his seminal paper of 1957. The flexibility of these classes allows us to also establish regularity of solutions to rather general nonlinear integral equations.'
title: |
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems:\
a unified approach via fractional De Giorgi classes
---
<span style="font-variant:small-caps;">Matteo Cozzi</span>
Introduction
============
The aim of the present paper is to provide a rather general and comprehensive basic regularity theory for various problems connected to the minimization of nonlocal elliptic energy functionals. The simplest class of problems that we address here can be briefly described as follows.
Description of the main results
-------------------------------
Let $n \in {\mathbb{N}}$, $s \in (0, 1)$ and $p \in (1, +\infty)$. Let $F: {\mathbb{R}}\to {\mathbb{R}}$ be a bounded measurable function. Given a bounded domain $\Omega \subset {\mathbb{R}}^n$, we consider a minimizer $u$ of the functional $$\label{Ffuncdef}
{\mathcal{G}}_s(v) := (1 - s) [v]_{W^{s, p}({\mathbb{R}}^n)}^p + \int_{{\mathbb{R}}^n} F(v(x)) \, dx,$$ within all functions $v \in W^{s, p}({\mathbb{R}}^n)$ that are equal to some given bounded function $u_0$ outside of $\Omega$.[^1] Here, $[ \, \cdot \,]_{W^{s, p}({\mathbb{R}}^n)}$ denotes the standard Gagliardo seminorm of the fractional Sobolev space $W^{s, p}({\mathbb{R}}^n)$. The main question that we positively answer in this work is the following: $$\mbox{\itshape Is it true that~$u$ is locally bounded and H\"older continuous inside~$\Omega$?}$$
In their important paper [@GG82], M. Giaquinta and E. Giusti studied the regularity of the minimizers of the energy $$\label{G1funcdef}
\mathcal{G}_1(v) := \| \nabla v \|_{L^p({\mathbb{R}}^n)}^p + \int_{{\mathbb{R}}^n} F(v(x)) \, dx,$$ and of more general functionals with $L^p$ gradient structure. In particular, they proved interior $L^\infty$ and $C^\alpha$ estimates for the minimizers of , when $F$ is bounded (or even subcritical at infinity). One of the most prominent aspects of their work is that the potential term $F$ may be discontinuous, and thus their theory can be applied to various non-differentiable functionals that arise for instance in connection with minimal surfaces, fluid dynamics, and free boundary problems.
Here, we provide an extension of these results to the nonlocal functional $\mathcal{G}_s$ displayed in and to more general variants. Following an idea of L. Caffarelli, C. H. Chan and A. Vasseur [@CCV11], we prove that the minimizers of $\mathcal{G}_s$ satisfy an *improved* fractional Caccioppoli inequality and therefore belong to a particular subset of $W^{s, p}$. We then show that the elements of this subset—which we call a *fractional De Giorgi class*, in honor of the one introduced by E. De Giorgi in [@DeG57]—are locally bounded, Hölder continuous functions that satisfy a Harnack estimate. By interpolating the technique of [@CCV11] with a suitable isoperimetric-type inequality (that is proved to hold in $W^{s, p}$ for large $s$), we obtain estimates which are uniform as the differentiability order $s$ goes to $1^-$. We stress that our results are new even in the quadratic case $p = 2$.
As said previously, one key point of our analysis is that no differentiability is assumed on the potential $F$. Instead, when $F$ is, say, of class $C^1$, we can differentiate the functional ${\mathcal{G}}_s$ and deduce that $u$ is a weak solution of the Euler-Lagrange equation $$\label{Lsp=F'}
{\mathcal{L}}_{s, p} u = f(u) \quad \mbox{in } \Omega,$$ where $f = F'$ and ${\mathcal{L}}_{s, p}$ is formally defined in the principal value sense by $${\mathcal{L}}_{s, p} u(x) := (1 - s) \, {\mbox{\normalfont P.V.}}\int_{{\mathbb{R}}^n} \frac{|u(x) - u(y)|^{p - 2} (u(x) - u(y))}{|x - y|^{n + sp}} \, dy,$$ up to an irrelevant positive factor.
The nonlinear singular integral operator ${\mathcal{L}}_{s, p}$ is often called *fractional p-Laplacian* and has been extensively studied in the last few years. A first, *localized* version of ${\mathcal{L}}_{s, p}$ has been originally introduced by H. Ishii and G. Nakamura in [@IN10], where the authors dealt with the solvability (in the viscosity sense) of the associated Dirichlet problem and established a connection with the classical $p$-Laplace operator in the limit as $s \rightarrow 1^-$. The first interior Hölder continuity results for ${\mathcal{L}}_{s, p}$ have been obtained by A. Di Castro, T. Kuusi and G. Palatucci in [@DKP14b], concerning weak solutions of equations with vanishing right-hand side, and by E. Lindgren in [@Lin14], for viscosity solutions of equations with a bounded right-hand side. Global regularity for weak solutions of the Dirichlet problem has been then proved by A. Iannizzotto, S. Mosconi and M. Squassina in [@IMS14]. The Hölder continuity of first derivatives is, to the best of our knowledge, still unknown. See the paper [@BL15] by L. Brasco and E. Lindgren for regularity results in fractional Sobolev spaces of higher order.
Of course, when $p = 2$ the operator ${\mathcal{L}}_{s, p}$ boils down to the well-known fractional Laplacian $$(-\Delta)^s u(x) := (1 - s) \, {\mbox{\normalfont P.V.}}\int_{{\mathbb{R}}^n} \frac{u(x) - u(y)}{|x - y|^{n + 2 s}} \, dy.$$ In this case, the theory is much more developed. See e.g. the works [@BK05; @Kas07a; @Kas09; @Kas11] of R. Bass and M. Kassmann and [@Sil06; @CS09; @CS11] of L. Caffarelli and L. Silvestre for regularity results for weak and viscosity solutions, respectively. See also [@RS14a; @RS14b] by X. Ros-Oton and J. Serra in relation to the fine boundary behavior of solutions to Dirichlet problems.
With the help of fractional De Giorgi classes, we can address here also the regularity of solutions to and to equations driven by more general integral operators. Under some mild hypotheses on the growth of $f$, we show that the solutions of are locally bounded, Hölder continuous, and satisfy a Harnack-type inequality. In this way, we provide a full extension of the De Giorgi regularity theory to this nonlocal nonlinear setting.
Motivations and applications
----------------------------
Our principal motivation for studying this regularity problem comes from the couple of papers [@CV15; @CV16] where E. Valdinoci and the author prove the existence of a particular class of minimizers for an energy similar to , connected to phase-separation phenomena. There, $p = 2$ and the term $F$ is a double-well potential essentially modeled upon the functions $$\label{Fexpl}
F_d(u) := |1 - u^2|^d, \quad \mbox{with } d > 0.$$ In local settings, potentials of this kind were considered for instance by L. Caffarelli and A. Cordoba in [@CC95; @CC06].
The strategy followed in [@CV15; @CV16] for the construction of those minimizers relies on several considerations that involve only the energy functional. The corresponding Euler-Lagrange equation is never used, if not for recovering minimal continuity properties of its solutions, via the already known regularity theory summarized in the previous subsection. This forced us there to stick with differentiable potentials, as such as $F_d$ with $d > 1$.
With the aid of the results obtained in the present paper, one could perform the same construction for a wider class of $F$’s, as for instance the whole range of explicit double-well potentials considered in .[^2]
As a matter of fact, the most adopted model in the context of phase-separation phenomena is that obtained by taking $d = 2$ in —which indeed gives a $C^\infty$ potential. This particular choice leads to the so-called fractional Allen-Cahn equation $$- (-\Delta)^s u = u - u^3,$$ and to questions related to the nonlocal version of a famous conjecture by E. De Giorgi. See [@CS05; @SV09; @CC10; @CC14; @CS15] for progresses in this direction and [@GG98; @AC00; @Sav09; @DKW11; @FV11; @FV16] for the state-of-the-art results on the conjecture in the classical case, formally obtained by taking $s = 1$. However, we believe that the analysis of the whole gamut of models given by might still be very interesting. It would be particularly important to understand the way in which the minimizers approach the pure states $\pm 1$—either asymptotically or with the formation of free-boundaries—in relation to the parameters $s$ and $d$. For $d = 2$ this has been accomplished in [@CS05; @PSV13; @CS15]. See instead [@CC06] for related results in the local case.
Of course, the regularity results provided here apply to much more general potentials $F$. Other important classes of examples are given by $$\label{Fexamples}
\begin{aligned}
F^{(1)}(u) & := \chi_{(0, +\infty)}(u),\\
F^{(2)}(u) & := \chi_{(-1, 1)}(u),\\
F^{(3)}(u) & := \lambda_1 \chi_{(-\infty, 0)}(u) + \lambda_2 \chi_{(0, +\infty)}(u),
\end{aligned}$$ with $\lambda_1 \ne \lambda_2$. All these choices lead to nonlocal variants of functionals connected to free boundary problems and often arising in fluid dynamics. In the classical case, they have been diffusely studied by H. Alt, L. Caffarelli and A. Friedman in e.g. [@AC81; @ACF84]. In nonlocal settings, functionals of this kind have been considered by L. Caffarelli, J.-M. Roquejoffre and O. Savin [@CRS10b].
Strategy of the proof
---------------------
Our approach to the proof of the regularity of minimizers of the energy introduced in follows, in its most general lines, the one developed by M. Giaquinta and E. Giusti in [@GG82] for functionals such as $\mathcal{G}_1$ in .
With the aid of Widman’s hole-filling technique [@Wid71] and of a suitable iteration lemma, the authors showed that any minimizer $u$ of ${\mathcal{G}}_1$ satisfies a family of Caccioppoli inequalities. More precisely, given any $k \in {\mathbb{R}}$, the upper truncation $(u - k)_-$ satisfies $$\label{caccintro}
\begin{aligned}
& \| \nabla (u - k)_- \|_{L^p(B_r(x_0))}^p \\
& \hspace{30pt} {\leqslant}H \left[ \frac{1}{(R - r)^p} \| (u - k)_- \|_{L^p(B_R(x_0))}^p + d^p \left| B_R(x_0) \cap \{ u < k \} \right| \right],
\end{aligned}$$ for any $x_0 \in \Omega$, any $0 < r < R < {{\mbox{\normalfont dist}}}(x_0, \partial \Omega)$ and some constants $d {\geqslant}0$, $H {\geqslant}1$. And analogously for the lower truncation $(u - k)_+$.
A slightly simpler version of was first obtained by E. De Giorgi in his pioneering work [@DeG57], where he singled out such inequality as the object encoding all the information about the Hölder continuity of the minimizers. The set of functions satisfying —and more general inequalities of the same type—is now typically called a *De Giorgi class*, in his honor.
The way one usually proceeds to prove that the elements of De Giorgi classes are Hölder continuous functions is through the application of a so-called *growth lemma*. In its most basic formulation, the growth lemma tells that there exists a small universal constant $\delta > 0$ such that $$\begin{dcases}
u \mbox{ satisfies } \eqref{caccintro},\\
u {\geqslant}0 \mbox{ in } B_R(x_0),\\
\dfrac{\left| B_R(x_0) \cap \{ u {\geqslant}1 \} \right|}{|B_R|} {\geqslant}\frac{1}{2},\\
d {\leqslant}\delta,
\end{dcases}
\quad \Longrightarrow \quad u {\geqslant}\delta \mbox{ in } B_{\frac{R}{2}}(x_0).$$ By scaling and iterating this result on concentric balls of halving radii, at each step one obtains that either the lower bound of $u$ increases of a small but universal quantity, or, thanks to a completely specular statement, the upper bound decreases of the same quantity. The quantification of this fact yields the desired Hölder continuity of $u$.
The proof of the growth lemma is usually split into two sublemmata. First, one shows that if the superlevel set $B_R(x_0) \cap \{ u {\geqslant}2 \delta \}$ occupies a large portion of the ball $B_R(x_0)$—say, having measure larger than $(1 - \eta) |B_R|$, with $\eta$ small and independent of $\delta$—, then $u {\geqslant}\delta$ in $B_{R/2}(x_0)$. At a second stage, one checks that the assumption on the size of the measure of the $2 \delta$-superlevel set is verified, provided $\delta$ is chosen small enough. This last step is essentially a consequence of the following isoperimetric inequality for level sets of Sobolev functions: $$\label{isoplevset}
\Big[ \left| B_1 \cap \{ v {\leqslant}0 \} \right| \left| B_1 \cap \{ v {\geqslant}1 \} \right| \Big]^{\frac{n - 1}{n}} {\leqslant}C \| \nabla v \|_{L^p(B_1)} \left| B_1 \cap \{ 0 < v < 1 \} \right|^{\frac{p - 1}{p}},$$ for some constant $C {\geqslant}1$ depending only on $n$ and $p$. We stress that holds for any $v \in W^{1, p}(B_1)$, not only for minimizers. This inequality is due to De Giorgi and is already contained in [@DeG57]. See also [@Giu03 Lemma 7.2 or 7.4], [@CV12] or Section \[sDGsec\] here for more elementary proofs.[^3]
In order to adapt this strategy to the minimizers of functional ${\mathcal{G}}_s$ in , we mainly need to establish two things: a Caccioppoli-type inequality and an isoperimetric lemma such as .
Caccioppoli inequalities for the solutions of have already been obtained by many authors (see for instance [@DKP14b; @KMS15a; @BP14]). In our setting, they may be written, e.g. for lower truncations, as $$\label{1scaccintro}
\begin{aligned}
& (1 - s) [(u - k)_-]_{W^{s, p}(B_r(x_0))}^p \\
& \hspace{20pt} {\leqslant}H \Bigg[ \frac{R^{(1 - s) p}}{(R - r)^p} \| (u - k)_- \|_{L^p(B_R(x_0))}^p + d^p \left| B_R(x_0) \cap \{ u < k \} \right| \\
& \hspace{20pt} \quad + (1 - s) \frac{R^{n + s p}}{(R - r)^{n + s p}} \| (u - k)_- \|_{L^1(B_R(x_0))} \int_{{\mathbb{R}}^n \setminus B_r(x_0)} \frac{(u(x) - k)_-^{p - 1}}{|x|^{n + s p}} \, dx \Bigg],
\end{aligned}$$ where, as before, we denote by $[\, \cdot \,]_{W^{s, p}(U)}$ the standard Gagliardo seminorm for fractional Sobolev functions over a measurable set $U \subseteq {\mathbb{R}}^n$, that is $$\label{GagspU}
[v]_{W^{s, p}(U)} := \left( \int_{U} \int_{U} \frac{|v(x) - v(y)|^p}{|x - y|^{n + s p}} \, dx dy \right)^{\frac{1}{p}}, \quad \mbox{for } v \in W^{s, p}(U).$$ It can be checked that inequality also holds true for the minimizers of . Notice the presence of an additional term on the third line of , that takes into account the nonlocality of the functional ${\mathcal{G}}_s$ or of the operator ${\mathcal{L}}_{s,p}$. Moreover, the constants $H$ and $d$ are independent of $s$ (the term $d$ is essentially the $L^\infty$ norm of $F$ or $F'$). Consequently, by the results of e.g. [@BBM01], we see that inequality correctly approaches , in the limit as $s \rightarrow 1^-$.
In light of these facts, one might be tempted to consider the functions that satisfy as elements of the fractional analogue of De Giorgi classes, and prove their Hölder continuity. This brings us to the second key ingredient: the De Giorgi isoperimetric inequality . As observed by L. Caffarelli and A. Vasseur [@CV12] in the case $p = 2$, formula *“may be considered as a quantitative version of the fact that a function with jump discontinuity cannot be in $H^1$”*. But functions with such discontinuity features may well belong to fractional Sobolev spaces: for instance, characteristic functions of sets with smooth boundaries are in $W^{s, p}$, if $s p < 1$. In fact, they play a central role in the theory of nonlocal perimeters recently developed by L. Caffarelli, J.-M. Roquejoffre and O. Savin in [@CRS10a]. Therefore, the hopes of finding an appropriate generalization of to fractional Sobolev spaces are low, at least for small $s$.
While for $s$ close to $1$ we are able to partially extend inequality to $W^{s, p}$ (see Proposition \[sDGlemprop\]) and therefore reproduce the strategy outlined before with no substantial modifications—thus proving a regularity result which is uniform as $s \rightarrow 1^-$—, the case of a small $s$ seems to require a different approach. Indeed, in this situation one needs to find an inequality providing the same information of , and holding at least for the minimizers of ${\mathcal{G}}_s$ and the solutions of , instead of all functions in a Sobolev space. Inspired by [@CCV11], where the authors develop a regularity theory for parabolic equations driven by nonlocal operators with general kernels, we propose the following definition: a function $u$ belongs to a *fractional De Giorgi class* in a bounded domain $\Omega$ if and only if it satisfies the improved Caccioppoli inequality $$\label{2scaccintro}
\begin{aligned}
& (1 - s) \Bigg[ [(u - k)_-]_{W^{s, p}(B_r(x_0))}^p + \iint_{B_r(x_0)^2} \frac{(u(y) - k)_+^{p - 1} (u(x) - k)_-}{|x - y|^{n + s p}} \, dx dy \Bigg] \\
& \hspace{20pt} {\leqslant}H \Bigg[ \frac{R^{(1 - s) p}}{(R - r)^p} \| (u - k)_- \|_{L^p(B_R(x_0))}^p + d^p \left| B_R(x_0) \cap \{ u < k \} \right| \\
& \hspace{20pt} \quad + (1 - s) \frac{R^{n + s p}}{(R - r)^{n + s p}} \| (u - k)_- \|_{L^1(B_R(x_0))} \int_{{\mathbb{R}}^n \setminus B_r(x_0)} \frac{(u(x) - k)_-^{p - 1}}{|x|^{n + s p}} \, dx \Bigg],
\end{aligned}$$ for any $k \in {\mathbb{R}}$, $x_0 \in \Omega$, $0 < r < R < {{\mbox{\normalfont dist}}}(x_0, \partial \Omega)$, and similarly for the upper truncations of $u$.
Observe that, unlike in , we have now two terms on the left-hand side. The newly added quantity $$\label{addquantity}
(1 - s) \iint_{B_r(x_0)^2} \frac{(u(y) - k)_+^{p - 1} (u(x) - k)_-}{|x - y|^{n + s p}} \, dx dy,$$ is precisely the one that carries with itself all the information that we are missing not having an inequality as at hand. Indeed, when e.g. $k = 1/2$, one can bound this term from below by $$\frac{1 - s}{C r^{n + s p}} \, |B_r(x_0) \cap \{ u {\leqslant}0 \}| |B_r(x_0) \cap \{ u {\geqslant}1 \}|,$$ with $C {\geqslant}1$ depending only on $n$ and $p$. The above quantity is similar to the one appearing on the left-hand side of . For functions in fractional De Giorgi classes, it can be bounded in terms of the right-hand side of , and this fact is the much needed replacement for inequality .
In addition, it is not hard to show that, for instance when $u \in W^{1, p}(B_r(x_0))$, the quantity in vanishes as $s \rightarrow 1^-$. Therefore, our definition of fractional De Giorgi classes as given by inequality is consistent with the classical notion, in the limit as $s \rightarrow 1^-$.
The main goal of the paper is to prove that the elements of fractional De Giorgi classes are locally bounded and Hölder continuous functions. Once we have this, the problem of establishing regularity for the minimizers of the functional ${\mathcal{G}}_s$ defined in and the solutions of equation is reduced to show that these critical points belong to those classes.
In the next section we give the definitions of the objects that we take into consideration, in their greater generality, and we present the rigorous statements of the results already discussed up to here for the simplified model governed by the energy functional .
Precise formulation of the setting and of the main results {#mainsec}
==========================================================
Let $n \in {\mathbb{N}}$, $s \in (0, 1)$ and $p \in (1, +\infty)$ be fixed parameters.
Let $K: {\mathbb{R}}^n \times {\mathbb{R}}^n \to [0, +\infty]$ be a measurable function satisfying $$\label{Ksimm}
K(x, y) = K(y, x) \quad \mbox{for a.a.~} x, y \in {\mathbb{R}}^n,$$ and $$\label{Kell}
\frac{(1 - s) \chi_{B_{r_0}}(x - y)}{\Lambda |x - y|^{n + s p}} {\leqslant}K(x, y) {\leqslant}\frac{(1 - s) \Lambda}{|x - y|^{n + s p}} \quad \mbox{for a.a.~} x, y \in {\mathbb{R}}^n,$$ for some $\Lambda {\geqslant}1$ and $r_0 > 0$.
Let $F: {\mathbb{R}}^n \times {\mathbb{R}}\to {\mathbb{R}}$ be a measurable function such that the composition $F \circ v$ is measurable, for any given measurable function $v: {\mathbb{R}}^n \to {\mathbb{R}}$.[^4] For any measurable set $\Omega \subseteq {\mathbb{R}}^n$ and measurable function $u: {\mathbb{R}}^n \to {\mathbb{R}}$, we consider the energy functional $$\label{Edef}
{\mathcal{E}}(u; \Omega) := \frac{1}{2p} \iint_{{\mathscr{C}}_\Omega} \left| u(x) - u(y) \right|^p K(x, y) \, dx dy + \int_{\Omega} F(x, u(x)) \, dx,$$ where $$\label{COmega}
{\mathscr{C}}_\Omega := {\mathbb{R}}^{2 n} \setminus \left( {\mathbb{R}}^n \setminus \Omega \right)^2.$$
Note that, if one takes as $K$ the standard kernel $$\label{K0}
K_0(x, y) := \frac{1 - s}{|x - y|^{n + s p}},$$ then ${\mathcal{E}}$ is the energy ${\mathcal{G}}_s$ considered in the introduction, up to a negligible factor. However, hypothesis allows for a much richer variety of kernels. Indeed, we can equivalently rewrite as $$K(x, y) = (1 - s) \frac{a(x, y)}{|x - y|^{n + s p}}, \quad \mbox{with } \, \frac{\chi_{B_{r_0}(x - y)}}{\Lambda} {\leqslant}a(x, y) {\leqslant}\Lambda.$$ Then one can choose for instance $$\label{aexamples}
\begin{array}{rll}
& a(x, y) = a_0 \left( \dfrac{x - y}{|x - y|} \right), & \mbox{with } a_0: S^{n - 1} \to \left[ \dfrac{1}{\Lambda}, \Lambda \right],\\
& a(x, y) = a_0 \left( x - y \right), & \mbox{with } a_0: {\mathbb{R}}\to \left[ \dfrac{1}{\Lambda}, \Lambda \right],\\
\mbox{or} & a(x, y) = \chi_{D}(x - y), & \mbox{with } D \subset {\mathbb{R}}^n, \mbox{ such that } B_{r_0} \subseteq D \subseteq B_{r_1}, \rule{0pt}{16pt}
\end{array}$$ with $0 < r_0 {\leqslant}r_1$, and even some more general non-translation-invariant kernels. The last possibility in , in particular, yields a truncated kernel.
We now introduce the concept of minimizers of ${\mathcal{E}}$ that we shall work with. Before, we need a few definitions of the functional spaces involved.
We denote by $L_s^{p - 1}({\mathbb{R}}^n)$ the space composed by the functions $u: {\mathbb{R}}^n \to {\mathbb{R}}$ that satisfy $$\int_{{\mathbb{R}}^n} \frac{|u(x)|^{p - 1}}{\left( 1 + |x| \right)^{n + s p}} \, dx < +\infty.$$ An object that will play an important role in encoding the behavior of functions in $L_s^{p - 1}({\mathbb{R}}^n)$ at large scales is the so-called *Tail* defined by $$\label{Tailudef}
\operatorname{Tail}(u; x_0, R) := \left[ (1 - s) R^{s p} \int_{{\mathbb{R}}^n \setminus B_R(x_0)} \frac{|u(x)|^{p - 1}}{|x - x_0|^{n + s p}} \, dx \right]^{\frac{1}{p - 1}},$$ for any fixed $x_0 \in {\mathbb{R}}^n$ and $R > 0$. The above scale-invariant quantity has already been considered in several papers, such as [@DKP14a; @DKP14b; @BP14; @KMS15b]. Observe in particular that it is finite, provided $u \in L^{p - 1}_s({\mathbb{R}}^n)$.
Another functional space that we will need is a modified fractional Sobolev space. Given an open set $\Omega \subseteq {\mathbb{R}}^n$, we denote as ${\mathbb{W}}^{s, p}(\Omega)$ the space of measurable functions $u: {\mathbb{R}}^n \to {\mathbb{R}}$ such that $$u|_\Omega \in L^p(\Omega) \quad \mbox{and} \quad (x, y) \longmapsto \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \in L^1({\mathscr{C}}_\Omega),$$ with ${\mathscr{C}}_\Omega$ as in . In the quadratic case $p = 2$, spaces of this kind have been for instance considered in [@SV13; @SV14]. Note that $W^{s, p}({\mathbb{R}}^n) \subseteq {\mathbb{W}}^{s, p}(\Omega) \subseteq W^{s, p}(\Omega)$.
We can now proceed to define the minimizers of the energy functional ${\mathcal{E}}$ in .
\[mindef\] Let $\Omega \subset {\mathbb{R}}^n$ be a bounded open set. A function $u \in {\mathbb{W}}^{s, p}(\Omega)$ is said to be a *minimizer* of ${\mathcal{E}}$ in $\Omega$ if $F(\cdot, u) \in L^1(\Omega)$ and $${\mathcal{E}}(u; \Omega) {\leqslant}{\mathcal{E}}(v; \Omega),$$ for any measurable function $v: {\mathbb{R}}^n \to {\mathbb{R}}$ such that $v = u$ a.e. in ${\mathbb{R}}^n \setminus \Omega$.
If $u \in {\mathbb{W}}^{s, p}(\Omega)$ is such that $F(\cdot, u) \in L^1(\Omega)$, then the energy ${\mathcal{E}}(u; \Omega)$ is finite, thanks to . Therefore, Definition \[mindef\] is meaningful.
On top of this, notice that $F(\cdot, u)$ is bounded, and thus integrable, whenever $u$ is bounded and $F$ is locally bounded in $u$, uniformly with respect to $x \in \Omega$. When $n < s p$, this is true in particular for any $u \in W^{s, p}(\Omega)$, thanks to the fractional Sobolev embedding (see e.g. [@DPV12]). On the contrary, when $n > sp$, we will often requre $F$ to satisfy $$\label{Fbounds}
|F(x, u)| {\leqslant}d_1 + d_2 |u|^q \quad \mbox{for a.a.~} x \in \Omega \mbox{ and any } u \in {\mathbb{R}},$$ with $1 {\leqslant}q < p^*_s$, where $p^*_s$ is the fractional Sobolev exponent given by $$\label{p*sdef}
p^*_s := \frac{n p}{n - s p}.$$ When $n = s p$, we simply assume $F$ to satisfy for some $1 {\leqslant}q < +\infty$, that is, we formally set $p^*_s := +\infty$ in such case. Note that, under , we have $F(\cdot, u) \in L^1(\Omega)$ for any $u \in W^{s, p}(\Omega)$.
In parallel to the energy ${\mathcal{E}}$ considered in and its minimizers, we take into account weak solutions to equations driven by integral operators with kernel $K$. For $K$ satisfying and , we formally define $$\begin{aligned}
{\mathcal{L}}u(x) := & \, {\mbox{\normalfont P.V.}}\int_{{\mathbb{R}}^n} |u(x) - u(y)|^{p - 2} (u(x) - u(y)) K(x, y) \, dy \\
= & \, \lim_{\delta \rightarrow 0^+} \int_{{\mathbb{R}}^n \setminus B_\delta(x)} |u(x) - u(y)|^{p - 2} (u(x) - u(y)) K(x, y) \, dy.\end{aligned}$$ When $K$ is of the form , we recover the operator ${\mathcal{L}}_{s, p}$ defined in the introduction and, in particular, the fractional Laplacian $(-\Delta)^s$ if $p = 2$.
Let $f: {\mathbb{R}}^n \times {\mathbb{R}}\to {\mathbb{R}}$ be a measurable function such that the composition $f(\cdot, v)$ is measurable whenever $v: {\mathbb{R}}^n \to {\mathbb{R}}$ is measurable. We are interested in studying weak solutions of the equation $$\label{Lu=f}
- {\mathcal{L}}u = f(\cdot, u),$$ in bounded domains of ${\mathbb{R}}^n$. Notice that, when $F$ is differentiable $u$, this is the Euler-Lagrange equation of ${\mathcal{E}}$, with $f = F_u$.
The precise notion of weak solution of that we take into account is as follows.
\[soldef\] Let $\Omega \subset {\mathbb{R}}^n$ be a bounded open set. A function $u \in {\mathbb{W}}^{s, p}(\Omega)$ is said to be a *weak solution* of in $\Omega$ if $$\begin{gathered}
- \frac{1}{2} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) K(x, y) \, dx dy \\
= \int_{{\mathbb{R}}^n} f(x, u(x)) \varphi(x) \, dx,\end{gathered}$$ for any $\varphi \in W^{s, p}({\mathbb{R}}^n)$ with $\operatorname{supp}(\varphi) \subset \subset \Omega$ and such that the right-hand side above is well-defined.
We have been rather sloppy in our definition of weak solutions, with respect to the choice of test functions that make the right-hand side of the identity written above converge. A sufficient condition to have it well-defined for any $\varphi \in W^{s, p}(\Omega)$ is that $f(\cdot, u) \in L^{(p^*_s)'}(\Omega)$ when $n > sp$, and simply $f(\cdot, u) \in L^1(\Omega)$ when $n < sp$, thanks to the fractional Sobolev embeddings. In the case $n < s p$, in particular, this last requirement on $f(\cdot, u)$ is fulfilled whenever $f$ is locally bounded in $u \in {\mathbb{R}}$, uniformly w.r.t. $x \in \Omega$, by employing the Sobolev embedding once again. On the other hand, when $n {\geqslant}s p$ we will almost always ask that $$\label{fbounds}
|f(x, u)| {\leqslant}d_1 + d_2 |u|^{q - 1} \quad \mbox{for a.a.~} x \in \Omega \mbox{ and any } u \in {\mathbb{R}},$$ for some $1 < q < p^*_s$, similarly to what we did in (again, with the understanding that $p^*_s = +\infty$ if $n = s p$). We remark that, with this choice, $f(\cdot, u) \varphi \in L^1(\Omega)$ for any $\varphi \in W^{s, p}(\Omega)$. We also notice that $f(\cdot, u) \varphi \in L^1(\Omega)$ for all such $\varphi$’s if $f$ is locally bounded in $u$, uniformly w.r.t. $x$, and $u$ is bounded, regardless to the values of $n$, $s$ and $p$.
After all these necessary premises, we can now proceed to state our main contributions to the regularity theory for the minimizers of ${\mathcal{E}}$ and the solutions of .
First, we have the following result concerning the local boundedness of these critical points. Of course, we can restrict ourselves to take $n {\geqslant}s p$, as otherwise the boundedness is warranted by the fractional Sobolev embedding.
\[boundmainthm\] Let $n \in {\mathbb{N}}$, $s \in (0, 1)$ and $p > 1$ be such that $n {\geqslant}s p$. Let $\Omega$ be an open bounded subset of ${\mathbb{R}}^n$. Suppose that the kernel $K$ satisfies hypotheses and . Let $u \in L^{p - 1}_s({\mathbb{R}}^n) \cap {\mathbb{W}}^{s, p}(\Omega)$ be either
1. a minimizer of ${\mathcal{E}}$ in $\Omega$, with $F$ satisfying , or
2. a weak solution of in $\Omega$, with $f$ satisfying .
Then, $u \in L^\infty_{{\rm loc}}(\Omega)$. In particular, for any $x_0 \in \Omega$ and $0 < R < {{\mbox{\normalfont dist}}}(x_0, \partial \Omega) / 2$, it holds $$\| u \|_{L^\infty(B_R(x_0))} {\leqslant}C,$$ for some constant $$C = C \left( n, s, p, q, \Lambda, d_1, d_2, R, r_0, \| u \|_{L^p(B_{2 R}(x_0))}, \operatorname{Tail}(u; x_0, 2 R), \| u \|_{L^\lambda(\Omega)} \right),$$ and $\lambda > p$. When $n > s p$, we can take $\lambda = p^*_s$, while when $n > p$, the constant $C$ does not blow up as $s \rightarrow 1^-$.
The estimate of Theorem \[boundmainthm\] is given in terms of an implicit constant $C$ that is meant to depend *at most* on the parameters listed. Indeed, in many specific cases (such as when $d_2 = 0$ or $1 {\leqslant}q {\leqslant}p$) the constant $C$ may be chosen to depend on fewer quantities. We refer the reader to Theorems \[minboundthm\] and \[solboundthm\]—respectively for minimizers and solutions—for a more detailed account of the dependencies of $C$.
We point out that, at least when $n > p$, the estimate provided in Theorem \[boundmainthm\] is independent of $s$, for $s$ close to $1$. This is not surprising at all, in view of the normalization of the kernel $K$ by means of the factor $(1 - s)$, implied by . Indeed, this is consistent with the fact that, for certain choices of kernels, the energy ${\mathcal{E}}$ approaches, as $s \rightarrow 1^-$, a local functional driven by a gradient-type Dirichlet term, such as ${\mathcal{G}}_1$ in . And similarly for the operator ${\mathcal{L}}$.
Theorem \[boundmainthm\] has been obtained in [@DKP14b] for the case of solutions of an equation like . In [@DKP14b], the result is stated assuming the right-hand side $f$ to be zero, although the techniques displayed there should apply also to more general situations. This is true, since the boundedness of $u$ may be recovered right from a standard Caccioppoli inequality of the form , and does not require its improved variant . We refer the interested reader to the proof of Proposition \[ulocboundprop\] for a verification of this fact.
Other results related to Theorem \[boundmainthm\] can be found for instance in [@KMS15b; @IMS14; @LPPS15; @BPV15].
Next is the main contribution of the present paper, ensuring the Hölder continuity of the minimizers of ${\mathcal{E}}$ and of the solutions of . Again, we only deal with the case $n {\geqslant}s p$. Moreover, by virtue of the boundedness result of Theorem \[boundmainthm\], we can now simply assume the potential $F$ and the right-hand side $f$ to be locally bounded functions.
The statement of the Hölder continuity result is as follows.
\[holmainthm\] Let $n \in {\mathbb{N}}$, $0 < s_0 {\leqslant}s < 1$ and $p > 1$ be such that $n {\geqslant}s p$. Let $\Omega$ be an open bounded subset of ${\mathbb{R}}^n$. Suppose that the kernel $K$ satisfies hypotheses and . Let $u \in L^{p - 1}_s({\mathbb{R}}^n) \cap {\mathbb{W}}^{s, p}(\Omega)$ be either
1. a minimizer of ${\mathcal{E}}$ in $\Omega$, with $F$ locally bounded in $u$, uniformly w.r.t. $x \in \Omega$, or
2. a weak solution of in $\Omega$, with $f$ locally bounded in $u$, uniformly w.r.t. $x \in \Omega$.
Then, $u \in C^\alpha_{{\rm loc}}(\Omega)$, for some $\alpha \in (0, 1)$. In particular, there exists a constant $C {\geqslant}1$ such that, for any $x_0 \in \Omega$ and $0 < R < \min \{ r_0, {{\mbox{\normalfont dist}}}(x_0, \partial \Omega) \} / 4$, it holds $$[u]_{C^\alpha(B_R(x_0))} {\leqslant}\frac{C}{R^\alpha} \Big( \| u \|_{L^\infty(B_{2 R}(x_0))} + \operatorname{Tail}(u; x_0, 2 R) + {\mathcal{F}}\Big),$$ where $$\label{Fdef}
{\mathcal{F}}:= \begin{dcases}
R^s \| F(\cdot, u) \|_{L^\infty(B_{2 R}(x_0))}^{1 / p} \vphantom{\frac{0}{0}} & \quad \mbox{if } (a) \mbox{ is in force}, \\
R^{\frac{s p}{p - 1}} \| f(\cdot, u) \|_{L^\infty(B_{2 R}(x_0))}^{1 / (p - 1)} \vphantom{\frac{0}{0}} & \quad \mbox{if } (b) \mbox{ is in force}.
\end{dcases}$$ The constants $\alpha$ and $C$ depend only on $n$, $s_0$, $p$ and $\Lambda$.
Analogously to Theorem \[boundmainthm\], the quantities determining the Hölder character of, say, a minimizer of ${\mathcal{E}}$ stay bounded as $s$ goes to $1$. Again, this is consistent with the local scenario (formally represented by the choice $s = 1$) where such results were proved in [@GG82].
In the case of a solution $u$ of , estimates like the one established in Theorem \[holmainthm\] have been obtained in [@DKP14b], for $f = 0$, and in [@IMS14], for Dirichlet problems driven by the specific operator ${\mathcal{L}}_{s, p}$ defined in the introduction (i.e. the operator ${\mathcal{L}}$ with kernel $K$ given by ). When $p = 2$, the literature is richer: similar results have been obtained in [@BK05; @Kas07a; @Kas09; @Kas11] and [@Sil06; @CS09; @CS11]. We also mention [@KMS15b], where the authors show the continuity of $u$ under very mild hypotheses on the right-hand side $f$. Their potential theoretic approach should also yield an Hölder modulus of continuity for $u$, once $f$ is chosen sufficiently regular.
As pointed out in the introduction, Theorems \[boundmainthm\] and \[holmainthm\] are, to the best of our knowledge, completely new for minimizers of ${\mathcal{E}}$, even if $p = 2$.
The verification of Theorem \[holmainthm\] is split between Section \[minsec\]—for minimizers—and Section \[solsec\]—for solutions. The $C^\alpha$ estimate in the two different situations is respectively given by Theorem \[minholdthm\] in Section \[minsec\] and Theorem \[solholdthm\] in Section \[solsec\]. We remark that the statements of these two results partially differ from that of Theorem \[holmainthm\], with respect to some limitations on the radius $R$. As for Theorem \[boundmainthm\] and Theorems \[minboundthm\]-\[solboundthm\], the result stated right above can be easily recovered from those of Sections \[minsec\]-\[solsec\] with the help of a straightforward covering argument.
A key ingredient of the proof of Theorem \[holmainthm\] is the improved Caccioppoli inequality . With the aid of this estimate—that holds, in a sometimes slightly weaker form, for both solutions and minimizers—we are able to prove a growth lemma, the crucial step for the Hölder continuity. In particular, we use the bound for the second member on the left-hand side of to replace the De Giorgi isoperimetric-type inequality , which may fail in the context of fractional Sobolev spaces.
Observe once again that the estimate provided in Theorem \[holmainthm\] is uniform in $s$, at least when $s$ is bounded away from $0$. On the other hand, the information represented by the upper bound for the second term on the left-hand side of tends to disappear as $s$ approaches $1$. As a result, one would naturally expect Hölder estimates which blow up as $s \rightarrow 1^-$. To resolve this apparent inconsistency, in Proposition \[sDGlemprop\] we obtain an estimate in the spirit of for functions belonging to (large regions of) the fractional Sobolev space $W^{s, p}$, with $s$ sufficiently close to $1$. The interpolation of this result with inequality yields $C^\alpha$ estimates uniform in $s$.
The last result that we present in this section is a Harnack-type inequality. Note that here we do not limit ourselves to $n {\geqslant}s p$, as the result is now meaningful for the full range of parameters. On the other hand, in place of , we take into account the following slightly more restrictive hypothesis on $K$: $$\label{Kell2}
\frac{1 - s}{\Lambda |x - y|^{n + s p}} {\leqslant}K(x, y) {\leqslant}\frac{(1 - s) \Lambda}{|x - y|^{n + s p}} \quad \mbox{for a.a.~} x, y \in {\mathbb{R}}^n,$$ with $\Lambda {\geqslant}1$. Observe that differs from in that the left-hand inequality—i.e. the ellipticity assumption on $K$—is now required to hold everywhere, instead of only in a neighborhood of the diagonal $\{ x = y\}$. Hypothesis formally corresponds to with $r_0 = +\infty$.
The statement of the Harnack inequality may now follow.
\[harmainthm\] Let $n \in {\mathbb{N}}$, $s \in (0, 1)$ and $p > 1$. Let $\Omega \subset {\mathbb{R}}^n$ be an open bounded subset of ${\mathbb{R}}^n$. Suppose that $K$ satisfies hypotheses and . Let $u$, $F$ and $f$ be as in Theorem \[holmainthm\], and assume in addition that $u {\geqslant}0$ in $\Omega$. Then, there exists a constant $C {\geqslant}1$ such that, for any $x_0 \in \Omega$ and $0 < R < {{\mbox{\normalfont dist}}}(x_0, \partial \Omega) / 2$, it holds $$\sup_{B_R(x_0)} u {\leqslant}C \left( \inf_{B_R(x_0)} u + \operatorname{Tail}(u_-; x_0, R) + {\mathcal{F}}\right),$$ with ${\mathcal{F}}$ as in . The constant $C$ depends only on $n$, $s$, $p$ and $\Lambda$. When $n \notin \{ 1, p \}$, the constant $C$ does not blow up as $s \rightarrow 1^-$.
Harnack inequalities for solutions to integral equations like have been obtained in [@Kas11; @DKP14a], both considering the homogeneous case (i.e. with no right-hand side). For minimizers, this is the first available result in this direction.
When comparing Theorem \[harmainthm\] to the classical Harnack inequalities for second-order partial differential equations (see e.g. [@M61; @DT84]), we immediately notice the presence here of an additional Tail term. As noted in [@Kas07b; @Kas11], this is the correct formulation of the Harnack inequality for nonlocal operators. Of course, when $u$ is non-negative on the whole of ${\mathbb{R}}^n$, one recovers the classical inequality.
Moreover, as the radius $R$ of the ball $B_R(x_0)$ over which the inequality is set can be chosen freely (as long as $B_{2 R}(x_0)$ is contained in $\Omega$), this is a global inequality. As a consequence, one can deduce from it a Liouville-type theorem for entire solutions of ${\mathcal{L}}u = 0$ which are bounded from above or below.
We do not know whether or not the stronger assumption , in place of , is necessary for the validity of Theorem \[harmainthm\]. The only place where we take full advantage of is in Theorem \[DGharthm\]: it is used to deduce formula , which gives a bound for the Tail of the positive part of $u$. We believe it to be an interesting problem to understand if a similar Harnack inequality might be obtained for more general kernels. To this aim, it would possibly be convenient to take into account a Tail term tailored on the kernel $K$ under consideration, rather than the canonical choice given by .
As anticipated in the introductory section, we obtain Theorems \[boundmainthm\], \[holmainthm\] and \[harmainthm\] by showing that the minimizers of ${\mathcal{E}}$ and the solutions of equally satisfy an improved Caccioppoli inequality of the form . We consider the set of all functions fulfilling and more general inequalities—which we call a *fractional De Giorgi class*—and prove that they are locally bounded, Hölder continuous and that, when non-negative, they satisfy a Harnack inequality.
In light of this, the present paper extends various results and techniques displayed in the classical references [@DeG57; @LU68; @GG82; @DT84; @Giu03] to a nonlocal setting.
The remaining part of the paper is organized as follows.
First, in Section \[notsec\] we fix some terminology that is often adopted in the paper.
In the preparatory Section \[prepsec\] we include a collection of numerical and functional inequalities that will be largely used in the subsequent sections.
Section \[sDGsec\] is devoted to the proof of a De Giorgi isoperimetric-type inequality for the level sets of functions that belong to fractional Sobolev spaces with large differentiability order $s$.
In Section \[DGsec\] we introduce fractional De Giorgi classes in the full generality needed for our applications. There, we also show that their elements are locally bounded, Hölder continuous functions that satisfy an Harnack-type theorem. These three facts are proved in Theorems \[DGboundthm\], \[DGholdthm\] and \[DGharthm\].
The conclusive Sections \[minsec\] and \[solsec\] contain the proofs of Theorems \[boundmainthm\], \[holmainthm\], \[harmainthm\]: these results are restated as Theorems \[minboundthm\], \[minholdthm\], \[minharthm\] for minimizers, in Section \[minsec\], and as Theorems \[solboundthm\], \[solholdthm\], \[solharthm\] for solutions, in Section \[solsec\].
Notation {#notsec}
========
In this brief section, we formally specify some of the notation that will be used more frequently in the remainder of the paper.
First of all, the dimension of the space in which we are set is always indicated by $n$, which is normally meant to be any natural number.
As we did in the two previous sections, we denote by $B_R(x_0)$ the open Euclidean ball of radius $R > 0$, centered at $x_0 \in {\mathbb{R}}^n$. That is, $$B_R(x_0) := \Big\{ x \in {\mathbb{R}}^n : |x - x_0| < R \Big\}.$$ When $x_0$ is the origin, we simply write $B_R$ in place of $B_R(0)$.
We use the symbol $\chi_\Omega$ to indicate the characteristic function of a set $\Omega \subseteq {\mathbb{R}}^n$, i.e. $$\chi_\Omega(x) := \begin{cases}
1 & \quad \mbox{if } x \in \Omega,\\
0 & \quad \mbox{if } x \in {\mathbb{R}}^n \setminus \Omega.
\end{cases}$$
For any two given parameters $s \in (0, 1)$, $p > 1$ and any measurable set $U \subseteq {\mathbb{R}}^n$, we have already introduced the fractional Sobolev space $W^{s, p}(U)$ as the subset of $L^p(U)$ made up by those functions that have finite Gagliardo seminorm $[\, \cdot \,]_{W^{s, p}(U)}$, as given by . For $n > s p$, the important fractional Sobolev exponent $p^*_s$ has been defined in . In section \[mainsec\], we also considered the modified Sobolev space ${\mathbb{W}}^{s, p}(U)$ and the weighted Lebesgue space $L^{p - 1}_s({\mathbb{R}}^n)$. For $u \in L^{p - 1}_s({\mathbb{R}}^n)$ and any $x_0 \in {\mathbb{R}}^n$, $R > 0$, we saw that the quantity $\operatorname{Tail}(u; x_0, R)$, as in , is well-defined and finite. For some later purposes, it is convenient to introduce also the related non-scaling-invariant Tail term $$\label{nsiTailudef}
\begin{aligned}
\overline{\operatorname{Tail}}(u; x_0, R) := & \, \left[ (1 - s) \int_{{\mathbb{R}}^n \setminus B_R(x_0)} \frac{|u(x)|^{p - 1}}{|x - x_0|^{n + s p}} \, dx \right]^{\frac{1}{p - 1}} \\
= & \, R^{- \frac{s p}{p - 1}} \, \operatorname{Tail}(u; x_0, R).
\end{aligned}$$
We adopt a short-hand notation for the level sets of functions. Given $u: {\mathbb{R}}^n \to {\mathbb{R}}$ and $k \in {\mathbb{R}}$, we denote the superlevel set of $u$ of level $k$ as $$\{ u > k \} := \Big\{ x \in {\mathbb{R}}^n : u(x) > k \Big\}.$$ Similarly, we write $\{ u < k \}$ for the sublevel set $\{ x \in {\mathbb{R}}^n : u(x) < k \}$. The other notations $\{ u = k \}$, $\{ u {\geqslant}k \}$ and $\{ u {\leqslant}k \}$ all have analogous meanings.
As it is customary, the positive and negative parts of a function (or a real number) $u$ are indicated by $u_+$ and $u_-$, respectively. This means that we have $u_+ := \max \{ u, 0 \}$ and $u_- := - \min \{ u, 0 \}$.
Of particular interest are also the lower truncation $(u - k)_+$ and the upper truncation $(u - k)_-$ of a function $u$ at level $k \in {\mathbb{R}}$. We often refer to their supports as $$\label{A+-def}
\begin{aligned}
A^+(k) := \operatorname{supp}((u - k)_+) = \{ u > k \}, \\
A^-(k) := \operatorname{supp}((u - k)_-) = \{ u < k \}.
\end{aligned}$$ The intersections of these sets with the ball $B_R(x_0)$ are denoted by $A^+(k, x_0, R)$ and $A^-(k, x_0, R)$, respectively. As before, we drop reference to $x_0$ when it is the origin, and simply write $A^+(k, R)$ and $A^-(k, R)$.
In Sections \[minsec\] and \[solsec\], we will frequently consider the measure element $$\label{dmudef}
d\mu = d\mu_K(x, y) := K(x, y) \, dx dy.$$ This terminology is used for the sole purpose of abbreviating several integral formulas.
Finally, we remark that we use several letters (roman or greek characters, in upper or lower cases) to denote constants and parameters. Sometimes—as in Theorem \[boundmainthm\] or Proposition \[minareDGprop\]—we write the quantities on which some constant depends between round brackets, right after the symbol used for said constant. We always use the letter $C$ to denote a general constant, greater or equal to $1$. The value of $C$ may change within the same statement, proof or even between different lines of the same formula. During proofs, we usually specify on which parameters a certain constant $C$ depends as soon as it appears in a formula; eventual other occurrences of $C$ in the same proof are supposed to depend on the same exact parameters, unless otherwise specified. When we need to be more precise on the value of some particular occurrence, we use subscripts, such as $C_\star, C_\sharp, C_1, C_2$, etc.
Some auxiliary results {#prepsec}
======================
Here we present several ancillary lemmata that will be used in the remainder of the paper. For their technical nature and rather general applicability, we preferred to collect them in this separate section.
The first four results are standard numerical inequalities. Most of them are probably well-known to the reader or very easy to be obtained. For the sake of completeness, we include their proofs in full details.
\[numestlem1\] Let $p {\geqslant}1$ and $a, b {\geqslant}0$. Then, $$\label{numest1}
(a + b)^p - a^p {\geqslant}\theta p a^{p - 1} b + (1 - \theta) b^p,$$ for any $\theta \in [0, 1]$.
Of course, it is enough to prove for $\theta = 0, 1$. Indeed, the case $\theta = 0$ plainly follows from the standard fact that the $p$-norm $|x|_p$ is monotone non-increasing in $p \in [1, +\infty)$, for any fixed $x \in {\mathbb{R}}^2$. On the other hand, $$(a + b)^p - a^p = p \int_a^{a + b} t^{p - 1} \, dt {\geqslant}p a^{p - 1} b,$$ which is for $\theta = 1$.
We observe that, when $p {\geqslant}2$, a simpler and stronger inequality holds true. Essentially, in this case one can replace both coefficients $\theta$ and $1 - \theta$ on the right-hand side of with $1$. However, for our applications the interpolation inequality of Lemma \[numestlem1\] will suffice.
Next are other three lemmata providing numerical estimates.
\[numestlem2\] Let $p {\geqslant}1$, $\mu \in [0, 1]$ and $a, b {\geqslant}0$. Then, $$\label{numest2}
|\mu a - b|^p - |a - b|^p {\leqslant}p b^{p - 1} a.$$
We consider separately the three possibilities $b {\geqslant}a$, $\mu a {\leqslant}b < a$ and $b < \mu a$. In the first case, $$|\mu a - b|^p - |a - b|^p = (b - \mu a)^p - (b - a)^p = p \int_{b - a}^{b - \mu a} t^{p - 1} \, dt {\leqslant}p b^{p - 1} a.$$ On the other hand, if $\mu a {\leqslant}b < a$, then $$\begin{aligned}
|\mu a - b|^p - |a - b|^p & = (b - \mu a)^p - (a - b)^p = p \int_{a - b}^{b - \mu a} t^{p - 1} \, dt \\
& {\leqslant}p (b - \mu a)^{p - 1} (2 b - (1 + \mu) a) {\leqslant}p b^{p - 1} a.\end{aligned}$$ Finally, when $b < \mu a$ the thesis is trivially verified, as the left-hand side of is negative.
\[numestlem3\] Let $p > 1$ and $a {\geqslant}b {\geqslant}0$. Then, $$a^p - b^p {\leqslant}\varepsilon a^p + \left( \frac{p - 1}{\varepsilon} \right)^{p - 1} (a - b)^p,$$ for any $\varepsilon > 0$.
We compute $$a^p - b^p = \left( b + (a - b) \right)^p - b^p = p \int_0^{a- b} (b + t)^{p - 1} \, dt {\leqslant}p a^{p - 1} (a - b).$$ For a fixed $\delta > 0$, we use Young’s inequality to deduce $$a^p - b^p {\leqslant}p \left( \delta^{\frac{1}{p}} a \right)^{p - 1} \left( \frac{a - b}{\delta^{\frac{p - 1}{p}}} \right) {\leqslant}(p - 1) \delta a^p + \delta^{1 - p} (a - b)^p,$$ and the conclusion follows by taking $\delta = \varepsilon / (p - 1)$.
\[numestlem4\] Let $p > 1$, $a \in {\mathbb{R}}$ and $b {\geqslant}0$. Then, $$\label{numest4}
(a - b)_+^{p - 1} {\geqslant}\min \{ 1, 2^{2 - p} \} a_+^{p - 1} - b^{p - 1}.$$
We consider separately the three possibilities $a {\geqslant}b$, $0 {\leqslant}a < b$ and $a < 0$. If $a {\geqslant}b$, it easy to see that $$(a - b)^{p - 1} + b^{p - 1} {\geqslant}\min \{ 1, 2^{2 - p} \} a^{p - 1},$$ which is . If $0 {\leqslant}a < b$, then $$(a - b)_+^{p - 1} - \min \{ 1, 2^{2 - p} \} a_+^{p - 1} = - \min \{ 1, 2^{2 - p} \} a^{p - 1} {\geqslant}- b^{p - 1},$$ and follows as well. In the case $a < 0$, inequality is also trivially true.
The next six results contain well-know functional inequalities in fractional Sobolev spaces. Our estimates are usually minor modifications of those available in the literature. We write them here in order to keep better track of the values of the constants involved and to have them ready for applications in the subsequent sections.
First is a weighted estimate related to the embeddings of fractional Sobolev spaces as the differentiability order varies.
\[sobinclem0\] Let $n \in {\mathbb{N}}$, $p {\geqslant}1$, $0 < \sigma {\leqslant}s < 1$ and $R > 0$. Then, for any $u \in W^{s, p}(B_R)$, it holds $$[u]_{W^{\sigma, p}(B_R)}^p {\leqslant}\delta^{(s - \sigma) p} [u]_{W^{s, p}(B_R)}^p + \frac{2^p |B_1|}{\sigma p} \, \chi_{(0, 2 R)}(\delta) \delta^{- \sigma p} \| u \|_{L^p(B_R)}^p,$$ for any $\delta > 0$.
The result is a weighted version of, say, [@DPV12 Proposition 2.1] and the proof follows the same lines of that presented there. However, we report it here for the reader’s convenience.
First of all, it is enough to deal with $R = 1$, as a scaling argument readily shows. Fix $\delta > 0$. On the one hand, we have $$\begin{aligned}
& \int_{B_1} \left[ \int_{B_1 \cap B_\delta(x)} \frac{|u(x) - u(y)|^p}{|x - y|^{n + \sigma p}} \, dy \right] dx \\
& \hspace{40pt} = \frac{1}{\delta^{n + \sigma p}} \int_{B_1} \left[ \int_{B_1 \cap B_\delta(x)} |u(x) - u(y)|^p \left( \frac{\delta}{|x - y|} \right)^{n + \sigma p} \, dy \right] dx \\
& \hspace{40pt} {\leqslant}\frac{1}{\delta^{n + \sigma p}} \int_{B_1} \left[ \int_{B_1 \cap B_\delta(x)} |u(x) - u(y)|^p \left( \frac{\delta}{|x - y|} \right)^{n + s p} \, dy \right] dx \\
& \hspace{40pt} {\leqslant}\delta^{(s - \sigma) p} \int_{B_1} \int_{B_1} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy.\end{aligned}$$ Of course, if $\delta {\geqslant}2$, we have already proved the claim. On the other hand, by Jensen’s inequality $$\begin{aligned}
\int_{B_1} \left[ \int_{B_1 \setminus B_\delta(x)} \frac{|u(x) - u(y)|^p}{|x - y|^{n + \sigma p}} \, dy \right] dx & {\leqslant}2^p \int_{B_1} |u(x)|^p \left[ \int_{{\mathbb{R}}^n \setminus B_\delta(x)} \frac{dy}{|x - y|^{n + \sigma p}} \right] dx \\
& = \frac{2^p |B_1|}{\sigma p} \, \delta^{- \sigma p} \int_{B_1} |u(x)|^p \, dx.\end{aligned}$$ These two formulas lead to the desired estimate.
In the following result we deal with Sobolev spaces having different orders of integrability and differentiability. Unlike in Lemma \[sobinclem0\], this estimate involves only the Gagliardo seminorms of these spaces, and no Lebesgue norms. Moreover, the inequality is stated for more general quantities than the Gagliardo seminorms, allowing for slightly more freedom in the choice of the domains of integration. This small tweak is of some importance for a future application in Lemma \[growthlem\].
\[sobinclem\] Let $n \in {\mathbb{N}}$, $1 {\leqslant}q < p$ and $0 < \sigma < s < 1$. Let $\Omega' \subseteq \Omega \subset {\mathbb{R}}^n$ be two bounded measurable sets. Then, for any $u \in W^{s, p}(\Omega)$, it holds $$\left[ \int_{\Omega} \int_{\Omega'} \frac{|u(x) - u(y)|^q}{|x - y|^{n + \sigma q}} \, dx dy \right]^{\frac{1}{q}} {\leqslant}C_0 |\Omega'|^{\frac{p - q}{p q}} \operatorname{diam}(\Omega)^{s - \sigma} \left[ \int_{\Omega} \int_{\Omega'} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy \right]^{\frac{1}{p}},$$ with $$C_0 := \left[ \frac{n (p - q)}{(s - \sigma) p q} |B_1| \right]^{\frac{p - q}{p q}}.$$
By Hölder’s inequality, we have $$\begin{aligned}
& \int_{\Omega} \int_{\Omega'} \frac{|u(x) - u(y)|^q}{|x - y|^{n + \sigma q}} \, dx dy \\
& \hspace{50pt} = \int_{\Omega} \int_{\Omega'} \frac{|u(x) - u(y)|^q}{|x - y|^{\frac{q}{p} (n + s p)}} \frac{1}{|x - y|^{\frac{p - q}{p} n - (s - \sigma) q}} \, dx dy \\
& \hspace{50pt} {\leqslant}\left( \int_{\Omega} \int_{\Omega'} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy \right)^{\frac{q}{p}} \left( \int_{\Omega} \int_{\Omega'} \frac{dx dy}{|x - y|^{n - \frac{(s - \sigma) p q}{p - q}}} \right)^{\frac{p - q}{p}}.\end{aligned}$$ Then, letting $d := \operatorname{diam}(\Omega)$ and changing variables appropriately, we compute $$\begin{aligned}
\int_{\Omega} \int_{\Omega'} \frac{dx dy}{|x - y|^{n - \frac{(s - \sigma) p q}{p - q}}} {\leqslant}\int_{\Omega'} \left( \int_{B_d} \frac{dz}{|z|^{n - \frac{(s - \sigma) p q}{p - q}}} \right) dx = \frac{n (p - q)}{(s - \sigma) p q} |B_1| |\Omega'| d^{\frac{(s - \sigma) p q}{p - q}},\end{aligned}$$ and the thesis follows.
Next is a fractional Poincaré inequality for functions having fat zero level sets. The corresponding Poincaré-Wirtinger-type inequality for functions with vanishing integral mean is due to [@BBM02; @P04]. Notice that the dependence of the constant on the parameter $s$ is explicit, at least when $s$ is far from $0$.
\[poinine\] Let $n \in {\mathbb{N}}$, $p {\geqslant}1$, $0 < s_0 {\leqslant}s < 1$ and $R > 0$. Let $u \in W^{s, p}(B_R)$ be such that $u = 0$ a.e. on a set $\Omega_0 \subseteq B_R$, with $|\Omega_0| {\geqslant}\gamma |B_R|$, for some $\gamma \in (0, 1]$. Then, $$\int_{B_R} |u(x)|^p \, dx {\leqslant}C_1 (1 - s) R^{s p} \int_{B_R} \int_{B_R} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy,$$ for some constant $C_1 {\geqslant}1$ depending only on $n$, $s_0$, $p$ and $\gamma$.
First, we restrict ourselves to take $R = 1$, as the general estimate follows then by scaling. Moreover, we may as well consider the unit cube $Q_1 = (-1/2, 1/2)^n$ instead of the ball $B_1$, by applying a suitable bi-Lipschitz diffeomorphism $T: \overline{Q}_1 \to \overline{B}_1$. More precisely, if $v := u \circ T$, then $v \in W^{s, p}(Q_1)$, with $$\label{Cstar1}
\begin{aligned}
C_\star^{-1} \| u \|_{L^p(B_1)}^p & {\leqslant}\| v \|_{L^p(Q_1)}^p {\leqslant}C_\star \| u \|_{L^p(B_1)}^p, \\
C_\star^{-1} [ u ]_{W^{s, p}(B_1)}^p & {\leqslant}[ v ]_{W^{s, p}(Q_1)}^p {\leqslant}C_\star [ u ]_{W^{s, p}(B_1)}^p,
\end{aligned}$$ and $$\label{Cstar2}
\left| \{ v = 0 \} \cap Q_1 \right| {\geqslant}\frac{\gamma}{C_\star} |Q_1|,$$ for some dimensional constant $C_\star {\geqslant}1$.
Applying for instance [@P04 Corollary 2.1], we know that there is a constant $C_\sharp {\geqslant}1$, depending only on $n$, $s_0$ and $p$, such that $$\label{MSpoinine}
\| v - v_{Q_1} \|_{L^p(Q_1)}^p {\leqslant}C_\sharp (1 - s) \, [v]_{W^{s, p}(Q_1)}^p,$$ where $$v_{Q_1} := \dashint_{Q_1} v(x) \, dx.$$ But, by , $$\| v - v_{Q_1} \|_{L^p(Q_1)}^p {\geqslant}\left| \{ v = 0 \} \cap Q_1 \right| |v_Q|^p {\geqslant}\frac{\gamma}{C_\star} |v_{Q_1}|^p,$$ and hence $$\| v \|_{L^p(Q_1)} {\leqslant}\| v - v_{Q_1} \|_{L^p(Q_1)} + |v_{Q_1}| {\leqslant}\left[ 1 + \left( \frac{C_\star}{\gamma} \right)^{1/p} \right] \| v - v_{Q_1} \|_{L^p(Q_1)}.$$ This, and yield the thesis.
We stress that a Poincaré-type inequality of this kind can be obtained with a simpler and more direct computation in the spirit of formula (4.2) in [@M03]. However, this strategy does not seem to yield a constant with the needed dependence on $s$.
We now have a couple of fractional Sobolev inequalities in balls. To deduce them, we report here below an analogous result by [@BBM02; @MS02], set in the whole Euclidean space.
\[sobinelem\] Let $n \in {\mathbb{N}}$, $p {\geqslant}1$ and $s \in (0, 1)$ be such that $n > s p$. Then, for any $W^{s, p}({\mathbb{R}}^n)$, it holds $$\left( \int_{{\mathbb{R}}^n} |u(x)|^{p^*_s} \, dx \right)^{\frac{p}{p^*_s}} {\leqslant}C_2 \frac{s (1 - s)}{(n - s p)^{p - 1}} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy,$$ for some constant $C_2 {\geqslant}1$ depending only on $n$ and $p$.
We recall that $p^*_s$ denotes the fractional Sobolev exponent defined in .
As a first corollary of Lemmas \[poinine\] and \[sobinelem\], we deduce the following homogeneous fractional Sobolev inequality in a ball.
\[nullsobcor\] Let $n \in {\mathbb{N}}$, $p {\geqslant}1$ and $0 < s_0 {\leqslant}s < 1$ be such that $n > s p$. Let $u \in W^{s, p}_0(B_R)$ and suppose that $u = 0$ on a set $\Omega_0 \subseteq B_R$ with $|\Omega_0| {\geqslant}\gamma |B_R|$, for some $\gamma \in (0, 1]$. Then, $$\left( \int_{B_R} |u(x)|^{p^*_s} \, dx \right)^{\frac{p}{p^*_s}} {\leqslant}C_3 \frac{1 - s}{(n - s p)^{p - 1}} \int_{B_R} \int_{B_R} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy,$$ for some constant $C_3 {\geqslant}1$ depending only on $n$, $s_0$, $p$ and $\gamma$.
Of course, we can reduce to the case $R = 1$, as the inequality is scaling invariant. In view of [@DPV12 Theorem 5.4], we know that there exists a function $\tilde{u} \in W^{s, p}({\mathbb{R}}^n)$ such that $\tilde{u}|_{B_1} = u$ and $\| \tilde{u} \|_{W^{s, p}({\mathbb{R}}^n)} {\leqslant}C' \| u \|_{W^{s, p}(B_1)}$, for some constant $C' {\geqslant}1$. A careful inspection of the several estimates leading to the proof of this result shows that, actually, one has $$\label{extine}
[ \tilde{u} ]_{W^{s, p}({\mathbb{R}}^n)}^p {\leqslant}C \left( [u]_{W^{s, p}(B_1)}^p + \frac{\| u \|_{L^p(B_1)}^p}{s(1 - s)} \right),$$ with $C {\geqslant}1$ depending only on $n$ and $p$. The conclusion of the corollary now follows by applying this with Lemmas \[poinine\] and \[sobinelem\].
When the zero level set of a function does not occupy a region as large as a fraction of the ball, but, instead, the function is supported well inside of that ball, we can still take advantage of Lemma \[sobinelem\] to get the following estimate.
\[sobinecor\] Let $n \in {\mathbb{N}}$, $p {\geqslant}1$ and $s \in (0, 1)$ be such that $n > s p$. Let $u \in W^{s, p}_0(B_R)$ be such that $\operatorname{supp}(u) \subseteq B_r$, with $0 < r < R$. Then, $$\begin{gathered}
\left( \int_{B_R} |u(x)|^{p^*_s} \, dx \right)^{\frac{p}{p^*_s}} \\
{\leqslant}C_4 \frac{1 - s}{(n - s p)^{p - 1}} \left[ \int_{B_R} \int_{B_R} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy + \frac{1}{(R - r)^{s p}} \int_{B_r} |u(x)|^p dx \right],\end{gathered}$$ for some constant $C_4 {\geqslant}1$ depending only on $n$ and $p$.
First, we apply Lemma \[sobinelem\] to obtain $$\begin{aligned}
\left( \int_{B_R} |u(x)|^{p^*_s} \, dx \right)^{\frac{p}{p^*_s}} & = \left( \int_{{\mathbb{R}}^n} |u(x)|^{p^*_s} \, dx \right)^{\frac{p}{p^*_s}}\\
& {\leqslant}C_2 \frac{s(1 - s)}{(n - s p)^{p - 1}} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy.\end{aligned}$$ Now, we essentially use [@DPV12 Lemma 5.1] to deduce a bound for the right-hand side of the inequality above in terms of quantities integrated over the ball $B_R$ alone. In fact, we redo the computation in order to keep track of the constants involved. By using that $\operatorname{supp}(u) \subseteq B_r$ and changing variables appropriately, we compute $$\begin{aligned}
& \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy \\
& \hspace{40pt} = \int_{B_R} \int_{B_R} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy + 2 \int_{B_r} |u(x)|^p \left[ \int_{{\mathbb{R}}^n \setminus B_R} \frac{dy}{|x - y|^{n + s p}} \right] dx \\
& \hspace{40pt} {\leqslant}\int_{B_R} \int_{B_R} \frac{|u(x) - u(y)|^p}{|x - y|^{n + s p}} \, dx dy + \frac{2 n |B_1|}{s p} \frac{1}{(R - r)^{s p}} \int_{B_r} |u(x)|^p dx,\end{aligned}$$ from which the thesis follows.
We conclude the section with an iteration lemma similar to e.g. [@GG82 Lemma 1.1]. For the reader’s convenience, we provide its simple proof in full details.
\[induclem\] Let $0 < r < R$ and $\Phi: [r, R] \to [0, +\infty)$ be a bounded function. Suppose that, for any $r {\leqslant}\rho < \tau {\leqslant}R$, it holds $$\label{frho<ftau}
\Phi(\rho) {\leqslant}\gamma \Phi(\tau) + A + \frac{B}{(\tau - \rho)^\alpha} + \frac{D}{(\tau - \rho)^\beta},$$ for some constants $A, B, D, \alpha, \beta > 0$ and $\gamma \in (0, 1)$. Then, for any $0 < r < R$, $$\label{induclemine}
\Phi(r) {\leqslant}C \left[ A + \frac{B}{(R - r)^\alpha} + \frac{D}{(R - r)^\beta} \right],$$ with $C {\geqslant}1$ depending only on $\alpha$, $\beta$ and $\gamma$.
Take $\theta \in (0, 1)$ to be later specified and consider the sequence $\{ \rho_i \}$ of positive numbers inductively defined by $$\begin{cases}
\rho_i - \rho_{i - 1} = (1 - \theta) \theta^i (R - r) & \quad \mbox{for } i \in {\mathbb{N}}\\
\rho_0 = r. &
\end{cases}$$ This sequence is obviously increasing and $\rho_i \rightarrow R$ as $i \rightarrow +\infty$. We claim that $$\label{induclemclaim}
\begin{aligned}
\Phi(r) & {\leqslant}\gamma^k \Phi(\rho_k) + A \frac{1 - \gamma^k}{1 - \gamma} + \frac{B \theta^{- \alpha}}{(1 - \theta)^\alpha (R - r)^{\alpha}} \frac{1 - \left( \frac{\gamma}{\theta^\alpha} \right)^k}{1 - \frac{\gamma}{\theta^\alpha}} \\
& \quad + \frac{D \theta^{- \beta}}{(1 - \theta)^\beta (R - r)^\beta} \frac{1 - \left( \frac{\gamma}{\theta^\beta} \right)^k}{1 - \frac{\gamma}{\theta^\beta}},
\end{aligned}$$ for any $k \in {\mathbb{N}}\cup \{ 0 \}$.
We prove by induction. Clearly, such inequality holds true for $k = 0$. Thus, we take $j \in {\mathbb{N}}$ and assume that is valid for $k = j - 1$. With the aid of we compute $$\begin{aligned}
\Phi(r) & {\leqslant}\gamma^{j - 1} \left[ \gamma \Phi(\rho_j) + A + \frac{B}{(\rho_j - \rho_{j - 1})^\alpha} + \frac{D}{(\rho_j - \rho_{j - 1})^\beta} \right] \\
&\quad + A \frac{1 - \gamma^{j - 1}}{1 - \gamma} + \frac{B \theta^{- \alpha}}{(1 - \theta)^\alpha (R - r)^{\alpha}} \frac{1 - \left( \frac{\gamma}{\theta^\alpha} \right)^{j - 1}}{1 - \frac{\gamma}{\theta^\alpha}} + \frac{D \theta^{- \beta}}{(1 - \theta)^\beta (R - r)^\beta} \frac{1 - \left( \frac{\gamma}{\theta^\beta} \right)^{j - 1}}{1 - \frac{\gamma}{\theta^\beta}} \\
& = \gamma^j \Phi(\rho_j) + A \left[ \frac{1 - \gamma^{j - 1}}{1 - \gamma} + \gamma^{j - 1} \right] + \frac{B \theta^{- \alpha}}{(1 - \theta)^\alpha (R - r)^{\alpha}} \left[ \frac{1 - \left( \frac{\gamma}{\theta^\alpha} \right)^{j - 1}}{1 - \frac{\gamma}{\theta^\alpha}} + \left( \frac{\gamma}{\theta^\alpha} \right)^{j - 1} \right] \\
& \quad + \frac{D \theta^{- \beta}}{(1 - \theta)^\beta (R - r)^{\beta}} \left[ \frac{1 - \left( \frac{\gamma}{\theta^\beta} \right)^{j - 1}}{1 - \frac{\gamma}{\theta^\beta}} + \left( \frac{\gamma}{\theta^\beta} \right)^{j - 1} \right] \\
& = \gamma^j \Phi(\rho_j) + A \frac{1 - \gamma^j}{1 - \gamma} + \frac{B \theta^{- \alpha}}{(1 - \theta)^\alpha (R - r)^{\alpha}} \frac{1 - \left( \frac{\gamma}{\theta^\alpha} \right)^j}{1 - \frac{\gamma}{\theta^\alpha}} \\
& \quad + \frac{D \theta^{- \beta}}{(1 - \theta)^\beta (R - r)^{\beta}} \frac{1 - \left( \frac{\gamma}{\theta^\beta} \right)^j}{1 - \frac{\gamma}{\theta^\beta}}, \end{aligned}$$ which is precisely with $k = j$. We can therefore conclude that holds for any $k {\geqslant}0$. Taking the limit as $k \rightarrow +\infty$ in , we are finally led to , provided we choose $\theta$ in such a way that $\gamma \theta^{-\alpha}$ and $\gamma \theta^{-\beta}$ are both strictly smaller than $1$.
A fractional De Giorgi isoperimetric-type inequality {#sDGsec}
====================================================
In this section, we establish an isoperimetric-type inequality for the level sets of functions belonging to $W^{s, p}$, when $s$ is close to $1$. This estimate will turn out to be crucial in the next section, where we use it to obtain $C^\alpha$ estimates uniform in the parameter $s$, as $s \rightarrow 1^-$.
The statement of this result is as follows.
\[sDGlemprop\] Let $n {\geqslant}2$ be an integer and $p > 1$. Fix $M > 0$ and $\gamma_1, \gamma_2 \in (0, 1)$. Then, there exist two constants $\bar{s} \in (0, 1)$ and $C > 0$ such that, given any $s \in [\bar{s}, 1)$ and $R > 0$, it holds $$\label{sDGine}
\begin{aligned}
& (k - h) \Big[ |B_R \cap \{ u {\leqslant}h \}| | B_R \cap \{ u {\geqslant}k \} | \Big]^{\frac{n - 1}{n}} \\
& \hspace{60pt} {\leqslant}C R^{n - 2 + s} (1 - s)^{1 / p} [u]_{W^{s, p}(B_R)} |B_R \cap \{ h < u < k \}|^{\frac{p - 1}{p}},
\end{aligned}$$ for any two real numbers $h < k$ and any $u \in W^{s, p}(B_R)$ satisfying $$\begin{gathered}
\| u \|_{L^p(B_R)}^p + (1 - s) R^{s p} [u]_{W^{s, p}(B_R)}^p {\leqslant}M R^{n} (k - h)^p, \\
|B_R \cap \{ u {\leqslant}h \}| {\geqslant}\gamma_1 |B_R| \quad \mbox{and} \quad |B_R \cap \{ u {\geqslant}k \}| {\geqslant}\gamma_2 |B_R|.\end{gathered}$$ The constant $C$ depends only on $n$ and $p$, while $\bar{s}$ may also depend on $M$, $\gamma_1$ and $\gamma_2$.
Inequality gives a bound from below for the measures of intermediate level sets of functions in fractional Sobolev spaces of differentiability order $s$ close to $1$. In particular, it provides a partial fractional counterpart to the classical result by E. De Giorgi that states that functions in $W^{1, p}$ cannot have jump discontinuities.
The constant $\bar{s}$, which unfortunately is not explicitly determined, acts as a threshold, separating *well-behaved* from *ill-behaved* functions in the scale $W^{s, p}$, as $s \in (0, 1)$. As noted in the introduction, characteristic functions may belong to $W^{s, p}$, if $s p < 1$. This suggests that $\bar{s} {\geqslant}1 / p$.
Since $\bar{s}$ may not depend solely on $n$ and $p$, we do not get a clear, global separation between *good* and *bad* Sobolev spaces. Instead, we identify a *transversal* region of, say, $\cup_{s \in (0, 1)} W^{s, p}(B_1)$, composed by functions that satisfy for various parameters $M$, $\gamma_1$ and $\gamma_2$. It would be interesting to understand whether or not $\bar{s}$ could be chosen independently from $M$, $\gamma_1$ or $\gamma_2$. See the brief discussion following the proof of Lemma \[growthlem\] in Section \[DGsec\] for some comments on the implications of such possible uniformity on the regularity of functions in fractional De Giorgi classes.
We now focus on the proof of Proposition \[sDGlemprop\]. Our aim is to deduce it from the already mentioned isoperimetric-type inequality obtained by E. De Giorgi in [@DeG57]. We recall here below this classical result and provide a short proof of it. Our argument follows the strategy outlined in [@Giu03], which is essentially based on the Poincaré-Sobolev inequality.
\[DGisolem\] Let $n {\geqslant}2$ be an integer and $p > 1$. Then, for any two real numbers $\ell < m$ and any $u \in W^{1, p}(B_1)$, it holds $$\Big[ | B_1 \cap \{ u {\leqslant}\ell \} | | B_1 \cap \{ u {\geqslant}m \} | \Big]^{\frac{n - 1}{n}} {\leqslant}\frac{C_\bullet}{m - \ell} \, \| \nabla u \|_{L^p(B_1)} | B_1 \cap \{ \ell < u < m \} |^{\frac{p - 1}{p}},$$ for some constant $C_\bullet {\geqslant}1$ depending only on $n$ and $p$.
Clearly, we can suppose that both sets $B_1 \cap \{ u {\leqslant}\ell \}$ and $B_1 \cap \{ u {\geqslant}m \}$ have positive measure, otherwise there is nothing to prove. Define $$w(x) := \begin{cases}
m - \ell & \quad \mbox{if } u(x) {\geqslant}m, \\
u(x) - \ell & \quad \mbox{if } \ell < u(x) < m, \\
0 & \quad \mbox{if } u(x) {\leqslant}\ell.
\end{cases}$$ By applying Poincaré-Sobolev inequality (see e.g. [@Giu03 Theorem 3.16]) to this function, we get $$\begin{aligned}
(m - \ell) |B_1 \cap \{ u {\geqslant}m \}|^{\frac{n - 1}{n}} & = \left( \int_{B_1 \cap \{ u {\geqslant}m \}} w(x)^{\frac{n}{n - 1}} \, dx \right)^{\frac{n - 1}{n}} {\leqslant}\| w \|_{L^{\frac{n}{n - 1}}(B_1)} \\
& {\leqslant}\frac{C_\bullet}{|B_1 \cap \{ u {\leqslant}\ell \}|^{\frac{n - 1}{n}}} \, \| \nabla w \|_{L^1(B_1)},\end{aligned}$$ for some $C_\bullet {\geqslant}1$ depending only on $n$ and $p$. Using then Hölder’s inequality, $$\| \nabla w \|_{L^1(B_1)} = \int_{B_1 \cap \{ \ell < u < m \}} | \nabla u(x) | \, dx {\leqslant}\| \nabla u \|_{L^p(B_1)} |B_1 \cap \{ \ell < u < m \}|^{\frac{p - 1}{p}}.$$ The combination of these two estimates leads to the thesis.
With this result at hand, we can deduce Proposition \[sDGlemprop\] via a contradiction argument. Here follow the details.
First of all, we suppose that $R = 1$, as the general case can then be obtained by scaling.
We claim that holds true with $$C := \frac{4 C_\bullet}{D_*},$$ where $C_\bullet$ is as in Lemma \[DGisolem\] and $D_*$ is defined by $$\label{Dstardef}
D_* := \left[ \frac{1}{p} \int_{S^{n - 1}} |e_1 \cdot \sigma|^p \, d{\mathcal{H}}^{n - 1}(\sigma) \right]^{\frac{1}{p}}.$$ To prove this fact, we argue by contradiction. Observe that we can limit ourselves to deal with the case of $h = 0$ and $k > 0$, since the inequality is invariant under translations in the dependent variable $u$. Therefore, we suppose that there exist three sequences $\{ s_j \}_{j \in {\mathbb{N}}} \subset (0, 1)$, $\{ k_j \}_{j \in {\mathbb{N}}} \subset (0, +\infty)$ and $\{ u_j \}_{j \in {\mathbb{N}}} \subset L^p(B_1)$, such that $$\begin{gathered}
\nonumber \lim_{j \rightarrow +\infty} s_j = 1, \\
u_j \in W^{s_j, p}(B_1), \, \, \mbox{with} \, \, \| u_j \|_{L^p(B_1)}^p + (1 - s_j) [u_j]_{W^{s_j, p}(B_1)}^p {\leqslant}M k_j^p, \\
| B_1 \cap \{ u_j {\leqslant}0 \} | {\geqslant}\gamma_1 |B_1|, \, \, | B_1 \cap \{ u_j {\geqslant}k_j \} | {\geqslant}\gamma_2 |B_1|,\end{gathered}$$ and $$\begin{aligned}
& k_j \Big[ | B_1 \cap \{ u_j {\leqslant}0 \} | | B_1 \cap \{ u_j {\geqslant}k_j \} | \Big]^{\frac{n - 1}{n}} \\
& \hspace{60pt} > 4 C_\bullet \, \frac{(1 - s_j)^{1 / p} [u_j]_{W^{s_j, p}(B_1)}}{D_*} \, | B_1 \cap \{ 0 < u_j < k_j \} |^{\frac{p - 1}{p}},\end{aligned}$$ for any $j \in {\mathbb{N}}$.
We now normalize the $u_j$’s over the sequence $\{ k_j \}$, i.e. we define $v_j := u_j / k_j$. Observe that $v_j \in W^{s_j, p}(B_1)$ satisfies $$\begin{gathered}
\label{vjleM} \| v_j \|_{L^p(B_1)}^p + (1 - s_j) [v_j]_{W^{s_j, p}(B_1)}^p {\leqslant}M, \\
\label{vjlev} | B_1 \cap \{ v_j {\leqslant}0 \} | {\geqslant}\gamma_1 |B_1|, \, \, | B_1 \cap \{ v_j {\geqslant}1 \} | {\geqslant}\gamma_2 |B_1|,\end{gathered}$$ and $$\label{sDGineM2}
\begin{aligned}
& \Big[ | B_1 \cap \{ v_j {\leqslant}0 \} | | B_1 \cap \{ v_j {\geqslant}1 \} | \Big]^{\frac{n - 1}{n}} \\
& \hspace{60pt} > 4 C_\bullet \, \frac{(1 - s_j)^{1 / p} [v_j]_{W^{s_j, p}(B_1)}}{D_*} \, | B_1 \cap \{ 0 < v_j < 1 \} |^{\frac{p - 1}{p}},
\end{aligned}$$ for any $j \in {\mathbb{N}}$.
Thanks to , we may apply [@BBM01 Corollary 7] or [@P04 Theorem 1.2] and deduce that, up to subsequences, $v_j$ converges in $L^p(B_1)$ to some function $v_\infty \in W^{1, p}(B_1)$, as $j \rightarrow +\infty$. Up to extracting a further subsequence, we may suppose that $v_j \rightarrow v_\infty$ a.e. in $B_1$ and that $(1 - s_j)^{1/p} [v_j]_{W^{s_j, p}(B_1)}$ has a limit as $j \rightarrow +\infty$, which, by [@P04 Theorem 1.2], necessarily satisfies $$\label{nablalesemi}
\lim_{j \rightarrow +\infty} (1 - s_j)^{1 / p} [v_j]_{W^{s_j, p}(B_1)} {\geqslant}D_* \| \nabla v_\infty \|_{L^p(B_1)},$$ with $D_*$ as in .
Since $v_\infty \in W^{1, p}(B_1)$, we know that $|B_1 \cap \{ v_\infty = t \}| = 0$, for a.a. $t \in {\mathbb{R}}$. To see this, one could apply e.g. [@MSZ03 Theorem 1.1] and conclude that almost all level sets of (a specific representative of) $v_\infty$ are countable $(n - 1)$-rectifiable sets, and thus have zero Lebesgue measure. Choose now $\varepsilon \in (0, 1/4]$ in a way that $$\label{uinftyzerolev}
|B_1 \cap \{ v_\infty = \varepsilon \}| = |B_1 \cap \{ v_\infty = 1 - \varepsilon \}| = 0.$$ It is not hard to see that $$\label{limlevsets}
\begin{aligned}
& \lim_{j \rightarrow +\infty} | B_1 \cap \{ v_j < \varepsilon \} | = |B_1 \cap \{ v_\infty < \varepsilon \}|,\\
& \lim_{j \rightarrow +\infty} | B_1 \cap \{ v_j > 1 - \varepsilon \} | = |B_1 \cap \{ v_\infty > 1 - \varepsilon \}|,\\
& \lim_{j \rightarrow +\infty} | B_1 \cap \{ \varepsilon < v_j < 1 - \varepsilon \} | = |B_1 \cap \{ \varepsilon < v_\infty < 1 - \varepsilon \}|.
\end{aligned}$$ We prove for instance the validity of the first limit in . Notice that $$\label{chiujtouinfty}
\lim_{j \rightarrow +\infty} \chi_{\{ v_j < \varepsilon \}} = \chi_{\{ v_\infty < \varepsilon \}} \quad \mbox{a.e.~in } B_1.$$ Indeed, for a.a. $x \in \{ v_\infty < \varepsilon \}$, we have that $v_j(x) < \varepsilon$ for all but a finite number of $j$’s, since $v_j \rightarrow v_\infty$ a.e. in $B_1$, as $j \rightarrow +\infty$. Therefore, $$\lim_{j \rightarrow +\infty} \chi_{\{ v_j < \varepsilon \}}(x) = 1 = \chi_{\{ v_\infty < \varepsilon \}}(x) \quad \mbox{for a.a.~} x \mbox{ in } B_1 \cap \{ v_\infty < \varepsilon \}.$$ Similarly, $$\lim_{j \rightarrow +\infty} \chi_{\{ v_j < \varepsilon \}}(x) = 0 = \chi_{\{ v_\infty < \varepsilon \}}(x) \quad \mbox{for a.a.~} x \mbox{ in } B_1 \cap \{ v_\infty > \varepsilon \},$$ and follows, thanks to . By , we may apply Lebesgue’s dominated convergence theorem to obtain that $$\lim_{j \rightarrow +\infty} \int_{B_1} \chi_{\{ v_j < \varepsilon \}}(x) \, dx = \int_{B_1} \chi_{\{ v_\infty < \varepsilon \}}(x) \, dx.$$ This gives the first formula in . Analogously, one gets the other two. In particular, we deduce from and the first two limits in that $$\label{uinftylev}
|B_1 \cap \{ v_\infty < \varepsilon \}| {\geqslant}\gamma_1 |B_1| \quad \mbox{and} \quad |B_1 \cap \{ v_\infty > 1 - \varepsilon \}| {\geqslant}\gamma_2 |B_1|.$$
In view of and , by letting $j \rightarrow +\infty$ in we immediately see that $$\begin{aligned}
& \Big[ | B_1 \cap \{ v_\infty < \varepsilon \} | | B_1 \cap \{ v_\infty > 1 - \varepsilon \} | \Big]^{\frac{n - 1}{n}} \\
& \hspace{60pt} {\geqslant}4 C_\bullet \| \nabla v_\infty \|_{L^p(B_1)} | B_1 \cap \{ \varepsilon < v_\infty < 1 - \varepsilon \} |^{\frac{p - 1}{p}}.\end{aligned}$$ By comparing this with the inequality of Lemma \[DGisolem\] and taking advantage of the fact that, by definition of $\varepsilon$, it holds $2 (1 - 2 \varepsilon ) {\geqslant}1$, we finally obtain that $$\begin{aligned}
& \Big[ | B_1 \cap \{ v_\infty < \varepsilon \} | | B_1 \cap \{ v_\infty > 1 - \varepsilon \} | \Big]^{\frac{n - 1}{n}} \\
& \hspace{50pt} {\geqslant}4 (1 - 2 \varepsilon) \Big[ | B_1 \cap \{ v_\infty {\leqslant}\varepsilon \} | | B_1 \cap \{ v_\infty {\geqslant}1 - \varepsilon \} | \Big]^{\frac{n - 1}{n}} \\
& \hspace{50pt} {\geqslant}2 \Big[ | B_1 \cap \{ v_\infty < \varepsilon \} | | B_1 \cap \{ v_\infty > 1 - \varepsilon \} | \Big]^{\frac{n - 1}{n}}.\end{aligned}$$ This is clearly a contradiction, since both sides are positive, by . We therefore conclude that holds true and the proposition is proved.
Fractional De Giorgi classes {#DGsec}
============================
In this section we introduce the notion of fractional De Giorgi classes that we take into consideration and prove that their elements are bounded, Hölder continuous functions. On top of this, we show that where they are non-negative, they also satisfy a nonlocal version of the Harnack inequality.
Definition and first properties
-------------------------------
In this first subsection, we give our definition of fractional De Giorgi classes and point out some elementary features of them. These classes are composed by functions that satisfy an improved Caccioppoli-type inequality, such as formula in the introduction. In fact, we consider here a broader family of inequalities, that depend on a number of parameters.
Let $n \in {\mathbb{N}}$, $s \in (0, 1)$ and $p > 1$. Let $\Omega$ be an open subset of ${\mathbb{R}}^n$.
Also fix $d {\geqslant}0$, $H {\geqslant}1$, $k_0 \in {\mathbb{R}}$, $\varepsilon \in (0, sp / n]$, $\lambda {\geqslant}0$ and $R_0 \in (0, +\infty]$.
Let $u \in L_s^{p - 1}({\mathbb{R}}^n) \cap W^{s, p}(\Omega)$ be a given function. We say that $u$ belongs to the *fractional De Giorgi class* $\operatorname{DG}_+^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0)$ if and only if it holds $$\label{DG+def}
\begin{aligned}
& [(u - k)_+]_{W^{s, p}(B_{r}(x_0))}^p + \int_{B_{r}(x_0)} (u(x) - k)_+ \left[ \int_{B_{2 R_0}(x)} \frac{ (u(y) - k)_-^{p - 1}}{|x - y|^{n + s p}} \, dy \right] dx \\
& \hspace{3pt} {\leqslant}\frac{H}{1 - s} \Bigg[ \left( R^\lambda d^p + \frac{|k|^p}{R^{n \varepsilon}} \right) |A^+(k, x_0, R)|^{1 - \frac{sp}{n} + \varepsilon} + \frac{R^{(1 - s) p}}{(R - r)^p} \| (u - k)_+ \|_{L^p(B_R(x_0))}^p \\
& \hspace{3pt} \quad + \frac{R^{n + s p}}{(R - r)^{n + s p}} \| (u - k)_+ \|_{L^1(B_R(x_0))} \overline{\operatorname{Tail}}((u - k)_+; x_0, r)^{p - 1} \Bigg],
\end{aligned}$$ for any point $x_0 \in \Omega$, radii $0 < r < R < \min \{ R_0, {{\mbox{\normalfont dist}}}\left( x_0, \partial \Omega \right) \}$ and $k {\geqslant}k_0$. Recall that the set $A^+(k, x_0, R)$ has been defined right below .
Analogously, $u \in \operatorname{DG}_-^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0)$ if and only if $$\label{DG-def}
\begin{aligned}
& [(u - k)_-]_{W^{s, p}(B_{r}(x_0))}^p + \int_{B_{r}(x_0)} (u(x) - k)_- \left[ \int_{B_{2 R_0}(x)} \frac{ (u(y) - k)_+^{p - 1}}{|x - y|^{n + s p}} \, dy \right] dx \\
& \hspace{3pt} {\leqslant}\frac{H}{1 - s} \Bigg[ \left( R^\lambda d^p + \frac{|k|^p}{R^{n \varepsilon}} \right) |A^-(k, x_0, R)|^{1 - \frac{sp}{n} + \varepsilon} + \frac{R^{(1 - s) p}}{(R - r)^p} \| (u - k)_- \|_{L^p(B_R(x_0))}^p \\
& \hspace{3pt} \quad + \frac{R^{n + s p}}{(R - r)^{n + s p}} \| (u - k)_- \|_{L^1(B_R(x_0))} \overline{\operatorname{Tail}}((u - k)_-; x_0, r)^{p - 1} \Bigg],
\end{aligned}$$ for any $x_0 \in \Omega$, $0 < r < R < \min \{ R_0, {{\mbox{\normalfont dist}}}\left( x_0, \partial \Omega \right) \}$ and $k {\leqslant}- k_0$.
Finally, we set $$\operatorname{DG}^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0) := \operatorname{DG}_+^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0) \cap \operatorname{DG}_-^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0).$$
With a slight abuse of notation, we denote by $\operatorname{DG}_+^{s, p}(\Omega; d, H, -\infty, \varepsilon, \lambda, R_0)$ the class of functions that satisfy for any $k \in {\mathbb{R}}$, and similarly for the spaces $\operatorname{DG}_-^{s, p}$ and $\operatorname{DG}^{s, p}$.
Notice that, with the above definitions, we have $$u \in \operatorname{DG}_+^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0) \quad \mbox{iff} \quad - u \in \operatorname{DG}_-^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0),$$ Furthermore, it is not hard to see that the following scaling properties hold true: $$\begin{aligned}
u \in \operatorname{DG}_+^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0) \quad & \mbox{iff} \quad u_{z, \rho} \in \operatorname{DG}_+^{s, p} \left( \rho \, \Omega + z; \rho^{- \frac{\lambda + n \varepsilon}{p}} d, H, k_0, \varepsilon, \lambda, \rho R_0 \right),\\
u \in \operatorname{DG}_+^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0) \quad & \mbox{iff} \quad u^{(\mu)} \in \operatorname{DG}_+^{s, p}(\Omega; \mu d, H, \mu k_0, \varepsilon, \lambda, R_0),\end{aligned}$$ where, for any $z \in {\mathbb{R}}^n$, $\rho, \mu > 0$, we set $$u_{z, \rho}(x) := u \left(\frac{x - z}{\rho}\right), \quad u^{(\mu)}(x) := \mu u(x),$$ and, as customary, we wrote $\rho \, \Omega + z = \{ \rho x + z : x \in \Omega \}$. Analogous statements clearly hold for the spaces $\operatorname{DG}_-^{s, p}$ and $\operatorname{DG}^{s, p}$.
We now proceed to inspect the regularity properties of the elements of the just defined classes.
Local boundedness
-----------------
We prove that the elements of fractional De Giorgi classes are locally bounded functions. Observe that here we only consider choices of parameters $n$, $s$, $p$ that satisfy the condition $n {\geqslant}sp$. Indeed, this is not at all a strong limitation, as when $n < s p$ the boundedness and the Hölder continuity of the functions in $\operatorname{DG}^{s, p}$—and, more generally, in $W^{s, p}$—is warranted by the fractional Morrey embedding (see e.g. [@DPV12]).
We begin with the following proposition, that establishes interior upper bounds for $u \in \operatorname{DG}^{s, p}_+$.
\[ulocboundprop\] Let $u \in \operatorname{DG}_+^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0)$, with $n {\geqslant}s p$, $k_0 {\geqslant}0$ and $0 < \varepsilon_0 {\leqslant}\varepsilon {\leqslant}s p / n$. Then, there exist $C {\geqslant}1$ and $\theta \in (0, \varepsilon_0 / 2]$, such that, for any $x_0 \in \Omega$ and $0 < R < \min \{ {{\mbox{\normalfont dist}}}\left( x_0, \partial \Omega \right), R_0 \} / 2$, it holds $$\label{ulocbound}
\begin{aligned}
\sup_{B_R(x_0)} \, (u - k_0)_+ & {\leqslant}C \, \frac{\delta^{- \frac{p - 1}{(\varepsilon - \theta) p}}}{(n - s p + n \theta)^{\frac{p - 1}{(\varepsilon - \theta) p}}} \left( \dashint_{B_{2 R}(x_0)} (u(x) - k_0)_+^p \, dx \right)^{\frac{1}{p}} \\
& \quad + \delta \operatorname{Tail}((u - k_0)_+; x_0, R) + \delta^{\frac{p - 1}{p}} \left( R^{\frac{\lambda + n \varepsilon}{p}} d + k_0 \right),
\end{aligned}$$ for any $\delta \in (0, 1]$. The constant $\theta$ depends on $n$, $p$ and $\varepsilon_0$, while $C$ also on $H$. When $n > s p$, we can even take $\theta = 0$.
Suppose without loss of generality that $x_0 = 0$.
Let $R {\leqslant}\rho < \tau {\leqslant}2 R$ and consider a cut-off function $\eta \in C^\infty_0({\mathbb{R}}^n)$ such that $0 {\leqslant}\eta {\leqslant}1$ in ${\mathbb{R}}^n$, $\operatorname{supp}(\eta) \subseteq B_{(\tau + 3 \rho) / 4}$, $\eta = 1$ in $B_\rho$ and $|\nabla \eta| {\leqslant}8 / (\tau - \rho)$ in ${\mathbb{R}}^n$. Fix $k {\geqslant}k_0 {\geqslant}0$ and set $w_k := (u - k)_+$, $v := \eta w_k$. Notice that $\operatorname{supp}(v) \subseteq B_{(\tau + 3 \rho) / 4}$. Let $\sigma \in [s - n \varepsilon_0 / (2 p), s]$ be given by $$\sigma := \begin{cases}
s & \quad \mbox{if } n > s p, \\
\max \left\{ 2 s - 1, s - \frac{n \varepsilon_0}{2 p} \right\} & \quad \mbox{if } n = s p.
\end{cases}$$ Observe that, with this choice, $n > \sigma p$. Also, $1 - \sigma {\leqslant}2 (1 - s)$. By Hölder’s inequality and Corollary \[sobinecor\], we have $$\label{locboundtech1}
\begin{aligned}
\| w_k \|_{L^p(B_\rho)}^p & {\leqslant}|A^+(k, \rho)|^{\frac{\sigma p}{n}} \| v \|_{L^{p^*_\sigma}(B_{(\tau + \rho) / 2})}^p \\
& {\leqslant}\frac{C (1 - \sigma)}{(n - \sigma p)^{p - 1}} |A^+(k, \rho)|^{\frac{\sigma p}{n}} \left[ [ v ]_{W^{\sigma, p}(B_{(\tau + \rho) / 2})}^p + \frac{\| v \|_{L^p(B_{(\tau + \rho) / 2})}^p}{(r - \rho)^{\sigma p}} \right] \\
& {\leqslant}\frac{C (1 - s)}{(n - \sigma p)^{p - 1}} \frac{|A^+(k, \rho)|^{\frac{\sigma p}{n}}}{(\tau - \rho)^{\sigma p}} \left[ (\tau - \rho)^{s p} [ v ]_{W^{s, p}(B_{(\tau + \rho) / 2})}^p + \| w_k \|_{L^p(B_\tau)}^p \right],
\end{aligned}$$ with $C {\geqslant}1$ depending only on $n$, $p$ and $\varepsilon_0$. Notice that, when $n = s p$ we also took advantage of Lemma \[sobinclem0\] (with $\delta = \tau - \rho$), to deduce the last inequality. Using then Young’s inequality and the definition of $\eta$, we compute $$\begin{aligned}
[ v ]_{W^{s, p}(B_{(\tau + \rho) / 2})}^p & {\leqslant}C \left[ [w_k]_{W^{s, p}(B_{(\tau + \rho) / 2})}^p + \int_{B_\tau} w_k(x)^p \left[ \int_{B_\tau} \frac{|\eta(x) - \eta(y)|^p}{|x - y|^{n + s p}} \, dy \right] dx \right] \\
& {\leqslant}C \left[ [w_k]_{W^{s, p}(B_{(\tau + \rho) / 2})}^p + \frac{1}{(\tau - \rho)^p} \int_{B_\tau} w_k(x)^p \left[ \int_{B_\tau} \frac{dy}{|x - y|^{n - p + s p}} \right] dx \right] \\
& {\leqslant}C \left[ [w_k]_{W^{s, p}(B_{(\tau + \rho) / 2})}^p + \frac{1}{1 - s} \frac{\tau^{(1 - s) p}}{(\tau - \rho)^p} \| w_k \|_{L^p(B_\tau)}^p \right].\end{aligned}$$ By combining this with and , we are led to the estimate $$\label{locboundtech2}
\begin{aligned}
\| w_k \|_{L^p(B_\rho)}^p & {\leqslant}\frac{C}{(n - \sigma p)^{p - 1}} |A^+(k, \rho)|^{\frac{\sigma p}{n}} \Bigg[ \left( \tau^\lambda d^p + \frac{k^p}{\tau^{n \varepsilon}} \right) \tau^{(s - \sigma) p} |A^+(k, \tau)|^{1 - \frac{sp}{n} + \varepsilon} \\
& \quad + \frac{\tau^{(1 - \sigma) p}}{(\tau - \rho)^p} \| w_k \|_{L^p(B_\tau)}^p + \frac{\tau^{n + s p}}{(\tau - \rho)^{n + \sigma p}} \| w_k \|_{L^1(B_\tau)} \overline{\operatorname{Tail}}(w_k; R)^{p - 1} \Bigg],
\end{aligned}$$ where $C$ now depends on $H$ too.
Fix $0 < h < k$. For $x \in A^+(k) \subseteq A^+(h)$, we have $$w_h(x) = u(x) - h {\geqslant}k - h,$$ and $$w_h(x) = u(x) - h {\geqslant}u(x) - k = w_k(x).$$ Accordingly, given any $r > 0$, $$\begin{aligned}
\| w_h \|_{L^p(B_r)}^p & {\geqslant}\int_{A^+(k, r)} w_h(x)^p \, dx {\geqslant}(k - h)^p |A^+(k, r)|,\\
\| w_h \|_{L^p(B_r)}^p & {\geqslant}\int_{A^+(k, r)} w_k(x)^p \, dx = \| w_k \|_{L^p(B_r)}^p,\\
\| w_h \|_{L^p(B_r)}^p & {\geqslant}(k - h)^{p-1} \int_{A^+(k, r)} w_k(x) \, dx = (k - h)^{p-1} \| w_k \|_{L^1(B_r)}.\end{aligned}$$ That is, $$|A^+(k, r)| {\leqslant}\frac{\| w_h \|_{L^p(B_r)}^p}{(k - h)^p}, \, \, \| w_k \|_{L^p(B_r)}^p {\leqslant}\| w_h \|_{L^p(B_r)}^p \, \mbox{ and } \,
\| w_k \|_{L^1(B_r)} {\leqslant}\frac{\| w_h \|_{L^p(B_r)}^p}{(k - h)^{p - 1}}.$$ With the aid of these estimates, inequality yields $$\begin{aligned}
\| w_k \|_{L^p(B_\rho)}^p & {\leqslant}\frac{C}{(n - \sigma p)^{p - 1}} \Bigg[
\frac{\tau^{\lambda + n \varepsilon} d^p + k^p}{(k - h)^p} \left( \frac{|A^+(k, \tau)|}{|B_\tau|} \right)^{\varepsilon - \frac{(s - \sigma)p}{n}} \\
& \quad + \left( \frac{|A^+(k, \tau)|}{|B_\tau|} \right)^{\frac{\sigma p}{n}} \left( \frac{\tau^p}{(\tau - \rho)^p} + \frac{\tau^{n + (s + \sigma) p} \, \overline{\operatorname{Tail}}(w_k; R)^{p - 1}}{(\tau - \rho)^{n + \sigma p} (k - h)^{p - 1}} \right) \Bigg] \| w_h \|_{L^p(B_\tau)}^p \\
& {\leqslant}\frac{C}{(n - \sigma p)^{p - 1}} \frac{\tau^{- n \varepsilon_\sigma}}{(k - h)^{\varepsilon_\sigma p}} \Bigg[
\frac{\tau^{\lambda + n \varepsilon} d^p + k^p}{(k - h)^p} + \frac{\tau^{p}}{(\tau - \rho)^p} \\
& \quad + \frac{\tau^{n + (s + \sigma) p} \, \overline{\operatorname{Tail}}(w_k; R)^{p - 1}}{(\tau - \rho)^{n + \sigma p} (k - h)^{p - 1}} \Bigg] \| w_h \|_{L^p(B_\tau)}^{(1 + \varepsilon_\sigma) p},\end{aligned}$$ with $\varepsilon_\sigma := \varepsilon - (s - \sigma) p / n {\leqslant}\sigma p / n$. Setting $$\varphi(\ell, \sigma) := \| w_\ell \|_{L^p(B_\sigma)}^p, \quad \mbox{for any } \ell, \, \sigma > 0,$$ we get $$\label{locboundtech5}
\begin{aligned}
\varphi(k, \rho) & {\leqslant}\frac{C}{(n - \sigma p)^{p - 1}} \frac{\tau^{- n \varepsilon_\sigma}}{(k - h)^{\varepsilon_\sigma p}} \left[
\frac{\tau^{\lambda + n \varepsilon} d^p + k^p}{(k - h)^p} + \frac{\tau^{p}}{(\tau - \rho)^p} \right. \\
& \quad \left. + \frac{\tau^{n + (s + \sigma) p} \, \overline{\operatorname{Tail}}(w_k; R)^{p - 1}}{(\tau - \rho)^{n + \sigma p} (k - h)^{p - 1}} \right] \varphi(h, \tau)^{1 + \varepsilon_\sigma}.
\end{aligned}$$
Consider now the two sequences of positive numbers $\{ k_i \}$ and $\{ \rho_i \}$, defined by $$k_i := k_0 + M (1 - 2^{-i}) \quad \mbox{and} \quad \rho_i := (1 + 2^{-i}) R,$$ for $i \in {\mathbb{N}}\cup \{ 0 \}$ and for some $M > 0$ to be determined. Note that $\{ k_i \}$ is increasing, while $\{ \rho_i \}$ is decreasing. Also set $\varphi_i := \varphi(k_i, \rho_i)$. Recalling definitions and , $$\begin{aligned}
\overline{\operatorname{Tail}}(w_{k_{i + 1}}; R)^{p - 1} & = (1 - s) \int_{A(k_{i + 1}) \setminus B_R} \frac{(u(x) - k_{i + 1})^{p - 1}}{|x|^{n + s p}} \, dx \\
& {\leqslant}(1 - s) \int_{A(k_0) \setminus B_R} \frac{(u(x) - k_0)^{p - 1}}{|x|^{n + s p}} \, dx = R^{- s p} \operatorname{Tail}(w_{k_0}; R)^{p - 1}.\end{aligned}$$ By this and , we compute $$\begin{aligned}
\varphi_{i + 1} & {\leqslant}\frac{C}{(n - \sigma p)^{p - 1}} \frac{\rho_i^{- n \varepsilon_\sigma}}{(k_{i + 1} - k_i)^{\varepsilon_\sigma p}} \left[ \frac{\rho_i^{\lambda + n \varepsilon} d^p + k_{i + 1}^p}{(k_{i + 1} - k_i)^p} + \frac{\rho_i^{p}}{(\rho_i - \rho_{i + 1})^p} \right. \\
& \quad \left. + \frac{\rho_i^{n + (s + \sigma) p} \, \overline{\operatorname{Tail}}(w_{k_{i + 1}}; R)^{p - 1}}{(\rho_i - \rho_{i + 1})^{n + \sigma p} (k_{i + 1} - k_i)^{p - 1}} \right] \varphi_i^{1 + \varepsilon_\sigma} \\
& {\leqslant}\frac{C}{(n - \sigma p)^{p - 1}} \frac{2^{(n + 3 p) i}}{R^{n \varepsilon_\sigma} M^{\varepsilon_\sigma p}} \left[ \frac{ R^{\lambda + n \varepsilon} d^p + k_0^p + M^p }{M^p} + 1 + \frac{\operatorname{Tail}(w_{k_0}; R)^{p - 1}}{M^{p - 1}} \right] \varphi_i^{1 + \varepsilon_\sigma} \\
& {\leqslant}\frac{C \delta^{- 1 + p}}{(n - \sigma p)^{p - 1}} \frac{2^{(n + 3 p) i}}{R^{n \varepsilon_\sigma} M^{\varepsilon_\sigma p}} \, \varphi_i^{1 + \varepsilon_\sigma},\end{aligned}$$ if we choose $$M {\geqslant}M_1 := \delta \operatorname{Tail}(w_{k_0}; R) + \delta^{\frac{p - 1}{p}} \left( R^{\frac{\lambda + n \varepsilon}{p}} d + k_0 \right),$$ for any fixed $\delta \in (0, 1]$. By applying for instance [@Giu03 Lemma 7.1], we infer that $\varphi_i \rightarrow 0$ as $i \rightarrow +\infty$, and hence $u {\leqslant}k_0 + M$ in $B_R$, provided it holds $$\varphi_0 {\leqslant}C \delta^{\frac{p - 1}{\varepsilon_\sigma}} (n - \sigma p)^{\frac{p - 1}{\varepsilon_\sigma}} R^{n} M^p,$$ that is $$M {\geqslant}M_2 := \frac{C \delta^{- \frac{p - 1}{\varepsilon_\sigma p}}}{(n - \sigma p)^{\frac{p - 1}{\varepsilon_\sigma p}}} \left( \frac{1}{R^n} \int_{B_{2 R}} w_{k_0}(x)^p \, dx \right)^{\frac{1}{p}}.$$ Estimate then follows by taking e.g. $M := M_1 + M_2$.
By applying Proposition \[ulocboundprop\] to both $u$ and $-u$, we get the desired two-sided boundedness result.
\[DGboundthm\]\
Let $u \in \operatorname{DG}^{s, p}(\Omega; d, H, k_0, \varepsilon, \lambda, R_0)$, with $n {\geqslant}s p$, $k_0 {\geqslant}0$ and $0 < \varepsilon_0 {\leqslant}\varepsilon {\leqslant}s p / n$. Then, $u \in L^\infty_{{\rm loc}}(\Omega)$. In particular, there exists a constant $C {\geqslant}1$, such that, for any $x_0 \in \Omega$ and $0 < R < \min \{ {{\mbox{\normalfont dist}}}(x_0, \partial \Omega), R_0 \} / 2$, $$\| u \|_{L^\infty(B_R(x_0))} {\leqslant}C R^{-\frac{n}{p}} \| u \|_{L^p(B_{2 R}(x_0))} + \operatorname{Tail}(u; x_0, R) + d R^{\frac{\lambda + n \varepsilon}{p}} + 2 k_0,$$ The constant $C$ depends on $n$, $s$, $p$, $\varepsilon_0$ and $H$. When $n > p$, it does not blow up as $s \rightarrow 1^-$.
We remark that up to now we have not fully exploited inequalities -, that define the fractional De Giorgi classes. Indeed, to obtain the previous results we only took advantage of the bounds that those inequalities provide for the Gagliardo seminorm of the truncations $(u - k)_\pm$, i.e. the first term appearing on the left-hand sides of and .
In the next subsection, on the contrary, we will make full use of such defining inequalities.
Hölder continuity
-----------------
We focus here on establishing Hölder continuity estimates for functions belonging to fractional De Giorgi classes. As before, we might well restrict ourselves to the case $n {\geqslant}s p$. However, since we will need some of the results obtained here in the following subsection, we do not make such assumption.
The fundamental step in recovering the Hölder regularity is made in the following *growth lemma*.
\[growthlem\] Let $u \in \operatorname{DG}_-^{s, p}(B_{4 R}; d, H, -1, \varepsilon, \lambda, R_0)$, for some values $R > 0$, $R_0 {\geqslant}4 R$ and $0 < \varepsilon_0 {\leqslant}\varepsilon {\leqslant}s p / n$. Suppose that $$\label{uge0}
u {\geqslant}0 \quad \mbox{in } B_{4 R},$$ and $$\label{1dens}
|B_{2 R} \cap \{ u {\geqslant}1 \}| {\geqslant}\gamma |B_{2 R}|,$$ for some $\gamma \in (0, 1)$. There exist a small constant $\delta \in (0, 1/8]$, such that, if $$\label{udecay}
R^{\frac{\lambda + n \varepsilon}{p}} d + \operatorname{Tail}(u_-; 4 R) {\leqslant}\delta,$$ then, $$\label{ugedelta}
u {\geqslant}\delta \quad \mbox{in } B_R.$$ The constant $\delta$ depends only on $n$, $p$, $\varepsilon_0$, $H$, $\gamma$ when $n {\geqslant}2$, and also on $s$, when $n = 1$.
First of all, by scaling, we can restrict ourselves to take $R = 1$. Let $\delta \in (0, 1/64]$ and $\tau \in (0, 2^{- n - 1}]$ to be specified later. Let then $\delta {\leqslant}h < k {\leqslant}2 \delta$ and $1 {\leqslant}\rho < r {\leqslant}2$.
We initially suppose that $$\label{deltadens}
|B_2 \cap \{ u < 2 \delta \}| {\leqslant}\tau |B_2|.$$ Under this additional assumption (and if $\tau$ is sufficiently small), we prove that holds true. Then, at a second stage, we will show that is in fact a consequence of the hypotheses made in the statement of the lemma, provided $\delta$ is chosen small enough.
By and the upper bound on $\tau$, we have $$\label{nullishalf}
\begin{aligned}
| B_\rho \cap \{ (u - k)_- = 0 \}| & = |B_\rho \setminus \{ u < k\}| {\geqslant}|B_\rho| - |B_2 \cap \{ u < 2 \delta \}| \\
& {\geqslant}\left[ 1 - \tau \left( \frac{2}{\rho} \right)^n \right] |B_\rho| {\geqslant}(1 - 2^n \tau) |B_\rho| \\
& {\geqslant}\frac{1}{2} \, |B_\rho|.
\end{aligned}$$ Let $\sigma \in (0, s)$ be defined by $$\sigma := \max \left\{ 2 s - 1, s - \frac{n \varepsilon_0}{2 p} \right\},$$ and notice that $n {\geqslant}1 > \sigma$. Also, $1 - \sigma {\leqslant}2 (1 - s)$ and $C^{-1} (1 - s) {\leqslant}s - \sigma {\leqslant}1 - s$, for some $C {\geqslant}1$ depending on $n$, $p$ and $\varepsilon_0$. By using , Corollary \[nullsobcor\] and Lemma \[sobinclem\], we find that $$\begin{aligned}
(k - h) |A^-(h, \rho)|^{\frac{n - \sigma}{n}} & {\leqslant}\left[ \int_{A^-(h, \rho)} (k - u(x))^{\frac{n}{n - \sigma}} \, dx \right]^{\frac{n - \sigma}{n}} {\leqslant}\left[ \int_{B_\rho} (u(x) - k)_-^{1^*_\sigma} \, dx \right]^{\frac{1}{1^*_\sigma}} \\
& {\leqslant}C (1 - \sigma) \int_{B_\rho} \int_{B_\rho} \frac{|(u(x) - k)_- - (u(y) - k)_-|}{|x - y|^{n + \sigma}} \, dx dy \\
& {\leqslant}C (1 - s) \int_{A^-(k, \rho)} \int_{B_\rho} \frac{|(u(x) - k)_- - (u(y) - k)_-|}{|x - y|^{n + \sigma}} \, dx dy \\
& {\leqslant}C (1 - s)^{\frac{1}{p}} |A^-(k, \rho)|^{\frac{p - 1}{p}} [(u - k)_-]_{W^{s, p}(B_\rho)}.\end{aligned}$$ By raising the above inequality to the $p$ power and applying , we then get $$\label{imppostech1}
\begin{aligned}
(k - h)^p |A^-(h, \rho)|^{\frac{n - \sigma}{n} p} & {\leqslant}C |A^-(k, \rho)|^{p - 1} \Bigg[ (d^p + k^p) |A^-(k, r)|^{1 - \frac{sp}{n} + \varepsilon} \\
& \quad + \frac{\| (u - k)_- \|_{L^p(B_r)}^p}{(r - \rho)^p} \\
& \quad + \frac{\| (u - k)_- \|_{L^1(B_r))} \overline{\operatorname{Tail}}((u - k)_-; \rho)^{p - 1}}{(r - \rho)^{n + s p}} \Bigg],
\end{aligned}$$ where $C$ may now depend on $H$ too. On the one hand, thanks to , $$\label{Lqcontrol}
\| (u - k)_- \|_{L^q(B_r)}^q = \int_{A^-(k, r)} \left( k - u(x) \right)^q \, dx {\leqslant}|A^-(k, r)| k^q,$$ for any $q {\geqslant}1$. On the other hand, using , once again and that $k {\geqslant}\delta$, we compute $$\begin{aligned}
\overline{\operatorname{Tail}}((u - k)_-; \rho)^{p - 1} & = (1 - s) \int_{{\mathbb{R}}^n \setminus B_\rho} \frac{(k - u(x))_+^{p - 1}}{|x|^{n + s p}} \, dx \\
& {\leqslant}C \left[ k^{p - 1} \int_{{\mathbb{R}}^n \setminus B_\rho} \frac{dx}{|x|^{n + s p}} + (1 - s) \int_{{\mathbb{R}}^n \setminus B_4} \frac{u_-(x)^{p - 1}}{|x|^{n + s p}} \, dx \right]
\\
& = C \left[ \frac{\rho^{- s p}}{n \varepsilon_0} \, k^{p - 1} + 4^{- s p} \operatorname{Tail}(u_-; 4)^{p - 1} \right] {\leqslant}C \left[ k^{p - 1} + \delta^{p - 1} \right] \\\
& {\leqslant}C k^{p - 1}.\end{aligned}$$ By exploiting the last two estimates in , together with the fact that, by assumption , $d {\leqslant}\delta {\leqslant}k$, we easily conclude that $$(k - h)^p |A^-(h, \rho)|^{\frac{n - \sigma}{n} p} {\leqslant}C (r - \rho)^{- n - s p} k^p |A^-(k, r)|^{p - \frac{s p}{n} + \varepsilon},$$ which, thanks to the definition of $\sigma$, in turn implies that $$\label{imppostech2}
(k - h)^{\frac{n}{n - \sigma}} |A^-(h, \rho)| {\leqslant}C (r - \rho)^{- \frac{n (n + p)}{(n - 1) p}} k^{\frac{n}{n - \sigma}} |A^-(k, r)|^{1 + \frac{\varepsilon_0}{2 p}},$$ at least when $n {\geqslant}2$.
Consider the sequences $\{ r_i \}$ and $\{ k_i \}$ defined by $r_i := 1 + 2^{-i}$ and $k_i := (1 + 2^{-i}) \delta$. Also set $\phi_i := |A^-(k_i, r_i)| / |B_{r_i}|$. By applying with $h = k_i$, $k = k_{i - 1}$, $\rho = r_i$ and $r = r_{i - 1}$, we obtain $$\phi_i {\leqslant}C 2^{\frac{n (n + 2 p)}{(n - 1) p} i} \phi_{i - 1}^{1 + \frac{\varepsilon_0}{2 p}}.$$ Note that, by , we know that $$\phi_0 = \frac{|A^-(2 \delta, 2)|}{|B_2|} {\leqslant}\tau.$$ Therefore, we may apply e.g. [@Giu03 Lemma 7.1] to deduce that holds true, at least if $\tau$ is chosen sufficiently small, in dependence of $n$, $p$, $\varepsilon_0$ and $H$ only.
Note that, when $n = 1$, one can deduce the same fact as above. But in this case $\tau$ would depend on $s$ too.
In order to conclude the proof, we now only need to show that the additional assumption can be deduced from the hypotheses of lemma, provided $\delta$ is small enough. To do so, we argue by contradiction and suppose that $$\label{deltadenscontra}
|B_2 \cap \{ u < 2 \delta \}| {\geqslant}\tau |B_2|,$$ with $\tau$ fixed as before.
We employ once again inequality . Notice that, up to here, we only took advantage of the estimate for the first term on the left-hand side of . Now we use it to obtain a bound for the second summand too. By arguing as before, we deduce from —applied with $r = 2$ and $R = 3$—that, for any $\ell \in [\delta, 1]$, $$\label{2summbound}
(1 - s) \left[ [(u - \ell)_-]_{W^{s, p}(B_2)}^p + \int_{B_2} \int_{B_2} \frac{(u(x) - \ell)_+^{p - 1} (u(y) - \ell)_-}{|x - y|^{n + s p}} \, dx dy \right] {\leqslant}C_1 \ell^p,$$ with $C_1 {\geqslant}1$ only depending on $n$, $p$, $\varepsilon_0$ and $H$.
We start by addressing the case $n {\geqslant}2$. Let $\bar{s} \in (0, 1)$ be the parameter found in Proposition \[sDGlemprop\], in correspondence to the choices $M = 8^p \left( |B_1| + C_1 \right)$, $\gamma_1 = \tau$ and $\gamma_2 = \gamma$. Observe that $\bar{s}$ depends only on $n$, $p$, $\varepsilon_0$, $H$ and $\gamma$.
Suppose that $s \in [\bar{s}, 1)$. Let $m {\geqslant}5$ be the unique integer for which $$\label{mdeltadef}
2^{- m - 1} {\leqslant}\delta < 2^{-m}.$$ Consider the decreasing sequence $k_i := 2^{-i}$, for $i \in \{ 0, \ldots, m \}$. Notice that $k_i \in (2 \delta, 1]$ for any $i \in \{ 1, \ldots, m - 1 \}$. Moreover, by , , and it is easy to see that $$\begin{aligned}
|B_2 \cap \{ (u - k_{i - 1})_- {\leqslant}2^{-i} \}| & = |B_2 \cap \{ u {\geqslant}k_i \}| {\geqslant}|B_2 \cap \{ u {\geqslant}1 \}| {\geqslant}\gamma |B_2|, \\
|B_2 \cap \{ (u - k_{i - 1})_- {\geqslant}3 \cdot 2^{- i - 1} \}| & = |B_2 \cap \{ u {\leqslant}k_{i + 1} \}| {\geqslant}|B_2 \cap \{ u < 2 \delta \}| {\geqslant}\tau |B_2|,\end{aligned}$$ and $$\label{Gagu-k}
\begin{aligned}
& \| (u - k_{i - 1})_- \|_{L^p(B_2)}^p + (1 - s) [(u - k_{i - 1})_-]_{W^{s, p}(B_2)}^p \\
& \hspace{50pt} {\leqslant}\left( |A^-(k_{i - 1}, 2)| + C_1 \right) k_{i - 1}^p {\leqslant}4^p \left( |B_1| + C_1 \right) 2^n (k_i - k_{i + 1})^p,
\end{aligned}$$ for any $i \in \{ 1, \ldots, m - 2 \}$. Consequently, we can apply Proposition \[sDGlemprop\] to the function $(u - k_{i - 1})_-$, with $h = k_{i - 1} - k_i = 2^{- i}$ and $k = k_{i - 1} - k_{i + 1} = 3 \cdot 2^{- i - 1}$. We easily get $$\begin{aligned}
& | B_2 \cap \{ u {\leqslant}k_{i + 1} \} |^{\frac{n - 1}{n}} {\leqslant}C \, 2^i (1 - s)^{1 / p} [(u - k_{i - 1})_-]_{W^{s, p}(B_2)} |B_2 \cap \{ k_{i + 1} < u < k_i \}|^{\frac{p - 1}{p}},\end{aligned}$$ for some $C {\geqslant}1$ depending only on $n$, $p$ and $\gamma$. By means of , we can control the Gagliardo seminorm of $(u - k_{i - 1})_-$ and deduce that, for any $i \in \{ 1, \ldots, m - 2 \}$, $$| B_2 \cap \{ u < 2 \delta \} |^{\frac{(n - 1) p}{n (p - 1)}} {\leqslant}| B_2 \cap \{ u {\leqslant}k_{i + 1} \} |^{\frac{(n - 1) p}{n (p - 1)}} {\leqslant}C |B_2 \cap \{ k_{i + 1} < u < k_i \}|,$$ where $C$ may now depend on $\varepsilon_0$ and $H$ too. By adding up the above inequality as $i$ ranges between $1$ and $m - 2$, we find $$(m - 2) | B_2 \cap \{ u < 2 \delta \} |^{\frac{(n - 1) p}{n (p - 1)}} {\leqslant}C \sum_{i = 1}^{m - 2} |B_2 \cap \{ k_{i + 1} < u < k_i \}| {\leqslant}C,$$ which in turn yields that $$| B_2 \cap \{ u < 2 \delta \} | {\leqslant}C \, m^{- \frac{n (p - 1)}{(n - 1) p}} {\leqslant}C |\log \delta|^{- \frac{n (p - 1)}{(n - 1) p}},$$ thanks to . But this is in contradiction with , if $\delta$ is sufficiently small.
On the other hand, when $s \in (0, \bar{s})$, we simply estimate $$\begin{aligned}
& (1 - s) \int_{B_2} \int_{B_2} \frac{(u(x) - 4 \delta)_+^{p - 1} (u(y) - 4 \delta)_-}{|x - y|^{n + s p}} \, dx dy \\
& \hspace{50pt} {\geqslant}\frac{1 - \bar{s}}{4^{n + p}} \int_{B_2 \cap \{ u {\geqslant}1 \}} ( u(x) - 4 \delta)^{p - 1} \, dx \int_{B_2 \cap \{ u < 2 \delta \}} (4 \delta - u(y)) \, dy \\
& \hspace{50pt} {\geqslant}\frac{1 - \bar{s}}{4^{n + p}} \frac{2 \delta}{2^{p - 1}} |B_2 \cap \{ u {\geqslant}1 \}| |B_2 \cap \{ u < 2 \delta \}| \\
& \hspace{50pt} {\geqslant}\frac{\delta}{C} \, |B_2 \cap \{ u < 2 \delta \}|,\end{aligned}$$ where we used , , that $\delta {\leqslant}1/8$ and the fact that $|x - y|^{n + s p} {\leqslant}4^{n + p}$, for any $x, y \in B_2$. By comparing this inequality with —used here with $\ell = 4 \delta$—, we readily get $$|B_2 \cap \{ u < 2 \delta \}| {\leqslant}C \delta^{p - 1},$$ which, again, contradicts , provided $\delta$ is chosen small enough.
The case $n = 1$ can be treated exactly in the same way as we just did, for $n {\geqslant}2$ and $s \in (0, \bar{s})$. Of course, this time $\delta$ may not be independent of $s$. The proof is therefore complete.
We remark that the proof just displayed makes complete use of inequality only when $s$ is smaller than the parameter $\bar{s}$ found in Proposition \[sDGlemprop\] (and when $n {\geqslant}2$). Indeed, when $s {\geqslant}\bar{s}$, we only needed estimate to control the first summand on its left-hand side. If $\bar{s}$ could be chosen to depend only on $n$ and $p$ in Proposition \[sDGlemprop\], then when $s {\geqslant}\bar{s}$ one would be able to prove Lemma \[growthlem\]—and thus, Hölder continuity, as we shall see momentarily—for a larger class of functions than $\operatorname{DG}^{s, p}$. Namely, one could drop the second term on the left-hand side of inequalities - and hence prove regularity for all functions that satisfy a more standard Caccioppoli-type inequality such as .
Also notice that Proposition \[sDGlemprop\] has been used for the sole purpose of having $\delta$ independent of $s$. This mainly implies that our $C^\alpha$ estimates will be independent of $s$ as well, for $s$ far from $0$. On the contrary, if one is not interested in obtaining uniform estimates, then the proof of Lemma \[growthlem\] simplifies, as the same argument that we adopted for $s {\leqslant}\bar{s}$ can be reproduced in the case of a general $s \in (0, 1)$.
Thanks to Lemma \[growthlem\], we are now in position to prove the Hölder regularity of functions in fractional De Giorgi classes.
\[DGholdthm\]\
Let $u \in \operatorname{DG}^{s, p}(\Omega; d, H, -\infty, \varepsilon, \lambda, R_0)$, with $0 < \varepsilon_0 {\leqslant}\varepsilon {\leqslant}s p / n$. Then $u \in C^\alpha_{{\rm loc}}(\Omega)$, for some $\alpha \in (0, 1)$. Moreover, given any $x_0 \in \Omega$ and $0 < R < \min \{ {{\mbox{\normalfont dist}}}\left( x_0, \partial \Omega \right), R_0 \} / 8$, it holds $$[ u ]_{C^\alpha(B_R(x_0))} {\leqslant}\frac{C}{R^\alpha} \Big( \| u \|_{L^\infty(B_{4 R}(x_0))} + \operatorname{Tail}(u; x_0, 4 R) + R^{\frac{\lambda + n \varepsilon}{p}} d \Big),$$ for some $C {\geqslant}1$. The constants $\alpha$ and $C$ depend only on $n$, $p$, $\varepsilon_0$, $H$ when $n {\geqslant}2$, and also on $s$ when $n = 1$.
Assume without loss of generality that $x_0 = 0$. Let $\delta \in (0, 1/8]$ be the constant found in Lemma \[growthlem\]—with $\gamma = 1/2$ and $4^p H$ instead of $H$. Take $$\label{alphadef}
0 < \alpha {\leqslant}\min \left\{ \frac{n \varepsilon_0}{2 p} , \log_4 \left( \frac{2}{2 - \delta} \right) \right\},$$ in such a way that $$\label{alphacond}
\int_4^{+\infty} \frac{(\rho^\alpha - 1)^{p - 1}}{\rho^{1 + n \varepsilon_0}} \, d\rho {\leqslant}\frac{\varepsilon_0 \delta}{8^{p + 1} p \max \{ 1, |B_1| \}}.$$ Observe that, by Lebesgue’s dominated convergence theorem, the integral appearing in can be made as small as desired, by taking $\alpha$ sufficiently small. Set $$\label{j0def}
j_0 := \left\lceil \frac{2}{n \varepsilon_0} \log_4 \left( \frac{8^{p + 1} p \max \{ 1, |B_1| \}}{\varepsilon_0 \delta} \right) \right\rceil.$$
We claim that there exist a non-decreasing sequence $\{ m_i \}$ and a non-increasing sequence $\{ M_i \}$ of real numbers, such that, for any $i \in {\mathbb{N}}\cup \{ 0 \}$, $$\label{miMidef}
\begin{aligned}
m_i {\leqslant}u {\leqslant}M_i \mbox{ in } B_{4^{1 - i} R} \quad \mbox{and} \quad M_i - m_i = 4^{- \alpha i} L,
\end{aligned}$$ with $$\label{Ldef}
L := 2 \cdot 4^{\frac{n \varepsilon_0}{2 p} j_0} \| u \|_{L^\infty(B_{4 R})} + \operatorname{Tail}(u; 4 R) + R^{\frac{\lambda + n \varepsilon}{p}} d.$$
We proceed by induction on the index $i$. Set $m_i := - 4^{- \alpha i} L / 2$ and $M_i := 4^{- \alpha i} L / 2$, for any $i = 0, \ldots, j_0$. Then, holds for these $i$’s, thanks to and . Now we fix an integer $j {\geqslant}j_0$ and suppose that the sequences $\{ m_i \}$ and $\{ M_i \}$ have been constructed up to $i = j$. Claim will be proved once we construct $m_{j + 1}$ and $M_{j + 1}$ appropriately.
Consider the function $$\label{vdef}
v := \frac{2 \cdot 4^{\alpha j}}{L} \left( u - \frac{M_j + m_j}{2} \right).$$ By and the monotonicity of $\{ m_j \}$, $\{ M_j \}$, we have that $$\label{meanjest}
\left| M_j + m_j \right| {\leqslant}\left( 1 - 4^{- \alpha j} \right) L.$$ Then, it is not hard to see that $$\label{vDG}
v \in \operatorname{DG}^{s, p} \left( B_{8 R}; \frac{2 \cdot 4^{\alpha j}}{L} \, d + R^{- \frac{n \varepsilon + \lambda}{p}} \left( 4^{\alpha j} - 1 \right), 2^p H, -\infty, \varepsilon, \lambda, R_0 \right),$$ and $$\label{|v|le1}
|v| {\leqslant}1 \quad \mbox{in } B_{4^{1 - j} R}.$$
Take now $x \in B_{4 R} \setminus B_{4^{1 - j} R}$ and let $\ell \in \{ 0, \ldots, j - 1 \}$ be the unique integer for which $x \in B_{4^{1 - \ell} R} \setminus B_{4^{- \ell} R}$. By and the monotonicity of $\{ m_i \}$ we have $$\begin{aligned}
v(x) & {\leqslant}\frac{2 \cdot 4^{\alpha j}}{L} \left[ M_\ell - m_\ell + m_\ell - \frac{M_j + m_j}{2} \right] {\leqslant}\frac{2 \cdot 4^{\alpha j}}{L} \left[ M_\ell - m_\ell + m_j - \frac{M_j + m_j}{2} \right] \\
& = \frac{2 \cdot 4^{\alpha j}}{L} \left[ M_\ell - m_\ell - \frac{M_j - m_j}{2} \right] = 2 \cdot 4^{\alpha (j - \ell)} - 1 \\
& {\leqslant}2 \left( 4^j \frac{|x|}{R} \right)^\alpha - 1.\end{aligned}$$ Similarly, one checks that $v(x) {\geqslant}- 2 \left( 4^j |x| / R \right)^\alpha + 1$, and hence $$\label{1pmvinsideest}
(1 \pm v(x))_-^{p - 1} {\leqslant}2^{p - 1} \left[ \left( 4^j \frac{|x|}{R} \right)^{\alpha} - 1 \right]^{p - 1} \quad \mbox{for a.a.~} x \in B_{4 R} \setminus B_{4^{1 - j} R}.$$ On the other hand, using we easily get that $$\label{1pmvoutsideest}
(1 \pm v)_-^{p - 1} {\leqslant}2^{p - 1} \left[ \left( \frac{2 \cdot 4^{\alpha j}}{L} \right)^{p - 1} |u|^{p - 1} + 4^{\alpha (p - 1) j} \right] \quad \mbox{a.e.~in } {\mathbb{R}}^n \setminus B_{4 R}.$$ With the help of , and changing variables appropriately, we compute $$\begin{aligned}
& \operatorname{Tail}((1 \pm v)_-; 4^{1 - j} R)^{p - 1} \\
& \hspace{10pt} {\leqslant}4^{- j s p + 2 p} R^{s p} \left[ \rule{0pt}{26pt} \int_{{\mathbb{R}}^n \setminus B_{4^{1 - j} R}} \frac{\left[ \left( 4^j \frac{|x|}{R} \right)^\alpha - 1 \right]^{p - 1}}{|x|^{n + s p}} \, dx \right. \\
& \hspace{10pt} \quad + \left. (1 - s) \left( \frac{4^{\alpha j}}{L} \right)^{p - 1} \int_{{\mathbb{R}}^n \setminus B_{4 R}} \frac{|u(x)|^{p - 1}}{|x|^{n + s p}} \, dx + 4^{\alpha (p - 1) j} \int_{{\mathbb{R}}^n \setminus B_{4 R}} \frac{dx}{|x|^{n + s p}} \rule{0pt}{28pt} \right] \\
& \hspace{10pt} {\leqslant}\frac{8^p p \max \{ 1, |B_1| \}}{\varepsilon_0} \left[ \int_{4}^{+ \infty} \frac{\left( \rho^\alpha - 1 \right)^{p - 1}}{\rho^{1 + n \varepsilon_0}} \, d\rho + 4^{\left( \alpha p - n \varepsilon_0 \right) j} \left( \frac{\operatorname{Tail}(u; 4 R)^{p - 1}}{L^{p - 1}} + 1 \right) \right].\end{aligned}$$ Recalling , , and , we are led to conclude that $$\label{Tailcontrol}
\operatorname{Tail}((1 \pm v)_-; 4^{1 - j} R) {\leqslant}\frac{\delta}{2}.$$
Now, we have that either $$\label{holderdico}
\left| B_{4^{1 - j} R / 2} \cap \{ v {\geqslant}0 \} \right| {\geqslant}\frac{1}{2} \left| B_{4^{1 - j} R / 2} \right| \mbox{ or } \left| B_{4^{1 - j} R / 2} \cap \{ v {\geqslant}0 \} \right| < \frac{1}{2} \left| B_{4^{1 - j} R / 2} \right|.$$ In the first of the two situations described by we set $w := 1 + v$, while in the second $w := 1 - v$. In any case, we obtain $$\left| B_{4^{1 - j} R / 2} \cap \{ w {\geqslant}1 \} \right| {\geqslant}\frac{1}{2} \left| B_{4^{1 - j} R / 2} \right|.$$ Furthermore, $$\begin{aligned}
w & \in \operatorname{DG}^{s, p} \left( B_{4^{1 - j} R}; \frac{2 \cdot 4^{\alpha j}}{L} \, d + R^{- \frac{n \varepsilon + \lambda}{p}} 4^{\alpha j}, 4^p H, -\infty, \varepsilon, \lambda, R_0 \right), \\
w & {\geqslant}0 \mbox{ in } B_{4^{1 - j} R},\end{aligned}$$ and $$\left( 4^{- j} R \right)^{\frac{\lambda + n \varepsilon}{p}} \left[ \frac{2 \cdot 4^{\alpha j}}{L} \, d + R^{- \frac{n \varepsilon + \lambda}{p}} 4^{\alpha j} \right] + \operatorname{Tail}(w_-; 4^{1 - j} R) {\leqslant}\delta,$$ thanks to , , , and . Therefore, we are in position to apply Lemma \[growthlem\] to $w$. We deduce that $$w {\geqslant}\delta \quad \mbox{in } B_{4^{- j} R}.$$ Assume for instance that the first alternative in is satisfied. By taking advantage of the above estimate, and , $$\begin{aligned}
u(x) & = \frac{M_j + m_j}{2} + \frac{L}{2 \cdot 4^{\alpha j}} \, v(x) = \frac{M_j + m_j}{2} + \frac{L}{2 \cdot 4^{\alpha j}} \left( w(x) - 1 \right) \\
& {\geqslant}M_j - \frac{M_j - m_j}{2} - \frac{L}{2 \cdot 4^{\alpha j}} (1 - \delta) \\
& {\geqslant}M_j - \frac{L}{4^{\alpha j}} \frac{2 - \delta}{2},\end{aligned}$$ for any $x \in B_{4^{- j} R}$. In view of , we finally conclude that $$M_j - 4^{ - (j + 1) \alpha} L {\leqslant}u {\leqslant}M_j \quad \mbox{in } B_{4^{- j} R},$$ that is, is true for $i = j + 1$, with $M_{j + 1} := M_j$ and $m_{j + 1} := M_{j + 1} - 4^{- (j + 1) \alpha} L$. Of course, if instead the second alternative in is valid, an analogous argument leads to the same conclusion, with $m_{j + 1} := m_j$ and $M_{j + 1} := m_{j + 1} + 4^{-(j + 1) \alpha} L$.
Claim holds therefore for any $i \in {\mathbb{N}}\cup \{ 0 \}$, and the Hölder continuity of $u$ follows in a standard way.
Harnack inequality {#harsubsec}
------------------
The conclusive part of this section is devoted to establishing a Harnack-type inequality for functions in fractional De Giorgi classes.
For simplicity of exposition, we restrict ourselves to assume $n {\geqslant}2$ throughout the whole subsection. In this way, the constants involved in the various propositions are independent of $s$, if $s$ is bounded away from $0$ (at least if $p \ne n$). When $n = 1$, all the arguments displayed are still valid, but several estimates may not be uniform in $s$.
As a first step towards the aforementioned goal, we have the following result, that slightly improves Lemma \[growthlem\].
\[2growthlem\] Let $t > 0$ and $u \in \operatorname{DG}_-^{s, p}(B_{4 R}(x_0); d, H, -t, \varepsilon, \lambda, R_0)$, for some $x_0 \in {\mathbb{R}}^n$ and $R > 0$, with $R_0 {\geqslant}4 R$ and $0 < \varepsilon_0 {\leqslant}\varepsilon {\leqslant}s p / n$. Suppose that $$u {\geqslant}0 \quad \mbox{in } B_{4 R}(x_0),$$ and $$|B_R(x_0) \cap \{ u {\geqslant}t \}| {\geqslant}\gamma |B_R|,$$ for some $\gamma \in (0, 1)$. There exists a constant $\delta > 0$, depending only on $n$, $p$, $\varepsilon_0$, $H$ and $\gamma$, such that, if $$R^{\frac{\lambda + n \varepsilon}{p}} d + \operatorname{Tail}(u_-; x_0, 4 R) {\leqslant}\delta t,$$ then, $$u {\geqslant}\delta t \quad \mbox{in } B_R(x_0).$$
Of course, we can assume $x_0$ to be the origin. We begin by addressing the case of $t = 1$. Set $$\tilde{\gamma} := 2^{- n} \gamma \in (0, 1),$$ and let $\delta$ be the constant found in Lemma \[growthlem\], corresponding to the above defined $\tilde{\gamma}$. It holds $$|B_{2 R} \cap \{ u {\geqslant}1 \}| {\geqslant}|B_R \cap \{ u {\geqslant}1 \}| {\geqslant}\gamma |B_R| = \tilde{\gamma} |B_{2 R}|.$$ Hence, we are in position to apply Lemma \[growthlem\] and deduce that $$u {\geqslant}\delta \quad \mbox{in } B_R.$$ The lemma is therefore proved, for $t = 1$.
The general case of $t > 0$ can be then easily deduced. Indeed, let $v := t^{-1} u$. The function $v$ belongs to $\operatorname{DG}_-^{s, p}(B_{4 R}; d/t, H, -1, \varepsilon, R_0)$ and fulfills the hypotheses of the lemma with $t = 1$ and $d/t$ in place of $d$. Thus, from the preceding argument we deduce that $u = t v {\geqslant}t \delta$, and the proof is complete.
Next, we use Lemma \[2growthlem\] to prove a *weak Harnack inequality*, which, together with Proposition \[ulocboundprop\], will lead to the proper Harnack inequality.
In order to do this, we first recall a classical covering lemma of Krylov and Safonov [@KS80]. We present it here in a version with balls in place of cubes, due to [@KS01].
\[KSlem\] Let $\gamma \in (0, 1)$, $R > 0$ and $E \subseteq B_R$ be a measurable set. Define $$E_\gamma := \bigcup \Big\{ B_R \cap B_{3 r}(x_0) : x_0 \in B_R, \, r > 0 \mbox{ and } |B_{3 r}(x_0) \cap E| {\geqslant}\gamma |B_r| \Big\}.$$ Then, either $E_\gamma = B_R$ or $$|E_\gamma| {\geqslant}\frac{1}{2^n \gamma} |E|.$$
With the aid of Lemma \[KSlem\] we can now prove the following result.
\[3growthlem\] Let $\gamma \in (0, 1)$, $t > 0$ and $k \in {\mathbb{N}}$. Let $u \in \operatorname{DG}_-^{s, p}(B_{16 R}; d, H, -t, \varepsilon, \lambda, R_0)$, with $R > 0$, $R_0 {\geqslant}16 R$ and $0 < \varepsilon_0 {\leqslant}\varepsilon {\leqslant}s p / n$. Suppose that $$\label{uge0B16}
u {\geqslant}0 \quad \mbox{in } B_{16 R},$$ and $$\label{supergegammak}
|B_R \cap \{ u {\geqslant}t \}| {\geqslant}\gamma^k |B_R|.$$ There exists a constant $\delta \in (0, 1/8]$, depending only on $n$, $p$, $\varepsilon_0$, $H$ and $\gamma$, such that, if $$\label{Tailledeltak}
R^{\frac{\lambda + n \varepsilon}{p}} d + \operatorname{Tail}(u_-; 16 R) {\leqslant}\delta^k t,$$ then $$u {\geqslant}\delta^k t \quad \mbox{in } B_{R}.$$
Set $\gamma_1 := 2^{- n} \gamma$ and $\gamma_2 := 3^{-n} \gamma_1 = 6^{-n} \gamma$. Let $\delta \in (0, 1/8]$ be the constant found in Lemma \[2growthlem\], in correspondence to $\gamma_2$. For any $i \in {\mathbb{N}}\cap \{ 0 \}$, write $$A^i := B_R \cap \left\{ u {\geqslant}\delta^i t \right\}.$$ Clearly, $$\label{Aincl}
A^{i - 1} \subseteq A^i \quad \mbox{for any } i \in {\mathbb{N}},$$ as $\delta {\leqslant}1$.
Notice that, in order to prove the lemma, it suffices to show that $$\label{Ak-1ge}
|A^{k - 1}| {\geqslant}\gamma_2 |B_R|,$$ since then an application of Lemma \[2growthlem\] would yield the thesis.
Let $i \in \{ 1, \ldots, k - 1 \}$ be fixed and suppose that, in the notation of Lemma \[KSlem\], it holds $B_R \cap B_{3 r}(x_0) \subseteq (A^{i - 1})_{\gamma_1}$, for some $x_0 \in B_R$ and $r > 0$. This implies that $$|B_{3 r}(x_0) \cap \{ u {\geqslant}\delta^{i - 1} t \}| {\geqslant}|B_{3 r}(x_0) \cap A^{i - 1}| {\geqslant}\gamma_1 |B_r| = \gamma_2 |B_{3 r}|.$$ Moreover, since we may suppose without loss of generality that $r {\leqslant}R/3$, we have $$\begin{gathered}
u {\geqslant}0 \quad \mbox{in } B_{12 r}(x_0), \\
(3 r)^{\frac{\lambda + n \varepsilon}{p}} d {\leqslant}R^{\frac{\lambda + n \varepsilon}{p}} d {\leqslant}\delta^k t {\leqslant}\frac{\delta^i t}{2}\end{gathered}$$ and $$\operatorname{Tail}(u_-; x_0, 12 r) = \left( \frac{12 r}{16 R} \right)^{\frac{s p}{p - 1}} \operatorname{Tail}(u_-; 16 R) {\leqslant}\operatorname{Tail}(u_-; 16 R) {\leqslant}\delta^k t {\leqslant}\frac{\delta^i t}{2},$$ thanks to , and the fact that $\delta {\leqslant}1/2$. Accordingly, an application of Lemma \[2growthlem\] gives that $$u {\geqslant}\delta^i t \quad \mbox{in } B_{3 r}(x_0).$$ We have therefore proved that $$\label{AgammainA}
(A^{i - 1})_{\gamma_1} \subseteq A^i \quad \mbox{for any } i \in \{ 1, \ldots, k - 1 \}.$$
We now claim that either $$\label{claimA}
\begin{aligned}
& \mbox{there exists } i \in \{ 1, \ldots, k - 1 \} \mbox{ such that } A^i = B_R, \\
& \mbox{or } |A^i| {\geqslant}\frac{1}{\gamma} |A^{i - 1}| \mbox{ for any } i \in \{ 1, \ldots, k - 1 \}.
\end{aligned}$$ Indeed, suppose that $\gamma |A^i| < |A^{i - 1}|$ for some index $i \in \{ 1, \ldots, k - 1 \}$. By , we deduce that $(2^n \gamma_1) |(A^{i - 1})_{\gamma_1}| = \gamma |(A^{i - 1})_{\gamma_1}| < |A^{i - 1}|$. But then Lemma \[KSlem\] yields that $(A^{i - 1})_{\gamma_1} = B_R$ and thus $A^i = B_R$, using once again . Consequently, claim holds true.
We now show that implies . As noted before, this will conclude the proof. Indeed, if $A^i = B_R$ for some $i \in \{ 1, \ldots, k - 1 \}$, then $A^{k - 1} = B_R$, thanks to , and follows trivially. On the other hand, if the other option in is verified, then $$|A^{k - 1}| {\geqslant}\frac{1}{\gamma} |A^{k - 2}| {\geqslant}\frac{1}{\gamma^2} |A^{k - 3}| {\geqslant}\ldots {\geqslant}\frac{1}{\gamma^{k - 1}} |A^0| {\geqslant}\gamma |B_R|,$$ where the last inequality is true in view of . Hence, we have verified the validity of the bound also in this case, since $\gamma = 6^n \gamma_2 {\geqslant}\gamma_2$. Thence, the proof is complete.
Starting from this result, the derivation of the weak Harnack inequality is rather straightforward.
\[weakHarprop\] Let $u \in \operatorname{DG}_-^{s, p}(B_{16 R}; d, H, - \infty, \varepsilon, \lambda, R_0)$, with $R > 0$, $R_0 {\geqslant}16 R$ and $0 < \varepsilon_0 {\leqslant}\varepsilon {\leqslant}s p / n$, and suppose that holds true. Then, there exist a small $q \in (0, 1)$ and a large $C {\geqslant}1$, both depending only on $n$, $p$, $\varepsilon_0$ and $H$, such that $$\label{weakHarine}
\left( \dashint_{B_R} u(x)^q \, dx \right)^{\frac{1}{q}} {\leqslant}C \left( \inf_{B_R} u + \operatorname{Tail}(u_-; R) + R^{\frac{\lambda + n \varepsilon}{p}} d \right).$$
Of course, we can assume that $u$ does not vanish identically on $B_R$, otherwise is obviously true. Let $\delta \in (0, 1/8]$ be the constant given by Lemma \[3growthlem\], for $\gamma = 1/2$. Set $$a := \frac{\log \gamma}{\log \delta} = \frac{1}{\log_{\frac{1}{2}} \delta} \in (0, 1).$$
We claim that $$\label{wHmainclaim}
\inf_{B_R} u + \operatorname{Tail}(u_-; 16 R) + R^{\frac{\lambda + n \varepsilon}{p}} d {\geqslant}\delta \left( \frac{|A^+(t, R)|}{|B_R|} \right)^{ \frac{1}{a} } t,$$ for any $t {\geqslant}0$. Notice that, if $u$ is bounded from above in $B_R$, then, by , inequality holds trivially for any $t {\geqslant}\sup_{B_R} u$. Thus, it suffices to verify for any $t \in [0, u^*)$, where $u^* \in (0, +\infty]$ denotes the supremum of $u$ in $B_R$.
Given $t \in [0, u^*)$, let $k = k(t)$ be the smallest integer for which $$\label{wHlevest}
|A^+(t, R)| {\geqslant}2^{-k} |B_R|,$$ i.e., $k$ is the only integer for which $$\log_{\frac{1}{2}} \frac{|A^+(t, R)|}{|B_R|} {\leqslant}k < 1 + \log_{\frac{1}{2}} \frac{|A^+(t, R)|}{|B_R|}.$$ Observe that, with this choice, it holds $$\label{wHdeltakbound}
\delta^k {\geqslant}\delta \left( \frac{|A^+(t, R)|}{|B_R|} \right)^{\frac{1}{a}}.$$ Furthermore, $$\label{wHclaim}
\inf_{B_R} u + \operatorname{Tail}(u_-; 16 R) + R^{\frac{\lambda + n \varepsilon}{p}} {\geqslant}\delta^k t.$$ Indeed, if $\operatorname{Tail}(u_-; 16 R) + R^{\frac{\lambda + n \varepsilon}{p}} {\geqslant}\delta^k t$, then is true, thanks to hypothesis . On the other hand, if $\operatorname{Tail}(u_-; 16 R) + R^{\frac{\lambda + n \varepsilon}{p}} < \delta^k t$, then this and enable us to apply Lemma \[3growthlem\] and deduce that $$u {\geqslant}\delta^k t \quad \mbox{in } B_{R}.$$ Again, follows. Putting together and , we see that is valid for any $t {\geqslant}0$.
Write now $$L := \inf_{B_R} u + \operatorname{Tail}(u_-; 16 R) + R^{\frac{\lambda + n \varepsilon}{p}},$$ and note that is equivalent to $$\frac{|A^+(t, R)|}{|B_R|} {\leqslant}\left( \frac{L}{\delta t} \right)^a.$$ Using this inequality and Cavalieri’s principle, for any $q > 0$ we compute $$\dashint_{B_R} u(x)^q \, dx = q \int_0^{+\infty} \frac{|A^+(t, R)|}{|B_R|} \, t^{q - 1} \, dt {\leqslant}q \left[ \int_0^L t^{q - 1} \, dt + \left( \frac{L}{\delta} \right)^a \int_{L}^{+\infty} t^{q - 1 - a} \, dt \right].$$ Choosing $q = a / 2$, we then get $$\dashint_{B_R} u(x)^q \, dx {\leqslant}(1 + \delta^{- a}) L^q,$$ that is $$\begin{aligned}
\left( \dashint_{B_R} u(x)^q \, dx \right)^{\frac{1}{q}} & {\leqslant}(1 + \delta^{- a})^{\frac{2}{a}} \left( \inf_{B_R} u + \operatorname{Tail}(u_-; 16 R) + R^{\frac{\lambda + n \varepsilon}{p}} d \right) \\
& {\leqslant}(1 + \delta^{- a})^{\frac{2}{a}} \left( \inf_{B_R} u + 16^{\frac{p}{p - 1}} \operatorname{Tail}(u_-; R) + R^{\frac{\lambda + n \varepsilon}{p}} d \right).\end{aligned}$$ This yields .
With this, we are now in position to prove the Harnack inequality for fractional De Giorgi classes with $r_0 = +\infty$.
\[DGharthm\]\
Let $u \in \operatorname{DG}^{s, p}(\Omega; d, H, -\infty, \varepsilon, \lambda, +\infty)$, with $0 < \varepsilon_0 {\leqslant}\varepsilon {\leqslant}s p / n$ and $0 {\leqslant}\lambda {\leqslant}\lambda_0$. Suppose that $u {\geqslant}0$ in $\Omega$. Then, for any $x_0 \in \Omega$ and $0 < R < {{\mbox{\normalfont dist}}}(x_0, \partial \Omega) / 2$, it holds $$\label{Harine}
\sup_{B_R(x_0)} u {\leqslant}C \left( \inf_{B_R(x_0)} u + \operatorname{Tail}(u_-; x_0, R) + R^{\frac{\lambda + n \varepsilon}{p}} d \right),$$ for some $C {\geqslant}1$ depending on $n$, $s$, $p$, $\varepsilon_0$, $\lambda_0$ and $H$. When $n \ne p$, the constant $C$ does not blow up as $s \rightarrow 1^-$.
We start supposing that $x_0$ and $R$ are such that $$\label{/64}
0 < R < \frac{{{\mbox{\normalfont dist}}}(x_0, \partial \Omega)}{64}.$$ At the end of the proof we will in fact show that this assumption is unnecessary.
We now prove that holds true under . Up to a translation, we may assume $x_0$ to be the origin. As a preliminary observation, we claim that, for any $z \in B_R$ and $0 < r {\leqslant}2 R$, $$\label{Tail+control}
\operatorname{Tail}(u_+; z, r) {\leqslant}C \left( \sup_{B_r(z)} u + \operatorname{Tail}(u_-; z, r) + r^{\frac{\lambda + n \varepsilon}{p}} d \right),$$ for some constant $C {\geqslant}1$ depending only on $n$, $p$, $\varepsilon_0$ and $H$. To check the validity of , we apply with $k := 2 M$ and $$M := \max \left\{ \sup_{B_r(z)} u, r^{\frac{\lambda + n \varepsilon}{p}} d \right\}.$$ Focusing on just the second term on the left-hand side of such inequality, we get $$\label{Tail+tech1}
\begin{aligned}
& \int_{B_{\frac{r}{2}}(z)} (u(x) - 2 M)_- \left[ \int_{{\mathbb{R}}^n} \frac{(u(y) - 2 M)_+^{p - 1}}{|x - y|^{n + s p}} \, dy \right] dx \\
& \hspace{20pt} {\leqslant}\frac{C}{1 - s} \bigg[ \left( r^\lambda d^p + \frac{M^p}{r^{n \varepsilon}} \right) |A^-(2 M, z, r)|^{1 - \frac{s p}{n} + \varepsilon} + r^{- s p} \| (u - 2 M)_- \|_{L^p(B_{r}(z))}^p \\
& \hspace{20pt} \quad + \| (u - 2 M)_- \|_{L^1(B_r(z))} \overline{\operatorname{Tail}} ((u - 2 M)_-; z, r/2)^{p - 1} \bigg]
\end{aligned}$$ with $C {\geqslant}1$ depending on $n$, $p$ and $H$.
We begin by dealing with the left-hand side. Observe that $|x - y| {\leqslant}2 |y - z|$, for any $x \in B_r(z)$ and $y \in {\mathbb{R}}^n \setminus B_r(z)$. Moreover, by Lemma \[numestlem4\], we know that $$(u(y) - 2 M)_+^{p - 1} {\geqslant}\min \{ 1, 2^{2 - p} \} u_+(y)^{p - 1} - 2^{p - 1} M^{p - 1}.$$ Using these two facts and that $u {\leqslant}M$ on $B_r(z)$, we compute $$\begin{aligned}
& \int_{B_{\frac{r}{2}}(z)} (u(x) - 2 M)_- \left[ \int_{{\mathbb{R}}^n} \frac{(u(y) - 2 M)_+^{p - 1}}{|x - y|^{n + s p}} \, dy \right] dx \\
& \hspace{30pt} {\geqslant}2^{- n - s p} M \int_{B_{\frac{r}{2}}(z)} \left[ \int_{{\mathbb{R}}^n \setminus B_r(z)} \frac{\min \{ 1, 2^{2 - p} \} u_+(y)^{p - 1} - 2^{p - 1} M^{p - 1}}{|y - z|^{n + s p}} \, dy \right] dx \\
& \hspace{30pt} {\geqslant}\frac{M r^{n - sp}}{C (1 - s)} \operatorname{Tail}(u_+; z, r)^{p - 1} - C r^{n - s p} M^p,\end{aligned}$$ where $C$ now depends on $\varepsilon_0$ too.
On the other hand, by taking advantage once again of Lemma \[numestlem4\] and of the fact that $u {\geqslant}0$ on $B_r(z)$, it is not hard to check that the right-hand side of can be bounded by $$\begin{aligned}
\frac{C r^{n - s p}}{1 - s} \Big( M^p + M \operatorname{Tail}(u_-; z, r)^{p - 1} \Big).\end{aligned}$$ By comparing these last two estimates with , we are easily led to .
We now proceed to prove the actual theorem. We consider separately the two cases $n {\geqslant}s p$ and $n < sp$.
If $n {\geqslant}s p$, we take advantage of the boundedness given by Proposition \[ulocboundprop\]. We notice that estimate implies in this setting that, for any $\delta_1 \in (0, 1]$, $$\label{Hineulocbound-1}
\sup_{B_r(z)} u {\leqslant}C \, \delta_1^{- \frac{p - 1}{\beta p}} \left( \dashint_{B_{2 r}(z)} u(x)^p \, dx \right)^{\frac{1}{p}} + \delta_1 \operatorname{Tail}(u_+; z, r) + \delta_1^{\frac{p - 1}{p}} r^{\frac{\lambda + n \varepsilon}{p}} d,$$ where $C {\geqslant}1$ and $\beta {\geqslant}\varepsilon_0 / 2$ depend on $n$, $s$, $p$, $\varepsilon_0$ and $H$. Both constants do not blow up as $s \rightarrow 1^-$, when $n > p$. By , this becomes $$\sup_{B_r(z)} u {\leqslant}C \left[ \delta_1^{ - \frac{p - 1}{\beta p}} \left( \dashint_{B_{2 r}(z)} u(x)^p \, dx \right)^{\frac{1}{p}} + \delta_1^{\frac{p - 1}{p}} \left( \sup_{B_r(z)} u + \operatorname{Tail}(u_-; z, r) + r^{\frac{\lambda + n \varepsilon}{p}} d \right) \right].$$ Fix now any $q \in (0, p)$. Using the weighted Young’s inequality, we have $$\begin{aligned}
\left( \dashint_{B_{2 r}(z)} u(x)^p \, dx \right)^{\frac{1}{p}} & {\leqslant}\left( \sup_{B_{2 r}(z)} u \right)^{\frac{p - q}{p}} \left( \dashint_{B_{2 r}(z)} u(x)^q \, dx \right)^{\frac{1}{p}} \\
& {\leqslant}\delta_2^{\frac{p}{p - q}} \sup_{B_{2 r}(z)} u + \delta_2^{- \frac{p}{q}} \left( \dashint_{B_{2 r}(z)} u(x)^q \, dx \right)^{\frac{1}{q}},\end{aligned}$$ for any $\delta_2 > 0$. By taking $\delta_1$, $\delta_2$ sufficiently small, we obtain $$\label{Hinelocbound2}
\sup_{B_r(z)} u {\leqslant}\frac{1}{2} \sup_{B_{2 r}(z)} u + C \left[ \left( \dashint_{B_{2 r}(z)} u(x)^q \, dx \right)^{\frac{1}{q}} + \operatorname{Tail}(u_-; z, r) + r^{\frac{\lambda + n \varepsilon}{p}} d \right].$$
On the other hand, when $n < s p$, we already know from the fractional Morrey embedding that $u$ is bounded. In particular, a careful analysis of the proofs of [@DPV12 Theorem 8.2] and [@Giu03 Lemma 2.2] reveals, together with the sharp Poincaré-Wirtinger-type estimate of [@BBM02; @P04] and the extension inequality , that $$\sup_{B_r(z)} u^p {\leqslant}\frac{C}{\left( 2^{\frac{s p - n}{p}} - 1 \right)^p} \left[ (1 - s) \delta_3^{s p - n} r^{s p - n} [u_+]_{W^{s, p}(B_r(z))}^p + \delta_3^{- n} r^{- n} \| u_+ \|_{L^p(B_r(z))}^p \right],$$ for any $\delta_3 \in [0, 1]$ and for some constant $C {\geqslant}1$ depending only on $n$ and $p$. To get this estimate, one could prove it first in the case $r = 1$ and then scale it. The arbitrary parameter $\delta_3$ is essentially the constant $R_0$ appearing in [@DPV12 formula (8.9)], while the denominator of the fraction in front of the square brackets comes from the proof of [@Giu03 Lemma 2.2]. By using —with $k = 0$, $x_0 = z$ and $R = 2 r$—to estimate the Gagliardo seminorm of $u_+$, we are led to $$\sup_{B_r(z)} u^p {\leqslant}C \left[ \delta_3^{- n} \dashint_{B_{2 r}(z)} u(x)^p \, dx + \delta_3^{s p - n} \Bigg( \operatorname{Tail}(u_+; z, r)^p + r^{\lambda + n \varepsilon} d^p \Bigg) \right],$$ where $C$ now depends on $n$, $s$, $p$ and $H$, but does not blow as $s \rightarrow 1^-$ if $p > n$ is fixed. By arguing as before, in the case $n {\geqslant}s p$, and starting from this last inequality instead of , we deduce once again.
We plan to take advantage of [@Giu03 Lemma 7.1]—or Lemma \[induclem\] here—to reabsorb the term $(1 / 2)\sup_{B_{2 r}(z)} u$ on the left-hand side of . To do it, we first need to perform an easy covering argument. Let $R{\leqslant}\rho < \tau {\leqslant}2 R$ be fixed. Note that $$B_\rho = \bigcup_{z \in B_{2 \rho - \tau}} B_{\tau - \rho}(z) \quad \mbox{and} \quad B_{2 (\tau - \rho)}(z) \subset B_\tau \mbox{ for any } z \in B_{2 \rho - \tau}.$$ Therefore, by using with $r = \tau - \rho$, we get $$\begin{aligned}
\sup_{B_\rho} u & = \sup_{z \in B_{2 \rho - \tau}} \sup_{B_{\tau - \rho}(z)} u \\
& {\leqslant}\frac{1}{2} \sup_{B_\tau} u + C \sup_{z \in B_{2 \rho - \tau}} \left[ \frac{\| u \|_{L^q(B_{2 (\tau - \rho)}(z))}}{(\tau - \rho)^{\frac{n}{q}}} + \operatorname{Tail}(u_-; z, \tau - \rho) + (\tau - \rho)^{\frac{\lambda + n \varepsilon}{p}} d \right] \\
& {\leqslant}\frac{1}{2} \sup_{B_\tau} u + C \left[ \frac{\| u \|_{L^q(B_{2 R})}}{(\tau - \rho)^{\frac{n}{q}}} + \operatorname{Tail}(u_-; R) + R^{\frac{\lambda + n \varepsilon}{p}} d \right],\end{aligned}$$ where we also used that $u {\geqslant}0$ in $B_R$. Now we are in position to apply [@Giu03 Lemma 7.1] and finally deduce that $$\sup_{B_R} u {\leqslant}C \left[ \left( \dashint_{B_{2 R}} u(x)^q \, dx \right)^{\frac{1}{q}} + \operatorname{Tail}(u_-; R) + R^{\frac{\lambda + n \varepsilon}{p}} d \right].$$ By choosing $q$ as in Proposition \[weakHarprop\] and combining the above estimate with , we conclude that follows, at least when is in force.
To finish the proof, we only need to show that the additional assumption is not necessary, and that in fact is true for any $x_0 \in \Omega$ and $0 < R < {{\mbox{\normalfont dist}}}(x_0, \partial \Omega) / 2$. Let $x_0$ and $R$ be as such, and take any $z \in B_R(x_0)$. Obviously, $B_R(z) \subset B_{2 R}(x_0) \subset \Omega$. Moreover, as holds under and $u {\geqslant}0$ in $B_{2 R}(x_0)$, we know that $$\begin{aligned}
\sup_{B_{R / 64}(z)} u & {\leqslant}C \left( \inf_{B_{R / 64}(z)} u + \operatorname{Tail}\left( u_-; z, \frac{R}{64} \right) + \left( \frac{R}{64} \right)^{\frac{\lambda + n \varepsilon}{p}} d \right) \\
& {\leqslant}C \left( \inf_{B_{R / 64}(z)} u + \operatorname{Tail}\left( u_-; x_0, R \right) + R^{\frac{\lambda + n \varepsilon}{p}} d \right).\end{aligned}$$ In particular, this implies that, for any $x', y' \in B_R(x_0)$ such that $|x' - y'| < R / 32$, we have $$\label{Harx'y'}
u(x') {\leqslant}C_\star \left( u(y') + \operatorname{Tail}\left( u_-; x_0, R \right) + R^{\frac{\lambda + n \varepsilon}{p}} d \right),$$ with $C_\star$ depending only on $n$, $s$, $p$, $\varepsilon_0$, $\lambda_0$ and $H$. Clearly, we can suppose that $C_\star {\geqslant}2$.
Let $x, y \in B_R(x_0)$ be any two points. We now show that is also true with $x$, $y$ in place of $x'$, $y'$, up to raising $C_\star$ to a universal power. Of course, we can suppose that $|x - y| {\geqslant}R / 32$, otherwise applies to them directly. Denoting by $\overrightarrow{xy}$ the oriented segment with end points $x$ and $y$, we consider $N$ consecutive points $\{ x_i \}_{i = 0}^N \subset \overrightarrow{xy}$ such that $x_0 = x$, $x_N = y$ and $R / 64 {\leqslant}|x_i - x_{i - 1}| < R / 32$ for any $i \in \{ 1, \ldots, N \}$. It is immediate to check that $N {\leqslant}128$. By applying with $x' = x_{i - 1}$ and $y' = x_i$, we get $$u(x_{i - 1}) {\leqslant}C_\star \left( u(x_i) + \operatorname{Tail}\left( u_-; x_0, R \right) + R^{\frac{\lambda + n \varepsilon}{p}} d \right).$$ Iterating such estimate over $i \in \{ 1, \ldots, N \}$, we easily find that $$\begin{aligned}
u(x) & {\leqslant}C_\star^N u(y) + \left( \operatorname{Tail}\left( u_-; x_0, R \right) + R^{\frac{\lambda + n \varepsilon}{p}} d \right) \sum_{i = 1}^N C_\star^i \\
& {\leqslant}C_\star^{129} \left( u(y) + \operatorname{Tail}\left( u_-; x_0, R \right) + R^{\frac{\lambda + n \varepsilon}{p}} d \right).\end{aligned}$$ This leads to , in its full generality.
Applications to minimizers {#minsec}
==========================
In this section, we show that the minimizers of the energy ${\mathcal{E}}$ introduced in are locally bounded and Hölder continuous functions that satisfy a nonlocal Harnack inequality wherever non-negative. That is, we prove Theorems \[boundmainthm\], \[holmainthm\] and \[harmainthm\] for minimizers. Those three results are rephrased—and sometimes more precisely stated—as follows.
\[minboundthm\]\
Let $n \in {\mathbb{N}}$, $s \in (0, 1)$ and $p > 1$ be such that $n {\geqslant}s p$. Let $\Omega$ be an open bounded subset of ${\mathbb{R}}^n$. Assume that $K$ and $F$ respectively satisfy hypotheses , and . If $u$ is a minimizer of ${\mathcal{E}}$ in $\Omega$, then $u \in L^\infty_{{\rm loc}}(\Omega)$. In particular, there exist four constants $C {\geqslant}1$, $R_0 \in (0, \min \{ 1, r_0 / 2 \}]$, $\varepsilon \in (0, sp / n]$ and $\kappa \in \{0, 1\}$ such that, given any $x_0 \in \Omega$ and $0 < R {\leqslant}\min \{ R_0, {{\mbox{\normalfont dist}}}\left( x_0, \partial \Omega \right) \} / 4$, it holds[^5] $$\| u \|_{L^\infty(B_R(x_0))} {\leqslant}C R^{- \frac{n}{p}} \| u \|_{L^p(B_{2 R}(x_0))} + \operatorname{Tail}(u; x_0, R) + d_1^{\frac{1}{p}} R^{\frac{n \varepsilon}{p}} + 2 \kappa.$$ Moreover, the constants can be chosen as follows:
1. if $d_2 = 0$, then $$C = C(n, s, p, \Lambda), \, \, R_0 = \frac{r_0}{2}, \, \, \varepsilon = \frac{s p}{n} \, \mbox{ and } \, \kappa = 0;$$
2. if $d_2 > 0$ and $1 {\leqslant}q {\leqslant}p$, then $$C = C(n, s, p, \Lambda, d_2), \, \, R_0 = \min \left\{ 1, \frac{r_0}{2} \right\}, \, \, \varepsilon = \frac{s p}{n} \, \mbox{ and } \, \kappa = 1;$$
3. if $d_2 > 0$ and $p < q < p^*_s$, then $$\begin{aligned}
& C = C(n, s, p, q, \Lambda, d_2), \, \, R_0 = R_0(n, s, p, q, \Lambda, d_2, r_0, \| u \|_{L^{p^*_\sigma}(\Omega)}), \\
& \varepsilon = 1 - \frac{q}{p^*_\sigma} \, \mbox{ and } \, \kappa = 0, \, \mbox{ for some } \, \sigma = \sigma(n, s, p, q) \in (0, s).\end{aligned}$$
When $n > s p$ we can even take $\sigma = s$, while when $n > p$ both constants $C$ and $R_0$ do not blow up as $s \rightarrow 1^-$.
\[minholdthm\]\
Let $n \in {\mathbb{N}}$, $0 < s_0 {\leqslant}s < 1$ and $p > 1$ be such that $n {\geqslant}s p$. Let $\Omega$ be an open bounded subset of ${\mathbb{R}}^n$. Assume that $K$ satisfies hypotheses , and that $F$ is locally bounded in $u \in {\mathbb{R}}$, uniformly w.r.t $x \in \Omega$. If $u$ is a minimizer of ${\mathcal{E}}$ in $\Omega$, then, $u \in C^\alpha_{{\rm loc}}(\Omega)$ for some $\alpha \in (0, 1)$. In particular, there exists a constant $C {\geqslant}1$ such that, given any $x_0 \in \Omega$ and $0 < R {\leqslant}\min \{ r_0 / 2, {{\mbox{\normalfont dist}}}\left(x_0, \partial \Omega \right) \} / 16$, it holds $$[ u ]_{C^\alpha(B_R(x_0))} {\leqslant}\frac{C}{R^\alpha} \left( \| u \|_{L^\infty(B_{4 R}(x_0))} + \operatorname{Tail}(u; x_0, 4 R) + R^s \| F(\cdot, u) \|_{L^\infty(B_{8 R}(x_0))}^{1 / p} \right).$$ The constants $\alpha$ and $C$ depend only on $n$, $s_0$, $p$ and $\Lambda$.
\[minharthm\]\
Let $n \in {\mathbb{N}}$, $s \in (0, 1)$ and $p > 1$. Let $\Omega$ be an open bounded subset of ${\mathbb{R}}^n$. Assume that $K$ satisfies hypotheses , and that $F$ is locally bounded in $u \in {\mathbb{R}}$, uniformly w.r.t $x \in \Omega$. Let $u$ be a minimizer of ${\mathcal{E}}$ in $\Omega$ such that $u {\geqslant}0$ in $\Omega$. Then, for any $x_0 \in \Omega$ and $0 < R < {{\mbox{\normalfont dist}}}(x_0, \partial \Omega) / 2$, it holds $$\sup_{B_R(x_0)} u {\leqslant}C \left( \inf_{B_R(x_0)} u + \operatorname{Tail}(u_-; x_0, R) + R^s \| F(\cdot, u) \|_{L^\infty(B_{2 R}(x_0))}^{1 / p} \right),$$ for some $C {\geqslant}1$ depending only on $n$, $s$, $p$ and $\Lambda$. When $n \notin \{1, p \}$, the constant $C$ does not blow up as $s \rightarrow 1^-$.
Theorems \[minboundthm\]-\[minharthm\] will be proved by showing that the minimizers of ${\mathcal{E}}$ belong to a fractional De Giorgi class, so that Theorems \[DGboundthm\],\[DGholdthm\] and \[DGharthm\] may be applied to them.
Before proceeding to the proof of this inclusion, we define the notions of sub- and superminimizers as one-sided generalizations of the minimizers introduced in Definition \[mindef\].
Let $\Omega \subset {\mathbb{R}}^n$ be a bounded measurable set. A function $u \in {\mathbb{W}}^{s, p}(\Omega)$ is said to be a *subminimizer* (*superminimizer*) of ${\mathcal{E}}$ in $\Omega$ if $$\label{minine}
{\mathcal{E}}(u; \Omega) {\leqslant}{\mathcal{E}}(u + \varphi; \Omega),$$ for any non-positive (non-negative) measurable function $\varphi: {\mathbb{R}}^n \to {\mathbb{R}}$ supported inside $\Omega$.
Recalling Definition \[mindef\], minimizers are, in particular, sub- and superminimizers. Conversely, it is not hard to see that if $u$ is at the same time a sub- and a superminimizer, then it is a minimizer as well.
Even if not immediately obvious, the minimality property introduced above behaves well with respect to set inclusion. That is, if $\Omega' \subset \Omega \subset {\mathbb{R}}^n$ and $u$ is a sub- or superminimizer in $\Omega$, then $u$ is also a sub- or superminimizer in $\Omega'$ (see e.g. [@CV15 Remark 1.2] for a more detailed explanation in the case of minimizers).
Furthermore, it is easy to check that $u$ is a subminimizer (superminimizer) for ${\mathcal{E}}$ if and only if $-u$ is a superminimizer (subminimizer) for the energy having the same interaction term and potential determined by $\widetilde{F}(x, u) = F(x, -u)$. Noticing that $\widetilde{F}$ satisfies the same growth assumption of $F$, this allows us to focus on subminimizers only.
We can now state the following result, where we prove that the minimizers of ${\mathcal{E}}$ belong to a suitable fractional De Giorgi class.
Unless otherwise specified, in what follows we assume $n \in {\mathbb{N}}$, $s \in (0, 1)$, $p > 1$ to be given parameters, and that $K$ and $F$ satisfy , and , respectively.
\[minareDGprop\] Let $u$ be a subminimizer of ${\mathcal{E}}$ in a bounded open set $\Omega \subset {\mathbb{R}}^n$. Then, there exist four constants $R_0 \in (0, r_0 / 2]$, $k_0 \in [-\infty, 1]$, $H {\geqslant}1$ and $\varepsilon \in (0, sp / n]$, such that $$\label{minareDGine}
\begin{aligned}
& [(u - k)_+]_{W^{s, p}(B_r(x_0))}^p + \int_{B_r(x_0)} (u(x) - k)_+ \left[ \int_{B_{2 R_0}(x)} \frac{(u(y) - k)_-^{p - 1}}{|x - y|^{n + s p}} \, dy \right] dx \\
& \hspace{15pt} {\leqslant}\frac{H}{1 - s} \Bigg[ \left( d_1 + \frac{|k|^p}{R^{n \varepsilon}} \right) |A^+(k, x_0, R)|^{1 - \frac{s p}{n} + \varepsilon} + \frac{R^{(1 - s) p}}{(R - r)^p} \| (u - k)_+ \|_{L^p(B_R(x_0))}^p \\
& \hspace{15pt} \quad + \frac{R^{n + s p}}{(R - r)^{n + s p}} \| (u - k)_+ \|_{L^1(B_R(x_0))} \overline{\operatorname{Tail}}((u - k)_+; x_0, r)^{p - 1} \Bigg],
\end{aligned}$$ for any $x_0 \in \Omega$, $0 < r < R {\leqslant}\min \{ R_0, {{\mbox{\normalfont dist}}}(x_0, \partial \Omega) \}$ and $k {\geqslant}k_0$. Consequently, $u$ belongs to the following fractional De Giorgi classes:
1. if $d_2 = 0$, then $$u \in \operatorname{DG}^{s, p}_+ \left( \Omega; d_1^{\frac{1}{p}}, H, -\infty, \frac{ s p }{n}, 0, \frac{r_0}{2} \right),$$ with $H = H(n, p, \Lambda)$;
2. if $d_2 > 0$ and $1 {\leqslant}q {\leqslant}p$, then $$u \in \operatorname{DG}^{s, p}_+ \left( \Omega; d_1^{\frac{1}{p}}, H, 1, \frac{ s p }{n}, 0, \min \left\{ 1, \frac{r_0}{2} \right\} \right),$$ with $H = H(n, p, \Lambda, d_2)$;
3. if $d_2 > 0$, $n {\geqslant}s p$ and $p < q < p^*_s$, then $$u \in \operatorname{DG}^{s, p}_+ \left( \Omega; d_1^{\frac{1}{p}}, H, 0, 1 - \frac{q}{p^*_\sigma}, 0, R_0 \right),$$ with $H = H(n, p, q, \Lambda, d_2)$, $R_0 = R_0(n, s, p, q, \Lambda, d_2, r_0, \| u \|_{L^{p^*_\sigma}(\Omega)})$ and for some constant $\sigma = \sigma(n, s, p, q) \in (0, s)$. When $n > sp$ we can even take $\sigma = s$, while when $n > p$ the constant $R_0$ does not blow up as $s \rightarrow 1^-$.
An analogous statement holds for superminimizers and the classes $\operatorname{DG}_-^{s, p}$.
Without loss of generality, we suppose $x_0 = 0$.
Let $r {\leqslant}\rho < \tau {\leqslant}R$ and consider a cut-off function $\eta \in C^\infty_0({\mathbb{R}}^n)$ satisfying $0 {\leqslant}\eta {\leqslant}1$ in ${\mathbb{R}}^n$, $\operatorname{supp}(\eta) = B_{(\tau + \rho) / 2}$, $\eta = 1$ in $B_\rho$ and $|\nabla \eta| {\leqslant}4 / (\tau - \rho)$ in ${\mathbb{R}}^n$. For any fixed $k {\geqslant}0$, we write $w_\pm := (u - k)_\pm$ and choose $v := u - \eta w_+$ as a test function in . Notice that $\operatorname{supp}(\eta w_+) = A^+(k, (\tau + \rho) / 2)$. Hence, and recalling as well as notation , we get $$\begin{aligned}
0 & {\leqslant}\frac{1}{2 p} \iint_{{\mathscr{C}}_{B_\tau}} \Big[ |v(x) - v(y)|^p - |u(x) - u(y)|^p \Big] d\mu \\
& \quad + \int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} \Big[ 2 d_1 + d_2 \left( |u(x)|^q + |v(x)|^q \right) \Big] \, dx.\end{aligned}$$ Recalling and , this is equivalent to $$\label{minareDG1}
\begin{aligned}
0 & {\leqslant}\int_{B_\tau} \int_{B_\tau} \Big[ |v(x) - v(y)|^p - |u(x) - u(y)|^p \Big] \, d\mu \\
& \quad + 2 \int_{B_\tau} \int_{{\mathbb{R}}^n \setminus B_\tau} \Big[ |v(x) - v(y)|^p - |u(x) - u(y)|^p \Big] \, d\mu \\
& \quad + 2 p \int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} \Big[ 2 d_1 + d_2 \left( |u(x)|^q + |v(x)|^q \right) \Big] \, dx.
\end{aligned}$$
We begin by dealing with the two terms involving double integrals. The following facts hold true: $$\begin{aligned}
\label{minareDGprop-1}
& \mbox{if either } x \notin A^+(k) \mbox{ or } y \notin A^+(k), \mbox{ then } |v(x) - v(y)|^p {\leqslant}|u(x) - u(y)|^p,\\
\label{minareDGprop1}
& \mbox{if } x, y \in A^+(k, \rho), \mbox{ then } |v(x) - v(y)|^p - |u(x) - u(y)|^p = - |w_+(x) - w_+(y)|^p,\\
\label{minareDGprop3}
\begin{split}
& \mbox{if } x \in A^+(k, \rho) \mbox{ and } y \notin A^+(k), \mbox{ then} \\
& \hspace{10pt} |v(x) - v(y)|^p - |u(x) - u(y)|^p {\leqslant}- \frac{1}{2} \, |w_+(x) - w_+(y)|^p - \frac{p}{2} \, w_-(y)^{p - 1} w_+(x),
\end{split}\\
\label{minareDGprop2}
\begin{split}
& \mbox{if } x, y \in A^+(k), \mbox{ then} \\
& \hspace{10pt} |v(x) - v(y)|^p {\leqslant}C \left[ (1 - \eta(x))^p |w_+(x) - w_+(y)|^p + w_+(y)^p \frac{|x - y|^p}{(\tau - \rho)^p} \right],
\end{split}\end{aligned}$$ for some constant $C {\geqslant}1$ depending only on $p$.
Indeed, and are immediate consequences of the definition of $v$. On the other hand, for general $x, y \in A^+(k)$ we may compute $$\begin{aligned}
|v(x) - v(y)|^p & = |(1 - \eta(x)) w_+(x) - (1 - \eta(y)) w_+(y)|^p \\
& {\leqslant}2^{p - 1} \Big[ (1 - \eta(x))^p |w_+(x) - w_+(y)|^p + w_+(y)^p |\eta(x) - \eta(y)|^p \Big],\end{aligned}$$ and follows in view of the properties of $\eta$. Finally, to get we observe that for $x \in A^+(k, \tau)$ and $y \notin A^+(k)$, $$\begin{aligned}
|v(x) - v(y)|^p - |u(x) - u(y)|^p = \left( (1 - \eta(x)) w_+(x) + w_-(y) \right)^p - \left( w_+(x) + w_-(y) \right)^p,\end{aligned}$$ and then apply Lemma \[numestlem1\] (with $\theta = 1/2$).
Thanks to properties -, we are in position to estimate the first double integral in . By , , , the symmetry assumption on $K$ and the properties of $w_\pm$, we get $$\begin{aligned}
& \int_{B_\rho} \int_{B_\rho} \Big[ |v(x) - v(y)|^p - |u(x) - u(y)|^p \Big] \, d\mu \\
& \hspace{50pt} {\leqslant}- \frac{1}{2} \int_{B_\rho} \int_{B_\rho} |w_+(x) - w_+(y)|^p \, d\mu - \frac{p}{2} \int_{B_\rho} \int_{B_\rho} w_-(y)^{p - 1} w_+(x) \, d\mu.\end{aligned}$$ On the other hand, after applying , , and , an immediate computation reveals that $$\begin{aligned}
& \iint_{B_\tau^2 \setminus B_\rho^2} \Big[ |v(x) - v(y)|^p - |u(x) - u(y)|^p \Big] \, d\mu \\
& \hspace{40pt} {\leqslant}C \left[ \iint_{B_\tau^2 \setminus B_\rho^2} |w_+(x) - w_+(y)|^p \, d\mu + \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \int_{B_R} w_+(x)^p \, dx \right] \\
& \hspace{40pt} \quad - \frac{p}{2} \int_{B_\rho} w_+(x) \left( \int_{B_\tau \setminus B_\rho} w_-(y)^{p - 1} K(x, y) \, dy \right) dx,\end{aligned}$$ with $C$ now depending also on $n$ and $\Lambda$. The combination of these last two estimates and another use of yield $$\label{minareDG2}
\begin{aligned}
& \int_{B_\tau} \int_{B_\tau} \Big[ |v(x) - v(y)|^p - |u(x) - u(y)|^p \Big] \, d\mu \\
& \hspace{20pt} {\leqslant}- \frac{1 - s}{C} \left[ [w_+]_{W^{s, p}(B_\rho)}^p + \int_{B_\rho} w_+(x) \left( \int_{B_\tau} \frac{w_-(y)^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx \right] \\
& \hspace{20pt} \quad + C \left[ (1 - s) \iint_{B_\tau^2 \setminus B_\rho^2} \frac{|w_+(x) - w_+(y)|^p}{|x - y|^{n + s p}} \, dx dy + \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p \right].
\end{aligned}$$
Now we deal with the second double integral in . We claim that $$\begin{aligned}
\label{minareDGprop0}
& \mbox{if } x, y \notin A^+ \left( k, \frac{\tau + \rho}{2} \right), \mbox{ then } |v(x) - v(y)|^p = |u(x) - u(y)|^p,\\
\label{minareDGprop4}
\begin{split}
& \mbox{if } x \in A^+(k) \mbox{ and } y \in A^+(k) \setminus B_\tau, \mbox{ then} \\
& \hspace{10pt} |v(x) - v(y)|^p - |u(x) - u(y)|^p {\leqslant}p w_+(y)^{p - 1} w_+(x).
\end{split}\end{aligned}$$ Since is an obvious consequence of the fact that $\operatorname{supp}(\eta w_+) = A(k, (\tau + \rho) / 2)$, we can concentrate on . Letting $x$ and $y$ as prescribed, we have $$\begin{aligned}
|v(x) - v(y)|^p - |u(x) - u(y)|^p = |(1 - \eta(x)) w_+(x) - w_+(y)|^p - |w_+(x) - w_+(y)|^p.\end{aligned}$$ The inequality stated in can then be obtained by employing Lemma \[numestlem2\].
In view of , , , again , and the fact that $$|x - y| {\geqslant}|y| - |x| {\geqslant}\left( 1 - \frac{\tau + \rho}{2 \tau} \right) |y| = \frac{\tau - \rho}{2 \tau} |y| {\geqslant}\frac{\tau - \rho}{2 R} |y|,$$ for any $x \in B_{(\tau + \rho) / 2}$ and $y \notin B_\tau$, it is not hard to see that $$\begin{aligned}
& \int_{B_\tau} \int_{{\mathbb{R}}^n \setminus B_\tau} \Big[ |v(x) - v(y)|^p - |u(x) - u(y)|^p \Big] \, d\mu \\
& \hspace{50pt} {\leqslant}C \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1} \\
& \hspace{50pt} \quad - \frac{1 - s}{C} \int_{B_\rho} w_+(x) \left( \int_{B_{r_0}(x) \setminus B_\tau} \frac{w_-(y)^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx.\end{aligned}$$ By putting together this last inequality with and , we obtain $$\label{minareDG3}
\begin{aligned}
& [w_+]_{W^{s, p}(B_\rho)}^p + \int_{B_\rho} w_+(x) \left( \int_{B_{r_0}(x)} \frac{w_-(y)^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx \\
& \hspace{15pt} {\leqslant}C \iint_{B_\tau^2 \setminus B_\rho^2} \frac{|w_+(x) - w_+(y)|^p}{|x - y|^{n + s p}} \, dx dy \\
& \hspace{15pt} \quad + \frac{C}{1 - s} \Bigg[ \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1} + \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p \\
& \hspace{15pt} \quad + d_1 |A^+(k, R)| + d_2 \int_{A^+(k, \tau)} \left( |u(x)|^q + |v(x)|^q \right) \, dx \Bigg],
\end{aligned}$$ for some $C {\geqslant}1$ depending only on $n$, $p$ and $\Lambda$.
We set $$\label{Phidef}
\Phi(t) := [w_+]_{W^{s, p}(B_t)}^p + \int_{B_t} w_+(x) \left( \int_{B_{r_0}(x)} \frac{w_-(y)^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx + \Psi(t),$$ with $$\Psi(t) := \begin{dcases}
0 & \mbox{if } d_2 = 0, \mbox{ or } d_2 > 0, \, n {\geqslant}sp \mbox{ and } 1 {\leqslant}q {\leqslant}p,\\
\frac{d_2}{1 - s} \| u \|_{L^q(A^+(k, t))}^q & \mbox{if } d_2 > 0, \, n {\geqslant}s p \mbox{ and } p < q < p^*_s,
\end{dcases}$$ for any $0 < t {\leqslant}R$, and we claim that $$\label{minareDG4}
\begin{aligned}
\Phi(\rho) & {\leqslant}C_\flat \left[ \Phi(\tau) - \Phi(\rho) \right] + \frac{C_\flat}{1 - s} \Bigg[ \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1} \\
& \quad + (1 + d_2) \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p + (d_1 + d_2 k^p R^{- n \varepsilon}) |A^+(k, R)|^{1 - \frac{s p}{n} + \varepsilon} \Bigg],
\end{aligned}$$ for some $C_\flat {\geqslant}1$ possibly depending on $n$, $p$, $q$, $\Lambda$, and $\varepsilon \in (0, sp / n]$.
Note that, in the case of $d_2 = 0$, claim follows immediately from , with $\varepsilon = s p / n$. To check the validity of when $d_2 > 0$, we consider separately the two possibilities $1 {\leqslant}q {\leqslant}p$ and $p < q < p^*_s$ (with $n {\geqslant}sp$ in the latter case).
Suppose that $1 {\leqslant}q {\leqslant}p$. Assuming $k {\geqslant}1$, for $x \in A^+(k)$ we have $$u(x) > k {\geqslant}1 \quad \mbox{and} \quad v(x) = (1 - \eta(x)) w_+(x) + k {\geqslant}k {\geqslant}1.$$ Therefore, $$\begin{aligned}
|u(x)|^q + |v(x)|^q & {\leqslant}|u(x)|^p + |v(x)|^p = |w_+(x) + k|^p + |(1 - \eta(x)) w_+(x) + k|^p \\
& {\leqslant}2^p \left( w_+(x)^p + k^p \right).\end{aligned}$$ By taking $R {\leqslant}1$, it follows that $$\begin{aligned}
\int_{A^+(k, \tau)} \left( |u(x)|^q + |v(x)|^q \right) \, dx & {\leqslant}2^p \int_{A^+(k, \tau)} \left( w_+(x)^p + k^p \right) dx \\
& {\leqslant}2^p \left[ \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p + k^p R^{- s p} |A^+(k, R)| \right].\end{aligned}$$ Using this with , we easily deduce , again with $\varepsilon = sp / n$.
We now deal with the case of $n {\geqslant}s p$ and $p < q < p^*_s$. Let $$\label{sigmamindef}
\sigma := \begin{cases}
s & \quad \mbox{if } n > s p, \\
\max \left\{ 2 s - 1, \dfrac{(2 q - p) n}{2 p q} \right\} & \quad \mbox{if } n = s p,
\end{cases}$$ and notice that $n > \sigma p$ and $q < p^*_\sigma$ in both cases $n > sp$ and $n = sp$. Also set $$\label{epsigmamindef}
\varepsilon_\sigma := 1 - \frac{q}{p^*_\sigma} \in \left( 0, \frac{s p}{n} \right).$$ We add to both sides of the quantity $d_2 (1 - s)^{-1} \| u \|_{L^q(A^+(k, \rho))}^q$. We obtain $$\label{minareDG5}
\begin{aligned}
\Phi(\rho) & {\leqslant}\frac{C_1}{1 - s} \Bigg[ (1 - s) \iint_{B_\tau^2 \setminus B_\rho^2} \frac{|w_+(x) - w_+(y)|^p}{|x - y|^{n + s p}} \, dx dy \\
& \quad + \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1} + \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p \\
& \quad + d_1 |A^+(k, R)| + d_2 \int_{A^+(k, \tau)} \left( |u(x)|^q + |v(x)|^q \right) \, dx \Bigg],
\end{aligned}$$ with $C_1 {\geqslant}1$ depending on $n$, $p$ and $\Lambda$. We proceed to estimate the last term of . By restricting ourselves to $k {\geqslant}0$, on $A^+(k)$ we have $$\label{u+vest}
\begin{aligned}
|u|^q + |v|^q & = |(1 - \eta) u + \eta w_+ + \eta k|^q + |(1 - \eta) u + \eta k|^q \\
& {\leqslant}3^q \Big( (1 - \eta)^q |u|^q + (\eta w_+)^q + k^q \Big).
\end{aligned}$$ On the one hand, it holds $$k^{p^*_{\sigma}} |A^+(k, R)| {\leqslant}\int_{A^+(k, R)} u(x)^{p^*_\sigma} \, dx {\leqslant}\int_{\Omega} |u(x)|^{p^*_\sigma} \, dx,$$ and thus $$\label{minarekest}
\begin{aligned}
3^q \int_{A^+(k, \tau)} k^q \, dx & = 3^q k^p \left( k^{p^*_\sigma} |A^+(k, R)| \right)^{\frac{q - p}{p^*_\sigma}} |A^+(k, R)|^{1 - \frac{q - p}{p^*_\sigma}} \\
& {\leqslant}3^q \| u \|_{L^{p^*_\sigma}(\Omega)}^{q - p} k^p |B_R|^{\frac{(s - \sigma) p}{n}} |A^+(k, R)|^{1 - \frac{s p}{n} + \varepsilon_\sigma} \\
& {\leqslant}k^p R^{- n \varepsilon_\sigma} |A^+(k, R)|^{1 - \frac{sp}{n} + \varepsilon_\sigma},
\end{aligned}$$ provided $$R {\leqslant}R_0 {\leqslant}\left[ \frac{1}{3^{q} |B_1|^{\frac{(s - \sigma) p}{n}} \| u \|_{L^{p^*_\sigma}(\Omega)}^{q - p}} \right]^{\frac{1}{n \varepsilon_\sigma + (s - \sigma) p}}.$$ On the other hand, with the help of Hölder’s inequality and Corollary \[sobinecor\], we compute $$\begin{aligned}
\int_{A^+(k, \tau)} (\eta(x) w_+(x))^q \, dx & {\leqslant}|B_R|^{\varepsilon_\sigma} \left[ \int_{B_\tau} w_+(x)^{p^*_\sigma} \, dx \right]^{\frac{q - p}{p^*_\sigma}} \left[ \int_{B_\tau} (\eta(x) w_+(x))^{p^*_\sigma} \, dx \right]^{\frac{p}{p^*_\sigma}} \\
& {\leqslant}C_2 \frac{(1 - \sigma) R^{n \varepsilon_\sigma}}{(n - \sigma p)^{p - 1}} \| u \|_{L^{p^*_\sigma}(\Omega)}^{q - p} \left[ [w_+]_{W^{\sigma, p}(B_\tau)}^p + \frac{\| w_+ \|_{L^p(B_R)}^p}{(\tau - \rho)^{\sigma p}} \right],\end{aligned}$$ with $C_2 {\geqslant}1$ depending only on $n$ and $p$. Notice that $1 - \sigma {\leqslant}2 (1 - s)$, by definition of $\sigma$. By taking $$R {\leqslant}R_0 {\leqslant}\min \left\{ \frac{1}{2}, \left[ \frac{(n - \sigma p)^{p - 1}}{4 C_1 C_2 d_2 3^q \| u \|_{L^{p^*_\sigma}(\Omega)}^{q - p}} \right]^{\frac{1}{n \varepsilon_\sigma}} \right\},$$ in the previous estimate and using Lemma \[sobinclem0\] (with $\delta = 1$), we get $$\begin{aligned}
\frac{3^q C_1 d_2}{1 - s} \int_{A^+(k, \tau)} (\eta(x) w_+(x))^q \, dx & {\leqslant}\frac{1}{2} \left( [w_+]_{W^{\sigma, p}(B_\tau)}^p + \frac{\| w_+ \|_{L^p(B_R)}^p}{(\tau - \rho)^{\sigma p}} \right) \\
& {\leqslant}\frac{1}{2} \left( [w_+]_{W^{s, p}(B_\tau)}^p + \frac{R^{(1 - s) p}}{(\tau - \rho)^{p}} \| w_+ \|_{L^p(B_R)}^p \right).\end{aligned}$$ By this, , and the definition of $\Phi$, we see that $$\begin{aligned}
&\frac{C_1 d_2}{1 - s} \int_{A^+(k, \tau)} \left( |u(x)|^q + |v(x)|^q \right) \, dx \\
& \hspace{15pt} {\leqslant}\frac{1}{2} \left( [w_+]_{W^{s, p}(B_\tau)}^p + \frac{R^{(1 - s) p}}{(\tau - \rho)^{p}} \| w_+ \|_{L^p(B_R)}^p \right) \\
& \hspace{15pt} \quad + \frac{C_1 d_2}{1 - s} \left( 3^{q} \int_{B_\tau \setminus B_\rho} |u(x)|^q \, dx + \frac{k^p}{R^{n \varepsilon_\sigma}} |A^+(k, R)|^{1 - \frac{sp}{n} + \varepsilon_\sigma} \right) \\
& \hspace{15pt} {\leqslant}\frac{1}{2} \Phi(\rho) + C \left[ \Phi(\tau) - \Phi(\rho) + \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p + \frac{d_2 k^p |A^+(k, R)|^{1 - \frac{sp}{n} + \varepsilon_\sigma}}{(1- s) R^{n \varepsilon_\sigma}} \right],\end{aligned}$$ for some $C {\geqslant}1$ depending on $n$, $p$, $q$ and $\Lambda$. This and yield $$\begin{aligned}
\Phi(\rho) & {\leqslant}\frac{1}{2} \Phi(\rho) + C \left[ \Phi(\tau) - \Phi(\rho) \right] + \frac{C}{1 - s} \Bigg[ \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1} \\
& \quad + \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p + (d_1 + d_2 k^p R^{- n \varepsilon_\sigma}) |A^+(k, R)|^{1 - \frac{s p}{n} + \varepsilon_\sigma} \Bigg],\end{aligned}$$ at least if $$R {\leqslant}R_0 {\leqslant}|B_1|^{-\frac{1}{n}},$$ since $\varepsilon_\sigma {\leqslant}sp / n$. After subtracting to both sides the term $\Phi(\rho) / 2$, we reach , with $\varepsilon = \varepsilon_\sigma$.
By adding the quantity $C_\flat \Phi(\rho)$ to both sides of and dividing by $1 + C_\flat$, we get $$\begin{aligned}
\Phi(\rho) & {\leqslant}\gamma \Phi(\tau) + \frac{\gamma}{1 - s} \Bigg[ \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1} \\
& \quad + (1 + d_2) \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p + (d_1 + d_2 k^p R^{- n \varepsilon}) |A^+(k, R)|^{1 - \frac{s p}{n} + \varepsilon} \Bigg],\end{aligned}$$ for any $0 < r {\leqslant}\rho < \tau {\leqslant}R$, with $\gamma = C_\flat / (1 + C_\flat) \in (0, 1)$. Inequality then follows after an application of Lemma \[induclem\].
In view of Proposition \[minareDGprop\] and the three main results of Section \[DGsec\], the validity of Theorems \[minboundthm\]-\[minharthm\] is easily deduced.
The local boundedness of $u$ claimed in Theorem \[minboundthm\] is an immediate consequence of Proposition \[minareDGprop\] and Theorem \[DGboundthm\] combined.
To check that also Theorem \[minholdthm\] is true, we first observe that, since we already know that $u$ is locally bounded, we may view it as a minimizer of ${\mathcal{E}}$ in $B_{8 R}(x_0) \subset \subset \Omega$, with potential $F$ now fulfilling with $d_1 = \| F(\cdot, u) \|_{L^\infty(B_{8 R}(x_0))}$ and $d_2 = 0$. Theorem \[minholdthm\] then follows by a straightforward application of Proposition \[minareDGprop\] and Theorem \[DGholdthm\].
Notice that the quantities $\alpha$ and $C$ are both claimed to be independent of $s$ in the statement of Theorem \[minholdthm\], even when $n = 1$. Apparently, this is in contradiction with Theorem \[DGholdthm\]. However, going through the proofs of Lemma \[growthlem\] it can be checked that $\alpha$ and $C$ can be chosen to be uniform in $s$, as long as $s$ is far from $1$. This is the case here, since, by assumption, $s {\leqslant}1 / p < 1$, if $n = 1$.
In an analogous way, we deduce Theorem \[minharthm\] from Theorem \[DGharthm\]. For the case $n = 1$, recall the opening remark of Subsection \[harsubsec\].
Applications to solutions {#solsec}
=========================
In the present section, we prove Theorems \[boundmainthm\], \[holmainthm\] and \[harmainthm\] in the case of solutions. Similarly to what we did in Section \[minsec\], we show that weak solutions of the integral equation are in a fractional De Giorgi class, so that their boundedness, Hölder continuity and Harnack property are readily deduced from Theorems \[DGboundthm\], \[DGholdthm\] and \[DGharthm\], respectively. The detailed statements of these results are reported here below.
\[solboundthm\]\
Let $n \in {\mathbb{N}}$, $s \in (0, 1)$ and $p > 1$ be such that $n {\geqslant}s p$. Let $\Omega$ be an open bounded subset of ${\mathbb{R}}^n$. Assume that $K$ and $f$ respectively satisfy hypotheses , and . If $u$ is a weak solution of in $\Omega$, then $u \in L^\infty_{{\rm loc}}(\Omega)$. In particular, there exist four constants $C {\geqslant}1$, $R_0 \in (0, r_0 / 2]$, $\varepsilon \in (0, sp / n]$ and $\kappa \in \{ 0, 1 \}$ such that, given any $x_0 \in \Omega$ and $0 < R {\leqslant}\min \{ R_0, {{\mbox{\normalfont dist}}}\left( x_0, \partial \Omega \right) \} / 4$, it holds $$\| u \|_{L^\infty(B_R(x_0))} {\leqslant}C R^{-\frac{n}{p}} \| u \|_{L^p(B_{2 R}(x_0))} + \operatorname{Tail}(u; x_0, R) + d_1^{\frac{1}{p - 1}} R^{\frac{n \varepsilon}{p} + \frac{s}{p - 1}} + 2 \kappa.$$ Moreover, the constants can be chosen as follows:
1. if $d_2 = 0$, then $$C = C(n, s, p, \Lambda), \, \, R_0 = \frac{r_0}{2}, \, \, \varepsilon = \frac{s p}{n} \, \mbox{ and } \, \kappa = 0;$$
2. if $d_2 > 0$ and $1 {\leqslant}q {\leqslant}p$, then $$C = C(n, s, p, \Lambda, d_2), \, \, R_0 = \min \left\{ 1, \frac{r_0}{2} \right\}, \, \, \varepsilon = \frac{s p}{n} \, \mbox{ and } \, \kappa = 1;$$
3. if $d_2 > 0$ and $p < q < p^*_s$, then $$\begin{aligned}
& C = C(n, s, p, q, \Lambda, d_2), \, \, R_0 = R_0(n, s, p, q, \Lambda, d_2, \| u \|_{L^{p^*_\sigma}(\Omega)}, r_0),\\
& \varepsilon = 1 - \frac{q}{p^*_\sigma} \, \mbox{ and } \, \kappa = 0, \, \mbox{ for some } \, \sigma = \sigma(n, s, p, q) \in (0, s).\end{aligned}$$
When $n > s p$ we can even take $\sigma = s$, while when $n > p$ both constants $C$ and $R_0$ do not blow up as $s \rightarrow 1^-$.
\[solholdthm\]\
Let $n \in {\mathbb{N}}$, $0 < s_0 {\leqslant}s < 1$ and $p > 1$ be such that $n {\geqslant}s p$. Let $\Omega$ be an open bounded subset of ${\mathbb{R}}^n$. Assume that $K$ satisfies hypotheses , and that $f$ is locally bounded in $u \in {\mathbb{R}}$, uniformly w.r.t. $x \in \Omega$. If $u$ is a weak solution of in $\Omega$, then $u \in C^\alpha_{{\rm loc}}(\Omega)$ for some $\alpha \in (0, 1)$. In particular, there exists a constant $C {\geqslant}1$ such that, given any $x_0 \in \Omega$ and $0 < R {\leqslant}\min \{ r_0 / 2, {{\mbox{\normalfont dist}}}\left(x_0, \partial \Omega \right) \} / 16$, it holds $$[ u ]_{C^\alpha(B_R(x_0))} {\leqslant}\frac{C}{R^\alpha} \left( \| u \|_{L^\infty(B_{4 R}(x_0))} + \operatorname{Tail}(u; x_0, 4 R) + R^{\frac{s p}{p - 1}} \| f(\cdot, u) \|_{L^\infty(B_{8 R}(x_0))}^{1 / (p - 1)} \right).$$ The constants $\alpha$ and $C$ depend only on $n$, $s_0$, $p$ and $\Lambda$.
\[solharthm\]\
Let $n \in {\mathbb{N}}$, $s \in (0, 1)$ and $p > 1$. Let $\Omega$ be an open bounded subset of ${\mathbb{R}}^n$. Assume that $K$ satisfies hypotheses , and that $f$ is locally bounded in $u \in {\mathbb{R}}$, uniformly w.r.t. $x \in \Omega$. Let $u$ be a weak solution of in $\Omega$ such that $u {\geqslant}0$ in $\Omega$. Then, for any $x_0 \in \Omega$ and $0 < R < {{\mbox{\normalfont dist}}}(x_0, \partial \Omega) / 2$, it holds $$\sup_{B_R(x_0)} u {\leqslant}C \left( \inf_{B_R(x_0)} u + \operatorname{Tail}(u_-; x_0, R) + R^{\frac{s p}{p - 1}} \| f(\cdot, u) \|_{L^\infty(B_{2 R}(x_0))}^{1 / (p - 1)} \right),$$ for some $C {\geqslant}1$ depending only on $n$, $s$, $p$ and $\Lambda$. When $n \notin \{ 1, p \}$, the constant $C$ does not blow up as $s \rightarrow 1^-$.
Observe that the statements of Theorems \[solboundthm\]-\[solharthm\] are almost completely identical to those of Theorems \[minboundthm\]-\[minharthm\], in Section \[minsec\]. The only notable difference resides in the diverse powers to which the quantities involving the potential and the forcing term—including $d_1$ in Theorems \[minboundthm\] and \[solboundthm\]—are raised. Of course, this is coherent with the different homogeneity properties of energy ${\mathcal{E}}$ and equation .
The notion of weak solutions of equation that we take into account has already been specified in Definition \[soldef\]. We now introduce the concepts of weak sub- and supersolutions.
Let $\Omega \subset {\mathbb{R}}^n$ be a bounded open set and $u \in {\mathbb{W}}^{s, p}(\Omega)$. The function $u$ is said to be a *weak subsolution* (*supersolution*) of in $\Omega$ if $$\label{weaksolineq}
\begin{multlined}
\int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) K(x, y) \, dx dy \\
{\leqslant}\int_{{\mathbb{R}}^n} f(x, u(x)) \varphi(x) \, dx,
\end{multlined}$$ for any non-negative (non-positive) $\varphi \in W^{s, p}({\mathbb{R}}^n)$ with $\operatorname{supp}(\varphi) \subset \subset \Omega$.
Of course, a function $u$ is a solution of if and only if it is at the same time a sub- and a supersolution. Moreover, $u$ is a subsolution (supersolution) of if and only if $-u$ is a supersolution (subsolution) of the same equation, but with right-hand side given by $\tilde{f}(x, u) := f(x, -u)$. Hence, we can restrict ourselves to, say, subsolutions, as supersolutions may be dealt with in a specular way.
In the next proposition, we show the crucial step in the proof of our regularity results, namely that solutions of are contained in a fractional De Giorgi class.
\[solareDGprop\] Let $u$ be a weak subsolution of in a bounded open set $\Omega \subset {\mathbb{R}}^n$. Then, there exist four constants $R_0 \in (0, r_0 / 2]$, $k_0 \in [-\infty, 1]$, $H {\geqslant}1$ and $\varepsilon \in (0, sp / n]$, such that $$\label{solareDGine}
\begin{aligned}
& [(u - k)_+]_{W^{s, p}(B_r(x_0))}^p + \int_{B_r(x_0)} \left[ (u(x) - k)_+ \int_{B_{2 R_0}(x)} \frac{(u(y) - k)_-^{p - 1}}{|x - y|^{n + s p}} \, dy \right] \\
& \hspace{60pt} {\leqslant}\frac{H}{1 - s} \Bigg[ \left( R^{\frac{s p}{p - 1}} d_1^{\frac{p}{p - 1}} + \frac{|k|^p}{R^{n \varepsilon}} \right) |A^+(k, x_0, R)|^{1 - \frac{s p}{n} + \varepsilon} \\
& \hspace{60pt} \quad + \frac{R^{(1 - s) p}}{(R - r)^p} \| (u - k)_+ \|_{L^p(B_R(x_0))}^p \\
& \hspace{60pt} \quad + \frac{R^{n + s p}}{(R - r)^{n + s p}} \| (u - k)_+ \|_{L^1(B_R(x_0))} \overline{\operatorname{Tail}}((u - k)_+; x_0, r)^{p - 1} \Bigg],
\end{aligned}$$ for any $x_0 \in \Omega$, $0 < r < R {\leqslant}\min \{ R_0, {{\mbox{\normalfont dist}}}(x_0, \partial \Omega) \}$ and $k {\geqslant}k_0$. Consequently, $u$ belongs to the following fractional De Giorgi classes:
1. if $d_2 = 0$, then $$u \in \operatorname{DG}^{s, p}_+ \left( \Omega; d_1^{\frac{1}{p - 1}}, H, -\infty, \frac{ s p }{n}, \frac{sp}{p - 1}, \frac{r_0}{2} \right),$$ with $H = H(n, p, \Lambda)$;
2. if $d_2 > 0$ and $1 < q {\leqslant}p$, then $$u \in \operatorname{DG}^{s, p}_+ \left( \Omega; d_1^{\frac{1}{p - 1}}, H, 1, \frac{ s p }{n}, \frac{sp}{p - 1}, \min \left\{ 1, \frac{r_0}{2} \right\} \right),$$ with $H = H(n, p, \Lambda, d_2)$;
3. if $d_2 > 0$, $n {\geqslant}sp$ and $p < q < p^*_s$, then $$u \in \operatorname{DG}^{s, p}_+ \left( \Omega; d_1^{\frac{1}{p - 1}}, H, 0, 1 - \frac{q}{p^*_\sigma}, \frac{sp}{p - 1}, R_0 \right),$$ with $H = H(n, p, q, \Lambda, d_2)$, $R_0 = R_0(n, s, p, q, \Lambda, d_2, r_0, \| u \|_{L^{p^*_\sigma}(\Omega)})$ and for some constant $\sigma = \sigma(n, s, p, q) \in (0, s)$. When $n > sp$ we can even take $\sigma = s$, while when $n > p$ the constant $R_0$ does not blow up as $s \rightarrow 1^-$.
An analogous statement holds for weak supersolutions and the classes $\operatorname{DG}_-^{s, p}$.
The argument is similar to that presented in Proposition \[minareDGprop\] for minimizers and the computations are simpler. Nevertheless, we report all the details, for the reader’s convenience.
Clearly, we can suppose $x_0 = 0$. Take $r {\leqslant}\rho < \tau {\leqslant}R$ and let $\eta \in C_0^\infty({\mathbb{R}}^n)$ be a cut-off function satisfying $0 {\leqslant}\eta {\leqslant}1$ in ${\mathbb{R}}^n$, $\operatorname{supp}(\eta) = B_{(\tau + \rho) / 2}$, $\eta = 1$ in $B_\rho$ and $|\nabla \eta| {\leqslant}4 / (\tau - \rho)$ in ${\mathbb{R}}^n$. Fix $k \in {\mathbb{R}}$ and write $w_\pm := (u - k)_\pm$. By testing formulation with $\varphi := \eta^p w_+$ and recalling notation , we obtain $$\label{solareDG1}
\iint_{{\mathscr{C}}_{B_\tau}} |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) d\mu {\leqslant}\int_{B_\tau} f(x, u(x)) \varphi(x) \, dx.$$
We begin to study the term on the left-hand side of . In particular, we now deal with the contributions to the double integral coming from the set $B_\tau \times B_\tau$. We claim that $$\begin{aligned}
\label{solareDGprop1}
& \mbox{if } x, y \notin A^+(k), \mbox{ then } |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) = 0,\\
\label{solareDGprop2}
\begin{split}
& \mbox{if } x \in A^+(k, \tau) \mbox{ and } y \in B_\tau \setminus A^+(k, \tau), \mbox{ then} \\
& \hspace{10pt} |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) \\
& \hspace{15pt} {\geqslant}\min \{ 2^{p - 2}, 1 \} \Big[ |w_+(x) - w_+(y)|^p + w_-(y)^{p - 1} w_+(x) \Big] \eta(x)^p,
\end{split} \\
\label{solareDGprop3}
\begin{split}
& \mbox{if } x, y \in A^+(k, \tau), \mbox{ then} \\
& \hspace{10pt} |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) \\
& \hspace{15pt} {\geqslant}\frac{1}{2} |w_+(x) - w_+(y)|^p \max \{ \eta(x), \eta(y) \}^p \\
& \hspace{15pt} \quad - C \max\{ w_+(x), w_+(y) \}^p |\eta(x) - \eta(y)|^p,
\end{split}\end{aligned}$$ for some $C {\geqslant}1$ depending only on $p$. Indeed, is a consequence of the fact that $\operatorname{supp}(\varphi) \subset A^+(k)$. Inequality is also almost immediate, since, if $x \in A^+(k, \tau)$ and $y \in B_\tau \setminus A^+(k, \tau)$, $$|u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) = \eta(x)^p (w_+(x) + w_-(y))^{p - 1} w_+(x),$$ and the conclusion follows by e.g. Lemma \[numestlem1\] (with $\theta = 0$) when $p {\geqslant}2$, and Jensen’s inequality when $p \in (1, 2)$. Finally, to prove we assume without loss of generality that $u(x) {\geqslant}u(y)$. As $x, y \in A^+(k, \tau)$, we have $$\begin{aligned}
& |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) \\
& \hspace{50pt} = \left( w_+(x) - w_+(y) \right)^{p - 1} \left( \eta(x)^p w_+(x) - \eta(y)^p w_+(y) \right).\end{aligned}$$ Notice that, if $\eta(x) {\geqslant}\eta(y)$, then $$|u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) {\geqslant}\left( w_+(x) - w_+(y) \right)^p \eta(x)^p,$$ and trivially follows. On the other hand, if $\eta(x) < \eta(y)$, we further compute $$\label{solareDGtech1}
\begin{aligned}
& |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) \\
& \hspace{20pt} = \left( w_+(x) - w_+(y) \right)^p \eta(y)^p - \left( w_+(x) - w_+(y) \right)^{p - 1} w_+(x) \left( \eta(y)^p - \eta(x)^p \right).
\end{aligned}$$ Then, we apply Lemma \[numestlem3\] with $a = \eta(y)$, $b = \eta(x)$ and $$\varepsilon = \frac{1}{2} \frac{w_+(x) - w_+(y)}{w_+(x)},$$ to obtain that $$\begin{aligned}
& \left( w_+(x) - w_+(y) \right)^{p - 1} w_+(x) \left( \eta(y)^p - \eta(x)^p \right) \\
& \hspace{50pt} {\leqslant}\frac{1}{2} \left( w_+(x) - w_+(y) \right)^p \eta(y)^p + \left[ 2 (p - 1) \right]^{p - 1} w_+(x)^p (\eta(y) - \eta(x))^p.\end{aligned}$$ This and lead to also when $\eta(x) < \eta(y)$.
By virtue of , , and , we estimate $$\label{solareDGtech2}
\begin{aligned}
& \int_{B_\tau} \int_{B_\tau} |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) d\mu \\
& \hspace{40pt} {\geqslant}\frac{1 - s}{C} \left[ [w_+]_{W^{s, p}(B_\rho)}^p + \int_{B_\rho} w_+(x) \left( \int_{B_\tau} \frac{w_-(y)^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx \right] \\
& \hspace{40pt} \quad - C (1 - s) \int_{B_\tau} \int_{B_\tau} \max \{ w_+(x), w_+(y) \}^p \, \frac{|\eta(x) - \eta(y)|^p}{|x - y|^{n + s p}} \, dx dy,
\end{aligned}$$ for some $C {\geqslant}1$ depending on $p$ and $\Lambda$. Then, recalling the properties of $\eta$, we get $$\begin{aligned}
& (1 - s) \int_{B_\tau} \int_{B_\tau} \max \{ w_+(x), w_+(y) \}^p \, \frac{|\eta(x) - \eta(y)|^p}{|x - y|^{n + s p}} \, dx dy \\
& \hspace{60pt} {\leqslant}2 (1 - s)\int_{B_\tau} w_+(x)^p \left( \int_{B_\tau} \frac{|\eta(x) - \eta(y)|^p}{|x - y|^{n + s p}} \, dy \right) dx \\
& \hspace{60pt} {\leqslant}\frac{4^{1 + p} (1 - s)}{(\tau - \rho)^p} \int_{B_\tau} w_+(x)^p \left( \int_{B_\tau} \frac{dy}{|x - y|^{n - (1 - s) p}} \right) dx \\
& \hspace{60pt} {\leqslant}C \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p,\end{aligned}$$ with $C$ now depending also on $n$. By this, inequality becomes $$\label{solareDG2}
\begin{aligned}
& \int_{B_\tau} \int_{B_\tau} |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) d\mu \\
& \hspace{20pt} {\geqslant}\frac{1 - s}{C} \left[ [w_+]_{W^{s, p}(B_\rho)}^p + \int_{B_\rho} w_+(x) \left( \int_{B_\tau} \frac{w_-(y)^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx \right] \\
& \hspace{20pt} \quad - C \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p.
\end{aligned}$$
We now turn our attention to the term integrated over ${\mathscr{C}}_{B_\tau} \setminus B_\tau^2$ on the left-hand side of . By , and the properties of $\eta$, we have $$\label{solaretech2.1}
\begin{aligned}
& \iint_{{\mathscr{C}}_{B_\tau} \setminus B_\tau^2} |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) d\mu \\
& \hspace{10pt} = 2 \int_{B_\tau} \eta(x)^p w_+(x) \left[ \int_{{\mathbb{R}}^n \setminus B_\tau} |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) K(x, y) \, dy \right] dx \\
& \hspace{10pt} {\geqslant}\frac{2 (1 - s)}{\Lambda} \int_{A^+(k, \rho)} w_+(x) \left[ \int_{ \left( B_{r_0}(x) \cap \{ u(x) {\geqslant}u(y) \} \right) \setminus B_\tau} \frac{\left( u(x) - u(y) \right)^{p - 1}}{|x - y|^{n + s p}} \, dy \right] dx \\
& \hspace{10pt} \quad - 2 (1 - s) \Lambda \int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} w_+(x) \left[ \int_{\{ u(y) > u(x) \} \setminus B_\tau} \frac{\left( u(y) - u(x) \right)^{p - 1}}{|x - y|^{n + s p}} \, dy \right] dx.
\end{aligned}$$ Now, on the one hand $$\label{solaretech2.2}
\begin{aligned}
& \int_{A^+(k, \rho)} w_+(x) \left[ \int_{ \left( B_{r_0}(x) \cap \{ u(x) {\geqslant}u(y) \} \right) \setminus B_\tau} \frac{\left( u(x) - u(y) \right)^{p - 1}}{|x - y|^{n + s p}} \, dy \right] dx \\
& \hspace{50pt} {\geqslant}\int_{B_\rho} w_+(x) \left[ \int_{ \left( B_{r_0}(x) \cap A^-(k) \right) \setminus B_\tau} \frac{\left( w_+(x) + w_-(y) \right)^{p - 1}}{|x - y|^{n + s p}} \, dy \right] dx \\
& \hspace{50pt} {\geqslant}\int_{B_\rho} w_+(x) \left( \int_{ B_{r_0}(x) \setminus B_\tau} \frac{w_-(y) ^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx.
\end{aligned}$$ On the other hand, for $x \in B_{(\tau + \rho) / 2}$ and $y \in {\mathbb{R}}^n \setminus B_\tau$, it holds $$|x - y| {\geqslant}|y| - |x| {\geqslant}|y| - \frac{\tau + \rho}{2 \tau} |y| = \frac{\tau - \rho}{2 \tau} |y| {\geqslant}\frac{\tau - \rho}{2 R} |y|,$$ and therefore, recalling definition , $$\begin{aligned}
& \int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} w_+(x) \left[ \int_{\{ u(y) > u(x) \} \setminus B_\tau} \frac{\left( u(y) - u(x) \right)^{p - 1}}{|x - y|^{n + s p}} \, dy \right] dx \\
& \hspace{70pt} {\leqslant}\left( \frac{2 R}{\tau - \rho} \right)^{n + s p} \int_{B_{\frac{\tau + \rho}{2}}} w_+(x) \left[ \int_{{\mathbb{R}}^n \setminus B_\tau} \frac{w_+(y)^{p - 1}}{|y|^{n + s p}} \, dy \right] dx \\
& \hspace{70pt} {\leqslant}\frac{C}{1 - s} \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1}.\end{aligned}$$ By this, and , it follows that $$\label{solareDG3}
\begin{aligned}
& \iint_{{\mathscr{C}}_{B_\tau} \setminus B_\tau^2} |u(x) - u(y)|^{p - 2} \left( u(x) - u(y) \right) \left( \varphi(x) - \varphi(y) \right) d\mu \\
& \hspace{70pt} {\geqslant}\frac{1 - s}{C} \int_{B_\rho} w_+(x) \left( \int_{ B_{r_0}(x) \setminus B_\tau} \frac{w_-(y) ^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx \\
& \hspace{70pt} \quad - C \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1}.
\end{aligned}$$
By putting together , and , we find that $$\label{solareDG4}
\begin{aligned}
& [w_+] _{W^{s, p}(B_\rho)}^p + \int_{B_\rho} w_+(x) \left( \int_{B_{r_0}(x)} \frac{w_-(y)^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx \\
& \hspace{20pt} {\leqslant}\frac{C}{1 - s} \left[ \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p + \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1} \right] \\
& \hspace{20pt} \quad + \frac{C}{1 - s} \int_{{\mathbb{R}}^n} f(x, u(x)) \varphi(x) \, dx.
\end{aligned}$$
To finish the proof, we now only need to control the term appearing on the third line of . By and the properties of $\eta$, we have $$\label{fest}
\int_{{\mathbb{R}}^n} f(x, u(x)) \varphi(x) \, dx {\leqslant}\int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} \Big( d_1 + d_2 |u(x)|^{q - 1} \Big) \eta(x)^p w_+(x) \, dx.$$ To estimate the term involving $d_1$, we simply apply weighted Young’s inequality to get $$\int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} d_1 \eta(x)^p w_+(x) \, dx {\leqslant}C \left( (\delta d_1)^{\frac{p}{p - 1}} |A^+(k, R)| + \frac{\| w_+ \|_{L^p(B_R)}^p}{\delta^p} \right),$$ for any $\delta > 0$. By choosing $\delta := (\tau - \rho) R^{s - 1} {\leqslant}R^s$, this yields in turn $$\label{d1est}
\begin{aligned}
& \int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} d_1 \eta(x)^p w_+(x) \, dx \\
& \hspace{50pt} {\leqslant}C \left( R^{\frac{s p}{p - 1}} d_1^{\frac{p}{p - 1}} |A^+(k, R)| + \frac{R^{(1 - s) p}}{(\tau - \rho)^{p}} \| w_+ \|_{L^p(B_R)}^p \right).
\end{aligned}$$ Note that, when $d_2 = 0$ we are already led to , with $\varepsilon = s p / n$. On the other hand, when $d_2 > 0$ the proof of is more involved. We consider separately the two possibilities $1 < q {\leqslant}p$ and $p < q < p^*_s$.
Suppose that $1 < q {\leqslant}p$. In this case, we take $k {\geqslant}1$. For $x \in A^+(k)$ we have that $u(x) > k {\geqslant}1$, and thus $$|u(x)|^{q - 1} {\leqslant}|u(x)|^{p - 1} = |w_+(x) + k|^{p - 1} {\leqslant}\max \left\{ 1, 2^{p - 2} \right\} \left( w_+(x)^{p - 1} + k^{p - 1} \right).$$ Accordingly, by Young’s inequality $$\begin{aligned}
\int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} d_2 |u(x)|^{q - 1} \eta(x)^p w_+(x) \, dx & {\leqslant}C \int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} \Big( w_+(x)^p + k^{p - 1} w_+(x) \Big) \, dx \\
& {\leqslant}C \left( \| w_+ \|_{L^p(B_R)}^p + k^p |A^+(k, R)| \right) \\
& {\leqslant}C \left[ \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p + \frac{k^p}{R^{s p}} |A^+(k, R)| \right],\end{aligned}$$ provided $R {\leqslant}1$ and with $C$ depending also on $d_2$. This and estimates , yield when $1 < q {\leqslant}p$, again with $\varepsilon = s p / n$.
Finally, we deal with the second case, when $p < q < p^*_s$, with $n {\geqslant}s p$. By adding the quantity $d_2 (1 - s)^{- 1} \| u \|_{L^q(A^+(k, \rho))}^q$ to both sides of and recalling , , we get $$\label{solareDG5}
\begin{aligned}
& [w_+]_{W^{s, p}(B_\rho)}^p + \frac{d_2}{1 - s} \| u \|_{L^q(A^+(k, \rho))}^q + \int_{B_\rho} w_+(x) \left( \int_{B_{r_0}(x)} \frac{w_-(y)^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx \\
& \hspace{20pt} {\leqslant}\frac{C_1}{1 - s} \Bigg[ \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p + \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1} \\
& \hspace{20pt} \quad + d_1^{\frac{p}{p - 1}} |A^+(k, R)| + d_2 \int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} |u(x)|^{q - 1} \Big( |u(x)| + \eta(x) w_+(x) \Big) dx \Bigg],
\end{aligned}$$ for some $C_1 {\geqslant}1$ depending only on $n$, $p$ and $\Lambda$. Then, we observe that, if $k {\geqslant}0$, $$\begin{aligned}
|u(x)|^q & = |(1 - \eta(x)) u(x) + \eta(x) w_+(x) + \eta(x) k|^q \\
& {\leqslant}3^{q - 1} \Big( (1 - \eta(x))^q |u(x)|^q + \left( \eta(x) w_+(x) \right)^q + k^q \Big),\end{aligned}$$ for any $x \in A^+(k)$. Hence, using once again Young’s inequality, $$\begin{aligned}
& d_2 \int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} |u(x)|^{q - 1} \Big( |u(x)| + \eta(x) w_+(x) \Big) dx \\
& \hspace{50pt} {\leqslant}2 d_2 \int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} \Big[ |u(x)|^q + \left( \eta(x) w_+(x) \right)^q \Big] dx \\
& \hspace{50pt} {\leqslant}4^q d_2 \int_{A^+(k, \tau)} \Big[ (1 - \eta(x))^q |u(x)|^q + \left( \eta(x) w_+(x) \right)^q + k^q \Big] \, dx.\end{aligned}$$ Consider now the two quantities $\sigma \in (0, s]$ and $\varepsilon_\sigma \in (0, sp / n)$ defined in and , respectively. By arguing as in the last part of the proof of Proposition \[minareDGprop\], we deduce from the estimate above that $$\begin{aligned}
& \frac{C_1 d_2}{1 - s} \int_{A^+ \left( k, \frac{\tau + \rho}{2} \right)} |u(x)|^{q - 1} \Big( |u(x)| + \eta(x) w_+(x) \Big) dx \\
& \hspace{40pt} {\leqslant}\frac{1}{2} [w_+]_{W^{s, p}(B_\tau)}^p + \frac{C d_2}{1 - s} \| u \|_{L^q(B_\tau \setminus B_\rho)}^q \\
& \hspace{40pt} \quad + \frac{C}{1 - s} \left[ \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p + d_2 k^p R^{- n \varepsilon_\sigma} |A^+(k, R)|^{1 - \frac{s p}{n} + \varepsilon_\sigma} \right],\end{aligned}$$ where $C$ may now depend on $q$ as well, and provided $R$ is smaller than a quantity $R_0$ depending only on $n$, $s$, $p$, $q$, $\Lambda$, $d_2$, $\| u \|_{L^{p^*_\sigma}}$. Combining this with and rearranging appropriately the summands, we find that $$\label{solareDG6}
\begin{aligned}
\Phi(\rho) & {\leqslant}C_\flat \left[ \Phi(\tau) - \Phi(\rho) \right] + \frac{C_\flat}{1 - s} \Bigg[ \left( d_1^{\frac{p}{p - 1}} + d_2 k^p R^{- n \varepsilon_\sigma} \right) |A^+(k, R)|^{1 - \frac{s p}{n} + \varepsilon_\sigma} \\
& \quad + \frac{R^{(1 - s) p}}{(\tau - \rho)^p} \| w_+ \|_{L^p(B_R)}^p + \frac{R^{n + s p}}{(\tau - \rho)^{n + s p}} \| w_+ \|_{L^1(B_R)} \overline{\operatorname{Tail}}(w_+; r)^{p - 1} \Bigg],
\end{aligned}$$ for some $C_\flat {\geqslant}1$ depending on $n$, $p$, $q$, $\Lambda$, and with $$\Phi(t) := [w_+]_{W^{s, p}(B_t)}^p + \frac{d_2}{1 - s} \| u \|_{L^q(B_t)}^q + \int_{B_t} w_+(x) \left( \int_{B_{r_0}(x)} \frac{w_-(y)^{p - 1}}{|x - y|^{n + s p}} \, dy \right) dx,$$ for any $0 < t {\leqslant}R$. Estimate (with $\varepsilon = \varepsilon_\sigma$) now follows by adding the quantity $C_\flat \Phi(\rho)$ to both sides of , dividing by $1 + C_\flat$ and applying Lemma \[induclem\].
With the aid of Proposition \[solareDGprop\] and Theorems \[DGboundthm\], \[DGholdthm\], the boundedness and Hölder continuity of the solutions of are readily established. Furthermore, the Harnack inequality for non-negative solutions follows by Theorem \[DGharthm\]. Thus, Theorems \[solboundthm\]-\[solharthm\] are proved. For more details, see the proofs of the analogous Theorems \[minboundthm\]-\[minharthm\] for minimizers, at the end of Section \[minsec\].
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is supported by a BGSMath Postdoctoral Fellowship and the MINECO grant MTM2014-52402-C3-1-P. The author also warmly thanks Xavier Cabré, Moritz Kassmann and Enrico Valdinoci for several inspiring conversations on the subject of this paper.
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[^1]: Note that any well-defined notion of minimizer of the functional ${\mathcal{G}}_s$ as defined in should involve some kind of restriction of ${\mathcal{G}}_s$ to the set $\Omega$, when comparing the energy of $u$ to those of its competitors. Otherwise, in fact, ${\mathcal{G}}_s$ might be always infinite, as a consequence of the prescribed values $u_0$ outside of $\Omega$. We will be more precise on the concept of minimizers that we take into account in Section \[mainsec\]. For the moment, we take the liberty of being slightly inaccurate and not worrying much about technicalities.
[^2]: Of course, the potentials $F_d$ introduced in are not bounded. However, the arguments presented in [@CV15; @CV16] always involve functions that assume values between $-1$ and $1$. Therefore, the local boundedness of $F_d$ is enough in this case.
[^3]: Notice that the exponents to which the two factors on the left-hand side of are raised may be slightly different in the works [@DeG57; @Giu03; @CV12]. We refer to Lemma \[DGisolem\] here for a proof of (and more general inequalities) in this exact fashion.
[^4]: As it is customary, one can take as $F$ any Carathéodory function, i.e. such that $F(\cdot, v)$ is measurable for any $v \in {\mathbb{R}}$ and $F(x, \cdot)$ is continuous for a.a. $x \in {\mathbb{R}}^n$. Indeed, when $F$ is a Carathéodory function, then $F \circ v$ is measurable every time $v$ is. However, we adopted a slightly broader definition in order to take into account for instance the examples listed in and many more.
[^5]: Here and in Proposition \[minareDGprop\] we continue to understand $p^*_s = +\infty$, when $n = s p$. The same convention will still be valid in Section \[solsec\].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differential structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in its exterior. If the Finsler function is fiberwise real analytic, it is determined uniquely. We also discuss the smoothness of the distance function between interior and boundary points.'
author:
- 'Maarten V. de Hoop $^{\diamond}$'
- 'Joonas Ilmavirta $^\dagger$'
- 'Matti Lassas $^\square$'
- 'Teemu Saksala $^{\diamond,\: \ast}$'
bibliography:
- 'bibliography.bib'
title: Inverse problem for compact Finsler manifolds with the boundary distance map
---
[^1]
Introduction {#Se:inverse_problem}
============
This paper is devoted to an inverse problem for smooth compact Finsler manifolds with smooth boundaries. We prove that the boundary distance map of such a manifold determines its topological and differential structures. In general, the boundary distance map is not sufficient to determine the Finsler function in those directions which correspond to geodesics that are either trapped or are not distance minimizers to terminal boundary points. To prove our result, we embed a Finsler manifold with boundary into a function space and use smooth boundary distance functions to give a coordinate structure and the Finsler function where possible.
This geometric problem arises from the propagation of singularities from a point source for the elastic wave equation. The point source can be natural (e.g. an earthquake as a source of seismic waves) or artificial (e.g. produced by the boundary control method or by a wave sent in scattering from a point scatterer). Due to polarization effects, there are singularities propagating at various speeds. We study the first arrivals and thus restrict our attention to the fastest singularities (corresponding to so-called *qP*-polarization, informally “pressure waves”). They follow the geodesic flow of a Finsler manifold, as we shall explain in more detail in section \[Se:elastic\].
An elastic body — e.g. a planet — can be modeled as a manifold, where distance is measured in travel time: The distance between two points is the shortest time it takes for a wave to go from one point to the other. If the material is elliptically anisotropic, then this elastic geometry is Riemannian. However, this sets a very stringent assumption on the stiffness tensor describing the elastic system, and Riemannian geometry is therefore insufficient to describe the propagation of seismic waves in the Earth. We make no structural assumptions on the stiffness tensor apart from the physically necessary symmetry and positivity properties, and this leads necessarily to Finsler geometry.
The inverse problem introduced above can be rephrased as the following problem in geophysics. Imagine that earthquakes occur at known times but unknown locations within Earth’s interior and arrival times are measured everywhere on the surface. Are such travel time measurements sufficient to determine the possibly anisotropic elastic wave speed everywhere in the interior and pinpoint the locations of the earthquakes? While earthquake times are not known in practice, this is a fundamental mathematical problem that underlies more elaborate geophysical scenarios. In the Riemannian realm the corresponding result [@Katchalov2001; @kurylev1997multidimensional] is a crucial stepping stone towards the results of [@belishev1992reconstruction; @deHoop1; @de2018inverse; @ivanov2018distance; @kurylev2010rigidity; @LaSa]. We expect that solutions to inverse problems for the fully anisotropic elastic wave equation rely on geometrical results similar to the ones presented in this paper.
Main results
------------
We let $(M,F)$ be a smooth compact, connected Finsler manifold with smooth boundary $\p M$ (For the basic of theory of Finsler manifolds see the appendix (Section \[Se:Appendix1\]) at the end of this paper.). We denote the tangent bundle of $M$ by $TM$ and use the notation $(x,y)$ for points in $TM$, where $x$ is a base point and $y$ is a vector in the fiber $T_xM$. The notations $T^\ast M$ and $(x,p)$ are reserved for the cotangent bundle and its points respectively.
We write $d_F:M\times M \to {{\mathbb R}}$ for the non-symmetric distance function given by a Finsler function $F$. For a given $x\in M$ the boundary distance function related to $x$ is $$r_x\colon\p M \to [0,\infty), \quad r_{x}(z)=d_F(x,z).$$ We denote $\mathcal{R}(M^{int})=\{r_x:x\in M^{int}\}$ the collection of all boundary distance functions. In this paper, we study the inverse problem with *boundary distance data* $$\label{eq:data}
(\mathcal{R}(M^{int}), \p M)$$
Do the boundary distance data determine $(M,F)$ up to isometry?
We emphasize that we do not assume $d_F$ to be symmetric and therefore data contain only information where the distance is measured from the points of $M^{int}$ to points in $\p M$. We note that for any $x \in M^{int}$, $$r_x(z)=d_{\stackrel{\leftarrow}{F}}(z,x), \quad z \in \p M,$$ where $\stackrel{\leftarrow}{F}$ is the Finsler function $$\label{eq:reversed_Fins}
\stackrel{\leftarrow}{F}(x,y):=F(x,-y).$$ Therefore, data are equivalent to the data $$(\{d_{\stackrel{\leftarrow}{F}}(\cdot,x)\colon \p M \to {{\mathbb R}}\: |\: x \in M^{int}\},\p M),$$ where the distance is measured from the boundary to the interior. In [@Katchalov2001; @kurylev1997multidimensional] it is shown that the data determine a Riemannian manifold $(M,g)$ up to isometry. In the Finsler case this is not generally true. Next we explain what can be obtained from the Finslerian boundary distance data .
\[De:good\_set\] For a Finsler manifold $(M,F)$ with boundary, we denote by\
$G(M,F)$ the set of points $(x,y)\in TM\setminus0, \: x \in M^{int}$ for which the geodesic starting at $x$ in direction $y$ reaches the boundary in finite time $t(x,y)$ and is minimizing between $x$ and $z(x,y):=\gamma_{x,y}(t(x,y))\in \partial M$, that is $$\gamma_{x,y}(0,t(x,y))\subset M^{int}.$$ We emphasize that for any interior point $x\in M^{int}$ and for any $y\in T_xM$ it holds that $t(x,y)>0$.
Since for any $(x,y) \in TM\setminus \{0\}$ and $a >0$ it holds that $\gamma_{x,ay}(t)=\gamma_{x,y}(at)$, we notice that $G(M,F)$ is a conic set. Let $(x,y) \in G(M,F)$, then $t(x,ay)=a^{-1}t(x,y)$ and $z(x,y)=z(x,ay)$ for any $a>0$. Moreover if $F(y)=1$, then $t(x,y)=d_F(x,z(x,y))$.
We show that the data determine the Finsler function in the closure of the set $G(M,F)$ and that the data are not sufficient to recover the Finsler function $F$ on $TM^{int} \setminus \overline{G(M,F)}$. The reason is that the data do not provide any information about the geodesics that are trapped in $M^{int}$ or do not minimize the distance between the point of origin and the terminal boundary point. Therefore, to recover the Finsler function $F$ globally we assume that for every $x \in M$ the function $F(x, \cdot)\colon T_x M \setminus\{0\} \to {{\mathbb R}}$ is real analytic. We call such a Finsler function *fiberwise real analytic*. For instance Finsler functions $F(x,y)=\sqrt{g_x(y,y)}$, where $g$ is a Riemannian metric, and Randers metrics are fiberwise real analytic. In Section \[Se:elastic\] we show that also the Finsler metric related to the fastest polarization of elastic waves is fiberwise real analytic.
Now we formulate our main theorems. If $(M_i,F_i), \: i \in \{1,2\}$ are smooth, connected, compact Finsler manifolds with smooth boundaries, we call a smooth map\
$\Phi\colon(M_1,F_1)\to(M_2,F_2)$ a *Finslerian isomorphism* if it is a diffeomorphism which satisfies $$F_1(x,y)=F_2(\Phi(x),\Phi_\ast y), \quad (x,y) \in TM_1.$$ Here $\Phi_\ast$ is the pushforward by $\Phi$. We say that the boundary distance data of manifolds $(M_i,F_i), \: i=1,2$ agree, if there exists a diffeomorphism $\phi\colon \p M_1\to \p M_2$ such that $$\label{eq:BDD_agree_start}
\{r_{x_1}:x_1\in M_1^{int}\}=\{r_{x_2}\circ \phi:x_2\in M_2^{int}\}\subset C(\p M_1).
$$ We emphasize that this is an equality of non-indexed sets and we do not know the point $x_1$ corresponding to the function $r_{x_1}$.
Our first main result shows that the boundary distance data determine a manifold upto a diffeomorphism and a Finsler function in an optimal set.
\[Th:smooth\] Let $(M_i,F_i), \: i=1,2$ be smooth, connected, compact Finsler manifolds with smooth boundaries. We suppose that there exists a diffeomorphism\
$\phi\colon\p M_1\to \p M_2$ so that holds. Then there is a diffeomorphism $\Psi\colon M_1\to M_2$ so that $\Psi|_{\p M_1}=\phi$. The sets $\overline{ G(M_1,F_1})$ and $\overline{ G(M_1,\Psi^*F_2)}$ coincide and in this set $F_1=\Psi^*F_2$, where the pullback $\Psi^*F_2:TM_1\to {{\mathbb R}}$ is the function $$\Psi^*F_2:TM_1\to {{\mathbb R}}, \quad \Psi^*F_2(x,y)=F_2(\Psi(x),\Psi_\ast y).$$
Moreover, for any $(x_0,y_0)\in TM_1^{int}\setminus \overline{G(M_1,F_1)}$ there exists a smooth Finsler function $F_3\colon TM_1\to[0,\infty)$ so that $d_{F_1}(x,z)=d_{F_3}(x,z)$ for all $x\in M_1$ and $z\in\partial M_1$ but $F_1\neq F_3$.
The set $G(M,F)$ can be large or small as the following examples illustrate. If every geodesic of $(M,F)$ is minimizing, then it holds that $\overline{
G(M,F)}=TM$. This holds for instance on simple Riemannian manifolds. If $M$ is any subset of $S^2$ larger than the hemisphere and if $F$ is given by the round metric, then $TM^{int} \setminus
\overline{G(M,F)}$ contains an open non-empty set $U$ whose canonical projection to $M$ is an open neighborhood of the equator.
Our second main result shows that the boundary distance data determine a fiberwise Finsler manifold upto isometry.
\[Th:analytic\] Let $(M_i,F_i), \: i=1,2$ be smooth, connected, compact Finsler manifolds with smooth boundary. We suppose that there exists a diffeomorphism\
$\phi\colon\p M_1\to \p M_2$ such that holds. If Finsler functions $F_i$ are fiberwise real analytic, then there exists a Finslerian isometry $\Psi \colon
(M_1,F_1)\to (M_2,F_2)$ so that $\Psi|_{\p M_1}=\phi$.
In Theorems \[Th:smooth\] and \[Th:analytic\] we measure distances from the interior to the boundary. If we measure in the opposite direction, from boundary to the interior, this corresponds to the data given with respect to the reversed Finsler function $\stackrel{\leftarrow}{F}(x,y)$. Our results give uniqueness for $\stackrel{\leftarrow}{F}$ and therefore $F$. That is, our main results hold no matter which way distances are measured.
### Outline of the proofs of the main results
Theorem \[Th:analytic\] essentially follows from Theorem \[Th:smooth\]. We split the proof of Theorem \[Th:smooth\] into four parts (subsections 3.1–3.4). In the first part, we show that the data determine $r_x$ for any $x \in M$. Then we study the properties of the map $ \mathcal{R}:M \ni x \mapsto r_x \in
L^\infty(\p M)$ and show that this map is a topological embedding. We use the map $\mathcal{R}$ to construct a map $\Psi: (M_1,F_1) \to
(M_2,F_2)$ that will be shown to be a homeomorphism. In the second part, we show that the map $\Psi$ is a diffeomorphism. In the third part we connect the set $G(M,F)$ to smoothness of the distance functions of the form $d_F(\cdot,z), \: z \in \p M$. In the final section we use this to prove that the map $\Psi$ is a Finslerian isometry.
We have included in this paper a supplemental Section \[Se:Appendix2\] and the appendix (Section \[Se:Appendix1\]), which contain necessary material for the proof of Theorem \[Th:smooth\]. We have also included some well-known results and properties in the Riemannian case while providing a detailed background of compact Finsler manifolds with and without boundary for the proof given in Section \[Se:proof\]. To the best of our knowledge, most of this material cannot be found in the literature.
Background and related work
---------------------------
### Geometric inverse problems
The claim and proof of Theorem \[Th:smooth\] are a modification of a similar result in a Riemannian case given in [@Katchalov2001; @kurylev1997multidimensional]. The Riemannian version was first proven in [@kurylev1997multidimensional]. In [@Katchalov2001] also the construction of smooth structure is considered. The Riemannian version of Theorem \[Th:smooth\] is related to many other geometric inverse problems. For instance, it is a crucial step in proving uniqueness for Gel’fand’s inverse boundary spectral problem [@Katchalov2001]. Gel’fand’s problem concerns the question whether the data $$\{\p M, (\lambda_j, \p_\nu \phi_j|_{\p M})_{j=1}^\infty\}$$ determine $(M,g)$ up to isometry. Above $(\lambda_j, \phi_j)$ are the Dirichlet eigenvalues and the corresponding $L^2$-orthonormal eigenfunctions of the Laplace-Beltrami operator. Belishev and Kurylev provide an affirmative answer to this problem in [@belishev1992reconstruction].
We recall that the Riemannian wave operator is a globally hyperbolic linear partial differential operator of real principal type. Therefore, the Riemannian distance function and the propagation of a singularity initiated by a point source in space time are related to one another. In other words, $r_x(z)=t(z)-s$, where $t(z)$ is the time when the singularity initiated by the point source $(s,x) \in
(0,\infty )\times M$ hits $z\in \p M$. If the initial time $s$ is unknown, but the arrival times $t(z), z\in \p M$ are known, then one obtains a boundary distance difference function $D_x(z_1,z_2):=r_x(z_1)-r_x(z_2), \: z_1,z_2 \in \p M$. In [@LaSa] it is shown that if $U\subset N$ is a compact subset of a closed Riemannian manifold $(N,g)$ and $U^{int}\neq \emptyset$, then *distance difference data* $\{(U,g|_{U}), \{D_x\colon U\times U
\to {{\mathbb R}}\:| \:x \in N\}\}$ determine $(N,g)$ up to isometry. This result was recently generalized to complete Riemannian manifolds [@ivanov2018distance] and for compact Riemannian manifolds with boundary [@de2018inverse].
If the sign in the definition of the distance difference functions is changed, we arrive at the distance sum functions, \[def: dist sum\] D\^+\_x(z\_1,z\_2)=d(z\_1,x)+d(z\_2,x),xM, z\_1,z\_2M . These functions give the lengths of the broken geodesics, that is, the union of the shortest geodesics connecting $z_1$ to $x$ and the shortest geodesics connecting $x$ to $z_2$. Also, the gradients of $D^+_x(z_1,z_2)$ with respect to $z_1$ and $z_2$ give the velocity vectors of these geodesics. The inverse problem of determining the manifold $(M,g)$ from the *broken geodesic data*, consisting of the initial and the final points and directions, and the total length, of the broken geodesics, has been considered in [@kurylev2010rigidity]. In [@kurylev2010rigidity] the authors show that broken geodesic data determine the boundary distance data and use then the results of [@Katchalov2001; @kurylev1997multidimensional] to prove that the broken geodesic data determine the Riemannian manifold up to isometry.
We let $u$ be the solution of the Riemannian wave equation with a point source at $(s,x) \in (0,\infty )\times M$. In [@duistermaat1996fourier; @greenleaf1993recovering] it is shown that the image, $\Lambda$, of the wavefront set of $u$, under the canonical isomorphism $T^\ast M \ni (x,p) \mapsto g^{ij}(x)p_i \in TM$, coincides with the image of the unit sphere $S_xM$ at $x$ under the geodesic flow of $g$. Thus $\Lambda \cap \p(S M)$, where $SM$ is the unit sphere bundle of $(M,g)$, coincides with the exit directions of geodesics emitted from $p$. In [@lassas2018reconstruction] the authors show that if $(M,g)$ is a compact smooth non-trapping Riemannian manifold with smooth strictly convex boundary, then generically the *scattering data of point sources* $\{\p M, R_{\p M}(M)\}$ determine $(M,g)$ up to isometry. Here, $R_{\p M}(x) \in R_{\p M}(M), \: x \in M$ stands for the collection of tangential components to boundary of exit directions of geodesics from $x$ to $\p M$.
A classical geometric inverse problem, that is closely related to the distance functions, asks: Does the Dirichlet-to-Neumann mapping of a Riemannian wave operator determine a Riemannian manifold up to isometry? For the full boundary data this problem was solved originally in [@belishev1992reconstruction] using the Boundary control method. Partial boundary data questions have been sudied for instance in [@lassas2014inverse; @milne2016codomain]. Recently [@kurylev2018inverse] extended these results for connection Laplacians. Lately also inverse problems related to non-linear hyperbolic equations have been studied extensively [@kurylev2014inverse; @lassas2018inverse; @wang2016inverse]. For a review of inverse boundary value problems for partial differential equations see [@LassasICM2018; @uhlmann1998inverse].
Another well studied geometric inverse problem formulated with the distance functions is the Boundary rigidity problem. This problem asks: Does the *boundary distance function* $d_F\colon\p M \times \p M \to {{\mathbb R}}$, that gives a distance between any two boundary points, determine $(M,F)$ up to isometry? For the best to our knowledge this problem has not been studied in Finsler geometry. For a general Riemannian manifold the problem is false: Suppose the manifold contains a domain with very slow wave speed, such that all the geodesics starting and ending at the boundary avoid this domain. Then in this domain one can perturb the metric in such a way that the boundary distance function does not change. It was conjectured in [@michel1981rigidite] that for all compact simple Riemannian manifolds the answer is affirmative. In two dimensions it was solved in [@pestov2005two]. For higher dimensional case the problem is still open, but different variations of it has been considered for instance in [@croke1991rigidity; @stefanov2016boundary; @stefanov2017local]. The Boundary distance data , studied in this paper, is much larger data than the knowledge of the boundary distance function. Therefore we can obtain the optimal determination of $(M,F)$, as explained in theorems \[Th:smooth\] and \[Th:analytic\], even though we pose no geometric conditions on $(M,F)$.
### Finsler and Riemannian geometry
We refer to the monographs [@bao2012introduction; @shen2001lectures] for the development of Finsler manifolds without boundaries. We point out that two major differences occur between the Riemannian and Finslerian realms that are related to the proof of Theorem \[Th:smooth\]. In Riemannian geometry the relation between $TM$ and $T^\ast M$ is simple; raising and lowering indices provides a fiberwise linear isomorphism. In Finsler geometry this is not possible, since the Legendre transform (see ) is not linear in the fibers. For this reason, we have to be more careful in the analysis of distance functions and the connection of their differential to the velocity fields of geodesics. Moreover the Finslerian gradient is not a linear operator.
The second issue arises from the lack of a natural linear connection compatible with $F$ on vector bundle $\pi\colon TM \to M,$ where $\pi(x,y)=x$ is the canonical projection to the base point. In Section \[Se:Appendix2\] we consider properties of Chern connection $\nabla$, which is a torsion free linear connection on the pullback bundle $\pi'\colon \pi^\ast TM \to TM$ (see [@bao1996notable; @chern1943euclidean; @chern1996finsler]). We derive natural compatibility relations for $\nabla$ and the fundamental tensor field $g$ on $\pi^\ast TM$ $g$ (see [@shen2001lectures Section 5.2] and Lemma \[Le:Chern\_prop\_2\]) in a special case. In [@shen1994connection] Shen proved the general version of the compatibility relations. We use Lemma \[Le:Chern\_prop\_2\] to formulate the initial conditions for so-called transverse vector fields (see [@Chavel Section III.6]) with respect to $\p M$ along boundary normal geodesics. After this we give a definition of an index form related to these vector fields and use it to prove results similar to classical theorems, originally by Jacobi, related to the minizing of geodesics after focal points (for the Riemannian case, see for instance [@Chavel Section III.6]). Inverse problems arising from elastic equations have been also extensively studied. See e.g. [@BAO2018263; @bal2015reconstruction; @bao2018inverse; @griesmaier2018uncertainty; @hu2015nearly; @nakamura1993identification; @nakamura1994global].
From Elasticity to Finsler geometry {#Se:elastic}
===================================
The main physical motivation of this paper is to obtain a geometric and coordinate invariant point of view to the inverse problems related to the propagation of seismic waves. The seismic waves are modelled by the anisotropic elastic wave equation in ${{\mathbb R}}^{1+3}$. This elastic system can be microlocally decoupled to 3 different polarizations [@stolk2002microlocal]. In this section, we introduce a connection between the fastest polarization (known as the quasi pressure polarization and denoted by qP) and the Finsler geometry. More over it turns out that the Finsler metric arising from elasticity is fiberwise real analytic. We use the typical notation and terminology of the seismological literature, see for instance [@cerveny2005seismic]. We let $c_{ijk\ell}(x)$ be the smooth stiffness tensor on ${{\mathbb R}}^3$ which satisfies the symmetry $$\label{eq:symmetry_of_elastic_tensor}
c_{ijk\ell}(x)=c_{jik\ell}(x)=c_{k\ell ij}(x), \quad x\in {{\mathbb R}}^3.$$ We also assume that the density $\rho(x)$ is a smooth function of $x$ and define density–normalized elastic moduli $$a_{ijk\ell}(x)=\frac{c_{ijk\ell}(x)}{\rho(x)}.$$ The elastic wave operator $P$, related to $a_{ijk\ell}$, is given by $$P_{i\ell}=\delta_{i\ell}\frac{\p^2}{\p t^2}-a_{ijk\ell}(x)\frac{\p}{\p x^j}\frac{\p}{\p x^k}+\hbox{lower order terms.}$$
For every $(x,p)\in {{\mathbb R}}^3\times {{\mathbb R}}^3$ we define a square matrix $\Gamma(x,p)$, by $$\label{eq:Chirstoffel_matrix}
\Gamma_{i\ell}(x,p):=a_{ijk\ell}(x)p^kp^j.$$ The matrix $\Gamma(x,p)$ is called the *Christoffel matrix*. Due to the matrix $\Gamma(x,p)$ is symmetric. One also assumes that $\Gamma(x,p)$ is positive definite for every $(x,p)\in {{\mathbb R}}^3\times ({{\mathbb R}}^3 \setminus \{0\})$.
The principal symbol $\delta(t,x,\omega, p)$ of the operator $P$ is then given by $$\delta(t, x,\omega, p)=\omega^2I-\Gamma(x,p), \quad (t, x,\omega, p)\in {{\mathbb R}}^{1+3}\times {{\mathbb R}}^{1+3}.$$ Since the matrix $\Gamma(x,p)$ is positive definite and symmetric, it has three positive eigenvalues $\lambda^m(x,p),\: m \in \{1,2,3\}$.
We assume that $$\label{eq:nonvanishing_lamda_der}
\lambda^1(x,p) > \lambda^{m}(x,p), \quad m \in \{2,3\}\hbox{, $(x,p) \in {{\mathbb R}}^3\times ({{\mathbb R}}^3\setminus \{0\})$}.$$ Then it follows from the Implicit Function Theorem that $\lambda^1(x,p)$ and a related unit eigenvector $q^1(x,p)$ are smooth with respect to $(x,p)$. See for instance [@Evans Chapter 11, Theorem 2] for more details. Moreover the function $\lambda^1(x,p)$ is homogeneous of degree $2$ with respect to $p$.
To keep the notation simple, we write from now on $\lambda:=\lambda^1(x,p)$ and $q:=q^1(x,p)$. We use $\Gamma q = \lambda q$ and to compute the Hessian of $\lambda(x,p)$ with respect to $p$. We obtain $$\label{eq:Hessian_of_g^1_1}
\hbox{Hess}_p(\lambda(x,p))= 2\bigg(\Gamma(q(x,p))+(Dq)^T(\lambda(x,p)I-\Gamma(x,p))Dq\bigg),$$ where $Dq$ is the Jacobian of $q(x,p)$ with respect to $p$ and the superscript $T$ stands for transpose. Since $\Gamma(q(x,p))$ is positive definite it follows from and that the Hessian of $\lambda(x,p)$ is also positive definite. We note that a similar result has been presented in [@antonelli2003geometrical] under the assumption the stiffness tensor is homogeneous and transversely isotropic.
We define a continuous function $f(x,p):=\sqrt{\lambda(x,p)}$, which is smooth outside ${{\mathbb R}}^3 \times \{0\}$. We conclude with summarizing the properties of the function $f$
- $f \colon {{\mathbb R}}^3\times({{\mathbb R}}^3\setminus \{0\})\to (0,\infty)$ is smooth, real analytic on the fibers;
- for every $(x,p) \in {{\mathbb R}}^3\times {{\mathbb R}}^3$ and $s\in {{\mathbb R}}$ it holds that $f(x,sp)=|s|f(x,p)$;
- for every $(x,p) \in {{\mathbb R}}^3\times({{\mathbb R}}^3\setminus \{0\})$ the Hessian of $\frac{1}{2}f^2$ is symmetric and positive definite with respect to $p$.
Therefore, $f$ is a convex norm on the cotangent space. Finally, we define a Finsler function $F$ to be the Legendre transform of $f$. Thus the bicharacteristic curves of Hamiltonian $\frac{1}{2}\big(\lambda(x,p)-1\big)$ are given by the co-geodesic flow of $F$. Moreover the $qP$ group velocities are given by the Finsler structure.
Another geometrical inverse problem on Finsler manifolds, using exterior geodesic sphere data, is presented in [@Finsler_Dix], extending an earlier result on Riemannian manifolds [@deHoop1].
Proof of theorem \[Th:smooth\] {#Se:proof}
==============================
In this section we provide a proof of Theorem \[Th:smooth\]. The proof is divided into four parts. In the first part, we consider the topology and introduce a homeomorphism $\Psi$ from $(M_1,F_1)$ onto $(M_2,F_2)$. The second part is devoted to proving that homeomorphism $\Psi$ is smooth and has a smooth inverse. In the third part, we study smoothness of a distance function $d_F(\cdot,z), \: z \in \p M$ in those interior points $x$ where a distance minimizing curve from $x$ to $z$ is a geodesic contained in the interior. Then, in the final part, we use the result obtained in the third part to prove that the Finsler functions $F_1, \: \Psi^\ast F_2$ coincide in the set $\overline{G(M_1, F_1)}$ (recall Notation \[De:good\_set\]), but not necessarily in its exterior.
Topology
--------
Here, we define a map $\Psi\colon(M_1,F_1)\to (M_2,F_2)$ that will be shown to satisfy the claim of Theorem \[Th:smooth\]. Whenever we do not need to distinguish manifolds $M_1$ and $M_2$ we drop the subindices.
We start with showing that data determine the function $r_x\colon\p M \to M$ for any $x \in \p M$. By the triangle inequality and the continuity of distance function $d_F(\cdot,z)$ on $M$ we have $$\label{eq:boundary_point_dist}
r_x(z):=d_F(x,z)=\sup_{q \in M^{int}}(d_F(q,z)-d_F(q,x))=\sup_{q \in M^{int}}(r_q(z)-r_q(x))$$ for all $z \in \p M$. Thus data determine $r_x$, moreover and imply $$\label{eq:BDD_agree_1}
\{r_{x_1}:{x_1}\in M_1\}=\{r_{x_2}\circ \phi:{x_2}\in M_2\}\subset C(\p M_1).$$ Since $\p M$ is compact it holds that for any ${x}\in M$ the corresponding boundary distance function $r_{x}$ belongs to $C(\p M) \subset L^\infty(\p M)$. By and we have recovered the mapping $$\label{eq:map_R}
\mathcal{R}\colon M \to C(\p M), \quad \mathcal{R}(x)=r_x.$$ In the next proposition, we study the properties of this map.
\[Pr:topology\] Let $(M,F)$ be a smooth compact Finsler manifold with smooth boundary. The map $\mathcal{R}$ given by is a topological embedding.
Since $M$ is compact, $d_F$ is a complete non-symmetric (path) metric, and by [@burago2001course Theorem 2.5.23] for any $x_1,x_2 \in M$ there exists a distance minimizing curve $\gamma\colon [0,d_F(x_1,x_2)]\to M$ from $x_1$ to $x_2$. Moreover, whenever $a,b \in [0,d_F(x_1,x_2)]$ are such that $\gamma((a,b))\subset M^{int}$, then $\gamma\colon [a,b]\to M$ is a geodesic.
Since the unit sphere bundle $SM:=F^{-1}\{1\}$ is compact there exists a universal constant $L>1$, such that for all $ x_1,x_2 \in M$ we have $$\label{eq:quasi}
\frac{1}{L} d_F(x_1,x_2)\leq d_F(x_2,x_1) \leq L d_F(x_1,x_2).$$ We let $x_1,x_2 \in M$ and $z\in \p M$. By triangular inequality we have $$|d_F(x_1,z)-d_F(x_2,z)| \leq L d_F(x_1,x_2).$$ Thus $\|r_{x_1}-r_{x_2}\|_\infty\leq L d_F(x_1,x_1)$, which proves that the map $\mathcal{R}$ is continuous.
We suppose then that $r_{x_1}=r_{x_2}$ for some $x_1,x_2 \in M$. We let $z$ be one of the closest boundary points to $x_1$. Then $z$ is also a closest boundary point to $x_2$. Denote $r_{x_1}(z)=h$. If $h=0$ then $x_1=z$ and thus $x_1=x_2$. We suppose then that $h>0$, which means that $x_1$ and $x_2$ are interior points of $M$. We let $\gamma$ be a unit speed distance minimizing curve from $x_1$ to $z$. Then $\gamma$ is a geodesic and $\dot{\gamma}(h)$ is an outward pointing normal vector to $\p M$ (see Lemma \[Le:normal\_geo\_is\_mini\] in the appendix for the details). Since $\gamma$ is also a distance minimizer from $x_2$ to $z$, we have proved $$x_1=\stackrel{\leftarrow}{\gamma}(h)=x_2,$$ where $\stackrel{\leftarrow}{\gamma}(t):=\gamma(h-t)$.
The injectivity of $\mathcal{R}$ implies that it is a topological embedding, as any continuous one-to-one map from a compact space to a Hausdorff space is an embedding.
Next we define maps $$\Phi\colon C(\p M_1) \to C(\p M_2), \quad \Phi(f)=f\circ \phi^{-1}$$ and $$\label{eq:map_Psi}
\Psi\colon M_1\to M_2, \quad \Psi=\mathcal R^{-1}_2\circ \Phi \circ \mathcal R_1$$ Here $\mathcal R_i$ is defined as $\mathcal{R}$ in . The main theorem of the section is the following
\[Th:topology\] Let $(M_i,F_i), \: i =1,2$ be as in Theorem \[Th:smooth\]. Then the map $\Psi\colon M_1\to M_2$ given by is a homeomoprhism. Moreover $\Psi|_{\p M_1}=\phi$.
By and Proposition \[Pr:topology\] it holds that $\Psi$ is well defined. Clearly the map $\Phi$ is a homeomorphism and therefore $\Psi$ is a homeomorphism.
We let $x_1 \in \p M_1$. Then $(\Phi \circ \mathcal R_1)(x_1)$ is $r_{x_2}$ for some $x_2 \in M_2$. Since $$r_{x_2}(\phi(x_1))=[(\Phi \circ \mathcal R_1)(x_1)](\phi(x_1))=r_{x_1}(x_1)=0.$$ This proves $\Psi(x_1)=\phi(x_1)$.
Differentiable structure
------------------------
Here, we show that the map $\Psi\colon M_1\to M_2$ is a diffeomorphism. We split the study in two cases, near the boundary and far from the boundary. We begin with the former one.
We extend $(M,F)$ to a closed Finsler manifold $(N,H)$ to facilitate the study of boundary points.
We let $\stackrel{\leftarrow}{\nu_{in}}$ be the inward pointing unit normal vector field to $\p M$ with respect to reversed Finsler function $\stackrel{\leftarrow}{F}$. We define the normal exponential map $\exp^\perp\colon \partial M\times{{\mathbb R}}\to N$ so that $$\exp^\perp(z,s)
:= \;
\stackrel{\leftarrow}{\exp_z}(s\stackrel{\leftarrow}{\nu_{in}}(z)),$$ where $\stackrel{\leftarrow}{\exp_z}$ is the exponential map of the reversed Finsler function $\stackrel{\leftarrow}{H}$.
\[Le:boundary\_normal\_coordinates\] There exists $h>0$ such that $M$ is contained in the image of the normal map. Moreover there exists $r>0$ such that $\exp^\perp\colon \p M\times [0,r) \to M$ is a diffeomorphism onto its image.
Define $$h=\max \{d_{F}(x,\p M): x\in M\}+c,$$ for any $c>0$. Since $N$ is compact the map $\exp^\perp\colon \p M\times [0,h) \to N$ is well defined. Moreover it holds that any interior point can be connected to any of its closest boundary points via distance minimizing geodesic that is normal to the boundary. Therefore we conclude that $M \subset \exp^\perp(\p M\times [0,h))$.
Notice that $$\exp^\perp(z,t)=\pi(\stackrel{\leftarrow}{\phi}_t(z, \stackrel{\leftarrow}{\nu_{in}}(z))),$$ where $ \stackrel{\leftarrow}{\phi_t}$ is the geodesic flow of $ \stackrel{\leftarrow}{H}$. Since $ \stackrel{\leftarrow}{\nu_{in}}$ is a smooth unit length vector field, this proves that $\exp^\perp$ is smooth.
We let $z\in \p M$. Give any local coordinates $(z',f)$ near $z$ such that $f|_{\p M}=0$ is a boundary defining function. Then with respect to coordinates $(z',t)$ for $(\p M \times (-h,h))$ we have $$D\exp^\perp (z,0)=
\left(\begin{array}{cc}
D_{z'}(z' \circ \exp^\perp) & \frac{\p}{\p t}(z' \circ \exp^\perp)
\\
&
\\
D_{z'}(f \circ \exp^\perp)& \frac{\p}{\p t}(f \circ \exp^\perp)
\end{array} \right)
=
\left(\begin{array}{cc}
id_{n-1} & \overline{a}
\\
\\
\overline 0^T & df(\stackrel{\leftarrow}{\nu_{in}})
\end{array} \right),$$ where $\overline{a}, \overline{0}\in {{\mathbb R}}^{n-1}$ and $df\big(\stackrel{\leftarrow}{\nu}_{in}\big)\neq 0$, since $\stackrel{\leftarrow}{\nu}_{in}$ is not tangential to $\p M$ and $f$ is a boundary defining function. Thus the Jacobian $\exp^\perp (z,0)$ is invertible and by the Inverse Function Theorem $\exp^\perp$ is a local diffeomorphism.
Next we show that there exists $r\in (0,h)$ such that $\exp^\perp\colon \p M\times [0,r) \to M$ is a diffeomorphism onto its image. If this does not hold, there exists a sequence $(x_j)_{j=1}^\infty \in M$ such that $$\exp^\perp (z_j^1,s^1_j)=x_j=\exp^\perp (z_j^2,s^2_j)$$ for some $s^i_j \to 0$, $i\in \{1,2\}$ as $j\to \infty$ and for some boundary points $z^1_j$ and $z^2_j$ such that $(z_1,s_1)\neq (z_2,s_2).$ Then $d_F(x_j,\p M)\to 0$ as $j\to \infty$ and by the compactness of $N$ we may assume that $x_j\to x \in \p M$. Let $\epsilon >0$ and choose $j\in {{\mathbb N}}$ so that $d_F(x_j,x), s^i_j <\epsilon$. Then for $i\in \{1,2\}$ it holds that $$d_F(x,z^i_j)\leq d_F(x,x_j)+d_F(x_j,z^i_j)<2L\epsilon$$ where $L$ is the constant of . Therefore, $z^i_j\to x$ as $j\to \infty$ for $i\in \{1,2\}$. This is a contradiction to the local diffeomorphism property of $\exp^\perp$. Thus there exists $r>0$ that satisfies the claim of this lemma.
We immediately obtain
\[Le:boundary\_coordinates\] Let $(M,F)$ be compact Finsler manifold with smooth boundary that is isometrically embedded into a closed Finsler manifold $(N,H)$. Let us denote $$U(\p M, \epsilon):=\{x\in M: d_F(x,\p M)<\epsilon\}.$$ There exists $\epsilon>0$ and a diffeomorphism $U(\p M, \epsilon) \ni x \mapsto (z(x),s(x)) \in (\p M \times [0,\epsilon))$, such that $$\label{eq:function_s}
d_F(x,z(x))=d_F(x,\p M)=s(x).$$
The claim follows from Lemma, \[Le:boundary\_normal\_coordinates\], if we denote $(z(x),s(x)):= (\exp^\perp)^{-1}(x)$.
We then consider points far from the boundary. Our goal is to show that for every $x_0 \in M^{int}$ there exists points $(z_i)_{i=1}^n \subset \p M$ and a neighborhood $U$ of $x_0$ such that the map $$U \ni x \mapsto (d_F(x,z_i))_{i=1}^n$$ is a coordinate map. To do this we need to set up some notation.
\[De:boundary\_cut\_points\] Let $z \in \p M$. We say that $$\tau_{\p M}(z):=\sup \{t>0: d_F(\exp^\perp(z,t),z)=d_F(\exp^\perp(z,t),\p M)=t\},$$ is the *boundary cut distance* to $z$. Then we define the collection of *boundary cut points* $\sigma(\p M)$ as follows $$\sigma(\p M)=\{\exp^\perp(z,\tau_{\p M}(z)): z \in \p M\}.$$
The set $\sigma(\p M)$ is not empty and the next lemma explains why we cannot use the coordinate structure given by Lemma \[Le:boundary\_coordinates\] far from $\p M$.
\[Le:prop\_boundary\_cut\] Let $z \in \p M$ and $t_0 =\tau_{\p M}(z)$. Then at least one of the following holds
1. The map $\exp^\perp$ is singular at $(z,t_0)$.
2. There exists $q \in \p M, \: q \neq z$ such that $ \exp^\perp(z,t_0)= \exp^\perp(q,t_0)$.
Moreover for any $t \in [0,t_0)$ the map $\exp^\perp$ is non-singular at $(z,t)$.
The proof of the first claim is a modification of the proof of [@do1992riemannian Chapter 13, Propostion 2.2]. The proof of the last claim is long. It is considered in detail in Section \[Se:Appendix2\].
\[Le:bcutlocus\_func\_cont\] The function $\tau_{\p M}\colon \p M \to {{\mathbb R}}$ is continuous.
The proof is a modification of the proofs of [@klingenberg Lemma 2.1.15] and [@do1992riemannian Chapter 13, Proposition 2.9].
Recall that the *cut distance function* of the extended manifold $(N,H)$ is defined as $$\label{eq:cut_dist_func}
\tau(x,v)=\sup \{t>0:d_H(x,{\gamma}_{x,v}(t))=t\}, \quad (x,v)\in TN, \quad F(x,v)=1.$$ We call a point $\gamma_{x,v}(\tau(x,v))$ *an ordinary cut point* to $x$. In the next Lemma we show that a boundary cut point always occurs before an ordinary cut point.
\[Le:boundary\_vs\_normal\_cut\_dist\] For any $z \in \p M$ it holds that $$\stackrel{\leftarrow}{\tau}(z,\stackrel{\leftarrow}{\nu}_{in}(z))>\tau_{\p M}(z),$$ where $\stackrel{\leftarrow}{\tau}$ is the cut distance function of the reversed Finsler metric $\stackrel{\leftarrow}{H}$.
The proof is a modification of the proof of [@Katchalov2001 Lemma 2.13].
Let $x_1 \in M$ and $z_{x_1} \in \p M$ be a closest boundary point to $x$. There exist neighborhoods $U \subset M$ of $x_1$ and $V$ of $z_{x_1}$ such that for every $(x,z)\in (U\times ( \p M \cap V))$ there exists the unique distance minimizing unit speed geodesic $\gamma_{x,z}$ from $x$ to $z$ and moreover $\gamma_{x,z}[0,d_F(x,y))) \subset M^{int}$.
The claim follows from the Implicit Function Theorem, Lemmas \[Le:cut\_dist\_func\] and \[Le:boundary\_vs\_normal\_cut\_dist\], and the fact that $\stackrel{\leftarrow}{\nu}_{in}$ is transversal to the boundary.
We let $z \in \p M$ and define an evaluation function $E_z\colon \mathcal{R}(M) \to {{\mathbb R}}$ by $E_z(r)=r(z)$. We note that the functions $E_z$ correspond to the distance function\
$d_F(\cdot,z)\colon M \to {{\mathbb R}}$ via the equation $$\label{eq:distance_func}
d_F(x,z)=(E_z \circ \mathcal{R})(x).$$ Since $z \in \p M$ was an arbitrary point we note that the function $d_F\colon M \times \p M\to {{\mathbb R}}$ is determined by the data in the sense of .
We define the *exit time function* $$\label{eq:exit_time_func}
\tau_{exit}\colon SM^{int} \to [0,\infty], \quad \tau_{exit}(x,v):=\inf\{t>0: \gamma_{x,v}(t) \in \p M\}.$$
If $(x,v)\in SM^{int}$ is such that $\tau_{exit}(x,v)<\infty$ and $\dot{\gamma}_{x,v}(\tau_{exit}(x,v))$ is transversal to $\partial M$ then there exists a neighborhood $U\subset SM$ of $(x,v)$ such that $\tau_{exit}|_{U}$ is well defined and $C^\infty$-smooth.
Since $\dot{\gamma}_{x,v}(t_0)$ is not tangential to $\p M$ the claim follows from the Implicit Function Theorem in boundary coordinates.
Take an interior point $x\in M$ near which we want to construct a system of coordinates. We let $v\in S_xM$ be such that the geodesic $\gamma_{x,v}$ emanating from $x$ to the direction $v$ is the shortest curve between $x$ and a terminal boundary point $z_x$. By Lemma \[Le:boundary\_vs\_normal\_cut\_dist\] these two points are not conjugate along $\gamma_{x,v}$.
We let $U \subset SM$ be so small neighborhood of $(x,v)$ that the exit time function $\tau_{exit}\colon U \to {{\mathbb R}}$ is defined and smooth. We have thus assumed that $x$ and $z_x=\gamma_{x,v}(\tau_{exit}(x,v))$ are connected minimally and without conjugate points by $\gamma_{x,v}$.
We let $\ell_x\colon T_xM\to T_x^*M$ be the Legendre transform, (to recall the definition see in the appendix). It and its inverse are smooth outside the origin. Thus the distance function $d_F(\cdot,z_x)$ is smooth near $x$ and its differential at $x$ is $\ell_x(v)\in T_x^*M$ (see Lemma \[Le:differential\_of\_dist\_func\]).
Pick any $u\in T_x^*M\setminus \{0\}$ with $\langle u,v\rangle=0$. For $s\in\mathbb R$, denote $$v_s=\frac{\ell_x^{-1}(\ell_x(v)+su)}{F^\ast(\ell_x(v)+su)}.$$ Here $F^\ast$ is the dual of $F$, (see ). The map $s\mapsto v_s\in S_xM$ is smooth.
Consider the geodesics $\gamma_{v_s}$ starting at $x$ in the direction $v_s$. Since $\gamma_{x,v}(\tau_{exit}(x,v))$ is transversal to $\partial M$, then $s\mapsto\gamma_{v_s}(\tau_{exit}(x,v_s))$ is smooth near $s=0$. Also since $x$ is not an ordinary cut point to $\gamma_{v_s}(\tau_{exit}(x,v_s))$ at $s=0$, it is not either an ordinary cut point to $\gamma_{v_s}(\tau_{exit}(x,v_s))$ when $\lvert s\rvert$ is small. Therefore, for $s$ sufficiently close to zero the distance function to $\gamma_{v_s}(\tau_{exit}(x,v_s))$ is smooth near $x$.
The differential of the distance function at $x$ amounts to $$\ell_x(v_s)=\frac{\ell_x(v)+su}{F^\ast(\ell_x(v)+su))}.$$ Therefore, for any $u$ with the required property there is a small non-zero $s$ so that there is a distance function to a boundary point which is smooth near $x$ and the differential at $x$ is $\frac{\ell_x(v)+su}{F^\ast(\ell_x(v)+su)}$.
We take $n-1$ covectors $u_1,\dots,u_{n-1} \in T^\ast_x M$ so that the set $$\{\ell_x(v),u_1,\dots,u_{n-1}\}\subset T_x^*M$$ is linearly independent and each $u_i\in T_x^*M$ is orthogonal to $v\in T_xM$. For each $i=1,\dots,n-1$ we take $s_i\neq0$ so that $$\frac{\ell_x(v)+s_iu_i}{F^\ast(\ell_x(v)+s_iu_i)}$$ is the differential of a distance function to a boundary point as described above.
This gives rise to distance functions to $n$ boundary points close to one another. These functions are smooth near $x$ and the differentials are $$\ell_x(v),\frac{\ell_x(v)+s_1u_1}{F^\ast(\ell_x(v)+s_1u_1)}, \ldots, \frac{\ell_x(v)+s_{n-1}u_{n-1}}{F^\ast(\ell_x(v)+s_{n-1}u_{n-1})}.$$ This set is linearly independent, so the distance functions give a smooth system of coordinates in a neighborhood of $x$. Thus we obtain
\[Le:interior\_coordinates\] Let $x_0 \in M^{int}$. There is a neighborhood $U$ of $x_0$ and points $z_1,\ldots,z_n \in \p M$, where $z_1$ is a closest boundary point to $x_0$, so that the mapping $U \ni x \mapsto (d_F(x,z_i))_{i=1}^n$ is a smooth coordinate map.
Moreover there exists an open neighborhood $V \subset \p M$ of $z_1$ such that the distance function $d_F\colon U\times V\to {{\mathbb R}}$ is smooth and the set $$\mathcal{V}:=
\bigg\{(z_i)_{i=2}^n \in V^{n-1}: \det (f_{z_2,\ldots, z_n}(x))\bigg|_{x=x_0}\neq 0\bigg\}.$$ is open and dense in $V^{n-1}:=V\times \cdots \times V$. Where $$\label{eq:determinant_of _distances}
f_{z_2,\ldots, z_n}(x):=D\widetilde f_{z_2,\ldots, z_n}(x),$$ and $D\widetilde f_{z_2,\ldots, z_n}$ stands for the pushforward of the map $$\widetilde f_{z_2,\ldots, z_n}(x):=(d_F(x,z_i))_{i=1}^n)\in {{\mathbb R}}^n, \quad x \in U.$$
It remains to show that the set $\mathcal{V}$ is open and dense in $V^{n-1}$. Clearly the function $$G:V^{n-1} \to {{\mathbb R}}, \quad G(z_2,\ldots,z_n)=\det(f_{z_2,\ldots, z_n}(x_0))$$ is continuous. Thus $\mathcal{V}=V^{n-1}\setminus G^{-1}\{0\}$ is open. Since the Legendre transform is an metric isometry between fibers, we have for every $z \in V$ $$d(d_F(\cdot,z))|_{x_0} \in S_{x_0}^\ast M:=\{p\in T_{x_0}^\ast M: F^\ast(p)=1\}$$ (For the details see Lemma \[Le:differential\_of\_dist\_func\] in the appendix.). We let $(e_i)_{i=1}^n$ be a basis of $T_{x_0}^\ast M$ and define a map $T\colon ( T_{x_0}^\ast M)^{n-1} \to {{\mathbb R}}$ by $$T((u_i))_{i=2}^{n} =\det(M(\ell_{x_0}(v), u_2,\ldots, u_{n})),$$ where $\ell_{x_0}(v)$ is as in the discussion before this lemma and $M(\ell_{x_0}(v), u_1,\ldots, u_{n-1})$ is a real $n\times n$ matrix with columns $\ell_{x_0}(v), u_1,\ldots, u_{n-1}$, with respect to basis $(e_i)_{i=1}^n$ of $T_{x_0}^\ast M$. Notice that $(T_{x_0}^\ast M)^{n-1} $ is a real analytic manifold and $T$ is a multivariable polynomial, and thus a real analytic function. Moreover by the discussion before this lemma we know that $T$ is not identically zero. Therefore, it follows from [@helgason2001differential Lemma 4.3] that $T^{-1}\{0\}\subset ( T_{x_0}^\ast M)^{n-1} $ is nowhere dense. Since determinant is multilinear and the map $V\ni z\mapsto d(d_F(\cdot,z))|_{{x_0}} \in S_{x_0}^\ast M$ is a smooth embedding, it follows that $\mathcal{V} \subset V^{n-1}$ is dense.
Now we are ready to prove the main theorem
\[Th:diffeo\] The mapping $\Psi$ given in is a diffeomorphism.
We let $x_0 \in M_1$. We suppose first that $x_0$ is an interior point. By the data it holds that $z \in \p M_1$ is a mimimizer of $d_{F_1}(x_0,\cdot)|_{\p M_1}$ if and only if $\phi(z)\in \p M_2$ is a minimizer of $d_{F_2}(\Psi(x_0),\cdot)|_{\p M_2}$. We let $z_1$ be a minimizer of $d_{F_1}(x_0,\cdot)|_{\p M_1}$. Since the map $\phi\colon \p M_1 \to \p M_2$ is a diffeomorphism it follows from the Lemma \[Le:interior\_coordinates\] that there exist points $z_2,\ldots,z_{n}\in \p M_1$ and a neighborhood $U\subset M_1$ of $x_0$ such that the maps $U \ni x_1 \mapsto (d_{F_1}(x_1,z_i))_{i=1}^n$ and $$\Psi(U) \ni x_2 \mapsto (d_{F_2}(x_2,\phi(z_i)))_{i=1}^n=(d_{F_1}(\Psi^{-1}(x_2),z_i))_{i=1}^n$$ are smooth coordinate maps. Thus with respect to these coordinates it holds that the local representation of $\Psi$ near $x_0$ is an indentity map of ${{\mathbb R}}^n$. This proves that $\Psi$ is a local diffeomorphism near any interior point of $M_1$.
We consider next the boundary case. We denote $$U=\{x_1 \in M_1: \hbox{ there exists precisely one minimizer for } d_{F_1}(x_1,\cdot)|_{\p M_1}\}^{int}.$$ By and since $\Psi$ is a homeomorphism, it holds that $$\label{eq:unique_boundary_distance_minimizers}
\Psi(U)=\{x_2 \in M_2: \hbox{ there exists precisely one minimizer for } d_{F_2}(x_2,\cdot)|_{\p M_2}\}^{int}.$$ By Lemma \[Le:boundary\_normal\_coordinates\] it follows that there exists $\epsilon>0$ and open neighborhood $V\subset U \subset M_1$ of $\p M_1$ such that the maps $$\begin{split}
&\: \exp^\perp_{F_1}\colon \p M_1 \times [0,\epsilon) \to V, \quad \hbox{and}
\\
& \: \exp^\perp_{F_2}\colon \p M_2 \times [0,\epsilon) \to \Psi(V)
\end{split}$$ are diffeomorphisms. Moreover due to Lemma \[Le:boundary\_coordinates\] for every $x \in V$ it holds that $x=\exp^\perp_{F_1}(z(x),s(x))$, where $z(x)$ is the minimizer of $d_{F_1}(x, \cdot)|_{\p M_1}$ and $s(x)=d_{F_1}(x, z(x))$. Therefore, by it holds that $$( \exp^\perp_{F_2})^{-1}(\Psi(p))
=(\phi(z(p)), s(p)).$$ Thus we have proved that with respect to coordinates $(V,( \exp^\perp_{F_1})^{-1})$ and\
$(\Psi(V),( \exp^\perp_{F_2})^{-1})$ the local representation of $\Psi$ is $$(\p M_1 \times [0,\epsilon))\ni(z,s)\mapsto (\phi(z),s) \in (\p M_2 \times [0,\epsilon)).$$ Since $\phi\colon \p M_1 \to \p M_2$ is a diffeomorphism we have proved that $\Psi$ is a local diffeomorphism near $\p M_1$.
By Theorem \[Th:topology\] the map $\Psi$ is one-to-one and we have proved that $\Psi$ is a diffeomorphism.
Smoothness of the boundary distance function
--------------------------------------------
Here we consider the smoothness of a boundary distance function and show that the closure of $$\label{eq:good_set}
\widehat{G}(M,F):=\{ (x,y)\in G(M,F): x \in M^{int}, \: d_F(\cdot,z(x,y)) \hbox{ is $C^\infty$ at } x\} \cup \p_{out} T M ,$$ where $\p_{out} T M \subset \p( TM)$ is the collection of outward pointing vectors (excluding tangential ones), coincides with the set $\overline{G(M,F)}$ (see Definition \[De:good\_set\]). This will be used in the next subsection to reconstruct $F$ in $G(M,F)$.
We note that if $(x,y)\in G(M,F)$, with $F(y)=1$ then $$\tau_{exit}(x,y)=t(x,y),$$ where $t(x,y)$ is as in Definition \[De:good\_set\]. We assume below in this section that all vectors are of unit length.
For those $(x,z)\in M^{int}\times \p M$ for which $d_F(\cdot,z)$ is smooth at $x$ we can use the differential of the distance function to determine the image of the distance minimizing geodesic from $x$ to $z$. In this sense our problem is related to the Finslerian version of Hilbert’s $4^{th}$ problem which is: To recover Finsler metric from the images of the geodesics. In this setting the problem has been studied for instance in [@bucataru2016funk; @bucataru2012projective; @bucataru2014finsler; @matveev2012projective; @paiva2005symplectic].
The main result in this section is
\[Le:closure\_of\_Gs\] For any smooth connected and compact Finsler manifold $(M,F)$ with smooth boundary it holds that $$\overline{\widehat{G}(M,F)}=\overline{G(M,F)}.$$
We need a couple of auxiliary results to prove this proposition. We state these auxiliary results below and prove them after the proof of Proposition \[Le:closure\_of\_Gs\].
\[Le:corner\_in\_the\_curve\_2\] Let $x_1,x_2\in M$ and let $c\colon [0,1]\to M$ be a rectifiable curve from $x_1$ to $x_2$. Let $t_0 \in (0,1)$ be such that $c(t_0)\in \p M$. If there exits $\delta>0$ such that $c|_{[t_0-\delta, t_0]}$ is a geodesic and $\lim_{t \nearrow t_0} \dot{c}(t)$ is transversal to $\p M$ then there exists a rectifiable curve $\alpha\colon [0,1]\to M$ from $x_1$ to $x_2$ such that $$\mathcal L (\alpha)<\mathcal{L}(c).$$
\[Le:transverse\_exit\_direction\_and\_smooth\_dist\] Suppose that $(x,v)\in G(M,F)$. If the exit direction is transversal to the boundary then for any $s\in (0,\tau_{exit}(x,v))$ the point $(x',v'):=(\gamma_{x,v}(s),\dot \gamma_{x,v}(s))\in \widehat{G}(M,F)$.
\[Le:final\_condition\] Suppose that $(x,v)\in G(M,F)$, and the exit direction $\eta$ is tangential to the boundary at $z$. Assume that there exists $h>0$ such that for any $h'\in (0,h)$ the geodesic $\stackrel{\leftarrow} {\gamma}_{z,\xi_{h'}}\colon [0,\tau_{exit}(x,v)]\to M$ is well defined, where $$\xi_{h'}:=\frac{-\eta+h'\stackrel{\leftarrow} {\nu}_{in}}{\stackrel{\leftarrow} {F}(z,-\eta+h'\stackrel{\leftarrow} {\nu}_{in})}\in T_zM.$$
Then there exist sequences $(h_j)_{j=1}^\infty, \: (\epsilon_j)_{j=1}^\infty\subset {{\mathbb R}}$ such that $h_j, \epsilon_j>0$, $h_j,\epsilon_j\to 0$ as $j\to \infty$ and moreover the geodesic $\stackrel{\leftarrow} {\gamma}_{z,\xi_{h_j}}$ is a distance minimizing curve of $(M,\stackrel{\leftarrow} {F})$ from $z$ to $\stackrel{\leftarrow} {\gamma}_{z,\xi_{h_j}}(\tau_{exit}(x,v)-\epsilon_j)$ for any $j\in {{\mathbb N}}$ that is large enough.
Since the sets $\widehat{G}(M,F)$ and $G(M,F)$ are conical it suffices to prove that $$\overline{\widehat{G}(M,F)} \cap SM =\overline{G(M,F)} \cap SM.$$ We first prove $\widehat{G}(M,F) \subset \overline{G(M,F)}$, which implies $\overline{\widehat{G}(M,F)} \subset \overline{G(M,F)}$. We let $(x,v) \in \widehat{G}(M,F)$. If $x \in M^{int}$, then clearly $(x,v)\in \overline{{G}(M,F)}$. If $(x,v) \in \p_{out}TM$ then due to transversality of $v$ and $T_x\p M$ there exists $\epsilon>0$ such that for every $t\in (0,\epsilon)$ we have $$(\gamma_{x,v}(-t),\dot \gamma_{x,v}(-t)) \in G(M,F).$$ Thus $(x,v) \in \overline{G(M,F)}$.
Next we show that $G(M,F) \subset \overline{\widehat{G}(M,F)}$. We let $(x,v) \in G(M,F)$. Lemma \[Le:transverse\_exit\_direction\_and\_smooth\_dist\] implies that for any $s \in (0,\tau_{exit}(x,v))$ we have $(\gamma_{x,v}(s),\dot \gamma_{x,v}(s))\in \widehat{G}(M,F)$ if $\dot \gamma_{x,v}(\tau_{exit}(x,v))$ is transversal to $\p M$. This implies, $(x,v)\in \overline{\widehat{G}(M,F)}.$
Therefore, we assume that $( \gamma_{x,v}(\tau_{exit}(x,v)),\dot \gamma_{x,v}(\tau_{exit}(x,v))):=(z,\eta)$ is tangential to $\p M$. We let $(N,H)$ be a smooth complete Finsler manifold without boundary that extends $(M,\stackrel{\leftarrow} {F})$ and $\Pi\subset T_zN$ be the two dimensional vector subspace spanned by $\{\eta,\stackrel{\leftarrow} {\nu_{in}}\}$. If $a\in (0,\tau_{exit}(x,v))$ is small enough, then $$S(a):=\{\stackrel{\longleftarrow} {\exp_z}(w)\in N: w\in \Pi, \:\stackrel{\leftarrow} {F}(z,w)<a \}$$ is a $C^1$-smooth hyper surface of $N$ with a coordinate system given by $\eta$ and $\stackrel{\leftarrow} {\nu_{in}}$.
We note that possible after choosing smaller $a$ the set $S(a)\cap \p M$ is given by a $C^1$-smooth graph $(s,c(s))\in S(a)$ such that for $s<0$ we have $c(s)<0$. This follows since $\p M$ is a smooth co-dimension 1 manifold and with respect to the coordinates $(\eta,\stackrel{\leftarrow} {\nu_{in}})$ of $S(a)$ we have $(-t,0)=\gamma_{x,v}(\tau_{exit}(x,v)-t)$ and $\gamma_{x,v}(\tau_{exit}(x,v)-t)$ does not hit $\p M$, if $t\in (0,a)$. Thus for any $h>0$ and $ t\in (0,a)$ the geodesic $\stackrel{\leftarrow} {\gamma}_{z,\xi_h}, $ $$\xi_h:=\frac{-\eta+h\stackrel{\leftarrow} {\nu_{in}}}{\stackrel{\leftarrow} {F}(z,-\eta+h\stackrel{\leftarrow} {\nu_{in}})}$$ of the Finsler function $H$, satisfies $\stackrel{\leftarrow} {\gamma}_{z,\xi_h}(t)\in M^{int}$.
Since the interval $[0,\tau_{exit}(x,v)-\frac{a}{2}]$ is compact and $\gamma_{x,v}([0,\tau_{exit}(x,v)-\frac{a}{2}])\subset M^{int}$. There exists $r>0$ such that for all $t\in [0,\tau_{exit}(x,v)-\frac{a}{2}]$ we have\
$d_F(\gamma_{x,v}(t),\p M)< r$. Therefore, the continuity of the exponential map implies that for any $h>0$ small enough and $t\in (0,\tau_{exit}(x,v)]$ we have $\stackrel{\leftarrow} {\gamma}_{z,\xi_h}(t)\in M^{int}$.
We note that Lemma \[Le:final\_condition\] implies that for any $h,\epsilon>0$ that are small enough we have $$(x',v'):=\bigg(\stackrel{\leftarrow} {\gamma}_{z,\xi_h}(\tau(x,v)-\epsilon),-\dot{\stackrel{\leftarrow} {\gamma}}_{z,\xi_h}(\tau(x,v)-\epsilon)\bigg)\in G(M,F).$$ Then Lemma \[Le:transverse\_exit\_direction\_and\_smooth\_dist\] implies $(x',v')\in \overline{ \widehat G(M,F)}$. Taking $h$ and $\epsilon$ to zero we finally obtain $(x,v)\in \overline{ \widehat G(M,F)}$.
Since $\lim_{t \nearrow t_0} \dot{c}(t)$ is transversal to $\p M$, there exists $\epsilon\in (0,\delta)$ such that $c|_{(t_0-\epsilon,t_0)}$ is a geodesic in $M^{int}$. We let $(\widetilde M, \widetilde F)$ be any compact Finsler manifold that extends $(M,F)$ and for which $c(t_0)$ is an interior point. Since $c_{|_{t_0-\epsilon,t_0}}$ is also a geodesic of the extended manifold $(\widetilde M, \widetilde F)$, it follows from [@shen2001lectures Proposition 11.3.1] that there exists $t_1\in (t_0-\epsilon,t_0)$ such that for $x:=c(t_1), \:z:=(c(t_0))$ we have $$(t_0-t_1)=d_{\widetilde F}(x,z)=d_{F}(x,z)=\tau_{exit}(x,\dot{c}(t_1)),$$ and the exponential map of $(\widetilde M, \widetilde F)$ is a $C^1$-diffeomorphism from $$\{y\in T_xM: F(y)<2(t_0-t_1)\}$$ onto a metric ball $B_{\widetilde F}(x,2(t_0-t_1))$ of $(\widetilde M, \widetilde F)$.
Since $\lim_{t \nearrow t_0} \dot{c}(t)$ is transversal to $\p M$ it follows from the Implicit Function Theorem that there exists a neighborhood $U\subset S_xM$ of $v:=\dot c(t_1)$ where the function $\tau_{exit}
$ is smooth and $\tau_{exit}(x,w)<2(t_0-t_1),$ whenever $w \in U$. We let $$C:=\{rw\in T_xM: w \in U,\: r\in [0,\tau_{exit}(x,w)]\}.
$$ Then $\exp_x(C)$ contains an open neighborhood of $z$ in $M$. Since path $c$ is continuous there exists $t_2>t_0$ such that $\exp_x^{-1}(c(t_2))\in C$, and moreover $$F(x,\exp_x^{-1}(c(t_2)))=d_{\widetilde F}(x,c(t_2))=d_{ F}(x,c(t_2)),$$ since for any $w \in U$ the radial geodesic $\gamma_{x,w}$ of $(\widetilde M, \widetilde F)$ has the minimal length among all curves connecting $x$ to $\gamma_{x,w}(t), t\in (0,2(t_0-t_1))$. If we denote by $\widetilde c$ the geodesic of $(\widetilde M, \widetilde F)$ that satisfies the initial condition $(\widetilde c(0),\dot{\widetilde c}(0))=(x,\dot{c}(t_1))$, then it leaves $M$ at $z$. Therefore, there exits a geodesic $\gamma$ of $(M,F)$ connecting $x=c(t_1)$ to $c(t_2)$ which satisfies $$\mathcal{L}(c|_{[t_1,t_2]})>\mathcal{L}(\gamma).$$ This implies the claim.
We note first that it follows from the Implicit Function Theorem that there exists a neighborhood $U\subset SM$ of $(x,v)$ such that function $\tau_{exit}$ is smooth in $U$. Therefore, the mapping $$U \ni (\widetilde x,w)\mapsto (z(\widetilde x,w),\eta(\widetilde x,w)):=(\gamma_{\widetilde x,w}(\tau_{exit}(\widetilde x,w)),\dot \gamma_{\widetilde x,w}(\tau_{exit}(\widetilde x,w)))$$ is smooth and without loss of generality we may assume $\eta(\widetilde x,w)$ is transverse to $\p M$ for any $(\widetilde x,w)\in U$.
We let $(N, H)$ be any compact Finsler manifold without boundary extending $(M,F)$. We set $(x',v'):=(\gamma_{x,v}(s),\dot \gamma_{x,v}(s))$ and it follows that the points $x'$ and $z:=z(x,v)$ are not conjugate along $\gamma_{x,v}$. Therefore, the exponential map $\exp_{x'}$, of Finsler function $H$ is a diffeomorphism in a neighborhood $V \subset T_{x'} N$ of $hv',$ for $h:=(\tau_{exit}(x,v)-s)$ onto some neighborhood of $z$ in $N$. Moreover $$h=d_F(x',z)\geq d_{H}(x',z).$$
To finish the proof, we show that we can find a smooth Finsler manifold $(\widetilde M, \widetilde F)$ so that $
M \subset \widetilde M \subset N, \: \widetilde F=H|_{\widetilde M}, \: z \in \widetilde M^{int}
$ and there exists a neighborhood $A \subset M^{int}$ of $x'$ so that $$d_F(\widetilde x,z)=\stackrel{\leftarrow}{\widetilde F}(z, \exp_z^{-1}(\widetilde x)), \quad \widetilde x \in A.$$ Above the exponential map is given with respect to $\stackrel{\leftarrow}{\widetilde F}$. This implies $(x',v')\in \widehat{G}(M,F)$ and since $s\in (0,\tau_{exit}(x,v))$ was arbitrary we have $(x,v)\in \overline{\widehat{G}(M,F)}$.
We let $W_0$ be the image of $V$ under the orthogonal projection $y\mapsto \frac{y}{F(x',y)}$ on $S_{x'}N$. We let $r_0\in (0, d_F(x',\p M))$ be so small that for any $w \in S_{x'}M$ geodesic $\gamma_{x',w}|_{[0,r_0]}$ is a distance minimizer of $H$ and contained in $M^{int}$. In addition we define $$\Gamma:=\{\exp_{x'}(r_0 w)\in M^{int}: w\in\overline{(S_{x'}N) \setminus W_0}\}.$$ Since this set is compact it follows from the triangle inequality that there exists $\epsilon_0>0$ which satisfies $$\label{eq:contradic_dist}
r_0+d_F(\Gamma,z)\geq d_F(x',z)+\epsilon_0,$$ as otherwise there would exist a $\stackrel{\leftarrow}{F}$-distance minimizing curve from $z$ to $x$ which is not $C^1$ at $x'$.
For $p\in N$ and $r>0$ we define $$\stackrel{\leftarrow}{B}_{H}(p,r):=\{q\in N: d_H(q,p)< r\}.$$ Since the points $x'$ and $z$ are not conjugate along $\gamma_{x,v}$ we can choose a neighborhood set $W_1\subset W_0$ of $v'$, and $2\epsilon_1< \epsilon_0$, $\delta >0$ such that $$\stackrel{\leftarrow}{B}_{H}(z,2\epsilon_1) \subset (\exp_{x'}((0, h+\delta) \times W_1))
\cap \stackrel{\leftarrow}{B}_{F}(z,\epsilon_0))$$ and the geodesic $\gamma_{x,v}$ is the shortest curve from $x'$ to $z$ contained in $\exp_{x'}([0, h+\delta) \times W_1))$.
We write $M_k:=M \cup \overline{ \stackrel{\leftarrow}{B}_{H}(z,k\epsilon_1)}$, for $k\in \{1,2\}$. Finally, we let $(\widetilde M, \widetilde F)$ be any smooth compact Finsler manifold with boundary such that $$M_1 \subset \widetilde M \subset M_2 \quad \hbox{and} \quad \widetilde F=H|_{\widetilde M}.$$
If $\beta$ is a distance minimizing curve of $(\widetilde M, \widetilde F)$ from $x'$ to $z$ it is a geodesic of $(M,F)$ for $t <r_0$. Therefore, we have that $\beta=\gamma_{x,v}$ if $\dot{\beta}(0)\in W_1$. If $\dot \beta(0)\in (W_0\setminus W_1)$, then $\beta$ hits $\p \widetilde M$ transversally outside $\stackrel{\leftarrow}{B}_{H}(z,2\epsilon_1)$, which cannot happen due to Lemma \[Le:corner\_in\_the\_curve\_2\]. If $\dot{\beta}(0) \in \overline{(S_{x'}N) \setminus W_0}$ then by we have $$r_0+d_F(\Gamma,z)-\epsilon_0\geq d_F(x',z)\geq d_{\widetilde F}(x',z)\geq r_0+s_1+2\epsilon_1,$$ where $s_1\geq 0$ is the time it takes to travel from $\Gamma$ to $(\stackrel{\leftarrow}{B}_{H}(z,2\epsilon_1)\cap \widetilde M)$ along the curve $\beta$. We note that $\beta(r_0+s_1)$ is contained in $M$ which implies $$s_1+\epsilon_0 \geq d_F(\Gamma,z).$$ Thus we arrive at a contradiction $0 \geq 2\epsilon_1$, and we have proven that $\gamma_{x,v}$ is the unique distance minimizing curve of $(\widetilde M, \widetilde F)$ connecting $x'$ to $z$.
Since $x'$ and $z$ are not conjugate points along $\gamma_{x,v}$, the exponential map of the reversed Finsler function $\stackrel{\leftarrow}{\widetilde F}$ is a diffeomorphism from a neighborhood of $-h\eta \in T_{z}\widetilde M, \: \eta:=\eta(x,v)$ to a neighborhood of $x'$. Thus the local distance function $$q \mapsto \rho(q,z):=\stackrel{\leftarrow}{\widetilde F}(z,\bigg(\!\stackrel{\leftarrow}{\widetilde \exp}_{z}\!\bigg)^{-1}(q))$$ is smooth near $x'$ and due to earlier part of this proof it coincides with $d_{\widetilde F}(\cdot,z)$ at $x'$.
We suppose that there exists a sequence $(x_j)_{j=1}^\infty\subset M$ that converges to $x'$ and for which it holds that $$\label{eq:local_dist_func}
d_{\widetilde F}(x_j,z) < \rho(x_j,z).$$ We let $\beta_j$ be a distance minimizing curve of $\stackrel{\leftarrow}{\widetilde F}$ from $z$ to $x_j$. Since $(\widetilde M, \widetilde F)$ is a compact (non-symmetric) metric space it follows form [@myers1945arcs] there exists a rectifiable curve $\beta_{\infty}$ connecting $z$ to $x'$, that is a uniform limit of $\beta_{j}$ and whose length is not greater than $d_{\widetilde F}(x',z)$. This implies $\beta_{\infty}(t)=\gamma_{x,v}(h-t),$ since $\gamma_{x,v}$ is the unique ${\widetilde F}$ distance minimizer from $x'$ to $z$.
Since $z\in \widetilde M^{int}$, there exists $R>0$ such that for every $j_k\in {{\mathbb N}}$ the curve $\beta_{j}(t)$ is a geodesic of $(\widetilde M, \stackrel{\leftarrow}{\widetilde F})$ if $t\in [0,R]$. Therefore, $$\label{eq:converg_of_initial_direct}
\dot{\beta}_j(0)\to -\eta \in S_{z}\widetilde M$$ and the continuity of the exit time function $\tau_{exit}$ implies that there exists $J\in {{\mathbb N}}$ such that for every $j>J$ the curve $\beta_{j}$ is a geodesic of $(M, \stackrel{\leftarrow}{ F})$. Thus and contradict with the assumption that $\stackrel{\leftarrow}{\widetilde \exp}_{z}$ is a diffeomorphism near $-h\eta$. Therefore, cannot hold and we have that $d_{\widetilde F}(\cdot, z)$ and the local distance function $ \rho(\cdot,z)$ coincide near $x'$. Hence there exists a neighborhood $A\subset M^{int}$ of $x'$ in which we have $$d_{F}(\cdot, z)=d_{\widetilde F}(\cdot, z)=\rho(\cdot,z),$$ due to continuity of the exit time function.
We let $(z,\eta)$ be the exit point and direction of $\gamma_{x,v}$. We let $(\widetilde M, \widetilde F)$ be any compact Finsler manifold for which $z\in M^{int}$, and choose $s\in (0,\tau_{exit}(x,v))$ and denote $\ell=\ell(s):=\tau_{exit}(x,v)-s$. Then for $x':=(\gamma_{x,v}(s))$ the point $z$ is not a conjugate point along $\gamma_{x,v}$. Since the conjugate distance function is lower continuous [@shen2001lectures Section 12.1] there exist neighborhoods $V\subset T_zM$ of $-\ell\eta$ and $U\subset \widetilde M$ of $\gamma_{x,v}([s,\tau_{exit}(x,v)])$ such that for any $y\in V$ the shortest curve that is contained in $U$ and connects $z$ to the point $x(y):=\stackrel{\leftarrow}{\exp}_z(y)\in U$, is the geodesic $t\mapsto \;\stackrel{\leftarrow}{\exp}_z(ty), \: t\in [0,1]$.
We let $(h_j)_{j=1}^\infty, \: (\epsilon_j)_{j=1}^\infty\subset {{\mathbb R}}$ be such that $h_j, \epsilon_j>0$, $h_j,\epsilon_j\to 0$ as $j\to \infty$. Denote $x_j:=\stackrel{\leftarrow} {\gamma}_{z,\xi_{h_j}}(\ell-\epsilon_j)$ , where $\xi_{h_j}:=\frac{-\eta+h_j\stackrel{\leftarrow} {\nu}_{in}}{\stackrel{\leftarrow} {F}(z,-\eta+h_j\stackrel{\leftarrow} {\nu}_{in})}$. Then $x_j\to x'$ as $j\to \infty$. We let $c_j:[0,\ell]\to M$ be a distance minimizing curve of $(M,\stackrel{\leftarrow} {F})$ from $z$ to $x_j$. Then we have $${{\mathcal L}}(c_j)\longrightarrow \ell \hbox{ as } j\longrightarrow \infty.$$ Due to [@myers1945arcs] we can without loss of generality assume that curves $c_j$ converge uniformly to a rectifiable curve $c_\infty$, from $z$ to $x'$, that satisfies $${{\mathcal L}}(c_\infty)\leq \ell.$$ Then [@shen2001lectures Proposition 11.3.1] implies $c_\infty=\stackrel{\leftarrow} {\gamma}_{z,-\eta}|_{[0,\ell]}$, since otherwise there would exist a distance minimizing curve of $(M,\stackrel{\leftarrow} {F})$ from $z$ to $x$ that is not $C^1$-smooth at $x'$. Since $c_j \to\stackrel{\leftarrow} {\gamma}_{z,-\eta|_{[0,\ell]}}$ uniformly in $[0,\ell]$ there exists $J\in {{\mathbb N}}$ such that for all $j\geq J $ we have $( \stackrel{\leftarrow}{\exp}_z)^{-1}(x_j)\in V$ and the image of $c_j$ is contained in $U$. Thus after unit speed reparametrization of $c_j$ we have, $c_j(t)=\stackrel{\leftarrow} {\gamma}_{z,\xi_{h_j}}(t)$ for any $t\in [0,\ell-\epsilon_j]$.
We recall that above we had $\ell=\tau_{exit}(x,v)-s$. The claim of this lemma follows using a diagonal argument for sequence $(\epsilon^j_i)_{i=1}^\infty,(h^j_i)_{i=1}^\infty\subset (0,1)$ which are chosen as above for $s_j\in (0,\tau_{exit}(x,v)),\: j \in {{\mathbb N}}$, such that $s_j \to 0$ when $j\to 0$.
Finsler structure
-----------------
In this section, we prove that the data determine the set $\widehat{G}(M,F)$, (see ) and the Finsler function $F$ on it. Again we deal separately with interior and boundary cases.
Let $x \in M^{int}$. The set $T_x M \cap \widehat{G}(M, F)$ contains an open non-empty set. Moreover the data determine the set $T_x M \cap \widehat{G}(M, F)$ and $F$ on it.
We let $z_x\in \p M$ be a closest boundary point to $x$. By Lemma \[Le:boundary\_vs\_normal\_cut\_dist\] the function $d_F(\cdot,z_x)$ is smooth at $x$. Moreover $$v:=-\dot{\stackrel{\leftarrow}{\gamma}}_{z_x,\stackrel{\leftarrow}{\nu}_{in}(z_x)}(d_F(x,z_x)) \in S_xM \cap \widehat{G}(M,F).$$ Thus the set $S_xM \cap \widehat{G}(M,F)$ is not empty. By Lemma \[Le:boundary\_vs\_normal\_cut\_dist\] there exists a neighborhood $U\subset S_xM$ of $v$ that is contained in $\widehat{G}(M,F)$.
Next, we prove the latter claim. We let $z \in \p M$ be such that the function $d_F(\cdot,z)$ is smooth at $x$. We let $v \in \stackrel{\leftarrow}{ S_x M}$. Then $$\label{eq:F_star }
d(d_{\stackrel{\leftarrow}{F}}(z,\cdot))\bigg|_x = \stackrel{\leftarrow}{g_{v}}(v,\cdot)= \stackrel{\leftarrow}{\ell_x}(v) \quad \hbox{if and only if} \quad \gamma_{x,-v}(d_F(x,z))=z,$$ where $g_{y}(\cdot,\cdot)$ is the hessian of $\frac 12 F^2(x,y)$ with respect to $y$ variables and $\stackrel{\leftarrow}{\ell_x}$ is the Legendre transform of Finsler function $\stackrel{\leftarrow}{F}$ at $x$. The property implies that the set $$\label{eq:F_star_2}
A(x):= \{d(d_{\stackrel{\leftarrow}{F}}(z,\cdot))\bigg|_x: z \in \p M, \: d_{\stackrel{\leftarrow}{F}}(z,\cdot) \hbox{ is $C^\infty$ at } x\}$$ satisfies $$A(x)=\:\stackrel{\leftarrow}{\ell_x}(\stackrel{\leftarrow}{ S_x M} \cap (-\widehat{G}(M,F))).$$ Since the Legendre transform is an isometry the dual map $\bigg(\!\!\stackrel{\leftarrow}{F}\!\!\bigg)^\ast$ is constant $1$ on $A(x)$. As the function $\bigg(\!\!\stackrel{\leftarrow}{F}\!\!\bigg)^\ast$ is positively homogeneous of order $1$ we have determined $\bigg(\!\!\stackrel{\leftarrow}{F}\!\!\bigg)^\ast$ on ${{\mathbb R}}_+A(x):=\{r p \in T^\ast_xM: p \in A(x), \: r>0\}$. Recall that the components of the Legendre satisfy $$\label{eq:recover_of_Legendre}
\bigg((\stackrel{\leftarrow}{\ell_x})^{-1}(p))\bigg)_j=\frac{1}{2}\bigg(\frac{\p}{\p p^i}\frac{\p}{\p p^j}\bigg(\bigg(\!\!\stackrel{\leftarrow}{F}\!\!\bigg)^\ast\bigg)^2(x,p)\bigg)p_i$$ for all $p \in T_x^\ast M.$ Since $\bigg(\!\!\stackrel{\leftarrow}{F}\!\!\bigg)^\ast$ is recovered on ${{\mathbb R}}_+A(x)$ the equation determines $(\stackrel{\leftarrow}{\ell_x})^{-1}$ on ${{\mathbb R}}_+A(x)$. Therefore, $$\begin{split}
&T_x M \cap (-\widehat{G}(M,F))=(\stackrel{\leftarrow}{\ell_x})^{-1}({{\mathbb R}}_+A(x)) \hbox{ and }
\\
&\stackrel{\leftarrow}{F}\; =\bigg(\bigg(\!\!\stackrel{\leftarrow}{F}\!\!\bigg)^\ast\bigg)^\ast \hbox{ on } T_x M \cap(- \widehat{G}(M,F)).
\end{split}$$ Finally, $$\begin{split}
&F(x,y)=\;\stackrel{\leftarrow}{F}(x,-y).
\end{split}$$ This concludes the proof.
\[Le:recovery\_of\_F\_in\_interior\] Let $x \in M^{int}_1$. Then $$\widehat{G}(M_1, F_1)\cap T_xM_1=\widehat{G}(M_1,\Psi^\ast F_2)\cap T_xM_1$$ and $$F_1(y)=F_2(\Psi_\ast y), \quad y \in (\widehat G(M_1, F_1)\cap T_xM_1).$$
The mapping $\Psi$ is a diffeomorphism that satisfies $$\label{eq:connection_of_distat_func}
d_{F_2}(\Psi(\cdot), \phi(\cdot))|_{M_1 \times \p M_1}=d_{F_1}(\cdot, \cdot)|_{M_1 \times \p M_1}.$$ Thus for any $z \in \p M_1$ the function $d_{F_2}(\Psi(\cdot), \phi(z))$ is smooth at $\Psi(x)$ if and only if $d_{F_1}(\cdot, z)$ is smooth at $x$. Therefore, the claims follow by applying the differential of $M_1$ to the both sides of and using –.
Next, we consider the boundary case.
Let $x \in \p M$. Then data determine $F$ on $$\p_{out}TM \cap T_xM=\{y \in T_xM: g_{\nu_{in}}(\nu_{in},y)<0\}.$$
We let $y \in T_xM \setminus \{0\}$ be an outward pointing vector that is not tangential to the boundary. We let $b>a\geq 0$ and choose any smooth curve $c\colon [a,b] \to M$ such that $$c((a,b)) \subset M^{int}, \quad c(b)=x, \: \dot{c}(b)=y.$$ Recal that with respect to the geodesic coordinates at $x$ we have $d_F(x,c(t))=F(\exp^{-1}_x(c(t)))$. Since $F$ is continuous we have $$\label{eq:F_on_boundary}
\lim_{t \to b}\frac{d_F(x,c(t))}{b-t}
=F(\dot c(b))=F(y).$$ Since $y \in T_xM \setminus \{0\}$ was an arbitrary outward pointing vector the data and determine $F$ on the set $\p_{out}TM \cap T_xM.$
\[Le:recovery\_of\_F\_on\_boundary\] For any $(x,y) \in \p_{out}TM_1$ it holds that $$F_1((x,y))=F_2(\Psi_\ast (x,y)).$$
The claim follows from and .
Now we are ready to give a proof of Theorem \[Th:smooth\].
By theorems \[Th:topology\] and \[Th:diffeo\] the map $\Psi\colon M_1 \to M_2$ is a diffeomorphism, and the pullback $\Psi^*F_2$ of $F_2$ gives a Finsler function on $M_1$. By Lemmas \[Le:recovery\_of\_F\_in\_interior\] and \[Le:recovery\_of\_F\_on\_boundary\] we have proved that $\overline{G(M_1,F_1)}$ and $\overline{ G(M_1,\Psi^*F_2)}$ coincide and in this set $F_1=\Psi^*F_2$.
We still have to show that the data are not sufficient to guarantee that $F_1$ and $F_2$ coincide in $TM_1^{int}\setminus \overline{G(M_1,F)}$. We denote a manifold $M_1$ by $M$ and a Finsler function $F_1$ by $F$. If $TM^{int}\setminus \overline{G(M,F)}$ is not empty, we choose $(x_0,v_0) \in TM^{int}\setminus \overline{G(M,F)}, \:F(v_0)=1$ and a neighborhood $V \subset TM^{int}\setminus \overline{G(M,F)}$ of $(x_0,v_0)$ such that $$\label{eq:V_is_far_from_pM}
\dist_{F}(\pi(V), \p M)>0.$$ We denote the orthogonal projection of $V$ to the unit sphere bundle of $(M,F)$ by $W$. We let $\alpha \in C_0^ \infty(W)$ be non-negative and define a function $$\label{eq:distorted_Fins}
H\colon {{\mathbb R}}\times TM \to {{\mathbb R}}, \quad H(s,y)=\bigg(1+s \alpha\bigg(\frac{y}{F(y)}\bigg)\bigg)F(y).$$ We show that there exists $\epsilon >0$ such that for any $s\in (-\epsilon,\epsilon)$ the function $H(s,\cdot)\colon TM \to {{\mathbb R}}$ is a Finsler function.
Since $\alpha$ is compactly supported it holds, for $|s|$ small enough, that $H(s,\cdot)$ is non-negative, continuous and $H(s,y)=0$ if and only if $y=0$. Moreover $H(s,\cdot)$ is smooth outside the zero section of $M$. Clearly also the scaling property $H(s,ty)=tH(s,y), \: t>0$ is valid.
We let $(x,y)$ be a smooth coordinate system of $TM$ near $(x_0,v_0)$. To prove that $H(s,\cdot)$ is a Finsler function, we have to show that for every $(x,y) \in TM \setminus \{0\}$ the Hessian $$\frac{1}{2}\frac{\p}{\p y^i}\frac{\p}{\p y^j}H^2(s,(x,y))=\frac{1}{2} \frac{\p}{\p y^i}\frac{\p}{\p y^j}\bigg[\bigg(1+s \alpha\bigg(\frac{y}{F(y)}\bigg)\bigg)^2\bigg(F(y)\bigg)^2\bigg]$$ is symmetric and positive definite. Since $H^2(s,(x,\cdot))\colon T_xM \to {{\mathbb R}}$ is smooth outside $0$ it follows that the Hessian is symmetric, and $\alpha \in C^\infty_0(W)$ implies $$\begin{split}
\frac{1}{2} \frac{\p}{\p y^i}\frac{\p}{\p y^j}H^2(s,(x,y))&
= g_{ij}(x,y) +\mathcal{O}(s)
\end{split}$$ where $g_{ij}$ is the Hessian of $\frac{1}{2}F^{2}$. Therefore, for $|s|$ small enough $H(s,\cdot)\colon TM \to {{\mathbb R}}$ is a Finsler function.
We let $\epsilon >0$ be so small that $H(s,\cdot)$ is a Finsler function for $s \in (0,\epsilon)$. We prove that for any $x \in M$ and $z \in \p M$ $$\label{eq:boundary_data_round_metric_and_distorted_metric}
d_{H(s,\cdot)}(x,z)= d_{F}(x,z).$$ This implies that the boundary distance data of $F$ and $H(s,\cdot)$ coincide.
If $c\colon [0,1] \to M$ is any piecewise $C^1$-smooth curve, implies $$\label{eq:lenghts_of_round_metric_and_distorted_metric}
\mathcal{L}_{F}(c)\leq \mathcal{L}_{H(s,\cdot)}(c).$$ We let $x \in M$ and $z \in \p M$. Since $M$ is compact there exists a $F$-distance minimizing curve $c\colon [0,d_{F}(x,z)] \to M$ from $x$ to $z$. We let $I, J \subset [0,d_{F}(x,z)] $ be a partition $[0,d_{F}(x,z)]$ such that $$c(t)\in M^{int} \quad \hbox{ if and only if } \quad t\in I.$$ Then $I$ is open in $[0,d_{F}(x,z)]$ and $J$ is closed. On set $I$ the curve $c$ is a union of distance minimizing geodesic segments of $F$ which have end points in $\p M$. Thus for any $t \in I$ we have $\dot{c}(t) \in G(M,F)$. This and imply $$d_{F}(x,z)=\mathcal{L}_{F}(c)= \mathcal{L}_{H(s,\cdot)}(c),$$ and the equation follows from .
The proof of Lemma \[Le:prop\_boundary\_cut\] {#Se:Appendix2}
=============================================
In this section, we denote by $(N,F)$ a compact, connected smooth Finsler manifold without boundary. We present the second variation formula in the case when the variation curves start from a smooth submanifold $S$ of $N$. We introduce the concept of a focal distance and connect it to the degeneracy of the normal exponential map $\exp^\perp$ of surface $S$. We use the results of this section to complete the proof of Lemma \[Le:prop\_boundary\_cut\] We define a pullback vector bundle $\pi^\ast TN$ over $TN\setminus \{0\}$ such that for every $(x,y)\in TN\setminus \{0\}$ the corresponding fiber is $T_xN$. Notice that $\pi^\ast TN$ is then defined by the following equation $$\pi^\ast TN=\{((x,y),(x,y'))\in (T_xN\setminus \{0\})\times T_xN: x \in N\}.$$ We let $(x,y)$ be a local coordinates for $TN$. We define a local frame $(\p_i)_{i=1}^n$ for $\pi^\ast TN$ by $$\label{eq:pullback_bundle_frame}
\p_i|_{(x,y)}:=\bigg((x,y),\frac{\p}{\p x^i}\bigg).$$ and a local co-vector field on $TN$ by $$\delta y^i:= dy^i+N^i_jdx^j, \quad N^i_j(x,y):=\frac{\p}{\p y^j} G^i(x,y).$$ Above the functions $G^i$ are the geodesic coefficients of $F$ in coordinates $(x,y)$ (see in \[Se:Appendix1\]). Notice that $\frac{\p}{\p y^i}$ is a dual vector to $\delta y^i$ and a dual vector $\frac{\delta }{\delta x^i}$ to $dx^i$ is given by $$\label{eq:horisontal_part}
\frac{\delta }{\delta x^i}:=\frac{\p }{\p x^i}-N^j_i\frac{\p }{\p y^j}.$$ Therefore, vectors $dx^i$ and $\delta y^j$ are linearly independet for all $i,j \in \{1, \ldots, n\}$ and it holds that $$\label{eq:horizontal_hertical_decomposition}
T^\ast(TN\setminus \{0\})=\span\{dx^i\}\oplus \span\{\delta y^i\}=:\mathcal{H}^\ast(TN) \oplus\mathcal{V}^\ast(TN).$$
We relate $\pi^\ast TN$ locally to $\mathcal{H}^\ast(TN)$ and to $\mathcal{V}^\ast(TN)$ by mappings $$\label{eq:relation_of_pullpack_H_V}
\p_i \mapsto dx^i \hbox{ and } \p_i \mapsto \delta y^i, \: i\in \{1,\ldots, n\}.$$
We denote the collection of smooth sections of $\pi^\ast TN$ by $\mathcal{S}( \pi^\ast TN)$. The *Chern connection* is defined on $ \pi^\ast TN$ by $$\label{eq:Chern_Connect}
\nabla\colon \mathcal{T}(TN)\times \mathcal{S}( \pi^\ast TN)\to \mathcal{S}(\pi^\ast TN), \quad \nabla_XU=\bigg\{dU^i(X)+U^j\omega^i_j(X)\bigg\}\p_i,$$ where $\mathcal{T}(TN)$ is the collection of all smooth vector fields on $TN\setminus \{0\}$ and the connection one forms $\omega^i_j$ on $TN\setminus \{0\}$ are given by $$\label{eq:Chern_forms}
\omega^i_j(x,y):=\Gamma^i_{jk}(x,y)dx^k,$$ and functions $\Gamma^i_{jk}(x,y)$ are defined by [@shen2001lectures equation (5.25)]. They satisfy $$\label{eq:props_of_Christof}
y^k\Gamma^i_{jk}(x,y)=N^i_j(x,y) \quad \hbox{ and } \quad \Gamma^i_{jk}= \Gamma^i_{kj}.$$ See [@shen2001lectures equations (5.24) and (5.25)].
Notice that any vector field $X$ on $TN$, that is locally given by $$X=X^i_x\frac{\delta}{\delta x^i}+X^i_y\frac{\p}{\p y^i}, \quad X^i_x,X^i_y\in C^\infty(TN)$$ defines a section $\widetilde X \in \mathcal{S}( \pi^\ast TN)$ by $$\widetilde X(x,y) =X^i_x(x,y)\p_i.$$
Let $X,Y,Z$ be vector fields on $TN$. Then $$\label{eq:Chern_prop_1}
\nabla_X\widetilde Y- \nabla_Y\widetilde X=\widetilde{[X,Y]}.$$
Equation follows from the definition of the Chern connection and .
The fundamental tensor $g$ on $\pi^\ast TN$ is defined by $$g(U,V):=g_{ij}(x,y)U^i(x,y)V^i(x,y), \quad U,V \in \mathcal{S}( \pi^\ast TN), \: (x,y) \in TN,$$ where $g_{ij}(x,y)=g_y(\frac{\p}{\p x^i},\frac{\p}{\p x^j})$.
Recall that if $X=X^i_x\frac{\delta}{\delta x^i}+X^i_y\frac{\p}{\p y^i}$ is in $T(TN)$, then $D\pi X=X^i_x\frac{\p}{\p x^i} \in TN$.
\[Le:Chern\_prop\_2\] Let $X, Y, Z$ be vector fields on $TN$. Then $$\label{eq:Chern_prop_2}
Yg(\widetilde X, \widetilde Z)\bigg|_{D\pi X}=\bigg[g(\nabla _Y\widetilde X,\widetilde Z)+g(\widetilde X,\nabla _Y\widetilde Z)\bigg]\bigg|_{D\pi X}.$$
The proof is a direct evaluation in coordinates, using that $g_{ij}$ is homogeneous of order zero with respect to directional variables. It is important to evaluate $Y g(\widetilde X, \widetilde Z)$ at $D\pi X$ as for an arbitrary direction, does not hold, since the Cartan tensor does not vanish identically.
If $ V(x)=V^i(x)\frac{\p}{\p x^i}$ is a vector field on $N$ then $\widehat V(x,y):=V^i(x)\frac{\delta}{\delta x^i}$ is a horizontal vector field on $TN$. We call $\widehat V$ a horizontal lift of $ V$. We define a covariant derivative $D_t$ of smooth vector field $V$ on geodesic $\gamma$ as $$\label{eq:covariant_der}
D_tV(t):= \bigg\{\dot{V}^i(t)+V^j(t)N_j^i(\dot{\gamma}(t))\bigg\}\frac{\p}{\p x^i}\bigg|_{\gamma(t)}.$$ In the next lemma we relate the covariant derivative to the Chern connection.
Let $t \mapsto c(t)$ be a geodesic on $(N,F)$ and $V$ be a smooth vector field on $c$ that is extendible. Write $V(x)=V^i(x)\frac{\p}{\p x^i}$. Then $\widetilde V(x,y):=V^i(x)\p_i|_{(x,y)}$ is a smooth section of $ \mathcal{S}( \pi^\ast TN)$ and $$\label{eq:Chern_prop_3}
\widetilde{D_t V}=\nabla_{\widehat{\dot c}} \widetilde V.$$
To prove the claim, we use and do a direct evaluation in coordinates.
We now consider variations of a geodesic $\gamma:[0,h]\to N$ normal to a hypersurface $S$ so that one endpoint stays on $S$ and the other one is fixed. We denote the starting point of $\gamma$ by $z_0\in S$. For a smooth curve $\sigma$ on $S$ we assume that a variation $\Gamma(s,t)$ satisfies $\Gamma(s,0)=\sigma(s)$, $\Gamma(s,h)=\gamma(h)$, and $\Gamma(0,t)=\gamma(t)$ for all values of the time $t\in[0,h]$ and the variation parameter $s$ near zero. The variation field $J(t):=\frac{\p}{\p s} \Gamma(s,t)|_{s=0}$ is a vector field along $\gamma$ and satisfies the boundary conditions $J(0)=\dot{\sigma}(0)$ and $J(h)=0$. We additionally assume that the variation is normal: $g_{\dot{\gamma}}(\dot{\gamma}, J)\equiv 0$.
The second variation formula (see [@shen2001lectures Chapter 10]) and equations – imply that $$\frac{\p^2}{\p s^2}\mathcal{L}(\Gamma(s,\cdot))\bigg|_{s=0}=\int_0^hg_{\dot
\gamma}(D_t^2 J(t)-\textbf{R}_{\dot{\gamma}}( J(t)), J(t))dt+
g(\nabla_{\widehat J} \widetilde\nu-\nabla_{\widehat \nu} \widetilde
J, \widetilde{J})\bigg|_{\nu},$$ where $\textbf{R}_{( \cdot)}( \cdot)$ is the *Riemannian curvature tensor* (see [@shen2001lectures Chapter 6]). If $J$ is a Jacobi field, the expression simplifies to $$\frac{\p^2}{\p s^2}\mathcal{L}(\Gamma(s,\cdot))\bigg|_{s=0}=
g(\nabla_{\widehat J} \widetilde\nu-\nabla_{\widehat \nu} \widetilde
J, \widetilde{J})\bigg|_{\nu}.$$ To discuss geodesic variations of $\gamma$, we consider normal Jacobi fields $J$ along $\gamma$ that satisfy $$\label{eq:Transverse_J_fields}
J(0)\in TS \hbox{ and } \bigg(\nabla_{\widehat J}
\widetilde\nu-\nabla_{\widehat \nu} \widetilde J\bigg)\bigg|_{\nu}=0.$$ We let $\mathcal{J}$ be the collection of all normal Jacobi fields on $\gamma$ that satisfy and $\mathcal
J_0=\{J\in\mathcal J:J(h)=0\}$. Following [@Chavel], we call $\mathcal{J}$ the space of *transverse Jacobi fields*.
\[Le:char\_of\_trans\_jacobi\] A vector field $J$ on $\gamma$ is a transverse Jacobi field if and only if $$\label{eq:char_of_trans_jacobi}
J(t)=D\exp^\perp\bigg|_{(z_0,t)} t \eta \quad \hbox{ for some } \eta =J(0)\in T_{z_0}S.$$
We let $\epsilon>0$ and $U\subset S$ be a neighborhood of $z_0$ be such that the normal exponential map $\exp^\perp\colon U \times (-\epsilon,\epsilon)\to N$ is a diffeomorphsim onto its image. We define a unit length vector field $W$, that is orthogonal to $S$, by $$W(x):=\frac{\p}{\p t}\exp^\perp(z,t)=\dot{\gamma}_{z,\nu(z)}(t), \quad (z,t)\in U \times (-\epsilon,\epsilon),$$ where $x=\exp^\perp(z,t)$.
For any $z\in U$ the geodesic $\gamma_{z,\nu(z)}$ of $F$ is also a geodesic of the local Riemannian metric $$g_W(x):=\hbox{Hess}_y\bigg(\frac 12 F(x,y)^2\bigg)\bigg|_{y=W(x)},$$ that is normal to $S$. This implies that the normal exponential maps of $F$ and $g_W$ coincide in $U \times (-\epsilon,\epsilon)$ and moreover $D_t =D^W_t$, where $D_t$ is the covariant derivative of $F$ (given in ) and $D^W_t$ is the covariant derivative of Riemannian metric $g_W$ on $\gamma_{z,\nu(z)}$.
Now we are ready to prove the claim of this lemma. We let $\sigma(s)\in S$ be a smooth curve with initial conditions $\sigma(0)=z_0$ and $\dot \sigma(0)=\eta=J(0)$. Define $\Gamma(s,t)=\exp^\perp(\sigma(s),t)$. Then $\Gamma(0,t)=\gamma(t)$ and all the variation curves $t \mapsto\Gamma(s,t)$ are geodesics. Therefore, the variation field $$V(t):=\frac{\p}{\p s}\Gamma(s,t)\bigg|_{s=0}, \quad t \in(-\epsilon,\epsilon)$$ is a Jacobi field of $F$ that satisfies $$V(0)=\frac{\p}{\p s}\Gamma(s,t)\bigg|_{s=t=0}=D\exp^\perp\bigg|_{(z_0,0)}\dot\sigma(0)=\eta.$$ Therefore, it suffices to show that $D_tJ(0)=D_tV(0)$. We note that since $g_W$ is a Riemannian metric the following symmetry holds true, $$\label{eq:symmetry_formula}
D_t \frac{\p}{\p s} =D_s\frac{\p}{\p t},$$ along any transverse curve $s\mapsto \Gamma(s,t)$. Above $D_s$ is a covariant derivative of $g_W$ along transverse curve $\Gamma(\cdot,t)$. We also assume that $ \dot{\sigma}$ can be extended to a smooth vector field $Y$ on $N$. Then the equation implies $$\begin{split}
D_tV(0)=
\overline\nabla_{Y}W\bigg|_{z_0}=\overline\nabla_{\eta}W,
\end{split}$$ where $\overline{\nabla}$ is the Riemannian connection of $g_W$. To end the proof we still have to show the first equation of the following $$\widetilde{\overline\nabla_{\eta}W}=\nabla_{\widehat \eta}\widetilde W=\nabla_{\widehat W}\widetilde \eta=\widetilde{ D_t J(0)},$$ where $\nabla$ is the Chern connection of $F$. The proof of this claim is a direct computations in local coordinates.
We obtain the following lemma as a direct consequence
Set $ \mathcal{J}$ is a real vector space of dimension $n-1$.
The claim follows since the dimension of $S$ is $n-1$ and the operator given by is linear in $T_{z_0}S$ and onto at $t=0$.
Similarly to the spaces $\mathcal J$ and $\mathcal J_0$ of Jacobi fields defined above, we denote by $\mathcal V$ the collection of piecewise smooth normal vector fields along $\gamma$ satisfying and by $\mathcal V_0$ the subspace vanishing at $\gamma(h)$. On $\mathcal{V}_0$ we define the index form $$\label{eq:index_form}
\begin{array}{rl}
I(V,W):=&\int_0^hg_{\dot \gamma}(D_t V(t),D_t W(t))-g_{\dot \gamma}(\textbf{R}_{\dot{\gamma}}( V(t)), W(t))dt
\\
-& g(\nabla _{\widehat W}\widetilde V\bigg|_{\dot \gamma(0)},\widetilde\nu ).
\end{array}$$
The index form $I$ on $\mathcal{V}_0$ is a symmetric bilinear form.
Clearly, $I$ is bilinear. It is proven in [@shen2001lectures Section 8.1] that for all $x\in N$ and $y,v,w \in T_xN$, it holds that $$g_{y}(\textbf{R}_{y}(v),w)=g_{y}(v,\textbf{R}_{y}(w)) .$$ Since $V, W$ are normal to $\dot{\gamma}$, the equation $$g(\nabla _{\widehat W}\widetilde V\bigg|_{\dot \gamma(0)},\widetilde\nu )
=g(\nabla _{\widehat V}\widetilde W \bigg|_{\dot \gamma(0)},\widetilde\nu)$$ follows from and the symmetry of the second fundamental form (see [@shen2001lectures Section 14.4]).
\[Le:Tangential\_variations\] Assume that $\gamma$ is not self-intersecting on $[0,h]$. We let $ V\in \mathcal{V}$. There exists $\delta>0$ and a variation $\Gamma(s,t)\colon(\delta,\delta) \times [0,h]\to N$ of $\gamma$ whose variation field $\frac{\p}{\p s}\Gamma(s,t)|_{s=0}$ is $ V$ and $\Gamma(s,0)$ is a smooth curve on $S$. Moreover if $t_1, \ldots, t_k \in [0,h]$ are the points where $ V$ is not smooth then $\Gamma\colon (\delta,\delta) \times (t_i,t_{i+1}) \to N$ smooth.
We let $W$ be a smooth vector field that is an extension of $\dot{\gamma}(t)$ in a neighborhood of $\gamma([0,h])$. Using the Fermi coordinates of $S$, with respect to the local Riemannian metric $g_W$, we can construct a Riemannian metric $\widetilde g$ in some neighborhood of $\gamma([0,h])$ such that $S$ is a geodesic submanifold of $\widetilde g$ and $\gamma$ is a geodesic of $\widetilde g$ that is $\widetilde g$-normal to $S$. Then we use the following variation to prove the claim of this lemma.
We let $\delta>0$ and define a variation of $\gamma$ with $$\label{eq:Variation_of_V}
\Gamma(s,t)=\exp_{\widetilde g}(\gamma(t),sV(t));\: t\in [0,h], s\in (-\delta,\delta),$$ where $\exp_{\widetilde g}$ is the exponential map of metric tensor $\widetilde g$. Since $S$ is a geodesic sub manifold with respect to $\widetilde g$ we have that $$\Gamma(s,0)=\exp_{\widetilde g}(\gamma(0),sV(0)) \in S, \quad s\in (-\delta,\delta).$$ Moreover $$\frac{\p}{\p s} \Gamma(s,t)\bigg|_{s=0}=D((\exp_{\widetilde g})_{\gamma(t)})\bigg|_0V(t)=V(t).$$ The claim is proven.
For a given vector field $V \in \mathcal{V}_0$ we call the variation of $\gamma(t)$ given by $\eqref{eq:Variation_of_V}$ the variation related to $V$.
We say that $\gamma(h)$ is a focal point of $S$ if the set $\mathcal{J}_0$ contains a non-zero Jacobi field.
\[Le:Char\_of\_trans\_Jacobi\] The point $\gamma(h)$ is a focal point of $S$ if and only if $D\exp^\perp$ is singular at $(z_0,h)$.
The claim follows from Lemma \[Le:char\_of\_trans\_jacobi\].
We define the quantities $\tau_S(z_0)$ and $\tau_f(z_0)$ as $$\label{eq:tau_S}
\tau_S(z_0)=\sup\{t>0:t=d_F(z_0,\gamma_{z_0,\nu}(t))=d_F(S,\gamma_{z_0,\nu}(t))\}$$ and $$\label{eq:tau_f}
\tau_f(z_0)=\inf\{t>0: \gamma(t) \hbox{ is a focal point to $S$}\}.$$
We note that $\tau_S$ is analogous to $\tau_{\p M}$ given in Definition \[De:boundary\_cut\_points\]. Our final goal is to show that $\tau_S(z_0) \leq \tau_f(z_0)$. This completes the proof of Lemma \[Le:prop\_boundary\_cut\]. To check the inequality, we still have to state one auxiliary result
If $\tau_f(z_0)> h$, then Index form $I$ is positive definite on $\mathcal V_0$. If $\tau_f(z_0)= h$, then $I$ is positive semidefinite on $\mathcal V_0$ and $I(V,V)=0$ if and only if $V\in \mathcal{J}_0$.
The proof is a modification of the proof of [@Chavel Theorem II.5.4].
Suppose that $\tau_f(z_0)< h$. Then there exists $W \in \mathcal{V}_0$ such that $$I(W,W)<0.$$ Moreover $$\label{eq:bnd_cut_poits_ocur_bf_focal}
\tau_S(z_0) \leq \tau_f(z_0).$$
Denote $\tau_f(z_0):=t_0 <h$. Choose a non-zero $J\in \mathcal{J}$ that vanishes at $t_0$. Define $$V(t)=
\left\lbrace \begin{array}{l}
J(t), \: t \leq t_0
\\
0, \: t \in [t_0,h).
\end{array}
\right.$$ By previous Lemma it holds that $I(V,V)=0$. Since $D_t J(t_0)\neq 0$ there exists a non-zero smooth vector field $X \in \mathcal{V}_0 $ on $\gamma$ that satisfies $$\hbox{supp}X\subset (0,h) \hbox{ and } X(t_0)=-D_tJ(t_0).$$ Therefore if $\epsilon >0$ is small enough $I(V+\epsilon X, V+\epsilon X)$ is negative.
Finally, we prove . We denote $W:=V+\epsilon X$. We let $\Gamma(s,t)$ be the variation of $\gamma(t)$ that is related to $W$. Since $\gamma$ is a geodesic we have $$\frac{d}{ds}\mathcal{L}(\Gamma(s,\cdot))=0 \hbox{ and } \frac{d^2}{ds^2}\mathcal{L}(\Gamma(s,\cdot))=I(W,W)<0.$$ Therefore, $\gamma$ cannot minimize the lenght from $S$ to $\gamma(h)$. Thus the inequality is valid.
Acknowledgements {#acknowledgements .unnumbered}
----------------
MVdH was supported by the Simons Foundation under the MATH + X program, the National Science Foundation under grant DMS-1815143, and by the members of the Geo-Mathematical Imaging Group at Rice University. JI was supported by the Academy of Finland (decision 295853). Much of the work was completed during JI’s visits to Rice University, and he is grateful for hospitality and support. ML was supported by Academy of Finland (decisions 284715 and 303754). TS was supported by the Simons Foundation under the MATH + X program. Part of this work was carried out during TS’s visit to University of Helsinki, and he is grateful for hospitality and support.
Basics of compact Finsler manifolds {#Se:Appendix1}
===================================
In this appendix, we summarize some basic theory of compact Finsler manifolds. This section is intended for the readers having background in imaging methods and elasticity. We follow the notation of [@shen2001lectures] and use it as a main reference. The main goal is to prove that if $x \in
M^{int}$ and $z_x \in \p M$ is a closest boundary point to $x$, that is the minimizer of $d_F(x,\cdot)|_{\p M}$ or $d_F(\cdot,x)|_{\p M}$, then the distance minimizing curve from $x$ to $z_x$ or from $z_x$ to $x$ respectfully is a geodesic that is perpendicular to the boundary. Readers who are not familiar with Finsler geometry are encouraged to read this section before embarking to the proof of Theorem \[Th:smooth\] presented in Section \[Se:proof\].
Most of the claims and the proofs given in this section are modifications of similar theorems in Riemannian geometry. We refer to the classical material where the Riemannian version is presented.
We let $N$ be a $n$-dimensional, compact, connected smooth manifold without boundary. We reserve the notation $TN$ for the tangent bundle of $N$ and say that a function $F\colon TN \to [0, \infty)$ is a *Finsler function* if
1. $F\colon TN\setminus \{0\} \to [0, \infty)$ is smooth
2. For each $x \in N$ the restriction $F\colon T_xN \to [0, \infty)$ is a *Minkowski norm*.
Recall that for a vector space $V$ a function $F\colon V \to [0, \infty)$ is called a Minkowski norm if the following hold
- $F\colon V\setminus \{0\} \to {{\mathbb R}}$ is smooth.
- For every $y \in V$ and $s >0$ it holds that $F(sy)=sF(y)$.
- For every $y \in V\setminus \{0\}$ the function $g_y\colon V \times V \to {{\mathbb R}}$ is a symmetric positive definite bilinear form, where $$\label{eq:Finsler_to_Riemannian}
g_y(v,w):=\frac{1}{2}\frac{\p}{\p s}\frac{\p}{\p t}\bigg[F^2(y+sv+tw)\bigg]\bigg|_{s=t=0}.$$
We call the pair $(N,F)$ a Finsler manifold. The length of a piecewise smooth curve $c\colon I\to N$, $I$ is an interval, is defined as $$\label{eq:lenght_of_curve}
\mathcal{L}(c):=\int_I F(\dot{c}(t)) dt.$$ For every $x_1,x_2 \in N$ we define $$d_F(x_1,x_2):=\inf_{c\in C_{x_1,x_2}} \mathcal{L}(c),$$ where $C_{x_1,x_2}$ is the collection of piecewise smooth curves starting at $x_1$ and ending at $x_2$. The function $d_F\colon N \times N \to [0,\infty)$ is a non-symmetric path metric related to $F$, meaning that for some $x_1,x_2\in N$ the distance $d_F(x_1,x_2)$ need not coincide with $d_F(x_2,x_1)$ (see [@bao2012introduction Section 6.2]).
We note that for all $x_1,x_2\in N$ it holds that $$\label{eq:dF_and_dtildef}
d_F(x_1,x_2)=d_{\stackrel{\leftarrow}{F}}(x_2,x_1),$$ where $\stackrel{\leftarrow}{F}$ is the reversed Finsler function $\stackrel{\leftarrow}{F}(x,y)=F(x,-y)$.
We use the notation $g_{ij}(x,y)$ for the component functions of the Hessian of $\frac{1}{2}F^2$ as in . A $C^1$ curve $\gamma\colon I \to N$, with a constant speed $F(\dot{\gamma}(t))\equiv c \geq 0$, is a geodesic of Finsler manifold $(N,F)$ if $\gamma(t)$ solves the system of *geodesic equations* $$\label{eq:geodesic_eq}
\ddot \gamma^i(t) + 2G^i(\dot{\gamma}(t))=0, \quad i \in \{1,\ldots, m\}.$$ Here, $G^i\colon TN \to {{\mathbb R}}$ is given in local coordinates $(x,y)$ by $$\label{eq:geodesic_coef_2}
G^i(x,y)=\frac{1}{4}g^{il}(x,y)\bigg\{2\frac{\p g_{jl}(x,y)}{\p x^k}-\frac{\p g_{jk}(x,y)}{\p x^l}\bigg\}y^jy^k.$$ Since $F^2(x,y)$ is positively homogeneous of degree two with respect to $y$ variables, it follows from that $G^i$ is positively homogeneous of degree two with respect to $y$, but not necessarily quadratic in $y$. Therefore, the geodesic equation is not preserved if the orientation of the curve $\gamma$ is reversed.
We define a vector field $\textbf{G}$, by $$\label{eq:geodesic_vector_field}
\textbf{G}(x,y):= y^i\frac{\p}{\p x^i}-2G^i(x,y)\frac{\p}{\p y^i}.$$ A curve $\gamma$ is a geodesic of $F$ if and only if $\gamma=\pi(c)$, where $c$ is an integral curve of $\textbf G$. Due to ODE theory for a given initial conditions $(x,y)\in TN$ there exists the unique solution $\gamma_{x,y}$ of , defined on maximal interval containing $0$. Thus by defining $\textbf{G}$ locally with , it extends to a global vector field on $TN$. We call $\textbf{G}$ the *geodesic vector field*.
\[Le:geodesic\_flow\_has\_const\_speed\] Let $c$ be an integral curve of geodesic vector field $\textbf G$, then $F(c(t))$ is a constant.
For the proof see [@shen2001lectures Section 5.4].
We use the notations $\phi_t$ for the geodesic flow of $F$ on $TN$ and $(x,v)$ for points in $SN$. By Lemma \[Le:geodesic\_flow\_has\_const\_speed\] we know that for $(x,v)\in SN$ and for any $t\in {{\mathbb R}}$ in the flow domain of $(x,v)$ it holds that $\phi_{t}(x,v)\in SN$. Since $SN$ is compact we have proven that $\phi$ on $SN$ is a global flow (see for instance [@Lee Theorem 17.11]), which means that the map $$\phi\colon {{\mathbb R}}\times SN \to SN$$ is well defined. Therefore, we can define the exponential mapping $\exp_x$, $x \in N$ by $$\label{eq:exponential_map}
\exp_x(y):=\pi(\phi_1(x,y))=\gamma_{x,y}(1), \quad y \in T_xN.$$ Moreover in [@shen2001lectures Section 11.4], it is shown that for any points $x_1,x_2 \in N$ there exists a globally minimizing geodesic from $x_1$ to $x_2$.
In the following, we relate the smoothness of a distance function to distance minimizing property of geodesics. This is done via the cut distance function $\tau\colon {SN} \to {{\mathbb R}}$, which is defined by $$\tau(x,v)=\sup \{t>0:d_F(x,{\gamma}_{x,v}(t))=t\}.$$ In the next lemma, we collect properties of the cut distance function.
\[Le:cut\_dist\_func\] Let $(x, v) \in \; {SN}$ and $t_0 =\tau(x,v)$. At least one of the following holds:
1. The exponential map $\exp_x$, of $F$, is singular at $t_0v$.
2. There exists $\eta \in \;S_xN, \: \eta \neq v$ such that $\exp_x( t_0v)= {\exp_x}(t_0\eta)$.
Moreover for any $t \in [0,t_0)$ the map ${\exp_x}$ is non-singular at $tv$. Also the map $\tau\colon {SN}\to {{\mathbb R}}$ is continuous.
See [@shen2001lectures Chapter 12] or [@bao2012introduction Chapter 8].
In the next lemma, we consider the regularity of the function $d_F$
\[Le:smoothness\_of\_the\_distance\_function\] Let $(x_1,v_1) \in SN$, $0<t_1 < \tau(x_1,v_1)$ and $x_2=\gamma_{x_1,v_1}(t_1)$. Then there exists neighborhoods $U$ of $x_1$ and $V$ of $x_2$ respectively such that the distance function $d_F\colon U\times V \to {{\mathbb R}}$ is smooth.
Since the cut distance function $\tau$ is continuous, there exist a neighborhood $U'\subset SM$ of $(x_1,v_1)$ and $\epsilon>0$ such that for any $t \in (t_1-\epsilon,t_1+\epsilon)$ and $(x,v) \in U'$ holds $t<\tau(x,v)$.
Consider a smooth function $$E\colon U' \times (t_1-\epsilon,t_1+\epsilon) \ni ((x,v),t) \to (x,\exp_xtv) \in N \times N.$$ Since for every $ ((x,v),t) \in U' \times (t_1-\epsilon,t_1+\epsilon)$ we have that the exponential map $\exp_x$ is not singular at $vt \in T_xN$, the Jacobian of $E$ is invertible in $U' \times (t_1-\epsilon,t_1+\epsilon)$. Thus the Inverse Function Theorem implies the existence of the neighborhood $U\times V \subset N\times N$ of $(x_1,x_2)$ such that $E$ is a diffeomorphism onto $U \times V$. Therefore the map $$U \times V \ni (x,y) \mapsto \exp^{-1}_xy \in TN$$ is smooth.
By the definition of the cut distance function and [@shen2001lectures Section 11.4], the following equation holds for any $(x,y) \in U \times V$, $$d_F(x,y)=F(x,\exp_x^{-1}y).$$ This implies the claim as $F$ is smooth outside the zero section.
The duality map between the tangent bundle and the cotangent bundle is given by the *Legendre transform* $\ell\colon TN\setminus\{0\}\to T^\ast N \setminus\{0\}$ which is defined by $$\label{eq:Legendre}
\ell(x,y)=\ell_x(y):=g_y(y,\cdot) \in T^\ast_xN, \quad y \in T_xN.$$ The Legendre transform is a diffeomorphism and for all $a>0$ and $(x,y)\in TN\setminus \{0\}$ we have $$\label{eq>legendre_scale}
\ell(x,ay)=a\ell(x,y).$$ (see [@shen2001lectures Section 3.1]). The dual $F^\ast$ of the Finsler function $F$, which is given by $$\label{eq:F_dual}
F^\ast(x,p):=\sup_{v\in S_xN}p(v), \quad (x,p) \in T^\ast N,$$ is a Finsler function on $T^\ast N$ and the Legendre transform $\ell_x$ satisfies $$F(x,v)=F^\ast(x,\ell_x(v)).$$
We let $S \subset N$ be a smooth submanifold of co-dimension $1$. It is shown in [@shen2001lectures Section 2.3] that for every $z \in S$ there exists precisely two unit vectors $\nu_1,\nu_2 \in S_zN$ such that $$T_zS=\{y \in T_zN: g_{\nu_i}(\nu_i,y)=0\}, \: i\in \{1,2\}.$$ Vectors $\nu_1,\nu_2 \in S_pN$ are called the unit normals of $S$. Notice that generally $\nu_1\neq -\nu_2$.
In the next lemma, we relate the Legendre transform of the velocity field of a distance minimizing geodesic to the differential of the distance function.
\[Le:differential\_of\_dist\_func\] Let $x_1 \in N$ and $x_2 \in N$ be such that $d_F(x_1,\cdot)$ is smooth at $x_2$. Then $$\label{eq:differential_of_dist_func}
d(d_F(x_1,\cdot))\bigg|_{x_2}=g_{\dot{\gamma}_{x_1,v}(t)}(\dot{\gamma}_{x_1,v}(t),\cdot)\bigg|_{t=d_F(x_1,x_2)}\in T^\ast_{x_2}N,$$ where $\gamma_{x_1,v}$ is the unique distance minimizing unit speed geodesic from $x_1$ to $x_2$.
Denote $t_0=d_F(x_1,x_2)$ and $$S(x_1,t_0)=\exp_{x_1}\{w \in T_{x_1}N: F(w)=t_0\}.$$ Recall that $$\label{eq:d_F=F}
d_F(x_1,\exp_{x_1}(tw))=F(tw)=t, \quad t>0, \: w\in S_{x_1}N$$ if $tw$ is close to $t_0 v$. We use a shorthand notation $d_F$ for the function $d_F(x_1,\cdot)$. We take a $t$-derivative from the both sides of to obtain $$\label{eq:d_F=F_2}
d(d_F)\bigg|_{\exp_{x_1}(tw)}(D\exp_{x_1}|_{tw}w)=d(d_F)\bigg|_{\exp_{x_1}(tw)}(\dot{\gamma}_{x_1,w}(t))=1.$$ Due to the set $S(x_1,t_0)$ is a regular level set of $d_F$ near $x_2$, and moreover implies $$T_{x_2}S(x_1,t_0)=\ker d(d_F)\bigg|_{x_2}.$$
Thus it suffices to prove that $$\ker g_{\dot{\gamma}_{x_1,v}(t_0)}(\dot{\gamma}_{x_1,v}(t_0),\cdot)=T_{x_2}S(x_1,t_0).$$ Notice that for any $w \in T_{x_1}M,$ such that $g_v(v,w)=0$ holds $$\label{eq:d_F=F_3}
0=g_{tv}(tv,tw)=\frac{1}{2}\frac{d}{ds}[F^2](t(v+sw))\bigg|_{s=0}=t_0 d(d_F)\bigg|_{\exp_{x_1}(tv)}(D\exp_{x_1}|_{tv}tw).$$ Therefore, $d(d_F)|_{\exp_{x_1}(tv)}(D\exp_{x_1}|_{tv}tw)=0$ and $(D\exp_{x_1}|_{t_0v}t_0w) \in T_{x_2}S({x_1},t_0)$. Recall that $J(t):= D\exp_{x_1}|_{tv}tw$ is the unique Jacobi field with initial conditions $J(0)=0, \: D_tJ(0)=w$. By Gauss’ Lemma [@shen2001lectures Lemma 11.2.1] we have $$0=g_{v}(v,w)=g_{\dot{\gamma}_{{x_1},v}(t_0)}(\dot{\gamma}_{{x_1},v}(t_0),D\exp_{x_1}|_{t_0v}w)).$$ In the above, we used the identity $$g_{tv_1}(tv_1,tv_2)=t^2g_{v_1}(v_1,v_2), \quad t>0;\: v_1,v_2 \in T_xN.$$ This implies that $$\ker g_{\dot{\gamma}_{{x_1},v}(t_0)}(\dot{\gamma}_{{x_1},v}(t_0),\cdot)=\{D\exp_{x_1}|_{t_0v}w: g_v(v,w)=0\}= T_{x_2}S({x_1},t_0),$$ since $\dim v^\perp =\dim T_{x_2}S({x_1},t_0)$ and $D\exp_{x_1}|_{t_0v}$ is not degenerate.
\[Le:normal\_geo\_is\_mini\] Let $S \subset N$ be a smooth closed submanifold of co-dimension $1$. Let $x \in N$. A distance minimizing curve from $S$ to $x$ (from $x$ to $S$) is a geodesic that is orthogonal to $S$ at the initial (terminal) point.
Since $S$ is compact there exists a closest point $z_x\in S$ to $x$. We denote $h=d_F(x,z_x)$. Since $(N,F)$ is complete there exists a distance minimizing geodesic $\gamma$ from $x$ to $z_x$.
We suppose first that $d_F(x,\cdot)$ is smooth at $z_x$. We denote $r(z)=d_F(x,z)$ for $z \in S$. Since $z_x$ is a minimal point of $r$ we have $d_S r(z_x)=0$. Here $d_S$ is the differrential operator of smooth manifold $S$. Then $d_Sr=\iota ^\ast d(d_F(x,\cdot))$, where $\iota: S \hookrightarrow N$. Thus $d (d_F(x,\cdot))$ vanishes on $T_{z_x}S$. By it holds that $$d(d_F(x,\cdot))|_{z_x}=g_{\dot \gamma(h)}(\dot \gamma(h),\cdot)\neq 0.$$ Thus $\dot \gamma(h)$ is normal to $S$ at $z_x$.
If $d_F(x,\cdot)$ is not smooth at $z_x$ there exists $\epsilon>0$ such that for any $t\in (\epsilon,h)$ $d_F(\gamma(t),\cdot)$ is smooth at $z_x$. By the first part of the proof it follows that $\dot \gamma(h)$ is perpendicular to $S$.
Due to the second claim for the reversed distance function can be proven in the same way, upon replacing $F$ by $\stackrel{\leftarrow}{F}$.
[^1]: $^\diamond$ Department of Computational and Applied mathematics, Rice University, USA\
$^\dagger$ Department of Mathematics and Statistics, University of Jyväskylä, Finland\
$^\square$ Department of Mathematics and Statistics, University of Helsinki, Finland,\
$^\ast$ **[email protected]**
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'John P. Ahlers'
- 'Jason W. Barnes'
- 'Sarah A. Horvath'
- 'Samuel A. Myers'
- 'Matthew M. Hedman'
bibliography:
- 'LASR\_paper\_arxiv.bib'
title: 'LASR-Guided Stellar Photometric Variability Subtraction: The Linear Algorithm For Significance Reduction'
---
Introduction
============
Recent observing programs such as the *Kepler* mission have demonstrated that stellar variability in high mass stars often renders transit light curves unusable. [@basri2010photometric]. Additionally, high-mass stars often rotate rapidly, inducing an oblate shape and a pole-to-equator luminosity gradient across the stellar surface [@barnes2009transit; @barnes2011measurement; @ahlers2015spin; @ahlers2016gravity]. These two effects add challenges to traditional light curve analysis, radial velocity measurements, Doppler tomography, and Rossiter McLaughlin measurements [@udry2007decade; @gimenez2006equations].
Approximately 60% of stars in the *Kepler* field of view display more stellar variability than the Sun [@mcquillan2013stellar]. Such variability produces effects in transit light curves that make traditional fitting challenging or impossible. Many of these targets are low-mass stars with variability caused by non-sinusoidal effects such as sunspots in their convective exteriors. Techniques for analyzing non-sinusoidal or non-periodic signals in the light curves such as the autocorrelation function [@mcquillan2014rotation] and Gaussian processes [@aigrain2016k2sc] produce strong results when applied to such stars. However, high-mass stars behave quite differently. At $\sim6250$K and hotter, stars invert to become radiative rather than convective at their surface [@winn2010hot]. These stars have weak or nonexistent sunspots, and commonly rotate rapidly as a mostly-rigid body throughout their lifetimes. High-mass stars in the classical instability strip pulsate with radial and nonradial modes at high amplitudes. Therefore, analysis of stellar variability in the light curves of high-mass stars comes with a unique set of challenges and must be handled differently than variability in low-mass stars.
For classical pulsators such as Delta Scuti ($\delta$-Scuti) and Gamma Doradus stars, the technique of “prewhitening" serves as the traditional method of asteroseismic analysis of transit light curves [e.g., @hernandez2009asteroseismic; @poretti2009hd]. Prewhitening fits sinusoids to photometry in an iterative process. This algorithm performs least-squares fits of several of the highest-amplitude oscillations in the time domain, determining approximate frequency, amplitude, and phase values for those oscillations. Prewhitening then fits several next-highest amplitude oscillations as a running total with the original fit, repeating this process until stellar variability has been resolved.
Prewhitening often serves as an adequate method for removing stellar variability and determining the frequencies of oscillation in classical pulsators. However, we explore an alternate route out of necessity: we found that prewhitening provided an inadequate fit of the complex signal of Kepler Object of Interest (KOI) 976. KOI-976 is a high-amplitude, rapidly rotating $\delta$-Scuti star with a complex variable signal. We tried to remove stellar variability from the transit light curve of KOI-976 by applying prewhitening, but we found that the complex signal contained too many oscillations for an accurate least-squares fit in the time domain. This roadblock led us to explore removing stellar variability in frequency-space instead through a new process that we call the Linear Algorithm for Significance Reduction (LASR).
In this paper we develop a frequency-domain method for removing the asteroseismic signal of high-mass pulsators. In $\S$\[sec:methods\] we detail the LASR technique. In $\S$\[sec:results\] we apply LASR to a synthetic dataset and to $\delta$-Scutis KIC 9700322 and KOI-976 to compare our technique against prewhitening. In $\S$4 we discuss how LASR compares with existing techniques for analyzing variability in photometric data.
Methods {#sec:methods}
=======
The Linear Algorithm for Significance Reduction (LASR) serves as an alternate method to prewhitening for signal reduction. It resolves a linear combination of oscillations in a time series by minimizing each oscillation’s significance in the frequency domain. The algorithm operates linearly: it reduces the highest-amplitude frequency, subtracts it from the time series, and then reduces the new highest-amplitude frequency.
The LASR technique has two primary advantages over traditional prewhitening. First, because it operates in frequency space, LASR’s fitting process treats every oscillation as independent and avoids the degeneracies and complex parameter space that prewhitening encounters for datasets containing many oscillation modes. Second, the computer code behind LASR is extremely simple to run and requires very little knowledge of signal processing, making it an accessible technique for inexperienced researchers. In $\S$\[sec:algorithm\] we detail the algorithm for significance reduction. In $\S$\[sec:interdependent\] we discuss handling interdependent frequencies including close frequency pairs and integer-multiple frequencies. In $\S$\[sec:error\] and $\S$\[sec:error\], we detail our best-fit error analysis. In Appendix \[app:deriv\] we mathematically derive that LASR’s straight-forward approach to subtracting stellar oscillations is robust for sinusoidal variability.
LASR Algorithm {#sec:algorithm}
--------------
LASR combines two well-known tools for signal processing: the Lomb-Scargle normalized periodogram, and the downhill simplex routine. To remove a single oscillation from a time series, LASR creates a window of the power spectrum of the time series around the peak of that oscillation (Figure \[fig:synthpeak\]). It then applies the downhill simplex routine to minimize that peak’s significance.
We use the traditional Lomb-Scargle normalized periodogram [@press2007numerical] for spectral analysis. The variations in brightness in high-amplitude $\delta$-Scuti stars correspond to $\sim$5% variations in uncertainty. We test whether these variations in uncertainty affect LASR by comparing the traditional Lomb-Scargle normalized periodogram with the generalized periodogram [e.g., @zechmeister2009generalised; @vio2010unevenly], which applies weights to each time bin according to its photometric uncertainty. We find no noticeable differences between the two methods for $\delta$-Scuti stars KOI-976 and KIC 9700322. In the case of different noise properties and instrument systematics, more computationally expensive techniques that better handle time bin uncertainty may provide better results in the significance reduction process. Such systematics do not appear in this analysis; we therefore favor the traditional Lomb-Scargle periodogram, $$P(\omega) =\frac{1}{2\sigma^2}\sum_{\phi=0,\frac{\pi}{2}} \frac{|\sum_j(h_j-\bar{h})\sin(\omega(t_j-\tau)+\phi)|^2}{\sum_j\sin^2(\omega(t_j-\tau)+\phi)}$$
where $h_j$ is the photometric flux value at time $t_j$, $\sigma$ is the standard deviation in the dataset, $\phi$ is a phase offset that includes the values $0$ and $\pi/2$, and $\omega=2\pi f$ is the angular frequency being sampled. The offset $\tau$ is defined as, $$\tan(2\omega\tau) = \frac{\sum_j\sin(2\omega t_j)}{\sum_j\cos(2\omega t_j)}$$
Several works exist to explain the statistical significance of frequency peaks in a periodogram [e.g., @press2007numerical; @baluev2008assessing]. We represent the statistical significance of an oscillation in our dataset by sampling a window of frequencies around the peak. LASR evenly samples frequencies in a window approximately three times the full width of the peak.
LASR’s adjustable window width depends on the width of frequency peaks; in general, a longer time series means narrower peaks in a periodogram. For KOI-976’s short cadence photometry discussed in $\S$3, we sample $P(\omega)$ 40 times in a window width of $0.7{\mathrm{\mu Hz}}$. We sum $P(\omega)$ values and use the resulting value to represent the significance $S_i(\omega,A,\delta)$ of the $i^{th}$ oscillation in our dataset as a function of frequency $\omega$, amplitude $A$, and phase $\delta$.
The goal of LASR is to find the ([$w$,$A$,$\delta$]{}) values that minimize $S(\omega,A,\delta)$. To do this, LASR uses a downhill simplex routine [@nelder1965simplex; @press2007numerical] to find the minimum of $S(\omega,A,\delta)$. We choose this minimization routine over more robust routines such as Powell’s Method [@brent2013algorithms] because of the well-behaved parameter space that results from minimizing oscillations in the frequency domain (see Appendix \[app:deriv\]). In Appendix \[app:inputs\] we list the necessary inputs to operate LASR and discuss computation times for running the algorithm and in \[app:code\] we provide pseudocode for writing this routine in any computer programming language.
LASR and Interdependent Frequencies {#sec:interdependent}
-----------------------------------
In general, LASR removes oscillations from a dataset one at a time because they are linearly independent from one another. However, two scenarios arise where this assumption fails: close frequency pairs and overtone frequencies.
We define close frequency pairs as oscillations close enough together in frequency space that spectral analysis cannot resolve them individually (Figure \[fig:synthclosefreqs\]), which can cause problems for prewhitening. If two frequencies exist close enough together that their periodogram windows (described in $\S$\[sec:algorithm\]) overlap, then $S_i(\omega,A,\delta)$ cannot be properly minimized. Close frequency pairs are common both for complex datasets with many independent oscillations and for short time series that yield wide frequency peaks in their periodograms. LASR’s solution, however, is simple: just remove both peaks at once. LASR can remove any number of peaks simultaneously by minimizing a combined $S_i(\omega,A,\delta)$ function that samples $P(\omega)$ values for all relevant peaks, and then minimizing that function within a single large simplex.
![A periodogram window for the highest-amplitude peak in the synthetic time series discussed in $\S$\[sec:synthetic\]. We set LASR to sample the spectral power of this peak 25 times. The black and red points show the $P_i(\omega)$ before and after LASR minimized the peak’s significance. The unreduced oscillation shows a clear peak with aliasing on either side. The dashed line marks the true frequency of this oscillation. LASR successfully reduces this oscillation and yields correct measurements of its ([$w$,$A$,$\delta$]{}) values (see Table \[table:synthresults\].)[]{data-label="fig:synthpeak"}](synthpeakplot.pdf)
Overtone frequencies can be more challenging because they are not mathematically independent. The power spectrum of an oscillation can be changed dramatically by integer multiple frequencies. Therefore, minimizing $S_i(\omega,A,\delta)$ for a single frequency in this scenario will result in a poor determination of its ([$w$,$A$,$\delta$]{}) values.
![Log-scale plot of the power spectrum of two close frequencies ($f_2$ and $f_3$ in Table \[table:synthresults\]). The two oscillations appear in frequency space as a single asymmetric peak (black). LASR minimizes the significance of both peaks simultaneously (red) and accurately determines the ([$w$,$A$,$\delta$]{}) of both oscillations. LASR samples frequencies of a periodogram window for each frequency; for close frequency pairs these two windows overlap.[]{data-label="fig:synthclosefreqs"}](synthclosefreqsplot)
[**Synthetic Time Series Quantity**]{} [**Value**]{}
---------------------------------------- ----------------------------------
Length of synthetic dataset $90~\mathrm{days}$
Photometric cadence $1~\mathrm{min}$
Flux normalization constant $1.0$
Transit gap start times $\sum_{n=1}^4 20n~\mathrm{days}$
Transit gap lengths $10~\mathrm{hr}$
Large gap start time $4.5\times10^6~\mathrm{s}$
Large gap length $3.0\times10^5~\mathrm{s}$
Gaussian uncertainty $5.0\times10^{-4}$
Lag correlation coefficient $0.5$
Injected transit depth $1.2\times10^{-4}$
Injected transit period $15~\mathrm{days}$
: Global parameters of the synthetic data we generate to test LASR. We choose these parameters based on typical quantities of *Kepler* short-cadence photometry. The lag correlation coefficient sets the lag-1 autocorrelation between successive time samples, transforming Gaussian noise to correlated noise [@haykin2006nonlinear].[]{data-label="table:synthparams"}
{width="\textwidth"}
[l|lllllllll]{} $f_\mathrm{\#}$ &
-----------------------
$f_\mathrm{start}$
(${\mathrm{\mu Hz}}$)
-----------------------
&
-----------------------
$f_\mathrm{result}$
(${\mathrm{\mu Hz}}$)
-----------------------
&
-----------------------
$f_\mathrm{actual}$
(${\mathrm{\mu Hz}}$)
-----------------------
&
----------------------
$a_\mathrm{start}$
($\mathrm{10^{-3}}$)
----------------------
&
----------------------
$a_\mathrm{result}$
($\mathrm{10^{-3}}$)
----------------------
&
----------------------
$a_\mathrm{actual}$
($\mathrm{10^{-3}}$)
----------------------
& $\delta_\mathrm{start}$ & $\delta_\mathrm{result}$ & $\delta_\mathrm{actual}$\
1 & 228.68 & 228.7$\pm$5e-6 & 228.7 & 10.0 & 20.0013$\pm$0.0014 & 20.0 & 3.000 & 3.99998$\pm$0.00014 & 4.000\
2 & 252.44 & 252.5$\pm$1.1e-5 & 252.5 & 1.0 & 7.5026$\pm$0.0015 & 7.5 & 3.000 & 2.9838$\pm$0.0003 & 3.00\
3 & 252.63 & 252.6$\pm$1.7e-5 & 252.6 & 1.0 & 4.9985$\pm$0.0016 & 5.0 & 3.000 & 2.0011$\pm$0.0005 & 2.00\
4 & 99.930 & 100.0$\pm$1.5e-5 & 100.0 & 1.0 & 7.4997$\pm$0.0014 & 7.5 & 3.000 & 0.0006$\pm$0.0004 & 0.000\
5 & 199.965 & 200.0$\pm$1.3e-5 & 200.0 & 1.0 & 7.4993$\pm$0.0014 & 7.5 & 3.000 & 0.9987$\pm$0.0004 & 1.000\
6 & 299.97 & 300.0$\pm$1.0e-5 & 300.0 & 1.0 & 7.4998$\pm$0.0014 & 7.5 & 3.000 & 2.0009$\pm$0.0003 & 2.000\
7 & 181.194 & 181.1994$\pm$0.0002 & 181.2 & 0.1 & 0.4975$\pm$0.0014 & 0.5 & 3.000 & 5.013$\pm$0.006 & 5.000\
The solution is again to simply remove both peaks simultaneously. The underlying challenge stems from recognizing that this behavior is occurring in the first place. In our analysis of KOI-976, we identify all relevant frequencies through spectral analysis before applying the LASR technique to search for such resonances. We find scant evidence of this scenario arising in KOI-976, but we test simultaneous subtraction in an idealized dataset in $\S$\[sec:synthetic\] and show its successful determination of ([$w$,$A$,$\delta$]{}) for overtone frequencies (Table \[table:synthresults\]).
Error Analysis {#sec:error}
--------------
We find the uncertainty in our best-fit parameters by calculating the covariance matrix of our dataset. Following @andrae2010error, we estimate that the likelihood function $\mathcal{L}$ of our model is nearly Gaussian at its maximum, allowing us to calculate the model’s covariance matrix using the Fisher information matrix $\mathcal{I}$, $$\mathcal{I}_{i,j} = \left(-\frac{\partial^2\log\mathcal{L}}{\partial\theta_i\partial\theta_j} \right)$$
where $\theta_i$ is the $i^{\mathrm{th}}$ model parameter and $\log\mathcal{L}\propto\chi^2$. We numerically approximate these second derivatives in each element of the Fisher matrix as,
$$\mathcal{I}_{i\neq j}= \frac{\chi^2_{i+,j+}+\chi^2_{i-,j-}-\chi^2_{i+,j-} -\chi^2_{i-,j+}}{4\Delta\theta_i\Delta\theta_j}$$
where $\chi^2_{i\pm,j\pm}=\chi^2(\theta_i\pm\Delta\theta_i,\theta_j\pm\Delta\theta_j)$ and $\Delta\theta_i$ is a very small step away from that parameter’s best-fit value. In our calculations we use a frequency step size of $\Delta f = 10^{-10}\mathrm{Hz}$, a normalized amplitude step size of $\Delta A = 10^{-6}$, and a phase step size of $\Delta p = 10^{-4}$.
We calculate the covariance matrix $\hat{\Sigma}$ of our best-fit model by taking the inverse of the Fisher matrix $\hat{\Sigma}= \mathcal{I}^{-1}$. We test whether $\hat{\Sigma}$ is positive definite by checking that all of its eigenvalues are positive. We take the diagonal elements of the covariance matrix to be the $1\sigma$ variance of our model parameters.
Results {#sec:results}
=======
We demonstrate the LASR technique by applying it to a synthetic time series and by applying it to $\delta$-Scuti KOI-976’s short-cadence *Kepler* photometry. We establish LASR’s ability to measure the frequency, amplitude, and phase ([$w$,$A$,$\delta$]{}) of oscillations in a time series containing Gaussian noise, time gaps, overtone frequencies, and close frequency pairs in $\S$\[sec:synthetic\]. We demonstrate that LASR provides a much better fit to KOI-976’s complex variable signal than traditional time-domain prewhitening in $\S$\[sec:koi976\].
LASR Subtraction of Synthetic Oscillations {#sec:synthetic}
------------------------------------------
We create a synthetic time series containing seven oscillation modes commonly seen in a $\delta$-Scuti variable star. We list the global parameters of the synthetic data in Table \[table:synthparams\]. We create a 90-day time series with a 1-minute cadence and add Gaussian noise and data gaps that would commonly occur in *Kepler* photometry. We include five data gaps: four periodic gaps that represent masked-out transits, and one large gap representing the gaps commonly seen in short-cadence *Kepler* photometry. The synthetic time series includes correlated noise commonly seen in *Kepler* data. We generate Gaussian noise using a Box-Muller transform [@box1958note; @press1992random] and weight each uncertainty with a lag-1 autocorrelation between successive time samples [@haykin2006nonlinear].
We include a small periodic transit in our synthetic time series to test its effects on our output code. The transit represents an Earth-radius planet orbiting a $\delta$-Scuti star with a transit depth smaller than *Kepler’s* detection limit. We list the transit parameters in Table \[table:synthparams\]. When testing its effects on our fitting process, we find this injected transit causes no significant influence on our results. In general, we find that transits do not influence the LASR algorithm until their transit depths grow larger than its photometric 1-$\sigma$ uncertainty. At that limit, transits are readily visible in the time series and should be removed. [**We include this transit to show that low-amplitude transits at or below the detection limit do not noticeably affect out fitting results. In our algorithm, we treat removing time bins affected by transits as a standalone prerequisite before applying our significance reduction routine.**]{}
We add seven oscillation modes to the synthetic data to test LASR’s capabilities. We include a single high-amplitude oscillation at $228.7{\mathrm{\mu Hz}}$ as an example of a stand-alone mode in the dataset. We add a close frequency pair at $252.5{\mathrm{\mu Hz}}$ and $252.6{\mathrm{\mu Hz}}$ to test LASR’s ability to reduce oscillations that cannot be individually subtracted through spectral analysis. Additionally, we include three frequencies at integer multiples of one another to test LASR’s ability to remove resonant, interdependent frequencies. We also add one oscillation whose photometric amplitude matches the synthetic data’s $1\sigma$ uncertainty value to test our algorithm’s ability to remove frequencies near the limit of statistical significance.
We subtract oscillations in order of highest-significance peak to lowest-significance (see Table \[table:synthresults\] and Figure \[fig:synthperiodo\]). LASR subtracts the single large peak ($f_1$) quickly and without difficulty. For the close frequency pair $f_2$ and $f_3$, we imitate a real time series by falsely identifying it as a single peak (see Figure \[fig:synthclosefreqs\]). We tried a single starting frequency value of $252.53{\mathrm{\mu Hz}}$ and could not reduce the window’s significance below 67.4%. We then set LASR to remove two close frequencies and immediately found the values listed in Table \[table:synthresults\]. LASR also yielded accurate ([$w$,$A$,$\delta$]{}) values for the resonant $f_4$, $f_5$, and $f_6$ frequencies in a simultaneous fit.
LASR Comparison to Prewhitening: KIC 9700322
--------------------------------------------
We subtract variability from the $\delta$-Scuti star KIC 9700322 and compare our results to @breger2011regularities, who fit stellar variability using the statistical package [period04]{}. Following @breger2011regularities, we use incorporate *Kepler’s* third quarter short cadence photometry in our fit and measure 76 frequencies. We show our best-fit of the stellar variability in Figure \[fig:KIC9700322\] and compare the five highest-amplitude and five lowest-amplitude oscillations to @breger2011regularities in Table \[table:KIC9700322\].
![Sample of KIC 9700322’s stellar variability (black) and our best-fit of the variable signal (red) using LASR. Our fit yeilds a reduced $\chi^2$ value of 1.13.[]{data-label="fig:KIC9700322"}](KIC9700322)
[l|llll]{} $f_\mathrm{\#}$ &
-----------------------
$f_\mathrm{LASR}$
(${\mathrm{\mu Hz}}$)
-----------------------
&
-----------------------
$f_\mathrm{PERIOD04}$
(${\mathrm{\mu Hz}}$)
-----------------------
&
----------------------
$a_\mathrm{LASR}$
($\mathrm{10^{-3}}$)
----------------------
&
-----------------------
$a_\mathrm{PERIOD04}$
($\mathrm{10^{-3}}$)
-----------------------
\
1 & 145.473$\pm$1.8e-5 & 145.472 & 29.391$\pm$0.003 & 29.463\
2 & 113.339$\pm$2e-5 & 113.339 & 27.268$\pm$0.003 & 27.266\
3 & 258.811$\pm$8e-5 & 258.812 & 4.899$\pm$0.003 & 4.902\
4 & 290.945$\pm$0.00015 & 290.945 & 2.665$\pm$0.003 & 2.663\
5 & 32.1338$\pm$0.0002 & 32.133 & 2.637$\pm$0.003 & 2.633\
... & ... & ... & ... & ...\
72 & 727.361$\pm$0.014 & 727.362 & 0.015$\pm$0.003 & 0.019\
73 & 263.59$\pm$0.02 & 263.588 & 0.014$\pm$0.003 & 0.015\
74 & 392.96$\pm$0.02 & 393.002 & 0.0141$\pm$0.003 & 0.015\
75 & 289.09$\pm$0.02 & 289.096 & 0.0135$\pm$0.003 & 0.016\
76 & 598.96$\pm$0.02 & 598.982 & 0.0126$\pm$0.003 & 0.014\
Our best-fit using LASR agrees almost completely with the frequencies and amplitudes found in @breger2011regularities. We successfully model KIC 9700322’s stellar variability throughout *Kepler’s* Q3 short cadence time series. We obtain a reduced $\chi^2$ value of 1.13 for our fit; KIC 9700322’s slightly reddened noise and its photometric outliers were the main causes for this value’s deviation from unity. In the case of relatively straightforward variability, LASR and prewhitening are equivalently reliable, with the added bonus for LASR of only ever fitting a small parameter space at any time.
LASR Subtraction of KOI-976 {#sec:koi976}
---------------------------
{width="\textwidth"}
We perform the same variability subtraction process as that described in $\S$\[sec:synthetic\] on Kepler Object of Interest (KOI) 976, a rapidly rotating $\delta$-Scuti star that hosts an eclipsing binary companion. KOI-976 displays typical seismic activity for a $\delta$-Scuti variable, possessing a few dominant nonradial modes between $\sim100{\mathrm{\mu Hz}}-300{\mathrm{\mu Hz}}$, as well as many low-amplitude oscillations spanning $\sim0{\mathrm{\mu Hz}}-500{\mathrm{\mu Hz}}$. In this analysis, we treat the star as a rigid rotator whose variable signal is well-modelled as a linear combination of sinusoids, which typically serves as an adequate assumption for $\delta$-Scuti stars.
We perform this analysis on KOI-976’s two available quarters of 1-minute *Kepler* photometry available on the Mikulski Archive for Space Telescopes. We use KOI-976’s presearch data conditioning data (PDC) available through the *Kepler* analysis pipeline [@smith2012kepler] and find no relevant differences between the PDC and raw-data versions of the photometry. KOI-976’s time series contains a single transit by its stellar companion, as well as several significant data gaps. We mask out the transit and treat it as another gap in the time series. These gaps produce significant aliasing in periodograms. As we show in Figure \[fig:synthpeak\], LASR subtraction of an oscillation removes both a peak and its aliases, so even very large data gaps are surmountable through this technique.
We follow the process detailed in $\S$\[sec:methods\] and remove KOI-976’s oscillations from the time series one at a time, except in the cases of close frequency pairs and overtone frequencies. We start with the highest-significance peak and work our way down to the significance detection limit based on KOI-976’s photometric uncertainty of $\sim10^{-4}$. In total, we subtract off 319 frequencies. Figure \[fig:periodo\] shows the original and reduced frequency power spectrum of this dataset.
[l]{} ![A comparison of variability subtraction using LASR (red) and traditional prewhitening (blue). We fit all KOI-976 short-cadence photometry and display a $\sim1$ day sample of KOI-976’s short-cadence photometry with best-fit (top) and its best-fit residuals (bottom). LASR produces a superior fit of KOI-976’s seismic activity with a log-likelihood ratio of $-2\log(\mathcal{L}_{PW}/\mathcal{L}_{LASR})=33806$, which illustrates our motivation to create this technique, as it better-reduces complex seismic signals of classical pulsators.[]{data-label="fig:fitplot"}](fitplot "fig:")\
LASR successfully minimizes all significant frequencies present in KOI-976’s short-cadence photometry. Figure \[fig:fitplot\] contrasts LASR with prewhitening and shows that LASR provides a better fit of KOI-976’s oscillations than prewhitening through linear regression. We obtain a log-likelihood ratio between the two methods of $-2\log(\mathcal{L}_{PW}/\mathcal{L}_{LASR})=33806$, indicating a superior resolution of KOI-976’s variability using our technique. We perform error analysis following $\S$\[sec:error\].
Discussion & Conclusion {#sec:discussion}
=======================
Our results show that LASR successfully removes stellar variability commonly seen in classical pulsators. Our technique can remove oscillations from photometry of arbitrary complexity so long as they are well-modelled as sinusoids. We find that for the rapidly-rotating $\delta$-Scuti KOI-976, LASR serves as a superior method for variability subtraction over the traditional prewhitening approach of linear regression in the time domain. In particular, we find that LASR more accurately fits the frequencies of individual oscillations. It also better-resolves close frequency pairs that can be very difficult to identify when fitting in the time domain. Combined with LASR’s reliability, relatively low computation cost, and ease-of-use, we consider our technique to be a useful tool for spectral analysis in asteroseismology.
We develop LASR out of necessity: we originally attempted to subtract variability from KOI-976 following traditional prewhitening methodology that has successfully resolved the oscillations of other $\delta$-Scuti stars [@breger2011regularities; @breger2012relationship]. We found, however, that for KOI-976, we could not obtain accurate frequencies using this technique. We observed several undesired aliasing effects occurring in the low-frequency range of our dataset due to these imperfect fits. We demonstrate that for this dataset, LASR successfully minimizes significant frequencies without producing aliasing effects (Figure \[fig:periodo\]) and provides a better fit of KOI-976’s variable signal than prewhitening (Figure \[fig:fitplot\]).
We put forth LASR as one of many tools available for analyzing photometry. Existing tools such as Period04 [@lenz2004period04] and others [@akritas1996linear; @vio2002joint; @rohlfs2013tools] provide robust and well-established techniques for signal processing of photometry. Additional techniques exist for analyzing stellar variability that contains non-sinusoidal or pseudo-periodic signals. These methods include modelling signals as multivariate random variables through Gaussian processes [@mackay1998introduction; @rasmussen2006gaussian; @aigrain2016k2sc], detecting signals through autocorrelation functions [@edelson1988discrete; @mcquillan2014rotation], and analyzing time-variable signals through wavelet analysis. These techniques are important tools for transit detection, denoising signals, and removing pseudo-periodic stellar signals or oscillations that vary with time. They are particularly useful for analysis of dwarf stars, where solar flares, sunspots, and non-rigid stellar rotation produce complex variable signals in photometry that must be modelled as random events. These techniques produce strong results in subtracting stellar variability, but often come with the complications of being computationally expensive or from treating stellar variability as a random. The LASR routine serves as an inexpensive and straightforward tool for analyzing high-mass stars, which typically do not possess sunspots or flares [@didelon1984stellar], whose surfaces behave as rigid rotators [@suarez2005modelling], and whose oscillations commonly remain constant over long timescales when on the main sequence [@breger1998period].
LASR currently subtracts stellar variability that is well-modelled as a linear combination of sinusoids. Future works can expand this technique to combine LASR’s significance reduction with Gaussian processes or wavelet analysis to analyze other forms of stellar variability. Such an approach could resolve a stellar signal without sacrificing information by treating seismic activity as a random process. Additionally, wavelet analysis could expand LASR’s purview to the variable signal of heartbeat stars [@hambleton2013physics; @smullen2015heartbeat] in the future.
LASR Algorithm {#lasr-algorithm}
==============
Inputs and Usage {#app:inputs}
----------------
LASR uses inputs to control its downhill simplex routine that customize its behavior to the target time series. We list the global parameters that typically remain constant throughout the subtraction process:
1. Frequency scale factor to set initial downhill step sizes. When working with KOI-976’s high-precision photometry, we use a scale factor of $10^{-3}{\mathrm{\mu Hz}}$.
2. Amplitude scale factor. Within an order of magnitude of the highest-significance oscillation’s amplitude is typically adequate. This value can be set smaller as LASR subtracts smaller oscillations in the dataset.
3. Phase scale, typically set to $1.0$.
4. Number of downhill steps for LASR to take. Convergence typically occurs within 50-100 steps for an independent oscillation, but requires about twice as many steps for close frequency pairs or overtone frequencies.
5. The half-width of the peak of the highest-significance frequency in the dataset. This value determines the width of the periodogram window for LASR to sample during each subtraction. For KOI-976 we used a halfwidth of $0.35{\mathrm{\mu Hz}}$.
6. The number of frequencies to sample in the periodogram window. We find that $10$ points provide an adequate sampling of the periodogram window in our analysis, but because of LASR’s relatively low computational cost we use $25$ points.
LASR requires the following inputs to subtract an oscillation:
1. Starting guess for frequency. Easily obtained via periodogram with sufficient accuracy.
2. Starting guess for amplitude. Order-of-magnitude values serve an adequate guess, as shown in Table \[table:synthresults\].
3. Starting guess for phase. Any value between 0 and $2\pi$ typically suffices.
LASR Pseudocode {#app:code}
---------------
We write this algorithm in [c++]{} using established techniques for periodograms and downhill simplex routines following [@press2007numerical]. Our program includes proprietary optimization code and personalized libraries that make direct sharing of this program impractical. However, the LASR routine is straightforward to create. We provide an outline below that uses a periodogram window as a black-box function in a downhill simplex routine to minimize significance and determine an oscillation’s ([$w$,$A$,$\delta$]{}) values. We also make an open-source version of our code available for download at <https://github.com/jpahlers/LASR>.
$S(G|\omega,A,\delta)$ calculates significance in periodogram window ($\S$\[sec:algorithm\]) centered on frequency to subtract.
Benchmark Results
-----------------
We list benchmark performance results of the LASR routine below using the KOI-976 short-cadence data set. The computation time scales with the size of the time series and the desired precision, driven mainly by the cost of calculating periodogram significance values. To speed up our computation time, we compute trigonometric recurrences of the periodogram [@press2007numerical] using two processors in parallel. In general, we find that $N=100$ downhill steps is typically enough to determine the best-fit ([$w$,$A$,$\delta$]{}) values of an oscillation to six significant figures and that 25 periodogram window samples robustly represents a frequency’s significance.
[**Benchmark Quantity**]{} [**Value**]{}
-------------------------------- --------------------
Number of downhill steps ($N$) 100
Time series data points 91235
Periodogram window samples 25
Computation time $12.58~\mathrm{s}$
: Quantities describing the LASR’s computation time using KOI-976’s short cadence photometry. Our algorithm typically fits the frequency, amplitude, and phase of a single oscillation in a timeseries in approximately 100 downhill steps that each take $\sim0.1$ when applied to KOI-976’s 91235 short-cadence time bins.
\[table:benchmark\]
The computation time is primarily dedicated to calculating periodograms. @press2007numerical shows how, depending on the choice of algorithm, Lomb-Scargle periodograms scale as $N\log(N)$, where $N$ is the number of points in the time series. The fitting precision is controlled by the LASR’s downhill simplex algorithm. The rate of convergence can vary largely between datasets, but we find that when fitting one oscillation (three parameters) in KOI-976’s short cadence photometry, we typically achieve four significance figures after $\sim50$ downhill steps and six significant figures after $\sim100$ downhill steps.
Derivation: One Minimum Per Parameter {#app:deriv}
=====================================
The oscillation significance $S(\omega,A,\delta)$ is a black box function that represents the significance of the residuals of a subtracted oscillation. In this section we provide a derivation showing that $S(\omega,A,\delta)$ has only one minimum per parameter (i.e. that $S(\omega,A,\delta)$ is U-shaped in each dimension over all values) so long as the actual frequency and subtracted frequency are relatively close ($|(\omega_1-\omega_2)/(\omega_1+\omega_2)|\lesssim0.1$).
We start with two sine waves $\psi_1=A_1\sin(\omega_1t+\delta_1)$ and $\psi_2=A_2\sin(\omega_2t+\delta_2)$, where $A_i$, $\omega_i$, and $\delta_i$ represent the amplitude, frequency, and phase of each function. A standard oscillation subtraction is therefore represented by, $$\psi=\psi_1-\psi_2 = A_1\sin(\omega_1t+\delta_1)-A_2\sin(\omega_2t+\delta_2)
\label{eq:subtraction}$$
Assuming $\omega_1\approx \omega_2$, equation \[eq:subtraction\] can be expanded as, $$\psi = A_1\sin(\omega_1t+\delta_1)-A_2\sin(\omega_1t+\delta_2)+A_2(\omega_1-\omega_2)t\cos(\omega_2t+\delta_2)+O((\omega_1-\omega_2)^3)$$
Utilizing the Harmonic Addition theorem [e.g., @nahin2001science], the zeroth-order terms $(A_1\sin(\omega_1t+\delta_1)-A_2\sin(\omega_1t+\delta_2))$ can be expressed as a single sine wave $A\sin(\omega_2t+\delta)$, where $$A=\sqrt{A_1^2+A_2^2-2A_1A_2\cos(\delta_1-\delta_2)}$$
and
$$\delta = \mathrm{atan}\left(\frac{A_1\cos(\delta_1)-A_2\sin(\delta_2)}{A_1\sin(\delta_1)-A_2\sin(\delta_2)} \right)$$
Therefore, this subtraction can be represented as, $$\psi = A\sin(\omega_2t+\delta)+A_2(\omega_1-\omega_2)t\cos(\omega_2t+\delta_2)+O((\omega_1-\omega_2)^3)
\label{eq:subtractfinal}$$
Because $S(\omega,A,\delta)$ is roughly proportional to the square of the amplitude of $\psi$, minimizing Equation \[eq:subtractfinal\] minimizes $S(\omega,A,\delta)$. The oscillation residual $\psi$ is minimized via minimizing $A$ and $\omega_1-\omega_2$. $A$ is minimized via minimizing $A_1-A_2$ and $\delta_1-\delta_2$. These are the *only* minima that appear in $\psi$; therefore, only one global minimum per parameter exists.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this note, we give an elementary proof of the lack of null controllability for the heat equation on the half line by employing the machinery inherited by the unified transform, known also as the Fokas method. This approach also extends in a uniform way to higher dimensions and different initial-boundary value problems governed by the heat equation, suggesting a novel methodology for studying problems related to controllability.'
address:
- 'Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK'
- 'Department of Mathematics, Izmir Institute of Technology, Izmir, Turkey'
author:
- Konstantinos Kalimeris
- 'Türker Özsar[i]{}'
title: An elementary proof of the lack of null controllability for the heat equation on the half line
---
Introduction
============
The Uniform Transform Method (UTM), also known as the Fokas method, is a powerful tool for obtaining solutions of initial - (inhomogeneous) boundary-value problems. This method was first introduced in [@Fok97] for the analysis of initial-boundary value problems for integrable nonlinear partial differential equations (PDEs). However, in later works it was proven to produce novel results for a general class of linear PDEs; see [@Fok02; @Fok08]. Recently researchers utilized the UTM to produce rigorous wellposedness results in Sobolev and Bourgain spaces for dispersive PDEs; see for instance [@Him17] and [@Ozs19] for the local and global wellposedness analysis of nonlinear Schrödinger type PDEs and [@Him19] for a similar analysis on the Korteweg-de Vries equation.
To date, there is no work on the boundary controllability of PDEs that utilizes the advantages of the UTM. This method has two basic elements: (i) the so-called *global relation*, an identity that relates the initial datum and a suitable time transform of known and unknown boundary values, and (ii) the *integral representation* of the solution. We illustrate a new methodology by making use of these two elements in order to provide an elementary proof of the lack of null controllability for the heat equation on the half line.
To this end, let us consider the following canonical initial-boundary value problem: $$\begin{aligned}
u_t &=& u_{xx},\quad x\in \mathbb{R}_+, \ \ t\in (0,T), \label{maineq1}\\
u(x,0) &=& u_0(x),\quad x\in \mathbb{R}_+, \label{maineq2}\\
u(0,t) &=& g(t),\quad t\in (0,T). \label{maineq3}\end{aligned}$$ We say - is *null controllable* in $[0,T]$ if given $u_0\in L^2(\mathbb{R}_+)$ there is $g\in L^2(0,T)$ such that $u(x,T)\equiv 0$.
It is well known that the above property does not hold for - for those solutions in $C([0,T];L^2(\mathbb{R}_+))$; see for example [@MicZua00] for a proof of this result. Our goal is to provide an alternate, yet very short proof of this fact. More precisely, we prove the following theorem.
\[mainthm\] There exists $u_0\in L^2(\mathbb{R}_+)$ such that $u(x,T)\not\equiv 0$ for any $g\in L^2(0,T)$ if $u\in C([0,T];L^2(\mathbb{R}_+))$ and it solves -.
Orientation {#orientation .unnumbered}
-----------
In Section 2, we provide a proof of Theorem \[mainthm\] via the global relation. In Section 3, we extend Theorem \[mainthm\] to the $N$-dimensional half space by outlining the straightforward and simple extension of the proof presented in Section 2 to $N$ dimensions. In Section 4, we discuss alternative pathways through the Fokas method, introducing also a characterisation for the null-controllability problem on the finite interval. In Section 5, we discuss the main results of this work, as well as its future implications.
Proof of Theorem \[mainthm\]
============================
By introducing the half-line Fourier $x$-transform, namely $$\label{half-F-1}
\hat{f}(\lambda)=\int_0^{\infty}e^{-i\lambda x}f(x)dx, \qquad \text{Im}\lambda\leq 0,$$ and $$\hat{F}(\lambda,t )=\int_{0}^{\infty }e^{-i\lambda x}F(x,t)dx, \hphantom{2a} \text{Im}\lambda\leq 0 ,$$ as well as the $t$-transform $$\label{t-tr-1}
\tilde f(\lambda,t)=\int_0^{t}e^{\lambda \tau}f(\tau)d\tau, \ \ t>0, \ \lambda \in \mathbb{C},$$ the global relation for -, given by the Fokas method (equation (12) in [@Fok08]) can be written in the following form: $$\label{GR-1-hl}
e^{\lambda^2t}\hat{u}(\lambda,t)=\hat{u}_0(\lambda)-\tilde{r}(\lambda^2,t)-i\lambda\tilde{g}(\lambda^2,t),\qquad \text{Im}\lambda\leq 0,$$ where $ r(t)=u_x(0,t)$ and $ g(t)=u(0,t),\ \ t>0 $. For matters of completeness we derive here the global relation using the half-Fourier transform. Indeed, through integration by parts we obtain $$\begin{aligned}
\hat{u}_t(\lambda ,t)&=\int_0^{\infty}e^{-i\lambda x}u_t(x,t) dx=\int_0^{\infty}e^{-i\lambda x}u_{xx}(x,t)dx\\
&=u_x(x,t)e^{-i\lambda x}\big|_{x=0}^{\infty}+i\lambda u(x,t)e^{-i\lambda x}\big|_{x=0}^{\infty}-\lambda^2 \hat{u}(\lambda,t).\end{aligned}$$ Thus, $$\hat{u}_t+\lambda^2\hat{u}=-r(t)-i\lambda g(t).$$ Integrating the above ordinary differential equation we obtain $$\hat{u}e^{\lambda^2 t}=\hat{u}_0-\int_0^t e^{\lambda^2\tau} [r(\tau )+i\lambda g(\tau )]d\tau,$$ which is .
Applying the condition $u(x,T)\equiv 0$ in , we obtain that $$\label{GR-2-hl}
0=\hat{u}_0(\lambda)-\tilde{r}(\lambda^2,T)-i\lambda\tilde{g}(\lambda^2,T),\qquad \text{Im}\lambda\leq 0.$$ Letting $\lambda\to -\lambda$ in and subtracting the resultant expression (which is valid for $\text{Im}\lambda\ge 0$) from we obtain the following equation: $$\label{main-eq-hl}
2i\lambda\tilde{g}(\lambda ^{2},T)=\hat{u}_{0}(\lambda )-\hat{u}_{0}(-\lambda ), \qquad \lambda\in{\mathbb R}.$$
Let $0\not\equiv u_0\in L^1\cap L^2(\mathbb{R}_+)$. Employing this assumption in along with the definition of $\tilde{g}$, we obtain the following uniform bound for some $M>0$: $$\label{impineq}
{\left|\int_0^Te^{\lambda ^2t}g(t)dt\right|}=\left|\frac{1}{2\lambda}[\hat{u}_{0}(\lambda )-\hat{u}_{0}(-\lambda )]\right|{<M}, \qquad \lambda^2>1.$$ Then ${g\equiv 0}$ due to the Lemma \[Yoslem\], below.
It is clear that if $g\equiv 0$, then ${\hat{u}_{0}(\lambda )}={\hat{u}_{0}(-\lambda )}$ for all $\lambda\in \mathbb{R}$, which would contradict with the assumption that $0\not\equiv u(0)=u_0$.
\[Yoslem\]([@Yosida], page 167, Lemma 2) Let $g\in L^2(0,T)$. If there is $M>0$ such that $\left|\int_0^Te^{\alpha t}g(t)dt\right|<M$ for every $\alpha>1$, then $g\equiv 0$.
We note that the proof in [@Yosida] is given for $g$ being a continuous function; the proof extends to $L^2$ functions via density, namely $g$ is vanishing almost everywhere.
The $N$-dimensional half space
==============================
In this section we extend Theorem \[mainthm\] to the higher dimensional half space $\mathbb{R}^N_+=\mathbb{R}^{N-1}\times \mathbb{R}_+$, $\ N>1$ (see also [@MicZua01]). The methodology we used previously for the proof of Theorem \[mainthm\] provides a straightforward path to study the (lack of) null controllability for $$\begin{aligned}
u_t &=& \Delta u,\quad x=(x',x_N)\in \mathbb{R}^N_+, \ \ t\in (0,T), \label{hmaineq1}\\
u(x,0) &=& u_0(x),\quad x\in \mathbb{R}^N_+, \label{hmaineq2}\\
u(x',0,t) &=& g(x',t),\quad x'\in \mathbb{R}^{N-1}, \ \ t\in (0,T). \label{hmaineq3}\end{aligned}$$
The relevant result can be obtained by using half space Fourier $x$-transform $$\displaystyle\hat{u}(\lambda)\doteq\int_{\mathbb{R}^{N-1}}\int_0^\infty e^{-i\lambda\cdot x}u(x)dx_ndx', \qquad \lambda=(\lambda',\lambda_N)\in\mathbb{R}^{N-1}\times \mathbb{C},\quad \text{Im}\lambda_N\le 0$$ and applying Fokas’s method only to the last variable $x_N$. Indeed, half space Fourier transform yields the global relation $$\label{HSGR}
e^{|\lambda|^2t}\hat{u}(\lambda,t)=\hat{u}_0(\lambda)-\tilde{h}(\lambda,t)-i\lambda_N\tilde{g}(\lambda,t), \qquad \text{Im}\lambda_N\le 0,$$ where $$\label{boundaryvalues}
\tilde{g}(\lambda,t)\doteq\int_0^te^{|\lambda|^2s}\widehat{g^{x'}}(\lambda',s)ds \ \ \ \text{ and } \ \ \ \tilde{h}(\lambda,t)\doteq\int_0^te^{|\lambda|^2s}\widehat{h^{x'}}(\lambda',s)ds,$$ with $h(x',t)\doteq u_{x_N}(x',0,t)$ and $\widehat{g^{x'}}$, $\widehat{h^{x'}}$ denoting Fourier transforms of $g$ and $h$ with respect to $x'$.
The proof of the lack of null controllability for solutions in the class $C([0,T];L^2(\mathbb{R}^N_+))$ follows the exact same steps with the proof of Theorem \[mainthm\]. Hence, is now replaced with $$\label{impineq2}
{\left|\int_0^Te^{\lambda_N^2t}F(\lambda',t)dt\right|}=\left|\frac{1}{2\lambda_N}[\hat{u}_{0}(\lambda',\lambda_N )-\hat{u}_{0}(\lambda',-\lambda_N)]\right|{<M}, \ \ \ \ \lambda_N^2>1,$$ where $F(\lambda',t):=e^{|\lambda'|^2t}\widehat{g^{x'}}(\lambda',t)$. Applying Lemma \[Yoslem\] for each fixed $\lambda'\in \mathbb{R}^{N-1}$, we conclude that $F\equiv 0$, which in turn implies that $g\equiv 0$.
Alternative Pathways
====================
In this section, we provide an alternative pathway to obtain a proof of Theorem \[mainthm\] via the integral representation of the Fokas method. Furthermore, this pathway provides a characterisation of the control for the finite interval problem given in . In this sense it suggests a more general viewpoint on studying controllability problems through this methodology.
The Half Line {#the-half-line .unnumbered}
-------------
The integral representation of the solution of - given by the Fokas method (equation (16) in [@Fok08]) takes the form: $$\begin{aligned}
\label{repform}
u(x,t)&=\frac{1}{2\pi}\int_{-\infty }^{\infty }e^{i\lambda x-\lambda ^{2}t}\hat{u}_{0}(\lambda )d\lambda \notag \\
&-\frac{1}{2\pi}\int_{\partial D^{+}}e^{i\lambda x-\lambda ^{2}t}
\left[ 2i\lambda \tilde{g}(\lambda ^{2},t)+\hat{u}_{0}(-\lambda ) \right] d\lambda,\end{aligned}$$ where $\partial D^+$ is depicted in Figure 1.
\[fig\] {width="0.4\linewidth"}
By applying $u(x,T)\equiv 0$, deforming $\partial D^+$ to the real line and taking the inverse Fourier transform of both sides in the resultant expression, we obtain . Then, the proof of Theorem \[mainthm\] follows by the exact same arguments of Section 2.
The Finite Interval {#the-finite-interval .unnumbered}
-------------------
It is well known that the null controllability is true, for instance in $C([0,T];L^2(\Omega))$, if one replaces the infinite domain $\mathbb{R}_+$ by the finite one $(0,L)$. Here, we wish to give a characterization of the set of suitable boundary controllers, say acting at the right Dirichlet boundary condition, using the integral representation obtained from the Fokas method. Thus, we consider the following problem: $$\begin{aligned}
\label{heat-fi}
\begin{cases}
u_t = u_{xx}, \qquad &x\in (0,L), \ t\in (0,T),\\
u(0,t)=0, \quad u(L,t)=h(t), \qquad &t\in (0,T)\\
u(x,0)=u_0(x), \qquad &x\in (0,L)
\end{cases}\end{aligned}$$ and the goal is to find a sufficient condition for the boundary controller $h$ so that it steers the given initial datum $u_0$ to $u_T\equiv 0$ at $t=T$.
In analogy with the half line problem, one introduces the following Fourier $x$-transform where the integral is taken over the given spatial domain $(0,L)$: $$\label{fourthree}
\hat{u}(\lambda ,t)=\int_{0}^{L}e^{-i\lambda x}u(x,t)dx, \qquad \lambda \in\mathbb{C}.$$ Then, the corresponding global relation (equation (2.10) in [@Fok08]) for the above problem evaluated at $t=T$ becomes $$\label{globalrel}
0=\hat{u}_0(\lambda)+i\lambda e^{-i\lambda L}\tilde{h}(\lambda^2,T)-\tilde{g}_1(\lambda^2,T)+e^{-i\lambda L}\tilde{h}_1(\lambda^2,T), \qquad \lambda\in \mathbb{C},$$ with $g_1(t)=u_x(0,t)$, $h_1(t)=u_x(L,t)$, and ${h}(t)=u(L,t)$.
Similarly, the integral representation of the solution (equation (2.6) in [@Fok08]) evaluated at $t=T$ becomes $$\begin{gathered}
\label{int-rep-fi}
0=u(x,T)=\frac{1}{2\pi}\int_{-\infty }^{\infty }e^{i\lambda x-\lambda ^{2}T}\hat{u}_{0}(\lambda )d\lambda
-\frac{1}{2\pi}\int_{\partial D^{+}}e^{i\lambda x-\lambda ^{2}T}
\tilde{g}_{1} (\lambda^2,T) d\lambda \\
-\frac{1}{2\pi}\int_{\partial D^{-}}e^{-i\lambda (L-x)-\lambda ^{2}T}
\left[ \tilde{h}_{1}(\lambda^2,T)+i\lambda \tilde{h}(\lambda^2,T) \right] d\lambda , \end{gathered}$$ for all $x\in(0,L)$, where the contours $\partial D^\pm$ are depicted in Figure 1.
We next utilise the standard approach of Fokas method: Using the invariances of the global relation under the transformation $\lambda\mapsto-\lambda$, the unknown boundary transforms ($\tilde{g}_1$ and $\tilde{h}_1$) can be eliminated from the integral representation (see equation (32) in [@Fok08]). Through short and straightforward calculations, and by employing the definition of $\tilde{h}$, equation yields the following relation: $$\label{main-eq-fi}
\int_{\partial D_+}R(\lambda;x,T,L)d\lambda+\int_{\partial D_-}R(\lambda; x,T,L)d\lambda=U_0(x;T), \ \ \ \forall \ x\in (0,L),$$ where the integrand $R(\lambda;x,T,L)$ is given by $$\label{R-fi}
R(\lambda;x,T,L):=\frac{i}{\pi}\frac{\lambda e^{i\lambda x-\lambda^2T}}{e^{i\lambda L}-e^{-i\lambda L}}\left[\int_0^Te^{\lambda^2 s}h(s)ds\right]$$ and the known $U_0(x;T)$ is given by $$\begin{gathered}
\label{U-fi}
U_0(x;T)= \frac{1}{2\pi}\int_{-\infty}^\infty e^{i\lambda x-\lambda^2T}\hat{u}_0(\lambda)d\lambda
\\ -\frac{1}{2\pi}\int_{\partial D_+}e^{i\lambda x-\lambda^2T}\left[\frac{e^{i\lambda L}\hat{u}_0(\lambda)-e^{-i\lambda L}\hat{u}_0(-\lambda)}{e^{i\lambda L}-e^{-i\lambda L}}\right]d\lambda
\\-\frac{1}{2\pi}\int_{\partial D_-}e^{-i\lambda (L-x)-\lambda^2T}\left[\frac{\hat{u}_0(\lambda)-\hat{u}_0(-\lambda)}{e^{i\lambda L}-e^{-i\lambda L}}\right]d\lambda,\end{gathered}$$ with the contours $\partial D^\pm$ depicted in Figure 1, and the red dots denoting the zeros of $\exp (i\lambda L)-\exp (-i\lambda L)$ on the real axis.
Thus, we obtain the following characterization for the problem of null controllability: The problem is null controllable at time $t=T$ if and only if there exists $h=h(t)$ which satisfies .
Discussion
==========
In this work we analyse a family of null-controllability problems governed by the heat equation, using the machinery provided by the Fokas method. In this connection we make the following three remarks:
- It is straightforward but more technical to generalise the proof of Theorem \[mainthm\], so that one constructs a function $u_0$ satisfying Theorem \[mainthm\], with $u_0\in L^2(\mathbb{R}_+)$, but not necessarily $u_0\in L^1(\mathbb{R}_+)$.
- The methodology appearing in the current work can be applied to boundary value problems of higher dimensions such as $(\mathbb{R}_+)^N, \ N >1$, where all the spatial coordinates are positive. The relevant proof, which will be presented elsewhere, is based on the analysis of the Fokas method presented in [@Fok02] for the case of $N=2$, namely the quarter plane.
- If $u_0\in L^2(\mathbb{R}_+)$ and $g\in L^2(0,T)$, then - possesses a solution $u\in C([0,T];L^2(\mathbb{R}_+))$ in the transposition sense, and moreover this solution can be represented as in . Therefore, Theorem \[mainthm\] concerns such solutions. If the condition $u\in C([0,T];L^2(\mathbb{R}_+))$ is removed, then one can recover the null controllability in a larger class of solutions. This was proved in [@Ros2000] for the linearized KdV, heat, and Schrödinger equations.
The Fokas method provides the basic tools which are needed for the extension of the methodology introduced in the current work to linear PDEs, other than the heat equation. Indeed, one could obtain the Global Relation of the initial and boundary conditions, as well as the Integral Representation of the solution for problems which are posed on the half line and the finite interval and satisfy evolution equations where the rhs of is substituted by a higher order linear differential operator with constant coefficients (see [@Fok08]). The possibility of applying this methodology to null-controllability problems governed by other linear evolution PDEs is currently under investigation.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors wish to thank A.S. Fokas (University of Cambridge), whose prolific works are an endless source of inspiration. KK acknowledges funding by EPSRC. TÖ research is funded by TUBITAK 1001 Grant [\#]{}117F449.
[10]{}
Athanassios S. Fokas, *A unified transform method for solving linear and certain nonlinear pdes*, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences **453** (1997), no. 1962, 1411–1443.
, *A new transform method for evolution partial differential equations*, IMA Journal of Applied Mathematics **67** (2002), no. 6, 559–590.
, *A unified approach to boundary value problems*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 78, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008.
Athanassios S. Fokas, A. Alexandrou Himonas, and Dionyssios Mantzavinos, *The nonlinear [S]{}chrödinger equation on the half-line*, Trans. Amer. Math. Soc. **369** (2017), no. 1, 681–709.
A. Alexandrou Himonas, Dionyssios Mantzavinos, and Fangchi Yan, *The [K]{}orteweg–de [V]{}ries equation on an interval*, J. Math. Phys. **60** (2019), no. 5, 051507, 26.
Sorin Micu and Enrique Zuazua, *On the lack of null-controllability of the heat equation on the half-line*, Trans. Amer. Math. Soc. **353** (2001), no. 4, 1635–1659.
, *On the lack of null-controllability of the heat equation on the half space*, Portugaliae Mathematica **58** (2001), no. 1, 1–24.
T[ü]{}rker [Ö]{}zsar[i]{} and Nermin Yolcu, *The initial-boundary value problem for the biharmonic [S]{}chr[ö]{}dinger equation on the half-line*, Commun. Pure Appl. Anal. **18** (2019), no. 6, 3285–3316.
Lionel Rosier, *Exact boundary controllability for the linear [K]{}orteweg-de [V]{}ries equation on the half-line*, SIAM J. Control Optim. **39** (2000), no. 2, 331–351.
Kôsaku Yosida, *Functional analysis*, sixth ed., Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\], vol. 123, Springer-Verlag, Berlin-New York, 1980.
| {
"pile_set_name": "ArXiv"
} |
\#1\#2[${}^{#1}$\#2]{} \#1\#2[\^[\#1]{}]{}
Introduction {#intro}
============
The Sun produces its energy by the nuclear fusion of four protons into an $\alpha$ particle, chains of reactions that yield two positrons and two neutrinos. Since these low-energy neutrinos are weakly interacting, it was assumed that they traverse the Sun and reach the Earth without change. Measurement of the neutrino energy spectrum should thus give information about the conditions under which the nuclear reactions take place in the Sun. All solar neutrino experiments, however, have observed considerably fewer neutrinos than are predicted by detailed models of the physical processes in the Sun that are based on the nuclear reaction chains. As a result of this neutrino deficit, the assumption that the neutrinos are unchanged during their passage from the Sun to the Earth is now seriously questioned. For such transformations to occur neutrinos must have mass, a hypothesis of far-reaching consequences.
The experimental study of solar neutrinos is now over 30 years old. The first experiment, a radiochemical detector based on chlorine [@DAV95; @DAV98], observed a capture rate of $2.55 \pm 0.17 \pm 0.18$ SNU, where 1 SNU = 1 interaction/s in a target that contains $10^{36}$ atoms of the neutrino absorbing isotope. Although different standard solar models (SSM’s) predict somewhat different rates for the chlorine experiment (for example, $7.7^{+1.2}_{-1.0}$ SNU [@BAH95; @BAH98] and 7.2 SNU [@TUR98]), all such models predict a rate significantly higher than observed.
[lcccd]{} & $\nu$ & $\nu$ energy & $\nu$ flux & Ga capture\
Reaction & branch & (MeV) & (cm$^{-2}$ s$^{-1}$) & rate (SNU)\
$p + p {\rightarrow} d + e^+ + \nu$ & $pp$ & 0–0.42 & $(5.94 \pm 0.06) \times 10^{10}$ & 69.6\
$^7\text{Be} + e^- {\rightarrow} ^8\text{B} + \nu$ & $^7$Be & 0.38, 0.86 & $(4.80 \pm 0.43) \times 10^{9}$ & 34.4\
$^8\text{B} {\rightarrow} ^8\text{Be}^* + e^+ + \nu$ & $^8$B & 0–14.1 & $(5.15^{+0.98}_{-0.72})\times 10^{6}$ & 12.4\
CNO reactions & CNO & 0–1.73 & $(1.1 \pm 0.2) \times 10^{8}$ & 9.8\
$p + e^- + p {\rightarrow} d + \nu$ & $pep$ & 1.44 & $(1.39 \pm 0.01) \times 10^{8}$ & 2.8\
For 20 years, until about 1985, the chlorine experiment was the only measurement. This experiment is primarily sensitive to high-energy neutrinos with a $\sim20$% contribution from other sources, mainly . The flux of neutrinos is very dependent on the central temperature of the Sun ($T_{\odot}^{24}$[@BAH96]). As a result many models were suggested that would slightly suppress $T_{\odot}$ and hence decrease the flux significantly. (See Ref. [@BAH89] for a description of a large number of such models.) Most of these models, however, run into difficulty with some other measured aspect of the Sun. An alternative solution to this discrepancy could be neutrino oscillations. The Cl experiment operates on the inverse beta decay reaction and thus is only sensitive to electron neutrinos. If the neutrinos were to change flavor on their trip from the solar core to the Earth, the Cl experiment would not observe them.
In the mid 1980s, the Kamioka nucleon decay experiment (Kamiokande) began to measure the solar neutrino flux. This large water Cherenkov detector was originally designed to look for high-energy signals from proton decay. After great effort, the energy threshold was reduced to a level to permit a sensitivity to recoil electrons from solar neutrino interactions. The path of the recoil electrons is in the direction of the initial neutrino trajectory, and thus this experiment demonstrated for the first time that neutrinos were coming from the Sun. The measured flux [@FUK96] of $(2.80 \pm 0.19 \pm 0.33) \times 10^6$/(cm$^2$ s) was less than half of the solar model prediction, and the solar neutrino problem was thus confirmed by a second experiment.
Because the high-energy solar neutrino flux was suppressed, it became very important to also determine the flux of low-energy neutrinos produced in the dominant proton-proton ($pp$) reaction. Exotic hypotheses aside, the rate of the $pp$ reaction is directly related to the solar luminosity and is insensitive to alterations in the solar model. In the early 1990s the Russian-American Gallium Experiment (SAGE) and then the Gallium Experiment (GALLEX) began to publish results. These experiments are based on the neutrino capture reaction $(\nu_e,e^-)$ [@KUZ65] and have the very low threshold of 233 keV [@TOI96]. They are thus sensitive to low-energy $pp$ neutrinos, whose end point energy is 423 keV [@BAH97], and provide the only feasible means at present to measure low-energy solar neutrinos. The SAGE result [@GAV98] of $66.9 ^{+7.1 +
5.4}_{-6.8 -5.7}$ SNU with a target of Ga metal and the GALLEX result [@HAM99] of $77.5 \pm 6.2 ^{+4.3}_{-4.7}$ SNU with a target of GaCl$_3$ are both well below the SSM prediction from the Bahcall-Pinsonneault solar model [@BAH98] of $129^{+8}_{-6}$ SNU. The insensitivity of Ga to the solar model is seen in the capture rate calculation from the model of Brun, Turck-Chièze, and Morel [@TUR98] of 127.2 SNU. The contributions of the components of the solar neutrino flux to the capture rate are given in Table \[theoretical\_fluxes\].
With the four experiments having three different thresholds, one can deduce some information concerning the $\nu$ energy spectral distribution. If one fits the data from all experiments with the neutrino fluxes as free parameters, the best fit is when the $\nu_{\text{Be}}$ flux is greatly reduced whereas there is an appreciable $\nu_{\text{B}}$ flux [@HEE97; @Smirnov98; @BAH982]. This is an apparent paradox as it is difficult to form in the Sun without forming .
In 1996 Super-Kamiokande began to take data. This 50 kton water Cherenkov detector is the first high-count-rate solar neutrino experiment. The present result [@Super-K] for the neutrino flux, assuming that neutrino transformations do not occur, is $(2.44 \pm 0.05 _{-0.07}
^{+0.09}) \times 10^6$/(cm$^2$ s), in agreement with its predecessor.
The purpose of this paper is to summarize all of the SAGE data for the last eight years. It is organized in the same way as the SAGE experiment is carried out: after presenting some general aspects of the experiment, we consider the chemical extraction of Ge from Ga and the subsequent Ge purification. Then we present how the Ge is counted, how events are identified, and how the data are analyzed to give the solar neutrino capture rate. Finally, we consider the sources of systematic uncertainty, give the overall results, and conclude with the implications for solar and neutrino physics.
In an attempt to make the material understandable to the general reader, but still useful to the specialist, each of these subjects is first discussed in a general way, followed by subsections that give more detail. The reader who wants a general overview need only read the beginning of each section. The reader who desires more information regarding a particular subject should read the appropriate subsection.
SAGE Overview {#overview}
=============
In this section we give some general information on the location of the experiment, its physical characteristics, and the division of the SAGE data into three experimental periods.
Baksan Neutrino Observatory {#bno}
---------------------------
The SAGE experiment is situated in a specially built underground laboratory at the Baksan Neutrino Observatory (BNO) of the Institute for Nuclear Research of the Russian Academy of Sciences in the northern Caucasus mountains. The main chamber of the laboratory is 60 m long, 10 m wide, and 12 m high. It is located 3.5 km from the entrance of a horizontal adit excavated into the side of Mount Andyrchi and has an overhead shielding of 4700 meters of water equivalent. To reduce neutron and gamma backgrounds from the rock, the laboratory is entirely lined with 60 cm of low-radioactivity concrete with an outer 6 mm steel shell. All aspects of the experiment are in this underground area, with additional rooms devoted to chemistry, counting, and a low-background solid-state Ge detector. Other facilities for subsidiary measurements are in a general laboratory building outside the adit.
Extraction history {#extrac_hist}
------------------
The data from SAGE span nearly a decade during which the experiment evolved a great deal. As a result, the data can be naturally divided into several periods characterized by different experimental conditions. Extractions on approximately 30 tons of Ga began in 1988; by late 1989 backgrounds were low enough to begin solar neutrino measurements. The data period referred to as SAGE I began in January 1990 and ended in May 1992 [@ABD94]. In the summer of 1991, the extraction mass was increased to nearly 60 tons. The SAGE I data were taken without digitized wave forms and the $L$ peak could not be analyzed because of high electronic noise at low energy. (The decay modes of are described below in Sec. \[counting\].) The solar neutrino capture rate determined from this data were published in Ref. [@NIC94].
Within a few months after SAGE I was completed, the experiment was greatly improved with respect to electronic noise. The following period of data, from September 1992 to December 1994, is referred to as SAGE II. It is distinguished by recording of the counter wave form in most runs which makes possible analysis of events in the low-energy $L$ peak.
Designation Included extractions Comments
------------- ------------------------------- ---------------------------
SAGE I Jan. 90 $\rightarrow$ May 92 Rise time from ADP
SAGE II.1 Sep. 92 $\rightarrow$ Oct. 93 Rise time from wave form,
begin to use $L$ peak
SAGE II.2 Nov. 93 $\rightarrow$ June 94 Ga theft period
SAGE II.3 July 94 $\rightarrow$ Dec. 94 Before Cr experiment
SAGE III.1 Jan. 95 $\rightarrow$ June 95 Some extractions
during Cr experiment
SAGE III.2 July 95 $\rightarrow$ present After Cr experiment
: Definition of the various segments of SAGE data.[]{data-label="Data_Assignment_Table"}
During SAGE II, there was a period (which we call SAGE II.2) in which 2 tons of gallium, approximately 3.6% of the total mass, was stolen from the experiment. The gallium was apparently removed in small quantities from November 1993 to June 1994. During this period a prototype gravity wave laser interferometer at BNO detected unapproved transport of materials from underground. After discovery of the theft, all of the gallium was cleaned, additional security controls for access to the gallium were instituted, and SAGE resumed operation. As this period of time has some uncertainty with respect to experimental control, it is singled out for separate treatment, and is not included in our best estimate for the neutrino capture rate.
An experiment using a 517 kCi $^{51}$Cr neutrino source [@ABD96; @ABD98] began in late December of 1994 and continued until May 1995. We refer to all data after January 1995 as SAGE III, with a special designation of SAGE III.1 for solar neutrino extractions during the Cr experiment.
Table \[Data\_Assignment\_Table\] summarizes the data period designations. The exposure times and other data for all runs of SAGE that are potentially useful for solar neutrino capture rate determination are given in Table \[run\_parameter\_table\].
[l c d c c c c c c c c c c]{}
& Mean & Exposure & Ga\
Exposure & exposure & time & mass & Extraction & Counter & Pressure & Percent & Operating & Counting & $K$-peak & $L$-peak & Peak\
date & date & (days) & (tons) & efficiency & name & (mm Hg) & GeH$_4$ & voltage & system & efficiency & efficiency & ratio\
Jan. 90 & 1990.040 & 42.0 & 28.67 & 0.78 & Ni 1 & 604 & 28.0 & 1230 & 2 & 0.333 & &\
Feb. 90 & 1990.139 & 30.0 & 28.59 & 0.79 & LA12 & 635 & 53.0 & 1450 & 5 & 0.249 & &\
Mar. 90 & 1990.218 & 26.0 & 28.51 & 0.81 & Ni 1 & 640 & 25.0 & 1238 & 2 & 0.343 & &\
Apr. 90 & 1990.285 & 19.0 & 28.40 & 0.76 & LA24 & 850 & 30.0 & 1430 & 5 & 0.335 & &\
July 90 & 1990.540 & 21.0 & 21.01 & 0.78 & Ni 1 & 524 & 19.3 & 1130 & 2 & 0.327 & &\
June 91 & 1991.463 & 53.0 & 27.43 & 0.82 & LA74 & 715 & 28.0 & 1300 & 2 & 0.334 & &\
July 91 & 1991.539 & 23.0 & 27.37 & 0.66 & LA77 & 710 & 24.0 & 1300 & 3 & 0.320 & &\
Aug. 91 & 1991.622 & 26.3 & 49.33 & 0.78 & RD2 & 570 & 34.0 & 1700 & 5 & 0.250 & &\
Sep. 91 & 1991.707 & 27.0 & 56.55 & 0.78 & LA40 & 935 & 40.0 & 1630 & 2 & 0.338 & &\
Nov. 91 & 1991.872 & 26.0 & 56.32 & 0.81 & LA46 & 108 & 30.0 & 1746 & 3 & 0.339 & &\
Dec. 91 & 1991.948 & 26.8 & 56.24 & 0.79 & LA51 & 870 & 27.0 & 1394 & 2 & 0.336 & &\
Feb. 92-1 & 1992.138 & 24.5 & 43.03 & 0.80 & LA71 & 666 & 12.0 & 1110 & 2 & 0.322 & &\
Feb. 92-2 & 1992.138 & 24.5 & 13.04 & 0.80 & LA50 & 640 & 30.0 & 1165 & 2 & 0.305 & &\
Mar. 92 & 1992.214 & 20.9 & 55.96 & 0.78 & LA46 & 810 & 20.5 & 1292 & 2 & 0.316 & &\
Apr. 92 & 1992.284 & 23.5 & 55.85 & 0.83 & LA51 & 815 & 23.0 & 1386 & 2 & 0.333 & &\
May 92 & 1992.383 & 27.5 & 55.72 & 0.67 & LA95 & 675 & 69.0 & 1620 & 2 & 0.282 & &\
Sep. 92 & 1992.700 & 116.8 & 55.60 & 0.53 & LA110 & 720 & 21.0 & 1311 & 3 & 0.338 & 0.322 &\
Oct. 92 & 1992.790 & 27.2 & 55.48 & 0.83 & LA111 & 725 & 25.0 & 1391 & 3 & 0.341 & 0.327 &\
Nov. 92 & 1992.871 & 26.7 & 55.38 & 0.81 & LA105 & 730 & 23.0 & 1351 & 3 & 0.315 & 0.297 &\
Dec. 92 & 1992.945 & 24.3 & 55.26 & 0.85 & LA116 & 740 & 26.0 & 1406 & 3 & 0.325 & 0.315 & 1.04\
Jan. 93 & 1993.039 & 32.3 & 55.14 & 0.76 & LA110 & 770 & 25.0 & 1412 & 3 & 0.342 & 0.314 &\
Feb. 93 & 1993.115 & 23.0 & 55.03 & 0.79 & LA107 & 730 & 24.0 & 1336 & 6 & 0.315 & &\
Apr. 93 & 1993.281 & 26.6 & 48.22 & 0.83 & LA111\* & 710 & 23.0 & 1352 & 3 & 0.322 & &\
May 93 & 1993.364 & 30.9 & 48.17 & 0.82 & LA116 & 705 & 16.0 & 1210 & 3 & 0.327 & & 1.04\
June 93 & 1993.454 & 30.4 & 54.66 & 0.80 & LA110 & 740 & 24.0 & 1352 & 3 & 0.338 & 0.313 &\
July 93 & 1993.537 & 27.9 & 40.44 & 0.80 & LA111 & 675 & 22.0 & 1266 & 3 & 0.353 & &\
Aug. 93-1 & 1993.631 & 34.0 & 40.36 & 0.79 & LA107 & 680 & 12.0 & 1210 & 6 & 0.317 & & 1.00\
Aug. 93-2 & 1993.628 & 63.8 & 14.09 & 0.51 & A9 & 765 & 12.0 & 1130 & 6 & 0.322 & & 1.20\
Oct. 93-1 & 1993.749 & 13.0 & 14.06 & 0.79 & A12 & 750 & 14.0 & 1224 & 6 & 0.333 & & 1.00\
Oct. 93-2 & 1993.800 & 34.7 & 14.10 & 0.80 & LA111\* & 710 & 15.0 & 1162 & 3 & 0.328 & 0.309 & 1.03\
Oct. 93-3 & 1993.812 & 24.6 & 14.02 & 0.84 & LA116 & 665 & 14.0 & 1184 & 3 & 0.323 & 0.299 & 1.04\
Nov. 93-1 & 1993.855 & 14.0 & 14.07 & 0.87 & LA119 & 665 & 13.0 & 1113 & 3 & 0.321 & 0.316 & 1.08\
Nov. 93-2 & 1993.844 & 53.4 & 26.16 & 0.52 & LA110 & 675 & 9.0 & 1094 & 3 & 0.340 & 0.326 & 1.00\
Dec. 93-1 & 1993.936 & 30.5 & 26.13 & 0.78 & A19 & 760 & 12.0 & 1287 & 3 & 0.336 & & 1.08\
Dec. 93-2 & 1993.939 & 39.9 & 28.05 & 0.80 & LA111 & 690 & 12.0 & 1230 & 3 & 0.345 & 0.331 & 1.02\
Jan. 94-1 & 1994.048 & 42.2 & 26.67 & 0.82 & LA107 & 760 & 12.0 & 1196 & 6 & 0.328 & & 1.00\
Jan. 94-2 & 1994.051 & 41.1 & 27.44 & 0.80 & LA111\* & 750 & 12.5 & 1065 & 3 & 0.308 & & 1.04\
Feb. 94 & 1994.137 & 28.0 & 54.01 & 0.64 & LA116 & 600 & 15.0 & 1090 & 3 & 0.312 & 0.326 & 1.04\
Mar. 94 & 1994.218 & 31.0 & 53.94 & 0.78 & LA105 & 625 & 10.0 & 1190 & 3 & 0.309 & 0.311 & 1.00\
Apr. 94 & 1994.283 & 22.5 & 53.88 & 0.73 & LA110 & 685 & 27.0 & 1331 & 3 & 0.328 & 0.335 & 1.00\
May 94-3 & 1994.374 & 32.9 & 26.99 & 0.85 & LA111 & 610 & 17.0 & 1215 & 3 & 0.329 & 0.343 & 1.00\
July 94 & 1994.551 & 31.3 & 50.60 & 0.80 & LA107 & 620 & 22.0 & 1236 & 3 & 0.301 & 0.269 & 1.00\
Aug. 94 & 1994.634 & 31.0 & 50.55 & 0.80 & LA105 & 655 & 13.0 & 1196 & 3 & 0.312 & 0.307 & 1.05\
Sep. 94-1 & 1994.722 & 33.2 & 37.21 & 0.76 & A13 & 695 & 18.0 & 1270 & 3 & 0.334 & 0.319 & 1.07\
Oct. 94 & 1994.799 & 28.8 & 50.45 & 0.76 & A19 & 695 & 25.0 & 1375 & 3 & 0.334 & & 1.06\
Nov. 94 & 1994.886 & 31.0 & 50.40 & 0.79 & LA113 & 685 & 28.5 & 1383 & 3 & 0.306 & 0.314 & 1.05\
Dec. 94 & 1994.951 & 21.0 & 13.14 & 0.80 & A12\* & 610 & 16.5 & 1184 & 6 & 0.310 & & 1.02\
Mar. 95 & 1995.209 & 42.5 & 24.03 & 0.92 & A28 & 690 & 18.5 & 1222 & 6 & 0.321 & & 1.00\
July 95 & 1995.538 & 19.9 & 50.06 & 0.86 & LA107 & 635 & 30.0 & 1333 & 3 & 0.298 & 0.317 & 1.01\
Aug. 95 & 1995.658 & 46.7 & 50.00 & 0.70 & A12 & 710 & 17.0 & 1260 & 3 & 0.325 & 0.312 & 1.01\
Sep. 95 & 1995.742 & 28.8 & 49.95 & 0.67 & LA46 & 645 & 37.0 & 1382 & 3 & 0.283 & 0.294 & 1.02\
Oct. 95 & 1995.807 & 18.7 & 49.83 & 0.49 & A19 & 680 & 18.5 & 1248 & 3 & 0.319 & 0.294 & 1.08\
Nov. 95 & 1995.875 & 25.8 & 49.76 & 0.89 & A9 & 685 & 33.0 & 1429 & 3 & 0.310 & 0.294 & 1.21\
Dec. 95-2 & 1995.962 & 32.7 & 41.47 & 0.73 & LA113 & 725 & 18.5 & 1271 & 3 & 0.319 & 0.278 & 1.00\
Jan. 96 & 1996.045 & 29.7 & 49.64 & 0.77 & A12 & 715 & 24.0 & 1340 & 3 & 0.321 & 0.310 & 1.00\
May 96 & 1996.347 & 49.9 & 49.47 & 0.75 & LA116 & 685 & 21.5 & 1295 & 3 & 0.320 & 0.319 & 1.00\
Aug. 96 & 1996.615 & 45.0 & 49.26 & 0.77 & A13 & 675 & 23.0 & 1332 & 3 & 0.327 & 0.330 & 1.09\
Oct. 96 & 1996.749 & 45.8 & 49.15 & 0.83 & LA116 & 635 & 15.0 & 1185 & 3 & 0.318 & 0.319 & 1.02\
Nov. 96 & 1996.882 & 48.7 & 49.09 & 0.78 & A12 & 720 & 21.5 & 1308 & 3 & 0.323 & 0.306 & 1.00\
Jan. 97 & 1997.019 & 49.8 & 49.04 & 0.85 & LA113 & 700 & 29.0 & 1372 & 3 & 0.308 & 0.295 & 1.00\
Mar. 97 & 1997.151 & 44.9 & 48.93 & 0.93 & A13 & 650 & 23.5 & 1339 & 3 & 0.323 & 0.335 & 1.08\
Apr. 97 & 1997.277 & 42.9 & 48.83 & 0.90 & LA116 & 670 & 29.0 & 1360 & 3 & 0.313 & 0.320 & 1.02\
June 97 & 1997.403 & 45.6 & 48.78 & 0.87 & A12 & 675 & 24.5 & 1320 & 3 & 0.314 & 0.314 & 1.00\
July 97 & 1997.537 & 45.9 & 48.67 & 0.91 & LA51 & 690 & 15.5 & 1242 & 3 & 0.321 & 0.312 & 1.03\
Sep. 97 & 1997.671 & 46.4 & 48.56 & 0.75 & A13 & 650 & 25.0 & 1318 & 3 & 0.322 & 0.335 & 1.04\
Oct. 97 & 1997.803 & 45.0 & 48.45 & 0.83 & LA116 & 635 & 23.5 & 1318 & 3 & 0.328 & 0.327 & 1.03\
Dec. 97 & 1997.940 & 47.0 & 48.34 & 0.88 & A12 & 710 & 27.0 & 1382 & 3 & 0.318 & 0.306 & 1.00
Extraction of G from G
======================
The extraction and concentration of germanium in the SAGE experiment consists of the following steps.
1. [Ge is extracted from the Ga metal into an aqueous solution by an oxidation reaction.]{}
2. [The aqueous solution is concentrated.]{}
1. [Vacuum evaporation reduces the volume of aqueous solution by a factor of 8.]{}
2. [Ge is swept from this solution as volatile GeCl$_4$ by a gas flow and trapped in 1 l of de-ionized water.]{}
3. [A solvent extraction is made from the water which concentrates the Ge into a volume of 100 ml.]{}
3. [The gas GeH$_4$ is synthesized, purified, and put into a proportional counter.]{}
The average extraction efficiency from the Ga metal to GeH$_4$ was 77% before 1997 and 87% thereafter. Each of these steps will now be briefly described and this section concludes with a description of the evidence that the extraction procedure does indeed remove germanium with high efficiency.
Chemical extraction procedure {#chem_extr_proc}
-----------------------------
### Extraction of Ge from metal Ga
A procedure for the extraction of Ge from metallic Ga was first investigated at Brookhaven National Laboratory [@BAH78]. It is based on the selective oxidation of Ge in liquid Ga metal by a weakly acidic H$_2$O$_2$ solution. This method was developed and fully tested in a 7.5-ton pilot installation at the Institute for Nuclear Research [@BAR84]. The final procedure extracts Ge with high efficiency and dissolves only a small amount of Ga [@IS90; @Vere91].
![Chemical reactor for extraction of Ge from Ga.[]{data-label="reactor"}](reactor.eps){width="3.375in"}
The Ga at BNO is contained in chemical reactors, each of which is able to extract from as much as 8 tons of Ga. The reactor (Fig. \[reactor\]) is a 2-m$^3$ Teflon tank with $\sim 40$-mm-thick walls to which band heaters are attached. The Teflon tank is placed inside a secondary stainless steel tank. The Ga can be stirred with a motor that can turn an internal mixer at up to 80 rpm. A specially designed set of vanes are attached to the inside cover of the reactor that serve to completely disperse the extraction reagents (density 1.0 kg/l) throughout the liquid Ga (density 6.1 kg/l). The vanes are made from Teflon and the stirrer and cover are Teflon lined. A glass viewport in the reactor cover enables one to see the extremely vigorous mixing action. Ten such reactors are installed at BNO which are connected with a system of heated Teflon tubing and a Teflon pump that can transfer Ga between reactors. A system of glass-Teflon dosing pumps can put a measured volume of reagents into any reactor, and a vacuum suction device made from Teflon, glass, and zirconium extends to the Ga surface to remove the reagents. The filling of a reactor with reagents and the stirring are controlled by an automated system.
Each measurement of the solar neutrino flux begins by adding to the Ga approximately 700 $\mu$g of stable Ge carrier (distributed equally among all of the reactors) in the form of a solid Ga-Ge alloy with known Ge content $(\sim 2 \times 10^{-4}$ mass %). The reactor contents are then stirred so as to thoroughly disperse the carrier throughout the Ga metal. After a typical exposure interval of 4–6 weeks, the Ge carrier and any additional Ge atoms produced by solar neutrinos or other processes are chemically extracted from the Ga.
The efficiency of Ge extraction depends on a number of parameters. Since the efficiency falls rapidly as the Ga temperature increases, we begin to extract with the Ga at 30.0–30.5, just slightly above its freezing temperature (29.8). The efficiency increases with an increase in the amount of oxidizing agent (H$_2$O$_2$), but this has the detrimental effect of dissolving more Ga. The efficiency also depends on the volume of aqueous phase which defines the time of later concentration of Ge, the most time consuming part of the entire extraction process. Taking into account all of these factors, a procedure was developed which extracts about 85% of the Ge and dissolves only 0.1% of the Ga.
The extraction solution for a reactor containing 7.5 tons of Ga consists of 200 l of de-ionized water, 5 l of 7 M HCl,[^1] and 16 l of a 30% solution of H$_2$O$_2$. All components of this solution are purified so their Ge content is negligible. Immediately after the reagents are added, reactor stirring starts at a speed of 70 rpm. As the mixture is intensively stirred, the gallium turns into fine droplets which are covered with a Ga oxide film. This film prevents fusion of the droplets and holds the Ga as an emulsion [@Vere87; @Rowley]. The dissolved Ge in the Ga migrates to the surface of the droplets where it is oxidized and incorporated into the oxide film. Because of the highly exothermic oxidation reaction, the Ga temperature rapidly rises. After approximately 25 min, the H$_2$O$_2$ has been consumed; the Ga temperature plateaus and the emulsion spontaneously breaks down. To dissolve the oxide containing Ge, the extraction procedure is finished by adding 45 l of 7 M HCl (cooled to $-15$) and stirring for 1–2 min. The Ga temperature at the end of this extraction process is $\sim 50$.
The extraction solution is immediately decanted and sent to the first step of concentration, which is evaporation. The Ga in each reactor is then washed by adding 20 l of 0.5 M HCl. This solution is stirred with the liquid Ga for about 1 min, is decanted out, and is added to the previous extraction solution. Finally, to prevent oxidation of the Ga during the interval between extractions, a solution of 0.5 M HCl is added to the reactor and left there until the next extraction.
### Vacuum evaporation of extraction solutions
Extraction is made sequentially from one reactor to the next. All the extraction solutions, whose total volume is 2200 l for 60 tons of gallium, are combined at the evaporation step, which is carried out in a glass recirculation apparatus with a steam-heated active volume of 70 l. As the evaporation proceeds, the acidity of the evaporated solution increases. Ge is volatile from concentrated chloride solutions, so the evaporation is stopped when the volume of solution reaches 250–270 l, before loss of Ge can begin. The average time for evaporation is 15 h.
### Sweeping
The next step is based on the volatility of GeCl$_4$ from a concentrated solution of HCl. The evaporated extraction solution, which contains 250 g of Ga/l in the form of chloride, is transferred to glass vessels with a volume of 200 l. These vessels are part of a sealed gas flow system. The HCl concentration is raised to 9 M by adding purified 12 M HCl and an air flow at 1.0 m$^3$/h is initiated. Ge is swept as GeCl$_4$ from this 50acid solution through a counter-current scrubber where the GeCl$_4$ is absorbed in a 1.0 l volume of de-ionized H$_2$O. The amount of Ge remaining in the solution $C(t)$ falls exponentially: $C(t) = C(0) \exp[-1.84 V(t)]$ where $V$ is the volume of sweep gas in m$^3$. The duration of sweeping is usually 2.5 h which gives 99% Ge extraction efficiency. At the end of the sweep the acidity of the absorber solution is in the range of 4.0 M to 4.2M, which excludes loss of Ge.
### Solvent extraction
A solvent extraction is then carried out to further concentrate the Ge. This procedure is based on the high distribution coefficient of Ge between an acidic water solution and an organic solvent, such as CCl$_4$. To achieve an optimal acidity (8.5 M), the appropriate amount of purified 12 M HCl is added to the solution obtained from sweeping. The Ge is first extracted into CCl$_4$ and then is back extracted into low-tritium H$_2$O. This process is repeated 3 times. To remove the residual CCl$_4$, a very small amount of hexane is added to the organic phase at the last step of the final back extraction. The final traces of hexane are removed by heating the solution at 90 for 40 min. This results in the Ge being concentrated in a volume of 100 ml of low-tritium H$_2$O.
### Germane synthesis
The final step of the extraction process is to synthesize germane (GeH$_4$) which is used as a 20%–30% fraction of the counting gas in a proportional counter. NaOH is added to the 100-ml water solution to adjust the $p$H to the range of 8–9, and the solution is placed in a reaction flask on a high-vacuum glass apparatus. Any air is swept out of the solution and the connecting piping with a He flow and 2 g of low-tritium NaBH$_4$ dissolved in 40 ml of low-tritium H$_2$O is added. The mixture is then heated to 70, at which temperature the Ge is reduced by the NaBH$_4$ to make GeH$_4$. The H$_2$ generated by the reaction and the flowing He sweep the GeH$_4$ onto a Chromosorb 102 gas chromatography column at $-196$ where it is trapped. When the reaction is finished the column temperature is raised to $-35$ and the GeH$_4$ is eluted with He carrier gas. It is then frozen on another Chromosorb 102 trap at $-196$ where most of the He is pumped away. The GeH$_4$ is then transferred with a mercury-filled Toepler pump to a glass bulb at $-196$ where any residual He is pumped away. The Toepler pump is used again to transfer the GeH$_4$ to a calibrated stem, where the GeH$_4$ volume is measured. During this transfer the temperature of the bulb is held at $-142$ so as to minimize Rn. A measured quantity of old low-background Xe is added and this gas mixture is inserted into a miniature proportional counter. The counter has been evacuated at $10^{-6}$ torr and baked at 100 for at least 6 h.
### Modified procedures for SAGE III
![Ga temperature using two extraction procedure begun in 1997. The extraction reagents are added 1 min before time zero. Extraction begins when the mixer is started. The two vertical lines about 15 min after the start of extraction are when the HCl is added to dissolve the oxide containing Ge.[]{data-label="reacextr"}](reacextr.eps){width="3.375in"}
#### Extraction from Ga.
At the beginning of 1997, the extraction procedure was modified to a two-step extraction process. In the first step the volume of reagents added to each reactor is reduced from the values given previously by a factor of 2. The remaining steps in the removal of Ge from the Ga proceed the same as previously described, but only require about 15 min because of the reduced H$_2$O$_2$ volume. This first step extracts about 75% of the Ge from the Ga, dissolves 0.05% of the Ga, and raises the Ga temperature to about 40 (Fig. \[reacextr\], lower curve). After the first extraction from each reactor, a second extraction is carried out in the same order using the same volume of reagents as in the first extraction. By the time the second extraction begins the Ga has cooled to 37 and an additional drop of 1.5–2 occurs when the new reagents are added. Since the initial Ga temperature is now elevated, the efficiency of the second extraction is less than the first, and averages 70%. Again 0.05% of the Ga is dissolved and the final Ga temperature is 49 at the end of this second extraction (Fig. \[reacextr\], upper curve). This modified procedure results in a total efficiency of Ge removal from the Ga in excess of 90%, but both procedures dissolve the same total amount of Ga (0.1%).
#### Evaporation of extraction solutions.
The vacuum evaporation was modified at the beginning of SAGE III. Instead of stopping the distillation before the Ge volatilizes, the Ge is allowed to evaporate, at which time collection of Ge in the condensate is begun. Evaporation is continued until all the Ge has been transferred to the condenser. The condensate is then further evaporated until its acidity is 4.5 M. This solution, whose volume is about 130 l, is transferred to the sweeping apparatus, 12 M hydrochloric acid is added to obtain 9 M acidity, and the Ge is swept out in the same manner as for SAGE I and II. An important advantage of this new method is that the solution that results from sweeping is pure 9 M HCl, free from Ga or Ge, so it can be used in later extractions. These chemical technology modifications in SAGE III increase the efficiency of Ge extraction by 6%–7%, decrease the average duration of concentration by 3–4 h, and reduce the consumption of concentrated HCl by 2.5 times.
Chemical extraction efficiency {#Chemical_extraction_efficiency}
------------------------------
The total efficiency of extraction of Ge is given by the ratio of the Ge content of the synthesized germane to the Ge present in the reactors at the beginning of the exposure interval. As a check, the amount of extracted Ge is also determined by atomic absorption analysis of a small fraction of the solution used in the GeH$_4$ synthesis. The extraction efficiency prior to 1997 was typically 80%. The modified extraction procedure initiated in 1997 gives about a 10% higher overall efficiency. The extraction efficiency for each run is given in Table \[run\_parameter\_table\].
Since each extraction leaves 10%–20% of the carrier Ge still present in the Ga, it is customary to make a second extraction within a few days after the first. This second extraction removes most of the residual Ge so that the Ge content of each reactor is well known after the carrier Ge is added. Occasionally a third extraction is made to totally deplete the Ge content. The extracts from these additional extractions are usually processed in the same manner as for the solar neutrino extraction, including counting of the synthesized GeH$_4$.
Tests of the extraction efficiency {#extr_effs}
----------------------------------
The Ga experiment relies on the ability to extract a few tens of atoms of from $5 \times 10^{29}$ atoms of Ga. To measure the efficiency of extraction, about 700 $\mu$g of stable Ge carrier is added to the Ga at the beginning of each exposure, but even after this addition, the separation factor of Ge from Ga is still 1 atom in $10^{11}$. In such a situation one can legitimately question how well the extraction efficiency is known. We have performed auxiliary measurements to verify that this efficiency is well established, and briefly describe these tests in this section.
### experiment {#cr_expt}
The most direct experiment of this type involved the irradiation of Ga with the 747-keV neutrinos from an artificial source of [@ABD96; @ABD98]. Eight exposures of 13 tons of Ga were made to a 517 kCi source. The atoms were extracted by our usual chemical procedure and their number determined by counting. The ratio $R$ of the measured neutrino capture cross section [@ABD96; @ABD98] to the theoretically calculated cross sections of Bahcall [@BAH97] and Haxton [@HAX98] was
$$\begin{aligned}
R & \equiv & \frac{\sigma_{\text{measured}}}
{\sigma_{\text{theoretical}}} \\
& = & \left\{ \begin{array}{l l}
0.95 \pm 0.12 \text{ (expt)}\ ^{+0.035}_{-0.027}
\text{ (theor)} & \text{ (Bahcall)}, \\
0.87 \pm 0.11 \text{ (expt)}\ \pm 0.09
\text{ (theor)} & \text{ (Haxton)}.
\end{array}
\right.
\nonumber\end{aligned}$$
With either of these theoretical cross sections, $R$ is consistent with 1.0, which implies that the extraction efficiency of atoms produced in Ga by the neutrinos from is the same as that of natural Ge carrier.
### Ga(n,${\gamma}$) experiment
To test the possibility that atomic excitations might tie up in a chemical form from which it would not be efficiently extracted, the radioactive isotopes and , which beta decay to and , were produced in liquid gallium by neutron irradiation. The Ge isotopes were extracted from the Ga using our standard procedure. The number of Ge atoms produced was determined by mass spectroscopic measurements [@ABD94] and was found to be consistent with the number expected based on the known neutron flux and capture cross section, thus suggesting that chemical traps are not present.
### Removal of
Further evidence that the extraction efficiency is well understood came from monitoring the initial removal from the Ga of cosmogenically produced . This nuclide was generated in the Ga as it resided outside the laboratory exposed to cosmic rays. When the Ga was brought underground, the reduction in the content in the initial extractions was the same as for the Ge carrier.
### carrier {#Ge_carrier}
A special Ge carrier was produced which contained a known number of atoms. This carrier was added to a reactor holding 7 tons of Ga, three successive extractions were carried out, and the number of atoms in each extraction was determined by counting. The results [@Abazov91] verified that the extraction efficiencies of the natural Ge carrier and track each other very closely.
Counting of $^{\bf71}$G {#counting}
=======================
General overview
----------------
Once the is isolated internally in the proportional counter, its decay must be identified. decays solely by electron capture to the ground state of $^{71}$Ga with a half-life of 11.43 days [@HAM85]. The probabilities of $K$, $L$, and $M$ capture are 88%, 10.3%, and 1.7%, respectively [@GEN71]. $K$ capture gives Auger electrons with an energy of 10.367 keV (41.5% of all decays), 9.2-keV x rays accompanied by 1.2-keV Auger electrons from the subsequent $M$–$L$ transition (41.2% of all decays), and 10.26-keV x rays accompanied by 0.12-keV Auger electrons (5.3% of all decays). $L$ and $M$ capture give essentially only Auger electrons with energies of 1.2 keV and 0.12 keV, respectively [@BRO86]. The proportional counter observes the Auger electrons and, with considerably less efficiency, the x rays emitted during the relaxation of the atomic electron shell. As a result, about the same fraction of events occur in the $L$ and $K$ peaks.
These low-energy Auger electrons and x rays produce a nearly pointlike ionization in the counter gas. This ionization will arrive at the anode wire of the proportional counter as a unit, resulting in a fast rise time for the pulse. In contrast, although a typical $\beta$ particle produced by a background process may also lose 1–15 keV in the counter gas, it will leave an extended trail of ionization. This ionization will arrive at the anode wire distributed in time according to its radial extent in the counter, which usually gives a pulse with a slower rise time than for a event. The identification of true events and the rejection of background events are thus greatly facilitated by using a two parameter analysis: a candidate event must not only fall within the appropriate energy region, but must also have a rise time consistent with pointlike ionization.
To properly determine the background rate it is necessary to count each sample for a long time after any has decayed. We endeavor to begin to count as soon as possible after extraction and to continue counting for at least 160 days. Since the number of high-quality low-background counters and of available counting channels is limited, runs are occasionally ended before the desired counting duration is met to permit another run to begin. Further, since many counters are measured in a common system, counting time is frequently lost for calibration or for counter installation or removal.
This section continues with a discussion of how the proportional counters are made, how their counting efficiency is determined, and how they are calibrated, and concludes with a description of the counting electronics.
Proportional counters {#prop_cntrs}
---------------------
The design and construction of the proportional counters are based on the experience gained in the Cl experiment. They are made only from materials that are radioactively clean, are assembled in a clean environment, are only exposed to high levels of radioactivity during efficiency measurement, and are always counted for background before use in a solar neutrino extraction.
![Schematic view of a proportional counter.[]{data-label="counter_view"}](counter.eps){width="3.375in"}
Although several different types of counters were used at the beginning of the experiment, all counters used since the extraction of September 1991 are of a common type, shown in Fig. \[counter\_view\]. The counter bodies are fabricated by a glassblower from Heraeus Amersil transparent synthetic fused silica (Suprasil). The main body is 10 cm long with an 8-mm outer diameter. One end is open for insertion of the cathode but can be sealed with a flared plug. Three tubes are attached to the other end — one tube with 2-mm inner diameter is used for insertion of the filling gas; the other two tubes are capillaries for the cathode and anode electrical feedthroughs. A 2-mm hole is made in the counter body near its center over which a very thin piece of blown silica is sealed. There is a corresponding hole in the cathode at this position so that x rays from external sources can pass through this window into the counter gas for calibration.
The main body of the counter is large enough to hold a snug-fitting zone-refined iron cathode sleeve, whose dimensions are approximately 5 mm diameter, 5 cm length, and 1/3 mm thickness. The cathodes are individually machined to fit each counter body, making sure that there is sufficient space between the cathode and the body to permit the counter to be heated to at least 100 for bakeout of impurities. The iron is drilled and cut to length using only new tools and ultrapure hexane as lubricant.
The major component of the counters, Suprasil, has a total metallic impurity content of $\leq 1$ ppm by weight and OH and equivalent H$_2$O contents of $\sim 10^3$ ppm. The cathode material typically has less than 1 ppm metallic impurities, except for copper which is present at $\sim 7$ ppm.
The first step in counter fabrication is a helium leak test of the seals and thin calibration window of the counter body. All parts of the counter are then thoroughly cleaned: the silica parts are soaked overnight in aqua regia, etched briefly in hydrofluoric acid, thoroughly washed in high-purity water, and dried in an oven at slightly above 100. The cathodes are washed in hexane in an ultrasonic bath, baked, and dried under vacuum for approximately 24 h at 500. After cleaning, all counter parts are handled only with gloves and clean tools.
The final steps of counter fabrication take place inside a laminar flow clean bench. Under a microscope, a 25-$\mu$m wire of high-purity tungsten is spot welded to the cathode and then threaded through a thin capillary to the outside of the counter, where an external lead pin is connected. Again under a microscope, a 12.5-$\mu$m tungsten anode wire is threaded through the second capillary, through the center of the cathode sleeve, and welded to a 50-$\mu$m tungsten spring wire held in place at the end of the counter by the Suprasil end plug. With the anode and cathode wires held taut in the capillaries, the electrical connections to the external leads are made with a small dab of conducting epoxy injected into the end of the capillary with a hypodermic needle. With the wires still held taut, the quartz end plug is gently welded in place by a glassblower, and then (with the counter filled with $\sim 0.1$ atm hydrogen to prevent oxidation of the thin wires), the capillaries are heated and sealed around the cathode and anode wires. The counters are then tested for gas tightness, evacuated and baked for $\geq 72$ h, purged, and filled for testing with P-10 counter gas (90% argon, 10% methane).
Counters are tested at the time of fabrication for stability, gain, and resolution. Counter background rates are measured at Baksan and are in the range of 0.1/day (0.07/day) in the $L$-peak ($K$-peak) candidate regions.
Measurement of proportional counter efficiency {#counter_efficiency}
----------------------------------------------
This section gives a general description of the methods of counter efficiency measurement, shows how these methods are applied to determine the $L$- and $K$-peak efficiencies of several typical counters, and presents how the counting efficiency of the solar neutrino extractions is determined.
### Measurement methods
Two different techniques and three different isotopes are employed: to measure volume efficiency, and and to measure the $L$- and $K$-peak efficiencies.
The first method uses to measure the volume efficiency, which we define as the probability that the decay of a radioactive atom in the gas phase in the volume of a proportional counter will produce a detectable pulse. The source is produced by the $(n,\alpha)$ reaction on using fast neutrons from the research breeder reactor of the Institute of Physics and Power Engineering in Obninsk. The extracted is purified on a Ti getter and then mixed with 90% Ar plus 10% CH$_4$. A small sample of this mixture is placed into the counter under test, the counter high voltage is set so that the $L$ peak is at least one-quarter scale on the energy analog-to-digital converter (ADC), and an energy spectrum is measured. The gas sample is then transferred with very high efficiency $E_{\text{transfer}}$ ($>99.5$%) to a counter that was specially constructed for these measurements. It is 20 cm in length with an internal diameter of 4 mm. It has a deposited carbon film cathode, shaped ends to minimize end effects, a volume of 2.5 cm$^3$, and a volume efficiency of $(99.5\pm 0.2)$%. Additional Ar–10% CH$_4$ is added to bring the pressure in this standardization counter to about the same value as in the test counter, and another energy spectrum is measured under similar conditions to that of the test counter. To find the position of the $L$ peak, these two spectra are fit to a Gaussian plus a constant background. An energy threshold is then set at one-third of the peak value, equivalent to about 80 eV, and the total number of counts above this threshold determined by summation. This gives the count rates in the counter under test, $R_{\text{test}}$, and in the standardization counter, $R_{\text{standard}}$. After making minor corrections for background rates, the volume efficiency of the test counter is given by $\epsilon_v = 0.995 R_{\text{test}}
E_{\text{transfer}} D /R_{\text{standard}}$ where $D$ is the decay factor of the between the times of measurement of the two spectra. Because of the high and well-known transfer efficiency and standardization counter efficiency, the total estimated uncertainty in the volume efficiency of the test counter using this method is only 0.005, or approximately 0.6%.
The second counter efficiency measurement method uses . A brief description is given here; for more details see Ref. [@Abdurashitov; @and; @Gusev; @and; @Yants; @95]. The source is made by the $(p,n)$ reaction on 99%-enriched with 7-MeV protons from the cyclotron of the Nuclear Physics Institute of Moscow State University. is extracted from the gallium target, synthesized into H$_4$, and added to a normal GeH$_4$-Xe counter filling. decays both by electron capture (64%) and by positron emission (36%). About 40% of the electron capture decays go to an excited state of the daughter which emits a coincident 1106-keV gamma ray. The measurements are made by placing a proportional counter with a filling on the axis of and 10 cm to 12 cm distant from a large Ge semiconductor detector which observes the gamma rays. Energy spectra are taken of the events produced by electron capture decays of by gating the signal from the proportional counter with the output of a single channel analyzer set on the 1106-keV gamma ray. The $K$, $L$, and volume efficiencies are defined as the ratio of the number of counts in the $K$ peak, $L$ peak, and total spectrum, respectively, to the number of 1106-keV gammas detected by the large germanium detector. In these calculations, small corrections are made to the raw number of observed events because of random coincidences and background in the Ge detector. The uncertainty in this measurement method is mainly from the partial detection of $M$-peak events. The $M$ peak in Ge is at $\sim 120$ eV, a higher energy than in Ar, but it contains a much larger fraction of the total number of decays (7% compared to 1.4% in Ar). Even though part of the $M$ peak is detected in these Ge spectra, a substantial correction for the missing fraction of events below threshold energy is still required. The estimated uncertainty in the peak efficiency is thus slightly less than 2.5% (or 0.008 in absolute efficiency) and 1.7% (0.015 in absolute efficiency) in the volume efficiency.
The final measurement method uses produced by neutron irradiation of . After extraction and purification of the Ge, H$_4$ is synthesized, and mixed with Xe-GeH$_4$. Measurements of the volume efficiency are then made using a similar technique to that described for . In addition, the $L$- and $K$-peak efficiencies are determined by integration over the peaks. The uncertainty in this measurement method is about the same as for the method.
### Application and test
[lccc]{}
Counter &\
\
name & & &\
LA51 & 0.887 $\pm$ 0.005 & &\
LA88 & 0.876 $\pm$ 0.005 & 0.854 $\pm$ 0.015 & 0.879 $\pm$ 0.015\
LA105 & 0.872 $\pm$ 0.005 & &\
LA107 & 0.874 $\pm$ 0.005 & &\
LA110 & 0.933 $\pm$ 0.005 & &\
LA111 & 0.948 $\pm$ 0.005 & &\
LA111\* & 0.897 $\pm$ 0.005 & 0.895 $\pm$ 0.015 & 0.908 $\pm$ 0.015\
LA113 & 0.875 $\pm$ 0.005 & &\
LA114 & 0.892 $\pm$ 0.005 & 0.918 $\pm$ 0.015 & 0.913 $\pm$ 0.015\
LA116 & 0.901 $\pm$ 0.005 & &\
A8 & 0.868 $\pm$ 0.005 & 0.867 $\pm$ 0.015 &\
A13 & 0.928 $\pm$ 0.005 & &\
A28 & 0.893 $\pm$ 0.005 & &\
A31 & 0.872 $\pm$ 0.005 & &\
Average & 0.894 $\pm$ 0.025 & &\
[l c c d d d d d]{}
& & & GeH$_4$ &\
Counter & Isotope & Pressure & fraction & &\
name & used & (mm Hg) &(volume %) & Measured & Calculated & Measured & Calculated\
LA88 & & 640 & 10.6 & 0.326 & 0.329 & 0.313 & 0.327\
LA88 & & 735 & 9.5 & 0.332 & 0.345 & 0.322 & 0.316\
LA111\* & & 710 & 15. & 0.347 & 0.345 & 0.334 & 0.330\
LA111\* & & 735 & 9.5 & 0.358 & 0.353 & 0.321 & 0.324\
LA114 & & 745 & 8. & 0.355 & 0.354 & 0.316 & 0.320\
LA114 & & 909 & 17.4 & 0.379 & 0.369 & 0.320 & 0.310\
A8 & & 800 & 12. & 0.348 & 0.349 & 0.310 & 0.308
Table \[volume\_efficiency\] gives the measured volume efficiencies for 14 counters using the measurement methods based on , , and . For those counters that were measured with more than one isotope, the agreement is very good and distributed in the expected statistical manner.
The efficiencies in the $L$ and $K$ peaks for four counters measured with the coincidence method and for three counters measured with are given in Table \[efficiency\_comparison\]. Because different gas compositions and pressures were used in these counter fillings, these measurements can only be compared if one has a procedure for correcting the efficiency for the gas filling. Since our solar neutrino runs also have different counter fillings, such a correction procedure is also essential for determining the counting efficiency for normal extractions.
The counting efficiency $\epsilon(P,G)$, before the application of energy or rise time cuts, can be written in the general form $$\label{efficiency_formula}
\epsilon(P,G) = \epsilon_v (1 - f_D) E(P,G),$$ where $\epsilon_v$ is the volume efficiency, $G$ is the fraction of the counting gas that is GeH$_4$, $P$ is the total counter pressure in standard atmospheres, and $f_D$ is the fraction of peak events that lie outside the $\pm 1$ full width at half maximum (FWHM) energy window, determined empirically for our counters from and spectra to be 0.063 for the $L$ peak and 0.202 for the $K$ peak. Monte Carlo simulations, based on our standard counter geometry, were made to determine the dependence of the efficiency on $P$ and $G$ [@Kouzes89]. Fits to these calculations with a polynomial function give $E(P,G) = A(G) + B(G) P +
C(G) P^2$, where $A(G) = A_0 + A_1 G$, $B(G) = B_0 + B_1 G$, and $C(G) = C_0
+ C_1 G$. This equation applies to both the $L$ and $K$ peaks with different constants in the expressions for $A$, $B$, and $C$. For the $L$ peak the constants are $A_0 = 51.0$, $A_1 = 5.51$, $B_0 = -15.7$, $B_1 = 1.58$, $C_0 =
3.0$, and $C_1 = 0.000113$, and for the $K$ peak the constants are $A_0 =
29.7$, $A_1 = -8.27$, $B_0 = 28.4$, $B_1 = -5.02$, $C_0 = -6.22$, and $C_1 =
2.27$. Over the range of counter fillings for usual extractions, the estimated uncertainty in $E$ from the Monte Carlo calculations is $\pm 1$%.
With the aid of this efficiency formula it is now possible to compare the measurements in Table \[efficiency\_comparison\]. Calculated efficiencies for these counters in the $L$ and $K$ peaks, using the volume efficiency measured with and Eq. (\[efficiency\_formula\]), are given in columns 6 and 8 of Table \[efficiency\_comparison\]. The total uncertainty in the calculated efficiencies is estimated to be 1.5%, consisting of 0.6% from uncertainty in $\epsilon_v$, 1.0% from uncertainty in $f_D$, and 1.0% from the uncertainty in the Monte Carlo simulations. The calculated efficiencies agree with the values measured with and within the errors of calculation and measurement.
### Counting efficiency for solar neutrino extractions
The counters used during the course of the experiment are listed in Table \[run\_parameter\_table\]. The counter type used for the majority of extractions is indicated by the designation “LA” or “A”. The second type was used for three extractions during 1990 and is indicated by “Ni”; the final type was used only for the August 1991 extraction and is indicated by the designation “RD”.
The counting efficiency used for each extraction is calculated by Eq. (\[efficiency\_formula\]) and is given in Table \[run\_parameter\_table\]. The volume efficiency of most counters has been directly measured with ; if a counter’s volume efficiency has not been measured, it is assumed to equal the average of all measured counters. Because the analysis reported in this section resulted in new counter efficiencies for SAGE I, these revised efficiencies are given in this table and are used in any combined fits which include SAGE I data.
Counter calibration {#ext_cals}
-------------------
Immediately after filling counters are calibrated through their side window with the 5.9-keV x rays from an source. They are recalibrated with after about 3 days of operation, and then again approximately every 2 weeks until counting ends. This usually gives more than ten calibrations, with at least four during the first month of counting while the is decaying. In addition, beginning with SAGE II, calibrations are usually made with a source whenever an calibration is done. The 22-keV Ag x rays that follow decay pass through the counter window and fluoresce the Fe cathode, giving the $K$ x-ray peak from Fe at 6.4 keV. Although these x rays originate near the counter window, they are absorbed throughout the counter volume, and thus give the average counter response. Beginning with the February 1993 extraction, a +Se source was periodically used. The Cd x rays fluoresce a Se target whose $L$ and $K$ x rays enter the counter through its side window and give peaks at 1.4 keV and 11.208 keV.
The energies of the peaks from these calibration sources are summarized in Table \[x\_ray\_energies\]. These various calibration lines have been used to check the linearity of the energy and amplitude of the differentiated pulse (ADP) counting channels and to determine offsets. There are also Xe escape peaks with the and +Se sources, but these lines are usually weak and not useful for energy scale determination.
[dcc]{}
Energy (keV) & Source & Origin\
\
\[-10pt\] 1.4 & +Se & Se $L$ x ray through window\
1.625 & & Xe escape peak through window\
5.895 & & Mn $K$ x ray through window\
6.4 & & $K$ x ray from Fe cathode\
11.208 & +Se & Se $K$ x ray through window\
The typical counter resolution measured with an source is in the range of 20%–23%. Scaling the resolution by the square root of the energy, this implies resolutions in the $L$ and $K$ peaks of 45%–50% and 15%–17%, respectively, values that are observed in -filled counters operated at low voltage.
Many calibrations are done on each counter. With each calibration a small fraction of the GeH$_4$ molecules are broken into fragments which can be deposited on the anode wire near the counter window. This process, which we call “polymerization”, gradually increases the anode diameter, reduces the electric field, and gives a depression of the apparent energy measured with an source or a +Se source. Polymerization (see, e.g., [@Vavra86]) occurs most readily at high count rates, so we maintain the rate below 10 events/s during calibration. A check for the presence of polymerization is made by comparing the peak positions of the 5.895-keV line from the source (which provides events only at the counter window) and the 6.4-keV line from the source (which provides events over a much larger fraction of the counter volume). If the counter anode is not polymerized near the window and the energy channel is linear, the ratio of peak positions will be 6.4/5.895 = 1.086. For each extraction the ratio of the 6.4-keV to 5.895-keV peak positions averaged over all calibrations is given in Table \[run\_parameter\_table\] relative to the unpolymerized value of 1.086. Most counters show little or no evidence of polymerization. For polymerized counters the peak ratio is greater than 1.00 and is used to correct the energy scale derived from each calibration.
Linearity of counter gain {#gain_linearity}
-------------------------
Calibrations with the +Se source have been used to check the predicted position and resolution of the $K$ peak from an calibration. Since the 11.208-keV peak energy with the +Se source is very close to the 10.367-keV energy of $K$-peak events, this method has the advantage that very little extrapolation of the peak position in energy is needed. Some departures from linearity are present in the region of the $K$ peak.
Measurements have been made as a function of GeH$_4$ fraction $G$, counter pressure $P$, and operating voltage $V$. The ratio of the peak positions is equal to the ratio of the energies (11.2/5.9) up to a critical voltage $V_{\text{crit}} = 10.5 G + 0.6 P + 588$. Above this critical voltage, the location of the $K$ peak $[P_K(\mnuc{71}{Ge})]$ can be inferred from the location of the peak $[P(\mnuc{55}{Fe})]$ using the formula
$$\label{Gain_Scaling_Formula}
\frac{P_K(\mnuc{71}{Ge})}{P(\mnuc{55}{Fe})} = \frac{10.367}{5.895}
[1 - (4.5G + 2.78)(V - V_{\text{crit}}) \times 10^{-6}],$$
where $G$ is expressed in percent, $P$ is in mm Hg, and $V$ is in volts. The typical correction due to the nonlinearity of the gain is a reduction in the predicted peak position of 2%.
This set of experiments also measured the resolution of the peaks from +Se and from . Below a critical voltage, the ratio of the resolutions was equal to the expected value of $\sqrt{5.9/11.2}$, but above this voltage, given by $V_{\text{crit}} = 6 G + P/3 + 824$, the +Se resolution was wider than predicted from the resolution. From these measurements the relationship between the $K$-peak resolution $[R_K(\mnuc{71}{Ge})]$ and the resolution $[R(\mnuc{55}{Fe})]$ was found to be
$$\label{Resolution_Scaling_Formula}
\frac{R_K(\mnuc{71}{Ge})}{R(\mnuc{55}{Fe})} = \sqrt{\frac{5.895}{10.367}}
[ 1 + 1.5 \times 10^{-3} (V - V_{\text{crit}}) ].$$
Note that the value for $V_{\text{crit}}$ for the resolution correction is not the same as for the gain correction. The typical correction for the $K$ peak results in an increase in the predicted resolution of 15%.
The correction to the gain and resolution predicted by these empirical formulas is accurate to about 30%. The nonlinearity in gain and resolution is only present at the higher energies. No corrections are required for the $L$ peak because the critical voltages are much higher than for the $K$ peak.
Electronic systems {#elec_sys}
------------------
As indicated in Table \[run\_parameter\_table\], SAGE has used several different counting systems as the experiment progressed. Most runs of SAGE I were counted in what we call system 2. Since the fall of 1992, during SAGE II and III, most first extractions were counted in system 3. System 6 measured a few first extractions, but most were from a low mass of Ga. The major specifications of these various counting systems are given in Table \[sys\_specs\]. Since this article focuses on SAGE II and III, we will mainly consider counting system 3 in the following. Some additional information concerning system 6 and previous systems is given in Appendix \[other\_counting\_systems\].
Specification Sys. 2 Sys. 6 Sys. 3
--------------------------------- -------------- --------- ---------
Number of channels 7 7 8
Number of channels with NaI 5 6 8
Counter dynamic range (keV) 0.4–13 0.5–18 0.3–18
NaI dynamic range (keV) 50–3000 50–3000 50–3000
Max. counting rate (s$^{-1}$) 5 1000 1.5
NaI coincidence window ($\mu$s) 8 4 5.2
Energy time constant ($\mu$s) 1 NA NA
ADP time constant (ns) 10 10–500 10
NaI time constant ($\mu$s) 1 0.5 1
Bandwidth, $-3$ dB (MHz) 90 45 90
Rise time, 10%–90% (ns) 3.5 8 4
Noise, peak to peak (mV) $< 10$ $< 12$ $< 10$
Dead time
in acquisition mode (ms) 200 1 600
in calibration mode (ms) 50 1 120
ADC resolution (mV/ch) 10 1 1
Energy offset (ch) 0 0 0
ADP offset of 4096 ch (ch) $-$45 to +25 0 70–120
: Specifications of counting systems 2, 6, and 3.[]{data-label="sys_specs"}
The counting systems reside in a specially designed air-conditioned room in the underground facility. To minimize the fast neutron and gamma ray flux, the walls are made from low-radioactivity concrete with an outer steel shell. The entire room is lined with sheets of 1 mm zinc-galvanized steel to reduce radio-frequency noise. Power to the counting electronics is supplied by a filtered uninterruptible power supply, with signal and power cables laid inside independent steel conduits. The data acquisition computers, which are in the counting room, are networked so that the systems can be monitored outside the underground laboratory. The counting room is kept locked and access is restricted to counting personnel.
System 3 was moved to BNO and installed in the underground counting room in 1988. It can record events from up to eight counters which are placed inside the well of a NaI crystal that serves as an active shield (crystal, 23 cm diameter by 23 cm height; well, 9 cm diameter by 15 cm height). There are two layers of passive shielding. An inner layer of square tungsten rods (10 mm$\times$10 mm) encloses the NaI and the photomultipliers are shielded by Pb. All components are made from low-radioactivity materials which were assayed prior to construction by a low-background solid-state Ge detector. The preamplifiers are mounted as close to the counters as possible, but are separated by a thick layer of copper. The counters are sealed nearly air tight inside the apparatus. Dry nitrogen gas from evaporation of liquid nitrogen flows continuously through the NaI well to remove Rn. The entire apparatus may be lowered with a hoist into an outer shield whose bottom and sides consist of 24–32 mm of copper, 210 mm of lead, and 55 mm of steel, and whose top has 34 mm of Cu and 250 mm of steel. The cavity between the inner and outer shields is also purged continuously by gas from evaporating liquid nitrogen; to preclude backstreaming the flow rate is kept below that in the inner shield.
{width="6.500in"}
To minimize the length of the signal cables, the rack of counting electronics is immediately adjacent to the outer passive shield. The electronics is in a single rack designed to reduce rf interference. The block diagram of a single channel of system 3 is illustrated in Fig. \[sage\_daq\]. Briefly, the analog signal processing proceeds as follows: the proportional counter anode is directly connected to a charge-sensitive preamplifier. After further amplification the signal is split, with one channel going to the digital logic to determine that an event from that counter has occurred, and a second channel going to a 90 ns cable delay and then to a gated multiplexer. The signals from all 8 counters are input to separate gates of this multiplexer and the appropriate gate is opened by the digital logic for whichever counter has seen an event. The multiplexed output is split into four channels: two go to a digital oscilloscope which records the counter wave form with 8-bit resolution for 800 ns after pulse onset at two different amplification ranges, one appropriate for the $L$ peak and the other appropriate for the $K$ peak. One of the other two signals goes to an integrating ADC to measure the total pulse energy; the second signal is differentiated with a time constant of 10 ns, stretched, and input to a peak-sensing ADC. This second ADC measures the amplitude of the differentiated pulse, called “ADP.” Acquisition can be run in calibration or event acquisition modes. For each event in acquisition mode, the energy, ADP value, time of event, NaI time and energy, and the two digitized wave forms (high- and low-gain channels) are written to disk.
Selection of Candidate $^{\bf71}$G Events
=========================================
The counting data consist of a set of events for each of which there is a set of measured parameters, such as wave form, energy, NaI coincidence, etc. The first step of analysis is to sort through these events and apply various selection criteria to choose those events that may be from . We will describe here the selection procedure for events measured in counting system 3; the procedure for system 6 is identical except there are no measured wave forms, so the energy is measured by an ADC and the ADP method is used for rise time determination.
Standard analysis description {#stand_anal}
-----------------------------
The various steps to select potential events are the following:
\(1) The first step of event selection is to examine the event wave form and identify two specific types of events: those that saturate the wave form recorder and those that originate from high-voltage breakdown. Saturated events are mostly produced by alpha particles from natural radioactivity in the counter construction materials or from the decay of that has entered the counter during filling. Such events are easily identified and labeled by looking at the pulse amplitude at the end of the wave form. Saturated pulses have amplitude greater than 16 keV and occur in an average run at a rate of approximately 0.5/day. Since most such pulses are seen after any initial has decayed, they are mainly from internal counter radioactivity. Events from high-voltage breakdown have a characteristic wave form which rises very steeply and then plateaus. A true pulse from decay, in contrast, rises more slowly and after this initial rise, has a slow, but steady, increase in amplitude as the positive ions are collected. Breakdown pulses are identified by determining the slope of the wave form between 500 and 1000 ns after pulse digitization begins.
\(2) To minimize the concentration of Rn, the air in the vicinity of the counters is continuously purged with evaporating liquid nitrogen. Counter calibrations, however, are done with the counter exposed to counting room air which contains an average of 2 pCi of Rn per liter. When the shield is closed and counting begins, a small fraction of the decays of the daughters of can make pulses inside the counter that mimic those of . To remove these false events, we delete 2.6 h of counting time after any opening of the passive shield, and estimate the background removal efficiency of this time cut to be nearly 100%. See Sec. \[ext\_rn\] for further details.
\(3) It is possible that the Xe-GeH$_4$ counter filling may have a small admixture of that enters the counter when it is filled. Most of the decays of Rn give slow pulses at an energy outside the peaks, but approximately 8% of the pulses from Rn and its daughters make fast pulses in the $K$ peak that are indistinguishable from those of . Since Rn has a half-life of only 3.8 days, these events will occur early in the counting and be falsely interpreted as events. Each decay is, however, accompanied by three $\alpha$ particles, which are detected with high efficiency and usually produce a saturated pulse in the counter. Since the radon decay chain takes on average only about 1 h from the initiating decay of to reach $^{210}$Pb with a 22-yr half-life, deleting all data for a few hours around each saturated event removes most of these false events. We choose to delete from 15 min prior to 3 h after each saturated pulse. The efficiency of this cut in time is 95%. Further details are given in Sec. \[int\_rn\].
\(4) All events whose pulse is coincident with a NaI detector response are then eliminated. Since has no $\gamma$ rays associated with its decay, this veto reduces background from natural radioactivity.
-------------------------- -------- ------ ------ ------
Live
time 2–15 $L$ $K$
Cut description (days) keV peak peak
None 4129 4209 1990 821
Shield open time cut 4040 3962 1864 785
Saturated event time cut 3862 3641 1733 728
NaI coincidence cut 3862 1275 1106 519
Rise time cut 3862 NA 408 314
-------------------------- -------- ------ ------ ------
: Effect of cuts on the experimental live time and events for all runs of SAGE II and SAGE III that were counted in both $L$ and $K$ peaks (except May 1996). The results of the cut on each row include the effect of all cuts on preceding rows. Because the rise time cut varies with energy, no entry can be given for the “2-15 keV” column.[]{data-label="Live_Time_Cut_table"}
\(5) The next step is to set the energy windows for the Ge $L$ and $K$ peaks. The measure of energy is the integral of the pulse wave form for 800 ns after pulse onset. The peak position for each window is based on the calibration with , with appropriate corrections for polymerization, as described in Sec. \[ext\_cals\], and for nonlinearity, as described in Sec. \[gain\_linearity\]. If the peak position changes from one calibration to the next, then the energy window for event selection is slid linearly in time between the two calibrations. The resolution at each peak is held constant and is set to be the average of the resolutions with for all counter calibrations, scaled to the $L$- or $K$-peak energy as described in Sec. \[ext\_cals\]. (In the rare cases that the resolution of the first calibration is larger than the average, the resolution of the first calibration is used throughout the counting.) Events are then accepted as candidates only if their energy is within $\pm1$ FWHM of the central peak energy.
\(6) Finally, events are eliminated unless their rise time is in the range of what is expected for decays. For runs with wave form recording, the rise time is derived from a fit to the pulse shape with an analytical function, as described below in Sec. \[rise\_time\_techs\]. For those runs without wave form recording, the $L$ peak is not analyzed and the ADP measure of rise time is used to set the acceptance window for $K$-peak events.
For the 30 runs of SAGE II and III that could be counted in both the $L$ and $K$ peaks, the effect on the live time of each successive cut and the total number of candidate events that survive is given in Table \[Live\_Time\_Cut\_table\]. (The run of May 1996 is excluded because the counter was slightly contaminated with residual which had been used to measure this counter’s efficiency.) Figure \[2d\_hist\] shows all events from these same runs that survive the first four cuts. Events that occurred early in the counting are shown in the upper panel and at the end of counting in the lower panel.
![Upper panel shows the energy rise time histogram of all events observed during the first 30 days after extraction for all runs that could be counted in both $L$ and $K$ peaks (except May 1996). The live time is 711.1 days. The expected location of the $L$ and $K$ peaks as predicted by the and calibrations is shown darkened. Lower panel shows the same histogram for all events that occurred during an equal live time interval at the end of counting.[]{data-label="2d_hist"}](2d.eps){width="3.375in"}
Several runs were compromised and some were completely lost due to operational failures. Failure of an electronic component made it impossible to use the $L$ peak in the extractions of April 1993, May 1993, July 1993, and October 1994. Similar problems made it impossible to make a rise time cut in the $K$ peak for the runs of June 1991, July 1993, October 1994, and October 1997. These runs thus have a larger than normal number of events. If an electronic component fails that deteriorates the rise time response and the failure occurs early in the counting, while the is decaying, our policy is to not use any rise time cut in the $K$ peak and to reject this run in the $L$ peak. If the failure occurs later, the rise time cut is retained and the interval of failure is removed from the data. Extractions in March 1993, January 1995, May 1995, and March 1996 were entirely lost due to counter failure. The extractions of September 1993, September 1994-2, and July 1996 were lost because either the counter stopcock failed or some other gas fill difficulty occurred. Electronic failures caused the loss of the extractions of September 1993-1, May 1994-2, and April 1995. Extractions in June 1995 were lost due to radioactive contamination of the counters with isotopes that were being used at this time for counter efficiency measurement. Finally, we exclude several extractions from one reactor that were systematic studies in preparation for the Cr source experiment. Since their mass was no more than 7.5 tons of gallium, less than one atom of is detected on the average in such runs in the combination of both the $L$ and $K$ peaks. Two-reactor extractions, however, whose mass is approximately 15 tons, give on the average 1.5 events, sufficient to determine the solar neutrino capture rate, albeit with a large error [@Bahcall85].
Rise time analysis techniques {#rise_time_techs}
-----------------------------
![ events in $L$ and $K$ peaks.[]{data-label="example_LK_candidate_waveforms"}](levent.eps "fig:"){width="3.375in"} ![ events in $L$ and $K$ peaks.[]{data-label="example_LK_candidate_waveforms"}](kevent.eps "fig:"){width="3.375in"}
As described in Secs. \[extrac\_hist\] and \[elec\_sys\], the data acquisition system electronics has evolved over the course of SAGE. The data from SAGE I relied entirely on a hardware measurement of the rise time. This ADP technique suffices well in studies of the $K$-peak counter response, but is not capable of adequately differentiating rare $L$-peak events from noise.
Wherever possible for SAGE II, and throughout SAGE III, we derive a parameter that characterizes the rise time from the wave form, and are thus able to present both $L$- and $K$-peak results. For those runs with only ADP data, the $L$ peak cannot be analyzed and we present only $K$-peak data. All wave form data come from counting system 3.
### Waveform rise time determination – $T_N$ {#T_N}
Figure \[example\_LK\_candidate\_waveforms\] shows typical pulses in the $L$ and $K$ peaks from a -filled counter as captured by the digitizing oscilloscope in system 3. There are 256 channels full scale on the y axis corresponding to 1.040 V (130 mV/div) for digitizer channel 1 and 0.160 V (20 mV/div) for channel 2. The x axis has 1024 digitization points each with 1 ns duration. The relevant features of the pulses are the base line from $t = 0$ to roughly 120 ns, the dc offset that occurs when the gate opens at 120 ns, and the fast onset of the pulse at about 180 ns. The exact values of these times and offsets vary depending on the counting channel and the run; they even vary slightly from pulse to pulse within a given run. When determining the energy and rise time of the pulse, it is therefore necessary to determine accurately the onset of the pulse both in time and dc voltage level.
![Background candidate event in $K$ peak. Note the much slower fall when the pulse begins at $\sim 200$ ns than for the true $K$-peak event in Fig. \[example\_LK\_candidate\_waveforms\].[]{data-label="example_K_background_waveform"}](bkgdeven.eps){width="3.375in"}
By treating the trail of ionization in the proportional counter as a collection of point ionizations and integrating over their arrival time at the anode, it can be shown [@ELL90] that the voltage output $V$ of an infinite bandwidth preamplifier as a function of time $t$ after pulse onset has the form
$$\begin{array}{rcll}
\label{Tn_Formula}
V(0<t<T_N) & = & V_0[ & \frac{t + t_0}{T_N}\ln(1 + \frac{t}{t_0})
- \frac{t}{T_N}], \\
V(t>T_N) & = & V_0[ & \ln(1 + \frac{t - T_N}{t_0}) - 1 \\
& & & -\frac{t + t_0}{T_N}\ln(1 - \frac{T_N}{t+t_0})],
\end{array}$$
with $V(t<0) = 0$, where $T_N$ is the time duration over which the ionization arrives at the anode, $t_0$ is a time inversely proportional to the ion mobility, and $V_0$ is proportional to the total amount of ionization deposited in the counter. The parameter $T_N$ characterizes the rise time of the wave form. For the case of true point ionization, $T_N$ should be near zero. When $T_N$ is zero, the function reduces to the Wilkinson form $V(t:T_N=0) = V_0 \ln(1 + t/t_0)$. When $T_N$ is large, the event is characteristic of extended ionization, and is most likely a background event from a high-energy $\beta$ particle traversing the counter. Figure \[example\_K\_background\_waveform\] is an example of such a slow pulse in the $K$ peak.
Because this form for the pulse shape has a sound physical basis and reasonable mathematical simplicity, we fit every pulse that is not identified as saturation or breakdown to Eq. (\[Tn\_Formula\]). To account for the fact that the pulse onset time $t_{\text{onset}}$ is not at time zero, we replace $t$ by $t-t_{\text{onset}}$, and since the pulse begins at a finite voltage $V_{\text{offset}}$, we replace $V$ by $V-V_{\text{offset}}$. The fit is made from 40 ns before the time of pulse onset to 400 ns after onset. Five parameters are determined by the fit: $t_{\text{onset}}$, $V_{\text{offset}}$, $V_0$ (a measure of the energy deposited during the event which is not used in analysis), $t_0$ (whose value of slightly less than 1 ns is approximately constant for all pulses), and $T_N$ (the rise time).
### Alternative wave form analysis methods
Although we use fits to $T_N$ as our standard analysis technique, we also developed two alternative methods to discriminate pointlike ionization from extended-track ionization in the proportional counter pulses. These serve as checks on the event selection based on $T_N$. One technique is based on a fast Fourier transform (FFT) of the digitized wave form. No specific functional form for the pulse is assumed and hence this method has the advantage that it is sensitive to potential alterations in the pulse shape. See Appendix \[fft\] for further information concerning the FFT method. The second method of wave form analysis that was investigated also assumes no particular form for the pulse. This method, called the “RST method,” deconvolutes the observed wave form to find the initial ionization pattern in the counter. See Appendix \[RST\] for further details.
Since these three techniques are sensitive to different characteristics of the wave form, their selection of events is, not unexpectedly, different. Nonetheless, when many data sets are considered in combination, their results for the overall production rate are in good agreement, which provides strong support for the validity of our wave form analysis procedure.
### Hardware rise rime measurement: ADP {#adp}
The amplitude of the differentiated pulse is proportional to the product of the original pulse amplitude and the inverse rise time. The quantity ADP/energy is thus proportional to the inverse rise time. Events due to low-energy Auger electrons and x rays that produce point ionization in the counter all have a fast rise time. Events with a slower rise time (small ADP) are due to background pulses that produce extended ionization. Events with a very fast rise time (large ADP) are due to electronic noise or high-voltage breakdown.
Inherent in an ADP analysis is the uncertainty that arises from an imprecise knowledge of the offset for a given run. Nonzero offset occurs when the gate is opened after an event trigger. The electronic components which process the pulse are subject to small drifts in their offsets that are functions of external parameters, such as temperature. These nonzero offsets contribute to the dc offset on which the event pulse rides. Our approach has been to extrapolate ADP vs energy plots from the calibrations using the 5.9-keV peak and the escape peak to obtain an offset for each calibration. Since the offsets are typically distributed in a Gaussian manner with a sigma of 1 or 2 channels, the average is a good approximation when determining the $K$-peak selection window. For the $L$ peak, however, uncertainties of a few channels lead to significant variations in event selection. Utilizing the digitized pulses, it is possible to eliminate this uncertainty by determining every offset on a pulse-by-pulse basis.
A further disadvantage of the ADP method is that it is only responsive to the initial rise of the pulse. Occasional small pulses from high-voltage breakdown have rise time the same as for true $L$-peak pulses, but after their initial rise they turn flat, rather than gently rise as the positive ions are collected as with a real event. A breakdown event of this type is not distinguished from a event by the ADP method, but is easily recognized by examining the recorded wave form long after pulse onset.
Calibration of rise time response {#rise_tim_cal}
---------------------------------
To determine the values of $T_N$ for true pulses, we have filled counters with typical gas mixtures (20% GeH$_4$ and 80% Xe at a pressure of 620 mm Hg), added a trace of active H$_4$, and measured the pulses in each of the system 3 counting slots. All events inside 2 FWHM of the $L$ and $K$ peaks are then selected and the rise time $T_N$ of each event calculated with Eq. (\[Tn\_Formula\]). The rise time values are arranged in ascending order and an upper rise time limit set such that 5% of the events are excluded. This leads to event selection limits on $T_N$ of 0.0–10.0 ns in the $L$ peak and 0.0–18.4 ns in the $K$ peak. The variation with electronics channel and with counter filling, over the range of our usual gas mixtures, was measured to be approximately 1.2 ns. We choose to fix the event selection limits at the values given above, and include in the systematic error an uncertainty in the efficiency of $\pm1$% due to channel and filling variations. A major advantage of using $T_N$ is that the rise time limits are fixed and are the same for all extractions. The purpose of the calibrations with and other sources is solely to determine the energy scale.
For those runs in which the ADP method of rise time discrimination is used, the limits for the ADP cut are determined separately for each run from the calibrations. Histograms of the values of ADP/energy for the events within 2 FWHM of the 5.9-keV energy peak are analyzed to determine the cut point for 1% from the fast region (to eliminate noise) and 4% from the slow region (to eliminate background). All calibrations from a run are analyzed and the ADP window for event selection is slid linearly in time from one calibration to the next.
Statistical Analysis and Results of Single Runs {#results_stat_anal}
===============================================
In this section we describe how the data are analyzed to determine the production rate. We then give the results for individual runs and for all runs in the $L$- and $K$-peak regions.
Single-run results
------------------
The above selection criteria result in a group of events from each extraction in both the $L$- and $K$-peak regions which are candidate decays. To determine the rate at which was produced during the exposure time, it is assumed in each peak region that these events originate from two sources: the exponential decay of a fixed number of atoms and a constant-rate background (different for each peak). Under this assumption the likelihood function [@CLE83] for each peak region is
$$\label{likelihood_function}
{\cal L} = e^{-m} \prod_{i=1}^{N} (b + a e^{-\lambda t_{i}}),$$
where $$\begin{aligned}
m & = & bT + a \Delta/\lambda, \nonumber \\
T & = & \sum_{k=1}^{n} (t_{ek} - t_{bk}), \nonumber \\
\Delta & = & \sum_{k=1}^{n} (e^{-\lambda t_{bk}}
- e^{-\lambda t_{ek}}). \nonumber\end{aligned}$$
Here $b$ is the background rate, $\lambda$ is the decay constant of , $t_i$ is the time of occurrence of each event with $t = 0$ at the time of extraction, and $N$ is the total number of candidate events. The production rate $p$ of is related to the parameter $a$ by
$$\label{prod_rate_def}
a = \epsilon p (1 - e^{-\lambda \Theta}),
\nonumber$$
where $\Theta = t_E - t_B$ is the exposure time (i.e., the time of end of exposure $t_E$ minus the time of beginning of exposure $t_B$), and $\epsilon$ is the total efficiency for the extraction (i.e., the product of extraction and counting efficiencies). The total counting live time is given by $T$ and is a sum over the $n$ counting intervals, each of which has a starting time $t_{bk}$ and ending time $t_{ek}$. The parameter $\Delta$ is the live time weighted by the exponential decay of . Its value would be unity if counting began at the end of extraction and continued indefinitely. We convert the production rate (in atoms produced per day) to the solar neutrino capture rate (in SNU) using the conversion factor $2.977 \times 10^{-4}$ atoms of produced/(SNU day ton of gallium), where the mass of gallium exposed in each extraction is given in Table \[run\_parameter\_table\].
Because of the eccentricity of the Earth’s orbit, the Earth-Sun distance, and thus the production rate, varies slightly during the year. We correct the production rate for this effect by multiplying $\epsilon$ by the factor $1 + C$ where $C$ is given by
$$\begin{aligned}
\label{earth_sun_corr_factor}
C = \bigg(\frac{2e}{S[1+r^2]}\bigg)[& \cos X_E & + r \sin X_E \\
- (1-S)(& \cos X_B & + r \sin X_B)], \nonumber\end{aligned}$$
with $$\begin{aligned}
r & = & \omega/\lambda, \nonumber \\
X_E & = & \omega(t_E - t_p), \nonumber \\
X_B & = & \omega(t_B - t_p), \nonumber \\
S & = & 1 - e^{-\lambda \Theta}. \nonumber\end{aligned}$$
Here $e$ is the eccentricity of the orbit ($e = 0.0167$), $\omega$ is the angular frequency ($\omega = 2\pi/365.25 \text{ day}^{-1}$), and $t_p$ is the moment of perihelion passage, which has been 2–5 January for the past number of years. We use $t_p = 3.5$ days.
The best estimate of the solar neutrino capture rate in each peak region is determined by finding the values of $a$ and $b$ which maximize $\cal L$. In doing so we exclude unphysical regions; i.e., we require $a>0$ and $b>0$. The uncertainty in the capture rate is found by integrating the likelihood function over the background rate to provide a likelihood function of signal only, and then locating the minimum range in signal which includes 68% of the area under that curve. This procedure is done separately for the $L$ and $K$ peaks and the results are given in Tables \[L\_peak\_table\] and \[K\_peak\_table\]. We call the set of events in each peak region a “data set.”
[l d d d d d d r @[–]{} d d d]{}
& Lead & Live & & Number of & Number & Best &\
Exposure & time & time & & candidate & fit to & fit & & & Probability\
date & (h) & (days) & Delta & events & $^{71}$Ge & (SNU) & & $Nw^2$ & (%)\
Sep. 92 & 29.0 & 103.8 & 0.811 & 7. & 4.0 & 109. & 46 & 174. & 0.039 & 82.\
Oct. 92 & 27.3 & 96.3 & 0.839 & 10. & 0.0 & 0. & 0 & 61. & 0.179 & 19.\
Nov. 92 & 30.7 & 66.7 & 0.688 & 12. & 0.0 & 0. & 0 & 62. & 0.238 & 12.\
Dec. 92 & 26.7 & 47.5 & 0.835 & 10. & 7.4 & 153. & 51 & 208. & 0.055 & 65.\
Jan. 93 & 29.9 & 23.4 & 0.518 & 4. & 4.0 & 135. & 35 & 181. & 0.092 & 67.\
June 93 & 33.3 & 120.7 & 0.699 & 9. & 1.1 & 29. & 0 & 107. & 0.490 & 2.\
Oct. 93-2 & 51.9 & 71.6 & 0.686 & 2. & 2.0 & 193. & 19 & 297. & 0.097 & 55.\
Oct. 93-3 & 31.6 & 102.7 & 0.772 & 3. & 3.0 & 287. & 88 & 428. & 0.078 & 68.\
July 94 & 45.6 & 136.0 & 0.782 & 10. & 2.2 & 65. & 11 & 131. & 0.026 & 95.\
Aug. 94 & 32.6 & 116.4 & 0.838 & 20. & 0.0 & 0. & 0 & 67. & 0.056 & 73.\
Sep. 94-1 & 40.5 & 120.0 & 0.729 & 20. & 4.7 & 171. & 54 & 300. & 0.087 & 32.\
Nov. 94 & 30.4 & 112.3 & 0.660 & 10. & 2.7 & 76. & 18 & 143. & 0.041 & 79.\
July 95 & 35.5 & 110.6 & 0.776 & 16. & 1.2 & 35. & 0 & 104. & 0.336 & 3.\
Aug. 95 & 35.2 & 108.9 & 0.698 & 16. & 3.9 & 113. & 42 & 200. & 0.095 & 28.\
Sep. 95 &124.3 & 80.4 & 0.561 & 23. & 0.2 & 8. & 0 & 179. & 0.160 & 23.\
Oct. 95 & 39.3 & 120.7 & 0.793 & 17. & 3.2 & 169. & 33 & 319. & 0.041 & 78.\
Nov. 95 & 37.2 & 149.9 & 0.759 & 19. & 8.4 & 214. & 124 & 310. & 0.064 & 45.\
Dec. 95-2 & 78.3 & 119.9 & 0.530 & 22. & 0.8 & 40. & 0 & 174. & 0.102 & 42.\
Jan. 96 & 33.9 & 141.2 & 0.767 & 21. & 0.0 & 0. & 0 & 61. & 0.065 & 66.\
May 96 & 35.2 & 117.8 & 0.628 & 25. & 3.5 & 104. & 23 & 200. & 0.038 & 82.\
Aug. 96 & 32.9 & 148.7 & 0.790 & 20. & 5.6 & 126. & 58 & 204. & 0.048 & 68.\
Oct. 96 & 33.6 & 155.5 & 0.785 & 11. & 0.0 & 0. & 0 & 48. & 0.119 & 39.\
Nov. 96 & 35.0 & 162.5 & 0.795 & 13. & 0.2 & 5. & 0 & 58. & 0.042 & 85.\
Jan. 97 & 34.5 & 160.0 & 0.816 & 16. & 1.2 & 24. & 0 & 68. & 0.581 & 1.\
Mar. 97 & 36.3 & 160.9 & 0.814 & 10. & 2.6 & 45. & 9 & 89. & 0.126 & 17.\
Apr. 97 & 35.0 & 167.4 & 0.791 & 12. & 0.0 & 0. & 0 & 27. & 0.108 & 45.\
June 97 & 35.8 & 173.4 & 0.797 & 16. & 4.5 & 95. & 40 & 161. & 0.089 & 30.\
July 97 & 37.0 & 140.0 & 0.752 & 13. & 0.7 & 14. & 0 & 61. & 0.238 & 10.\
Sep. 97 & 33.6 & 166.6 & 0.826 & 12. & 1.1 & 24. & 0 & 77. & 0.059 & 64.\
Oct. 97 & 34.2 & 149.5 & 0.780 & 19. & 4.7 & 99. & 42 & 167. & 0.041 & 76.\
Dec. 97 & 34.4 & 137.1 & 0.726 & 15. & 3.1 & 69. & 18 & 131. & 0.045 & 73.\
& 433. & 64.3 & 55. & 43 & 68. & 0.020 & $>99$.\
[l d d d d d d r @[–]{} d d d]{}
& Lead & Live & & Number of & Number & Best &\
Exposure & time & time & & candidate & fit to & fit & & & Probability\
date & (hours) & (days) & Delta & events & $^{71}$Ge & (SNU) & & $Nw^2$ & (%)\
Jan. 90 & 25.0 & 57.4 & 0.849 & 8. & 0.0 & 0. & 0 & 64. & 0.367 & 4.\
Feb. 90 & 25.0 & 57.3 & 0.886 & 2. & 2.0 & 95. & 18 & 159. & 0.164 & 26.\
Mar. 90 & 25.0 & 47.5 & 0.839 & 9. & 2.8 & 107. & 0 & 224. & 0.053 & 66.\
Apr. 90 & 29.8 & 90.4 & 0.881 & 9. & 0.0 & 0. & 0 & 112. & 0.104 & 40.\
July 90 & 22.6 & 59.3 & 0.870 & 15. & 0.0 & 0. & 0 & 213. & 0.142 & 28.\
June 91 & 20.5 & 108.3 & 0.904 & 10. & 0.4 & 13. & 0 & 119. & 0.211 & 14.\
July 91 & 26.1 & 59.2 & 0.877 & 1. & 1.0 & 55. & 0 & 115. & 0.159 & 26.\
Aug. 91 & 73.8 & 94.4 & 0.651 & 16. & 9.8 & 412. & 243 & 577. & 0.036 & 83.\
Sep. 91 & 35.3 & 68.9 & 0.827 & 8. & 3.5 & 73. & 20 & 126. & 0.041 & 79.\
Nov. 91 & 40.8 & 112.6 & 0.822 & 14. & 2.4 & 48. & 0 & 102. & 0.095 & 30.\
Dec. 91 & 26.2 & 111.8 & 0.917 & 10. & 10.0 & 180. & 99 & 217. & 0.063 & 77.\
Feb. 92-1 & 21.5 & 192.7 & 0.900 & 14. & 0.0 & 0. & 0 & 43. & 0.057 & 74.\
Feb. 92-2 & 43.0 & 43.2 & 0.800 & 1. & 1.0 & 101. & 0 & 192. & 0.085 & 88.\
Mar. 92 & 26.0 & 167.8 & 0.840 & 21. & 10.1 & 245. & 155 & 342. & 0.043 & 72.\
Apr. 92 & 21.5 & 144.9 & 0.717 & 15. & 2.3 & 55. & 13 & 111. & 0.143 & 18.\
May 92 & 54.0 & 114.9 & 0.843 & 4. & 0.0 & 0. & 0 & 74. & 0.134 & 30.\
Sep. 92 & 29.0 & 103.8 & 0.811 & 6. & 2.1 & 55. & 12 & 104. & 0.108 & 25.\
Oct. 92 & 27.3 & 134.2 & 0.840 & 11. & 2.7 & 52. & 13 & 98. & 0.046 & 71.\
Nov. 92 & 30.7 & 123.4 & 0.695 & 16. & 5.1 & 130. & 57 & 210. & 0.046 & 68.\
Dec. 92 & 26.7 & 140.7 & 0.871 & 18. & 9.1 & 176. & 107 & 250. & 0.075 & 36.\
Jan. 93 & 29.9 & 119.1 & 0.816 & 13. & 5.6 & 111. & 45 & 181. & 0.130 & 14.\
Feb. 93 & 26.2 & 169.6 & 0.839 & 3. & 0.0 & 0. & 0 & 48. & 0.116 & 41.\
Apr. 93 & 25.0 & 155.3 & 0.820 & 7. & 2.9 & 71. & 25 & 124. & 0.041 & 82.\
May 93 & 33.4 & 126.8 & 0.411 & 8. & 1.4 & 64. & 5 & 153. & 0.073 & 51.\
June 93 & 33.3 & 120.7 & 0.699 & 9. & 2.1 & 51. & 3 & 111. & 0.154 & 11.\
July 93 & 27.5 & 124.5 & 0.761 & 28. & 7.6 & 224. & 114 & 348. & 0.040 & 78.\
Aug. 93-1 & 26.8 & 129.0 & 0.877 & 4. & 2.5 & 66. & 20 & 116. & 0.048 & 79.\
Aug. 93-2 & 53.8 & 53.0 & 0.769 & 1. & 1.0 & 120. & 0 & 227. & 0.093 & 67.\
Oct. 93-1 & 26.7 & 54.5 & 0.733 & 0. & 0.0 & 0. & 0 & 158. & &\
Oct. 93-2 & 51.9 & 72.6 & 0.694 & 2. & 0.8 & 69. & 0 & 198. & 0.048 & 86.\
Oct. 93-3 & 31.6 & 103.7 & 0.782 & 4. & 0.3 & 27. & 0 & 192. & 0.024 & 99.\
July 94 & 45.6 & 136.7 & 0.783 & 12. & 1.1 & 30. & 0 & 88. & 0.056 & 68.\
Aug. 94 & 32.6 & 117.2 & 0.841 & 7. & 3.0 & 71. & 25 & 123. & 0.042 & 78.\
Sep. 94-1 & 40.5 & 120.8 & 0.751 & 10. & 2.6 & 87. & 22 & 165. & 0.043 & 76.\
Oct. 94 & 55.4 & 120.3 & 0.681 & 44. & 4.8 & 136. & 27 & 257. & 0.075 & 45.\
Nov. 94 & 30.4 & 112.3 & 0.660 & 13. & 5.6 & 164. & 79 & 259. & 0.035 & 84.\
Dec. 94 & 29.3 & 100.0 & 0.803 & 9. & 0.0 & 0. & 0 & 236. & 0.184 & 19.\
Mar. 95 & 29.3 & 151.4 & 0.772 & 23. & 3.7 & 147. & 47 & 266. & 0.042 & 77.\
July 95 & 35.5 & 110.6 & 0.776 & 17. & 4.3 & 128. & 39 & 229. & 0.114 & 19.\
Aug. 95 & 35.2 & 108.9 & 0.698 & 8. & 3.6 & 100. & 38 & 168. & 0.058 & 59.\
Sep. 95 &124.3 & 80.4 & 0.561 & 10. & 1.0 & 48. & 0 & 201. & 0.144 & 19.\
Oct. 95 & 39.3 & 120.7 & 0.793 & 9. & 3.3 & 160. & 51 & 286. & 0.060 & 54.\
Nov. 95 & 37.2 & 149.9 & 0.759 & 13. & 2.7 & 66. & 18 & 125. & 0.039 & 83.\
Dec. 95-2 & 78.3 & 119.9 & 0.530 & 18. & 0.0 & 0. & 0 & 127. & 0.044 & 85.\
Jan. 96 & 33.9 & 141.2 & 0.767 & 14. & 4.6 & 117. & 45 & 193. & 0.091 & 29.\
May 96 & 35.2 & 117.8 & 0.628 & 6. & 2.3 & 66. & 13 & 126. & 0.028 & 95.\
Aug. 96 & 32.9 & 148.9 & 0.800 & 6. & 0.0 & 0. & 0 & 51. & 0.102 & 45.\
Oct. 96 & 33.6 & 155.5 & 0.785 & 10. & 5.0 & 107. & 55 & 165. & 0.066 & 47.\
Nov. 96 & 35.0 & 162.5 & 0.795 & 15. & 1.9 & 40. & 0 & 88. & 0.110 & 29.\
Jan. 97 & 34.5 & 160.0 & 0.816 & 8. & 1.4 & 29. & 0 & 70. & 0.123 & 23.\
Mar. 97 & 36.3 & 160.9 & 0.814 & 14. & 3.6 & 64. & 20 & 116. & 0.058 & 52.\
Apr. 97 & 35.0 & 167.4 & 0.791 & 10. & 4.2 & 84. & 38 & 137. & 0.052 & 63.\
June 97 & 35.8 & 160.2 & 0.797 & 11. & 5.8 & 121. & 65 & 183. & 0.033 & 86.\
July 97 & 37.0 & 127.3 & 0.751 & 9. & 0.0 & 0. & 0 & 37. & 0.204 & 16.\
Sep. 97 & 33.6 & 124.5 & 0.794 & 5. & 2.6 & 61. & 22 & 107. & 0.109 & 29.\
Oct. 97 & 34.2 & 149.5 & 0.780 & 7. & 0.3 & 6. & 0 & 42. & 0.429 & 3.\
Dec. 97 & 34.4 & 108.4 & 0.519 & 9. & 3.0 & 90. & 31 & 159. & 0.044 & 77.\
& 604. & 143.7 & 73. & 64 & 82. & 0.110 & 25.\
[l d d d r @[–]{} d l d]{}
& Number of & Number & Best &\
Exposure & candidate & fit to & fit & & &\
date & events & $^{71}$Ge & (SNU)& & $Nw^2$ &\
Sep. 92 & 13. & 6.0 & 79. & 44 & 123. & 0.097 & 25.\
Oct. 92 & 21. & 3.3 & 32. & 4 & 67. & 0.105 & 26.\
Nov. 92 & 28. & 4.3 & 56. & 10 & 111. & 0.047 & 70.\
Dec. 92 & 28. & 16.8 & 168. & 115 & 229. & 0.057 & 53.\
Jan. 93 & 17. & 10.0 & 124. & 81 & 177. & 0.089 & 32.\
June 93 & 18. & 3.3 & 42. & 4 & 92. & 0.557 & $<1$.\
Oct. 93-2 & 4. & 3.0 & 141. & 60 & 245. & 0.049 & 83.\
Oct. 93-3 & 7. & 4.0 & 185. & 80 & 303. & 0.052 & 77.\
July 94 & 22. & 3.4 & 47. & 9 & 94. & 0.027 & 95.\
Aug. 94 & 27. & 3.9 & 46. & 15 & 85. & 0.075 & 52.\
Sep. 94-1 & 30. & 6.5 & 112. & 50 & 188. & 0.082 & 39.\
Nov. 94 & 23. & 8.0 & 116. & 66 & 176. & 0.015 &$>99$.\
July 95 & 33. & 5.0 & 74. & 19 & 138. & 0.063 & 55.\
Aug. 95 & 24. & 7.4 & 105. & 60 & 161. & 0.061 & 56.\
Sep. 95 & 33. & 1.2 & 28. & 0 & 142. & 0.058 & 73.\
Oct. 95 & 26. & 6.5 & 163. & 75 & 270. & 0.019 &$>99$.\
Nov. 95 & 32. & 10.2 & 127. & 78 & 185. & 0.032 & 88.\
Dec. 95-2 & 40. & 0.5 & 12. & 0 & 95. & 0.068 & 62.\
Jan. 96 & 35. & 3.5 & 45. & 0 & 101. & 0.047 & 76.\
May 96 & 31. & 5.3 & 78. & 31 & 136. & 0.039 & 90.\
Aug. 96 & 26. & 4.5 & 51. & 14 & 96. & 0.089 & 35.\
Oct. 96 & 21. & 5.4 & 58. & 28 & 95. & 0.046 & 74.\
Nov. 96 & 28. & 1.9 & 21. & 0 & 57. & 0.103 & 37.\
Jan. 97 & 24. & 2.6 & 26. & 0 & 60. & 0.190 & 13.\
Mar. 97 & 24. & 6.1 & 54. & 24 & 90. & 0.134 & 15.\
Apr. 97 & 22. & 2.7 & 27. & 3 & 57. & 0.037 & 86.\
June 97 & 27. & 10.4 & 109. & 71 & 155. & 0.078 & 35.\
July 97 & 22. & 0.0 & 0. & 0 & 24. & 0.333 & 7.\
Sep. 97 & 17. & 4.3 & 49. & 22 & 84. & 0.043 & 80.\
Oct. 97 & 26. & 3.4 & 36. & 9 & 72. & 0.083 & 49.\
Dec. 97 & 24. & 6.2 & 80. & 40 & 128. & 0.031 & 89.\
Combined & 753. & 152.1 & 64. & 56 & 72. & 0.033 & 93.\
The overall likelihood function for a single extraction is the product of the separate likelihood functions for the $L$- and $K$-peak regions. The best fit capture rate is found by maximizing this function, allowing the independent background rates in the $L$ and $K$ peaks to be free variables. The uncertainty in this result is determined by finding the values of the capture rate at which the logarithm of the likelihood function decreases by 0.5, again choosing the background rates at these two points to be those which maximize the likelihood function. The results for all extractions that could be analyzed in both peaks are given in Table \[K\_plus\_L\_peak\_table\].
The capture rate for each extraction of all runs of SAGE is plotted in Fig. \[All\_extraction\_results\]. These results are derived from the $K$ peak plus $L$ peak wherever possible, otherwise from the $K$ peak alone. For those readers who may be interested in looking for temporal phenomena, the beginning time $t_B$ and ending time $t_E$ for each run can be inferred from the mean exposure date $t_m$ and total exposure time $\Theta$ given in Table \[run\_parameter\_table\] by the relationships $$\begin{aligned}
t_B & = & t_m - \frac{1}{\lambda}\ln\bigg(\frac{1 +
e^{\lambda\Theta}}{2}\bigg), \\
t_E & = & \Theta - t_B. \nonumber\end{aligned}$$
{width="5.000in"}
Global fits {#global_fits}
-----------
The combined likelihood function for any set of extractions is the product of the overall likelihood functions for each extraction. The best fit capture rate for the set of extractions is determined by maximizing this function, requiring the production rate per unit mass of Ga to be the same for each extraction, and allowing the background rates in both the $L$ and $K$ peaks to be different for each extraction. The uncertainty is found in the same way as for the $L$- and $K$-peak combination for a single extraction. There are a number of other techniques for estimating the uncertainties in addition to the two described and used here. When many runs are combined, the likelihood as a function of capture rate approaches a Gaussian, and the difference between the results of these techniques becomes slight.
The results of global fits to our data are given in Sec. \[Results\].
Systematic Uncertainties {#systematics}
========================
There are four basic sources of systematic error in SAGE: uncertainty in the chemical extraction efficiency, uncertainty in the counting efficiency, uncertainty due to nonsolar neutrino production of (such as by cosmic rays), and uncertainty due to nonconstant events which mimic (such as may be made by ). Table \[uncertainty\_table\] summarizes the results of our consideration of all these effects and additional information regarding each of these items follows.
[l d @ d]{}
&\
Origin of uncertainty & in percent & in SNU\
\
Ge carrier mass & $\pm$2.1% & $\pm$ 1.4\
Mass of extracted Ge & $\pm$2.5% & $\pm$ 1.7\
Residual Ge carrier & $\pm$0.8% & $\pm$ 0.5\
Ga mass & $\pm$0.3% & $\pm$ 0.2\
Total (extraction) & $\pm$3.4% & $\pm$ 2.3\
\
Volume efficiency & $\pm$1.4% & $\pm$ 0.9\
End losses & $\pm$0.5% & $\pm$ 0.3\
Monte Carlo interpolation & $\pm$1.0% & $\pm$ 0.7\
Shifts of gain & $-$3.1% & $+$2.1\
Resolution & $+$0.5%,$-$0.7% & $-$0.3,+0.5\
Rise time limits & $\pm$1.0% & $\pm$ 0.7\
Lead and exposure times & $\pm$0.8% & $\pm$ 0.5\
Total (counting) & $+$2.3%,$-$3.9% & $-$1.5,+2.6\
\
Fast neutrons & & $<-$ 0.02\
& & $<-$ 0.04\
& & $<-$ 0.7\
Cosmic-ray muons & & $<-$ 0.7\
Total (nonsolar) & & $<-$ 1.0\
\
Internal & & $<-$ 0.2\
External & & $ $ 0.0\
Internal & & $<-$ 0.6\
Total (background events)& & $<-$ 0.6\
Total & & $-$3.0,+3.5
Chemical extraction efficiency {#chem_extrac_eff}
------------------------------
The way in which the chemical extraction efficiency is determined was described in Sec. \[Chemical\_extraction\_efficiency\]. There are four sources of uncertainty.
### Mass of Ge carrier
The extraction efficiency is measured by adding to the Ga metal several slugs of Ga-Ge alloy which contain a known mass of Ge. This alloy is produced in large batches by reduction of Ge by Ga metal from chloride solution, and then divided into several hundred small slugs, each of which weighs 18–20 g and contains about 40 $\mu$g of Ge. The equality of Ge content was measured by extracting the Ge from a few dozen slugs. The standard deviation of these measurements was 2.1%, which we take as the uncertainty in the mass of added Ge carrier.
### Mass of extracted Ge
There is also an uncertainty in how much carrier has been synthesized into GeH$_4$. This is determined by the accuracy to which the GeH$_4$ volume can be determined and is estimated to be 2.5%.
### Residual Ge carrier
Since the extraction of carrier Ge is not complete, residual Ge carrier from preceding extractions will contribute to the extraction efficiency measurement. Each extraction for solar neutrino data is followed by a second extraction to remove this surplus carrier. The amount removed during the first two extractions is at least 95%, but is uncertain as described above, which leads to an uncertainty in the amount of remaining carrier. The extraction efficiency uncertainty due to the uncertainty in the residual carrier is $\pm 0.8$%.
### Mass of Ga
The total mass of Ga has been weighed periodically with a precision of 0.3%. The amount removed during each extraction is small (typically 0.1%) and is known well (2%). We take the uncertainty in the Ga mass for all runs to be $\pm 0.3$%.
Counting efficiency {#cnting_effs}
-------------------
The counter efficiency is calculated using Eq. \[efficiency\_formula\]. There are thus three sources of uncertainty: the volume efficiency $\epsilon_v$, the end effects (or equivalently the fraction of degraded events), and the gas efficiency.
### Volume efficiency {#vol_eff}
As described in Sec. \[counter\_efficiency\], the volume efficiency of seven counters of the “LA” type has been directly measured with an uncertainty of 0.6%. These seven counters were used for 48 of our 88 data sets. The uncertainty in $\epsilon_v$ for counters of this same type used in 36 other data sets is estimated from the spread in the measured $\epsilon_v$ for the measured counters, which is $\pm 2.3$% relative uncertainty. The uncertainty in $\epsilon_v$ for counters of the “Ni” and “RD” types, which were used for four data sets, is taken as $\pm 3$%. Averaging over the different types of counters used for all extractions, the uncertainty assigned to volume efficiency is taken as $\pm 1.4$%.
### End effects
The reduced electric field near the ends of the counter cathode results in a fraction of events that lie outside the $\pm1$ FWHM energy windows. Uncertainties in these end effects are due to variations in the physical dimensions of the counters. Based on measurements of various “LA” counters, these dimensional differences lead to an uncertainty of $\pm 4.1$% in the end effect. This gives $\pm 0.5$% relative uncertainty in the factor $(1 - f_D)$. This should be valid for measurements made with the “LA” counters, which were used for most of our data, and this value is taken for the entire data set.
### Monte Carlo interpolation of measured gas parameters {#mc_gas}
The uncertainties in the gas efficiency consist of three components: uncertainty in the Monte Carlo calculations, uncertainty in the measured gas pressure, and uncertainty in the measured percentage of GeH$_4$. The limited statistics used in the Monte Carlo calculations to determine the constants in the gas efficiency formula leads to an uncertainty of 1.0% in the determination of the gas efficiency. The uncertainty in the gas pressure measurements is $\pm 5$ Torr, which corresponds to an uncertainty in the gas efficiency for a typical counter filling (710 Torr at 24% GeH$_4$) of $\pm
0.2$% relative change. The uncertainty in the measured percentage of GeH$_4$ is taken to be $\pm 1$%, which corresponds to an uncertainty in the gas efficiency for an average counter filling of $\pm0.2$% relative change. Adding these three contributions in quadrature yields a relative total uncertainty in the gas efficiency of $\pm 1.0$%.
### Gain shifts {#gain_shifts}
If the calibration mean shifts between two calibrations, there is an error made in the efficiency estimate. This error has been minimized by two features of our standard analysis. First, we use a two FWHM wide energy window. Since the peak is relatively Gaussian and the window limits are far out on the tail, uncertainties in the location of the centroid of the peak do not greatly affect the efficiency. Second, by sliding the energy window between calibrations we hope to minimize any error in estimating the centroid due to the observed gain shifts. Although the correction for nonlinearity of the counter response \[Eq. (\[Gain\_Scaling\_Formula\])\] results in an additional uncertainty in the gain of 0.7%, the total uncertainty in the gain is dominated by the shifts.
To estimate the error generated by using an incorrect centroid, we computed the area under a Gaussian between two integration limits which are shifted by an amount $\delta$. We then compared this number to the 0.9815 number expected from integration limits of $\pm 2$ FWHM. Using a typical $K$-peak resolution of 20%–23% we calculated the true efficiency for various values of $\delta$ expressed as a fraction of the true mean.
Typical gain shifts are of the order of a few percent. This results in an uncertainty of approximately $-3.1$% in the efficiency. Note that this effect can only decrease our efficiency so it is a one sided systematic uncertainty.
### Energy resolution
As a result of the statistics of our calibration spectra, the resolution is known to about 2.1%. For the $K$ peak, there is an additional uncertainty due to the counter nonlinearity \[Eq. (\[Resolution\_Scaling\_Formula\])\] of $\pm 4.5$%. Adding these in quadrature, the uncertainty in the resolution results in an uncertainty in the efficiency of about +0.5%, $-0.7$%. Again, because the energy window is so wide, the uncertainty in the efficiency due to the resolution uncertainty is not large.
### Rise time limits {#adp_lims}
As described in Section \[rise\_tim\_cal\], when the wave form method of rise time determination is used, there is an uncertainty in the efficiency of $\pm1$% that arises from changes in the rise time limits due to counting channel and filling variations. For those runs that used the ADP method of rise time determination, we can find the uncertainty of the ADP cut as follows: Usually a calibration has between 1000 and 5000 events in the peak. We base our lower ADP threshold on 4% of those or 40–200 events. This small number of events is subject to statistical fluctuations. For the most extreme case, we take the square root of 40 (6.4) and notice that the efficiency due to the ADP cut could actually be between 94.4% and 95.6% instead of the 95% we believe it to be. Thus this is a $\pm 0.6$% uncertainty. Since the vast majority of our data are based on wave form analysis, we use $\pm1$% for all runs.
### Lead and exposure times {#lead_exp_tims}
Because extraction usually occurs from several reactors over the course of 6–10 h, there is an uncertainty in the exposure time and in the time from extraction to the start of counting (which we call the “lead time”) of roughly 3 h. The lead time is typically 36 h and the exposure time is typically 34 days. These small uncertainties make a small contribution to the uncertainty associated with the solar neutrino flux. By Eq. (\[prod\_rate\_def\]), the solar neutrino production rate $p$ is proportional to the quantity $[e^{-\lambda t_{\text{lead}}}(1 - e^{-\lambda
\Theta})]^{-1}$, where $\lambda$ is the decay constant, $t_{\text{lead}}$ is the lead time, and $\Theta$ is the exposure time. By differentiation one finds that $\delta p/p$ due to $t_{\text{lead}}$ is about $\pm 0.8$% and due to $\Theta$ is about $\pm 0.11$%.
Nonsolar neutrino contributions to the $^{\bf71}$G signal {#bkgds}
---------------------------------------------------------
In addition to solar neutrinos, can also be produced from Ga by the reaction $(p,n)$. The protons that initiate this reaction can be secondaries made by the $(n,p)$ reaction of fast neutrons or by the $(\alpha,p)$ reaction where the $\alpha$’s are from radioactive decay or may arise from photonuclear reactions initiated by cosmic-ray muons. The yields of these reactions have been measured with neutrons from radioactive sources, $\alpha$’s from a Van de Graaff generator, and high-energy muons from accelerators (see, e.g., [@BAH78]). Based on these results, great care was taken in the design and construction of SAGE to minimize these potential background sources. A major advantage of using Ga metal as the solar neutrino target (as opposed to an aqueous solution, such as the GaCl$_3$ target of GALLEX) is that the target contains no free protons, and thus the production rates of all these reactions are low.
Any one of these processes could produce a background effect that must be subtracted from our measured solar neutrino signal, but as will be seen below, our best estimates for all of these effects are very small and have large errors. Thus, rather than making a background subtraction, we include these effects here as systematic uncertainties.
Other Ge isotopes that may be misidentified as can be produced in similar reactions: can be made by $(p,n)$ and the spallation reaction on Ga by throughgoing cosmic-ray muons can make and . Since the production rate of by the spallation reaction is comparable to that of and its half-life is long (271 days), the decay rate is much less than that of and can be neglected. The short-lived isotope has a greater potential to give events that mimic and is considered below in Sec. \[69Ge\_background\].
### External neutrons {#ext_neutrons}
Fast neutrons mainly arise from the walls of the Ga chamber by $(\alpha,n)$ reactions where the $\alpha$ particles are from decay. This background is expected to be small due to the low cross sections for $(n,p)$ reactions on Ga isotopes and to the low fast neutron background, which is because the Ga chamber is lined with low-radioactivity concrete and steel. The fast neutron flux in the gallium chamber was measured by extracting from a tank with 187 kg of dry CaC$_2$O$_4$ and counting in a proportional counter. Fast neutrons above 3 MeV produce through the $(n,\alpha)$ reaction. The flux over 3 MeV is $(4.6 \pm 1.6) \times 10^{-3}$ neutrons/(cm$^2$ day) [@Gavrin91b]. The number of atoms produced by this flux in 60 tons of Ga metal is $<2.9 \times 10^{-4}$/day [@BAR79; @BAR87; @Korn98], which, using the conversion factor 56 SNU/( atom produced per day in 60 tons of Ga), corresponds to $< 0.016$ SNU.
[l d d d d d]{} &\
\
& & & Pb & &\
Prob. false $L$ event & 0. & 0.0042 & 0.0110 & 0.0072 & 0.0016\
Prob. false $K$ event & 0.00004 & 0.0490 & 0.0061 & 0.0018 & 0.0186\
Survival probability & 0. & 0. & 0.011 & 0.035 & 0.035\
### Internal radioactivity {#int_radio}
The second possible background source is due to $\alpha$ radioactivity in the gallium. The only appreciable sources of high-energy $\alpha$’s are from decays in the U and Th chains, mainly the 8.8-MeV $\alpha$ from at the end of the Th chain and the 7.8-MeV $\alpha$ from near the end of the U chain. The concentrations of radioactive impurities in the Ga have been measured in two ways: by direct counting in a low-background Ge detector by the Institute for Nuclear Research (INR) [@GAV86] and by glow discharge mass spectrometry by both Charles Evans Associates [@Evans] and Shiva Technologies [@Shiva]. No U, Th, or Ra was detected in any of these measurements. Expressed in grams of impurity per gram of Ga, the limits are U $<2.0\times 10^{-10}$ (Evans) and $<1.2\times 10^{-10}$ (Shiva); Th $<8.0\times 10^{-10}$ (INR), $<1.7\times
10^{-10}$ (Evans), and $<1.2\times 10^{-10}$ (Shiva); $<1.1
\times 10^{-16}$ (INR). We take the INR limit for and the Shiva limit for Th. Using the measured yields [@BAH78] in metallic Ga, the number of atoms produced per day in 60 tons of Ga is $<0.001$/day from and $< 0.013$/day from , which correspond to $<0.04$ SNU and $<0.7$ SNU, respectively.
### Cosmic-ray muons {#cr_muons}
The third possible background source is production of Ge isotopes by cosmic-ray muons. The global cosmic-ray muon flux in the gallium laboratory at BNO has been measured [@Gavrin91] to be $(3.03 \pm 0.10) \times 10^{-
9}$ muons/(cm$^2$ s). This flux can be converted to a production rate in 60 tons of Ga metal in two ways: (1) using cross sections measured at accelerators [@BAH78; @Cribier97] and scaling by the average muon energy (which is $\sim 381$ GeV) to the 0.73 power [@Rya65], one predicts 0.012/day [@Korn98], or (2) using cross sections for production of calculated in [@GAV87], one predicts 0.013/day. Both of these convert to rates of 0.7 SNU. Since the error of these estimations is about 100% [@Korn98], we consider this background as a systematic uncertainty for the muon background rate.
Background events that mimic $^{\bf71}$G
----------------------------------------
As is evident from Tables \[L\_peak\_table\] and \[K\_peak\_table\], a large fraction of the events that we select as candidates are not . In the $L$ ($K$) peak we select 433 (604) events as candidates, but most of these events occur late in the counting, and thus the best fit to plus constant background assigns only 64.3 (143.7) of them to be . These late events that we cannot discriminate from are produced by background processes, such as $\beta$ rays whose path through the counter either is very short or is parallel to the anode wire. As long as these background events occur at a constant rate, they only deteriorate our signal-to-background ratio, but do not change our extracted signal rate. If these background events mainly occur early or late in the counting, however, the extracted signal rate will be incorrect, too high or too low, respectively. Particularly insidious in this regard is the ubiquitous naturally occurring isotope .
### Internal radon {#int_rn}
Because it has a short half-life of only 3.8 days, and can produce events that mimic , any that enters the counter at the time it is filled will produce events early in the counting period that may be falsely interpreted as , and thus give an incorrectly high signal rate. To understand this process, let us first consider the principal decay sequence, which is
\#1 \#1\#2[\#1em1.1ex-.60em]{} [ $^{222}\text{Rn} \mapright{3.82\text{ d}} ^{218}\text{Po} + \alpha_1$\
$ \decayright{5.9}{3.05\text{ m}} ^{214}\text{Pb} + \alpha_2 $\
$ \decayright{9.4}{26.8\text{ m}} ^{214}\text{Bi} + \beta_1 $\
$ \decayright{12.9}{19.7\text{ m}} ^{214}\text{Po} + \beta_2 $\
$ \decayright{16.4}{10^{-4}\text{ s}} ^{210}\text{Pb} + \alpha_3. $ ]{} Events that are falsely identified as can be produced by one of the $\alpha$’s (this happens rarely as they are heavily ionizing), by one of the $\beta$’s (this occurs more frequently), by the recoil nucleus from the $\alpha$ decay if the initial nucleus is on the counter wall, or by a low-energy x ray emitted by one of the heavy elements in the chain. Fortunately, the start of this chain is easily recognized as at least one of the first two heavily ionizing $\alpha$ particles usually produces a pulse that saturates the energy scale. Thus, since this chain takes on average about 1 hour from the initiating decay of to reach 22-yr , if one makes a time cut of a few hours after each saturated event, then most of these false events will be removed.
We choose to eliminate all events that occur from 15 min before to 180 min after each detected saturation event (energy greater than 16 keV). To determine the effect of this time cut, we filled a counter with a typical mixture of Xe and GeH$_4$ to which had been added and measured it in system 3 under conditions identical to those of solar runs. Based on these measurements and Monte Carlo modeling, the spectrum of pulses in the counter was determined by each element in the Rn chain. The probability of a false event can then be directly calculated and the results are given in Table \[intrntable\].
Folding the probability of a false event with the probability of survival after the time cut (Table \[intrntable\]), we obtain the probability of observing a false event after the time cut to be 0.00043 in the $L$ peak and 0.00078 in the $K$ peak. The resolution of this counter was better than for the average solar neutrino extraction. If we use the average resolution, the number of false events in a typical solar neutrino run that satisfy all our event selection criteria divided by the number of detected saturated events due to is calculated to be 0.0006 and 0.0012 for the $L$ and $K$ peaks, respectively.
To estimate the number of false events that survive the time cut, we next calculate how many saturated events are present in our data that can be attributed to Rn. This is done by taking the data for each run, making the usual time cut after shield openings, and selecting all saturated events. The time sequence of these events is then fit to a decaying component with the 3.82-day half-life of plus a constant background. For the periods of SAGE II and III, this yields 294 saturated events initiated by , with 192.1 (294.0) events in the 31 (57) data sets that give the $L$- ($K$-) peak results. Since SAGE I did not have the capability to detect saturated events, we scale the number for the $K$ peak by the number of additional extractions (16) to make the $K$-peak total 376.5.
Combining these results, the number of false events that remain after the time cut is then given by $0.0006 \times 192.1 = 0.11$ in the $L$ peak and $0.0012 \times 376.5 = 0.44$ in the $K$ peak. Since we have observed a total of 64.3 (143.7) events in the $L$ ($K$) peaks, the fraction of false events is 0.2% (0.3%), which translates to a false signal rate of 0.1 (0.2) SNU. Combining the $L$- and $K$-peak results gives a total false signal rate of 0.2 SNU. Because the uncertainty in this correction is comparable with the magnitude of the correction itself, we choose to treat this effect as a systematic uncertainty, rather than as a background to be subtracted from the signal.
### External radon {#ext_rn}
that is external to a proportional counter can also produce false events. The radon levels in the counting room vary with external conditions, and usually fall within the range of $(2.0 \pm
1.0)$ pCi/l. To reduce the level of Rn in the vicinity of the counters, all passive shields are equipped with purge lines from evaporating liquid nitrogen and the shields have been made fairly hermetic. Each time a counter is calibrated, however, the shield must be opened, and some mine air will enter the volume around the counters. Under normal circumstances calibrations occur regularly every 2 weeks. Any false events that are produced by external Rn will thus occur more or less constantly in time, and will be treated by the maximum likelihood analysis as a constant background. Nevertheless, we minimize the effect of external Rn by making a time cut on the data for 2.6 h after any shield opening.
A special counter was constructed to give information on the false events that are produced by external . This consisted of one of our usual counters enclosed within a cylindrical quartz capsule (20 mm diameter). The sealed volume of the capsule was filled with air to which was added to make the total activity 3.5 nCi. This counter was measured in counting system 6 which uses the ADP method of rise time determination. In the $K$ peak of , after cuts for energy, ADP, and NaI, the measured count rate was $0.81 \pm 0.11$ events/min.
Because the internal volume of the proportional counter is shielded by the Fe cathode of 1/3 mm thickness, the false events are mainly produced by the $\beta$ particles from the decay of which have sufficient energy to penetrate to the active volume of the counter. We use the measurements from the internal Rn section, our Monte Carlo model for the counter response, make some reasonable assumptions regarding the location of Rn-daughter products, take into account the reduction of Rn in the vicinity of the counters due to the N$_2$ purge, and calculate the number of false events to be 0.005 in the sum of the $L$ and $K$ peaks per run. These calculations were made without taking into account the effect of the 2.6-h time cut after each shield opening, which reduces the number of events even further. Thus the number of false events produced by external is negligible.
This conclusion is verified by analyzing our full $L$ + $K$ data set without making the shield opening time cut. The result is 67.9 SNU, nearly equal to the result of 67.2 SNU when the time cut is used.
### Internal {#69Ge_background}
Because it can be produced by the same background reactions that make , has a short half-life of 39 h, and 64% of its decays are by electron capture, can produce events that will be misidentified as . We can estimate the production rate of from known data. The cosmic-ray production rate of can be determined in the same way as was done in Sec. \[cr\_muons\] for . Using the measured muon flux in the laboratory, the cross section for production of of 100 $\mu$b measured in GaCl$_3$ at CERN with 280 GeV muons [@Cribier97], the factor of 2 greater production rate of in Ga metal compared to GaCl$_3$ measured at FNAL with 225 GeV muons [@BAH78], and scaling as the muon energy to the 0.73 power, we estimate a production rate of 0.036 atoms of per day in 60 tons of Ga. The production rate of by $\alpha$ particles and neutrons is comparable to that of , viz., 0.015 atoms/day. We must add to this the production rate of by neutrinos, which we estimate as being comparable to that of , i.e., 5.8 SNU, thus making a total estimated production rate of 0.21 /day in 60 tons of Ga. Since the usual exposure interval is at least 30 days, will be fully saturated, and the total number of atoms at the end of exposure will be approximately 0.5. When counting starts, on the average 36 h after extraction, nearly half of these atoms will have decayed. Fortunately 86% of the decays of have a coincident $\beta^+$ or $\gamma$ and will be vetoed by the surrounding NaI detector with approximately 90% efficiency. Including the 14% of decays that occur by electron capture to the ground state, the total efficiency for detection will be no more than 25%. Approximately 70% of these decays will appear in the $L$ and $K$ peaks, leaving a total of 0.045 observed decays per run, or 1 event in every 44 data sets. Since the 211 events that we have assigned to in our 88 data sets correspond to 67 SNU, this implies that the false background is approximately 0.6 SNU. This very small value illustrates the desirability of siting the SAGE detector at great depth and the advantage of a NaI veto on all channels during counting.
Results {#Results}
=======
If we combine SAGE I with SAGE II (minus part 2) and SAGE III, the global best fit capture rate for the 88 separate counting sets is $67.2
^{+7.2}_{-7.0}$ SNU, where the uncertainty is statistical only. In the windows that define the $L$ and $K$ peaks there are 1037 counts with 211.15 assigned to (the total counting live time is 28.7 yr). If we were to include the data from SAGE II part 2, the overall capture rate would decrease by 7.1 SNU. The systematic control of the experiment was suspect during the period of the gallium theft (see Sec. \[extrac\_hist\]), and thus we exclude that data interval from our result.
The total systematic uncertainty is determined by adding in quadrature all the contributions given in Table \[uncertainty\_table\] and is $-$3.0, +3.5 SNU. Our overall result is thus $67.2 ^{+7.2 +3.5}_{-7.0 -3.0}$ SNU. If we combine the statistical and systematic uncertainties in quadrature, the result is $67.2 ^{+8.0}_{-7.6}$ SNU.
This section continues with the evidence that we are truly counting , considers how well the observed data fit the models that are assumed in analysis, and concludes with consideration of the internal consistency of the SAGE results.
Evidence for $^{71}$G
---------------------
The most direct visual evidence that we are really observing is in Fig. \[2d\_hist\]. The expected location of the $L$ and $K$ peaks is shown darkened in this figure. These peaks are apparent in the upper panel, but missing in the lower panel because the has decayed away. Events outside the two peak regions occur at about the same rate in both panels because they are mainly produced by background processes.
A quantitative indication that is being counted can be obtained by allowing the decay constant during counting to be a free variable in the maximum likelihood fit, along with the combined production rate and all the background rates. The best fit half-life to all selected events in both $L$ and $K$ peaks is then $10.5_{-1.9}^{+2.3}$ days, in good agreement with the measured value [@HAM85] of 11.43 days.
Consistency of the data with analysis hypotheses
------------------------------------------------
### Energy and rise time window positions
To test whether or not the energy and rise time windows are properly set, the windows can be made wider and the data reanalyzed. If the rise time window for accepted events is increased by 30%, i.e., from 0–10 ns to 0–13 ns in the $L$ peak and from 0–18.4 ns to 0–24.0 ns in the $K$ peak, then the overall result of all runs of SAGE II and III that were counted in system 3 is 68.3 SNU. This change is entirely consistent with the $\sim 3$% increase in counting efficiency due to the increased size of the rise time acceptance window. Similarly, if the energy window in both $L$ and $K$ peaks is opened from the usual 2 FWHM to 3 FWHM, then the overall result of all runs of SAGE II and III becomes 69.1 SNU. This increase from the value of 67.2 SNU in the 2 FWHM energy window is because some of the decays occur at the ends of the counter and their detected energy is reduced from the full peak value. This results in an increase in the counting efficiency in the wider energy window of 2%–3%. If this efficiency increase is included in the analysis, then the results in the two energy windows agree to better than 1%.
### Time sequence
A major analysis hypothesis is that the time sequence of observed events for each run consists of the superposition of events from the decay of a fixed number of atoms plus background events which occur at a constant rate. The quantity $Nw^2$ and the goodness of fit probability inferred from it provide a quantitative measure of how well the data fit this hypothesis (see [@CLE98] for the definition and interpretation of $Nw^2$). These numbers are evaluated for each data set and are given in Tables \[L\_peak\_table\], \[K\_peak\_table\], and \[K\_plus\_L\_peak\_table\]. There are occasional runs with rather low probability of occurrence, but no more of these are observed than are expected due to normal statistical variation.
This method can also be used to determine the goodness of fit of the time sequence for any combination of runs. These numbers are given in the various tables; for the combined time sequence of all $L$ plus $K$ events from all runs, this test yields $Nw^2 = 0.074$, with a goodness-of-fit probability of $(58 \pm 5)$%. A visual indication of the quality of this fit is provided in Fig. \[cntrate\] which shows the count rate for all events in the $L$ and $K$ peaks vs time after extraction. As is apparent, the observed rate fits the hypothesis quite well.
![Count rate for all runs in $L$ and $K$ peaks. The solid line is a fit to the data points with the 11.4-day half-life of plus a constant background. The vertical error bar on each point is proportional to the square root of the number of counts and is shown only to give the scale of the error. The horizontal error bar is $\pm 5$ days, equal to the 10-day bin size.[]{data-label="cntrate"}](cntrate.eps){width="3.375in"}
[l c d d d d r @[–]{} d d d]{}
& & & Number of & Number & & & &\
Data & & Number of & candidate & fit to & Best fit & & & Probability\
segment & Peak & data sets & events & $^{71}$Ge & (SNU) & & $Nw^2$ & (%)\
SAGE I & $K$ & 16. & 157. & 41.2 & 81. & 63 & 101. & 0.097 & 24.\
SAGE II & $L+K$ & 33. & 342. & 85.5 & 79. & 66 & 92. & 0.105 & 32.\
SAGE III & $L+K$ & 39. & 538. & 87.0 & 56. & 47 & 66. & 0.040 & 90.\
\
All & $L$ & 31. & 433. & 64.3 & 55. & 43 & 68. & 0.020 & $>99$.\
All & $K$ & 57. & 604. & 143.7 & 73. & 64 & 82. & 0.110 & 25.\
All & $L+K$ & 88. & 1037. & 211.1 & 67. & 60 & 74. & 0.074 & 58.\
### Production rate sequence
Another analysis hypothesis is that the rate of production is constant in time. By examination of Fig. \[All\_extraction\_results\], it is apparent that, within the large statistical uncertainty for each run, there are no substantial long-term deviations from constancy.
![Distribution of capture rate from 10000 simulations of SAGE III assuming true production rate of 67.2 SNU. The probability of a rate less than or equal to the observed rate of 56 SNU is 11% and is shown shaded.[]{data-label="sage3kl"}](sage3kl.eps){width="3.375in"}
To quantitatively test whether or not it is reasonable to assume that the production rate is constant, we can consider the three segments of SAGE data, whose results are given in Table \[Segment\_table\]. A test of the consistency of any data segment with the overall result of 67 SNU can be made by Monte Carlo simulation. For the purposes of illustration, we choose the most deviant segment, SAGE III, whose overall result is 56 SNU. We then simulate all 39 data sets of SAGE III assuming that the true production rate is 67 SNU. To ensure that these simulations parallel the real data as closely as possible, all parameters of the simulation, such as background rates, efficiencies, exposure times, and counting times, are chosen to be the same as for the real data. From the sequence of simulated event times, the combined production rate is calculated in exactly the same manner as for the real data. This process is repeated 10000 times and a histogram of the combined rate is produced. From the position of the observed rate for the real data in this histogram, we can calculate the probability that the real data are produced by the assumed initial production rate. As shown in Fig. \[sage3kl\], we find that $(11 \pm 0.3)$% of the 10000 simulations of SAGE III have a value that is lower than the observed value of 56 SNU. Since this probability is one tailed (maximum of 50%), this is the most aberrant of the three sections of SAGE data, and no systematic uncertainties were included in the simulations, a value of 11% is not extremely unusual, and there is thus no statistically significant evidence for production rate variation. The same analysis applied to SAGE I and SAGE II yields probabilities of 35% and 38%, respectively, highly consistent with the assumption of constant production rate.
Another way to consider this question is to use the cumulative distribution function of the production rate $C(p)$, defined as the fraction of data sets whose production rate is less than $p$. Figure \[prod\] shows this distribution for all data sets and the expected distribution from simulation, assuming a constant production rate of 67 SNU. The two spectra parallel each other closely and can be compared by calculating the $Nw^2$ test statistic [@CLE98]. This gives $Nw^2$ = 0.343 whose probability is 10%.
![Measured capture rate for all SAGE data sets (jagged curve) and the expected distribution derived by 1000 Monte Carlo simulations of each set (smooth curve). The capture rate in the simulations was assumed to be 67.2 SNU.[]{data-label="prod"}](prod.eps){width="3.375in"}
[c d d r @[–]{} d]{} & & Best &\
Exposure & Number of & fit &\
interval & data sets & (SNU) &\
1990 & 5. & 43. & 2 & 78.\
1991 & 6. & 112. & 82 & 145.\
1992 & 13. & 76. & 59 & 95.\
1993 & 15. & 84. & 65 & 105.\
1994 & 10. & 73. & 51 & 98.\
1995 & 13. & 101. & 77 & 128.\
1996 & 10. & 49. & 32 & 68.\
1997 & 16. & 46. & 35 & 58.\
\
Jan & 7. & 47. & 24 & 74.\
Feb & 6. & 41. & 20 & 63.\
Mar & 3. & 198. & 137 & 266.\
Apr & 5. & 41. & 22 & 63.\
May & 6. & 83. & 58 & 111.\
Jun & 3. & 37. & 3 & 80.\
Jul & 9. & 40. & 22 & 62.\
Aug & 9. & 79. & 57 & 102.\
Sep & 12. & 63. & 47 & 82.\
Oct & 11. & 64. & 42 & 90.\
Nov & 9. & 73. & 52 & 96.\
Dec & 8. & 123. & 95 & 153.\
\
Jan+Feb & 13. & 44. & 28 & 60.\
Mar+Apr & 8. & 70. & 48 & 94.\
May+Jun & 9. & 71. & 50 & 95.\
Jul+Aug & 18. & 60. & 45 & 77.\
Sep+Oct & 23. & 64. & 50 & 79.\
Nov+Dec & 17. & 95. & 77 & 113.\
\
Feb+Mar & 9. & 69. & 48 & 92.\
Apr+May & 11. & 60. & 44 & 78.\
Jun+Jul & 12. & 39. & 23 & 59.\
Aug+Sep & 21. & 70. & 57 & 84.\
Oct+Nov & 20. & 69. & 54 & 86.\
Dec+Jan & 15. & 88. & 70 & 106.
Although these statistical tests are consistent with a constant production rate, they can never exclude the possibility of a cyclic time variation whose magnitude is comparable with the statistical uncertainty. We thus give in Table \[combinations\] the capture rate result for several of the possible temporal combinations of SAGE data. Each of these data divisions fits well to the constant rate of 67 SNU, as is verified by $\chi^2$/degree of freedom = 8.2/7 (yearly), 14.6/11 (monthly), 4.9/5 (January + February bimonthly), and 3.9/5 (February + March bimonthly), which have probabilities of 32%, 20%, 43%, and 56%, respectively. We remind those readers who are interested in short-term periodicity that the known variation due to the change in Earth-Sun distance has been removed from our reported capture rate \[see Eq. (\[earth\_sun\_corr\_factor\])\].
Internal consistency of SAGE results
------------------------------------
The combined results for all runs in the $L$ and $K$ peaks are given in Table \[Segment\_table\]. The $L$-peak result is 12 SNU below the overall value of 67 SNU and the $K$-peak result is 6 SNU above. The statistical $1\sigma$ error of these results, however, extends upward to 68 SNU in the $L$ peak and downward to 64 SNU in the $K$ peak. Both $L$- and $K$-peak results thus overlap the overall value, and there is no evidence for inconsistency between the results in the $L$ and $K$ peaks.
As noted in Sec. \[Chemical\_extraction\_efficiency\], so as to remove most of the residual Ge carrier from the Ga metal, it is customary to make a second extraction 2 or 3 days after each solar neutrino extraction. Although these second extractions are usually counted, until recently they were often measured in counters which did not have the lowest background rates, and were rarely counted in electronic system 3 with the wave form recorder. Further, these runs were seldom counted for a long time. As a consequence, it was not possible for us to give a result for the production rate from these second extractions. This situation changed at the beginning of 1996, however, because SAGE then switched to a 6-week extraction schedule, which freed some better low background counters and made it possible to measure these samples from second extractions in system 3. Ten such extractions have been measured since 1996. Taking into account the delay between the first and second extractions and the extraction and counting efficiencies, in these ten extractions we expect to detect three atoms that are leftover from the first extraction, and seven atoms that are produced by solar neutrinos during the interval between extractions. The total number of atoms detected in these ten extractions was 1.1 with a 68% confidence range from 0.0 to 8.7. The number observed is statistically consistent with the number expected, thus confirming our extraction efficiency. Further, it establishes that the we detect is not an artifact of the extraction process and that our counting and data analysis do not find a significant quantity of if it is not present.
Summary and Conclusions {#conclusions}
=======================
We have presented the methods and procedures of the SAGE experiment: the extraction of Ge from Ga, the subsequent Ge purification, the counting of , the identification of candidate events, and the analysis of the counting data to obtain the solar neutrino production rate. Eight years of measurement of the solar neutrino flux give the capture rate result $67.2 ^{+7.2}_{-7.0}$ SNU, where the uncertainty is statistical only. Analysis of all known systematic effects indicates that the total systematic uncertainty is $^{+3.5}_{-3.0}$ SNU, considerably smaller than the statistical uncertainty. Finally, we have examined the counting data and shown that there is good evidence that is being counted, that the counting data fit the analysis hypotheses, and that the counting data are self-consistent.
The SAGE result of 67.2 SNU represents from 52% [@BAH98] to 53% [@TUR98] of SSM predictions. Given the extensive systematic checks and auxiliary measurements that have been performed, especially the neutrino source experiment [@ABD96; @ABD98], this $7\sigma$ reduction in the solar neutrino flux compared to SSM predictions is very strong evidence that the solar neutrino spectrum below 2 MeV is significantly depleted, as was previously shown for the flux by the Cl and Kamiokande experiments. If we take into account the results of all experiments, astrophysical solutions to the solar neutrino deficit can now nearly be excluded [@Berezinsky96; @Berezinsky97; @Dar98]. This conclusion is indeed implied by the SAGE result itself, as it lies $2.5 \sigma$ below the capture rate prediction of $88.1 _{-2.4} ^{+3.2}$ SNU obtained by artificially setting the rate of the $(\alpha,\gamma)$ reaction to zero and $1.5 \sigma$ below the astrophysical minimum capture rate of $79.5
_{-2.0} ^{+2.3}$ SNU [@BAH97]. The solar neutrino problem is now a model-independent discrepancy [@BAH982; @HEE97] that does not depend on the details of solar models or their inputs.
![Allowed regions of neutrino parameter space for two-flavor oscillations into active neutrino species. The analysis uses the results of all solar neutrino experiments, including the constraints from the energy spectrum and zenith-angle dependence measured by Super-Kamiokande. The black circles are the best fit points and the shading shows the allowed regions at 99% confidence. Thee figure is based on calculations in Ref. [@BAH982].[]{data-label="neutosc"}](oscill.eps){width="3.375in"}
More credible explanations for the solar neutrino deficit involve either matter-enhanced Mikheyev-Smirnov-Wolfenstein (MSW) neutrino oscillations, in which the solar $\nu_e$ oscillates into other flavor neutrinos or a sterile neutrino [@BAH982; @CAL98; @Bilenky99; @HAT97], or vacuum oscillations [@Krastev-Petcov; @BAH982; @GelbRosen98]. For both of these possibilities, the allowed regions of $\Delta m^2 - \sin^2 2 \theta$ parameter space determined from solar neutrino experiments for two-flavor oscillations into active neutrino species are shown in Fig. \[neutosc\]. The fit quality is about the same in both regions. There is also a fit with similar quality for MSW oscillations into sterile neutrinos, whose allowed region approximately coincides with the region shown for MSW oscillations with active neutrinos.
There are now very strong indications that the solar neutrino deficit has a particle physics explanation and is a consequence of neutrino mass. To fully unravel the solar neutrino story, however, will require more experiments, especially those with sensitivity to low-energy neutrinos or to neutrino flavor. SAGE continues to perform regular solar neutrino extractions every 6 weeks with $\sim 50$ tons of Ga and will continue to reduce its statistical and systematic uncertainties, thus further limiting possible solutions to the solar neutrino problem.
Acknowledgments {#acks .unnumbered}
===============
We thank J. N. Bahcall, M. Baldo-Ceolin, P. Barnes, L. B. Bezrukov, S. Brice, L. Callis, A. E. Chudakov, A. Dar, G. T. Garvey, W. Haxton, V. N. Kornoukhov, V. A. Kuzmin, V. A. Matveev, L. B. Okun, V. A. Rubakov, R. G. H. Robertson, N. Sapporo, A. Yu. Smirnov, A. A. Smolnikov, A. N. Tavkhelidze, and many members of GALLEX for their continued interest and for fruitful and stimulating discussions. We acknowledge the support of the Russian Academy of Sciences, the Institute for Nuclear Research of the Russian Academy of Sciences, the Ministry of Science and Technology of the Russian Federation, the Russian Foundation of Fundamental Research under Grant No. 96-02-18399, the Division of Nuclear Physics of the U.S. Department of Energy, the U.S. National Science Foundation, and the U.S. Civilian Research and Development Foundation under award No. RP2-159. This research was made possible in part by Grant No. M7F000 from the International Science Foundation and Grant No. M7F300 from the International Science Foundation and the Russian Government.
Other Counting Systems {#other_counting_systems}
======================
The counting systems have been designated by the numbers 1–6. The initial developmental work on system 1 [@BAR83; @GOG83], which used the amplitude of the differentiated pulse (ADP) method [@DAV72] to separate events from background, was done in Russia during the early 1980s. Based on this work, system 2 was developed at BNO during the years 1985–1988. System 2 was completed in 1989 and counted all but two first extractions through May 1992 (SAGE I). Counting system 5, which used the ADP method of rise time measurement, was used to count the other two first extractions during 1990 and 1991. During the summer of 1992, system 3, which has the capability to record the counter wave form, was brought on line; since that time, it has been used to count almost all first extractions. After the implementation of system 3 as the primary counting system, extensive upgrades to reduce backgrounds were performed on system 2 to enable SAGE to have low-noise counting capability in more than eight channels. The upgraded system is referred to as system 6. It has counted seven first extractions during SAGE II and III, mostly from low-mass samples of Ga, and has been used mainly for developmental work, such as testing proportional counters and counting cleanup extractions of gallium.
### Counting system 2 {#sys_2}
System 2 was a seven-channel system where each PC was counted in an independent passive shield; five of those channels had active shielding with NaI crystals. The passive shield consisted of an internal wall of tungsten (40–80 mm thick) or copper (20–30 mm thick) surrounded by lead (150 mm thick). The NaI events were recorded in coincidence mode with events from the proportional counters. Several of the performance characteristics of system 2 are given in Table \[sys\_specs\].
### Counting system 6 {#sys_6}
Modifications to system 2 began during 1992 when system 3 became the primary acquisition system; the improvements were so extensive that it was redesignated as system 6. The counting system has seven channels of acquisition with independent passive shields for each proportional counter. Six channels have an active shield, which operates in coincidence mode with events in the proportional counter. A modified ADP method with the application of several differentiation time constants is used for rejection of point ionization events from backgrounds. System 6 became fully operational in early 1993. To give this system wave form recording capability a digitizing oscilloscope was added, but this improvement has never been fully implemented.
Fourier transform of the wave form – $\bbox{R}$ {#fft}
===============================================
![Determination of the time and dc offset of a candidate event.[]{data-label="offset_det"}](offset.eps){width="3.375in"}
In contrast to the $T_N$ method, the pulse offset is determined independently from the wave form. One uses the intersection point of two lines, the zero-slope line of the offset and the initial slope of the pulse, to obtain the onset position in time and voltage. The initial slope is defined as a certain number of points before and after the point at 20% of the maximum pulse height. The exact number of points to fit is determined individually for each pulse since the number of points available will depend on the pulse height. This region is chosen so that the points are sufficiently linear. Figure \[offset\_det\] illustrates graphically how the onset point is determined.
Two data runs with counters filled with were used to check the energy offset from this wave form analysis determination. The runs were separated by three years and used different digitizer settings. The Gaussian centroid of each $L$ peak and $K$ peak was calculated, with each peak containing a few thousand counts. The extrapolated intercepts in energy are 0.005(6) keV and 0.022(8) keV using $L$- and $K$-peak energies of 1.17 keV and 10.37 keV. Given the energy resolution of our counters, the energy offset is effectively zero.
The algorithm for determining the pulse onset was checked using computer-simulated pulses, both with and without Gaussian noise. It correctly identifies the time offset to within 1 ns and the dc offset to within one channel. Those limits are, of course, dependent on the noise levels, but the levels used were approximately the same as for typical data. Thus, if each pulse is properly normalized to both zero time and dc offset, there is no need to apply an energy offset correction.
This technique uses the zero- and lowest-frequency values from a FFT to obtain measures of the energy and rise time of a pulse. The determination of the energy is straightforward from the definition of the Fourier transform,
$$\label{Fourier}
F(\omega) = \int_{-\infty}^{+\infty}{f(t)e^{-i \omega t} dt}.$$
At $\omega = 0$, $F(\omega)$ equals the area under the curve $f(t)$, which in this case is the digitized wave form of the event convoluted with a Hanning windowing function. We select an integration time of 800 ns, which is the maximum time allowable given the variation in the time of pulse onset. In effect, this technique is equivalent to summing channels used with $T_N$ and is analogous to an ADC that integrates for 800 ns.
In a Fourier analysis, the rise time behavior of a typical pulse clearly will be a very-low-frequency component. Studies with actual pulses and computer-simulated pulses generated with Eqs. (\[Tn\_Formula\]) show that one can accurately identify several distinct features of the wave form. As expected, the dominant components are the lowest frequencies along with the random noise that spans all frequencies. One can identify structure as well; most of it originates from the intrinsic properties of the oscilloscope, such as dithering and the finite digitization size. One of the advantages of a Fourier analysis is that such structure appears at high frequencies and is well separated from the rise time information. The lowest, nonzero, real component $F(1)$ scales similarly to an ADP value but is independent of electronic offsets and high-frequency noise contributions to the pulse. Dividing it by the energy $F(0)$ of the pulse produces a parameter $R$ that is proportional to the inverse rise time. Thus, one can perform a complementary analysis of the data that is analogous to the ADP method but is based solely on the digitized pulse and is independent of any underlying assumptions of its functional form.
RST method {#RST}
==========
In the standard analysis of our data we use the $T_N$ method and fit the observed pulse to Eq. \[Tn\_Formula\]. This function gives the correct description of the shape of the voltage pulse as recorded by the digital oscilloscope when the ionization produced in the proportional counter consists of a set of point ionizations evenly distributed along a straight track. Since events are usually a single cluster of ionization, this method works satisfactorily to select candidate events. It is, however, restricted to the particular form of ionization that is assumed, and gives a poor fit to other types of charge deposit in the counter, such as the combination of a point event from $K$-electron capture followed by capture of the 9.3-keV x ray at some other location in the counter. To give us the capability to investigate all possible events that may occur in the counter, we have also developed a more general method which can analyze an event produced by ionization with an arbitrary distribution of charge. We call this the “restored pulse method” or “RST method” for short.
We begin with the measured voltage pulse $V(t)$ as recorded by the digitizer. For an ideal point charge that arrives at the counter anode wire, $V(t)$ has the Wilkinson form $V(t) = W(t) = V_0 \ln(1 + t/t_0)$, provided the counter is ideal and the pulse processing electronics has infinite bandwidth. For a real event from the counter, with unknown charge distribution, $V(t)$ can in general be expressed as the convolution of the Wilkinson function with a charge collection function $G(t)$: $$\label{G_definition}
V(t) = W(t) \otimes G(t).$$ The function $G(t)$ contains within it the desired information about the arrival of charge at the counter anode, coupled with any deviations of the counter or electronics from ideal response. Equation (\[G\_definition\]) can be considered as the definition of $G(t)$.
To get the desired function $G(t)$, one must deconvolute Eq. (\[G\_definition\]). To perform this deconvolution, we have found it mathematically convenient to use the current pulse $I(t)$, which is obtained by numerical differentiation of $V(t)$: $$\begin{aligned}
I(t) & = & \frac{dV}{dt} = \frac{d}{dt} [W(t) \otimes G(t)] \\ \nonumber
& = & \frac{dW}{dt} \otimes G(t) = W^{'}(t) \otimes G(t),\end{aligned}$$ where $W^{'}(t)$ is normalized over the observed time of pulse measurement, $T_{\text{obs}},$ such that $\int_0^{T_{\text{obs}}} W^{'}(t)dt = 1$.
To deconvolute, we Fourier transform to the frequency domain and then use the theorem that convolution in the time domain becomes multiplication in the frequency domain [@NUMREP]. This simply gives $I(f) = W^{'}(f)
G(f)$, which can be solved for $G(f)$. We then Fourier transform $G(f)$ back to the time domain to get the desired function $G(t)$. The energy of the event is given by $\int_0^{T_{\text{obs}}} G(t) dt$. The duration of the collection of ionization is given by the width of $G(t)$, which can be used as a measure of the rise time.
![Analysis of typical pulse by the RST method. See text for explanation.[]{data-label="RST_example"}](rst.eps){width="3.375in"}
An example of this procedure as applied to a typical $K$-peak event is given in Fig. \[RST\_example\]. This pulse has $T_N =
3.9$ ns. The recorded voltage pulse after inversion and smoothing is given by $V(T)$ in the lower panel. The current pulse, obtained by numerical differentiation of the voltage pulse, is given by $I(t)$ in the upper panel. The deduced function $G(t)$ is also shown in the upper panel. It has a FWHM of about 15 ns, found to be typical for true $K$-peak events. The integrated current pulse, which records the pulse energy, is given by $\int G(t)dt$ in the lower panel.
This method has the advantage that it can reveal the basic nature of the ionization in the counter for an arbitrary pulse. It is also capable of determining the pulse energy over a wider range than the $T_N$ method. A problem that has been found with this method in practice, however, is that when data are analyzed one obtains multiple collection functions \[i.e., $G(t)$ has several distinct peaks separated in time\] more often than is expected from the known physical processes that take place in the counter. These multiple peaks are due to noise on the pulse and cutoff of the system frequency response at about 100 MHz. Attempts have been made to remove these extraneous peaks by filtering and smoothing the original pulse, but they have not been fully successful. Evidently we need faster electronics and a reduction in the noise level to be able to fully exploit this pulse shape analysis technique. As a result, we have only been able to use this method to select events on the basis of energy.
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[^1]: The symbol M stands for the amount of substance concentration in moles per liter.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Aurélien Bajolet, Yuri Bilu[^1]'
title: 'Computing Integral Points on $X_{{\mathop{\mathrm{ns}}}}^+(p)$'
---
Introduction
============
In his celebrated work of 1978 Mazur [@Ma78] completely described possible rational points on the modular curves $X_0(p)$, where $p$ is a prime number. In particular, he showed that the set $X_0(p)({{\mathbb Q}})$ consists only of the cusps if ${p>163}$, and of the cusps and the CM-points if ${37<p\le 163}$.
The curve $X_0(p)$ associated to the Borel subgroup of ${{\mathop{\mathrm{GL}}}}_2({{\mathbb F}}_p)$ It is natural to ask the same question on the modular curves associated to two other important maximal subgroups of ${{\mathop{\mathrm{GL}}}}_2({{\mathbb F}}_p)$, the normalizers of a split or a non-split Cartan subgroups (see [@Se97 Appendix A.5] or [@Ba10 Section 2] for the definitions). We shall denote these curves $X_{{\mathop{\mathrm{sp}}}}^+(p)$ and $X_{{\mathop{\mathrm{ns}}}}^+(p)$, respectively. This problem is not only interesting by itself, but is also motivated by applications; for instance, Serre’s uniformity problem about Galois representations [@BP11] would be solved if one show that for large $p$ the sets $X_{{\mathop{\mathrm{sp}}}}^+(p)({{\mathbb Q}})$ and $X_{{\mathop{\mathrm{ns}}}}^+(p)({{\mathbb Q}})$ consist only of the cusps and the CM-points (points corresponding to elliptic curves with complex multiplication). For the convenience of the reader, we reproduce the full list of the 13 rational CM $j$-invariants in Table \[tacm\].
$$\begin{gathered}
\begin{array}{c|ccccccccccccc}
df^2&
-3&
-3\cdot 2^2 &
-3\cdot3^3&
-4&
-4\cdot2^2&
-7&
-7^\cdot2^2&
-8\\
j&
0 &
2^43^35^3&
-2^{15}3\cdot5^3&
2^63^3&
2^33^311^3&
-3^35^3&
3^35^317^3&
2^65^3
\end{array}\\
\begin{array}{c|ccccccccccccc}
df^2 &
-11 &
-19 &
-43 &
-67 &
-163\\
j&
-2^{15}&
-2^{15}3^3 &
-2^{18}3^35^3 &
-2^{15}3^35^311^3 &
-2^{18}3^35^323^329^3
\end{array}
\end{gathered}$$
together with the discriminant of the CM order
Rational points on the curves $X_{{\mathop{\mathrm{sp}}}}^+(p)$ were determined recently [@BP11a; @BPR12] for all ${p\ne 13}$; in particular, it is shown in [@BPR12] that for ${p\ge 17}$ the set $X_{{\mathop{\mathrm{sp}}}}^+(p)({{\mathbb Q}})$ consists only of the cusps and the CM-points.
Unfortunately, the methods of [@BP11a; @BPR12] completely fail for the curve $X_{{\mathop{\mathrm{ns}}}}^+(p)$. To the best of our knowledge, the set $X_{{\mathop{\mathrm{ns}}}}^+(p)({{\mathbb Q}})$ is not known for every prime ${p\ge 13}$.
More is known about *integral* points on the curves $X_{{\mathop{\mathrm{ns}}}}^+(p)$, that is, points ${P\in X_{{\mathop{\mathrm{ns}}}}^+(p)({{\mathbb Q}})}$ such that ${j(P)\in {{\mathbb Z}}}$, where $j$ is the modular invariant. Kenku [@Ke85] determined the integral points on the curve $X_{{\mathop{\mathrm{ns}}}}^+(7)$; in fact, he found the $7$-integral points, that is, such that the denominator of $j(P)$ is a power of $7$. He used in an essential way the fact that the curve is of genus $0$.
More recently, Schoof and Tzanakis [@ST12] determined the integral points on $X_{{\mathop{\mathrm{ns}}}}^+(11)$, using the fact that this curve is of genus $1$. They showed that the only integral points on this curve are the CM-points. See also [@CC04].
We may also mention that integral points on the curve $X_{{\mathop{\mathrm{ns}}}}^+(N)$ of certain composite levels $N$ were determined much earlier by Heegner and Siegel [@He52; @Si68] in the context of the Class Number $1$ problem; see [@Se97 Appendix A.5] for more details. More recently, composite levels were examined by Baran [@Ba09; @Ba10]. Non of these methods seems to extend to higher prime levels either.
In [@BS13] Bajolet and Sha, using Baker’s method, obtained a fully explicit upper bound for the size of an integral point $P$ on $X_{{\mathop{\mathrm{ns}}}}^+(p)$ for an arbitrary prime ${p\ge 7}$. They showed that in general $$\label{ebsh}
\log|j(P)| <41993\cdot 13^{p} \cdot p^{2p+7.5}(\log{p})^{2},$$ and this bound can be substantially refined if ${p-1}$ is divisible by a small odd prime or by $8$. Sha [@Sh13; @Sh14] extended the result of [@BS13] to $S$-integral points on rather general modular curves over arbitrary number fields, giving an explicit version of the “effective Siegel’s theorem for modular curves” [@Bi02; @BI11].
Using bound , one can, in principle, enumerate all integral points on $X_{{\mathop{\mathrm{ns}}}}^+(p)$. However, this bound is too huge to perform this enumeration in reasonable time.
It turns out that the huge bound can be reduced using the numerical Diophantine approximation techniques, which go back to the work of Baker and Davenport [@BD69]. The idea of Baker and Davenport was elaborated in [@BH96; @BH98; @BH99; @BHV01; @Ha00; @PS87; @TW89] in the context of the Diophantine equations of Thue and of related types, providing practical methods for solving these equations.
In the present article we adapt these techniques to modular curves and develop an algorithm for finding integral points on the modular curve $X_{{\mathop{\mathrm{ns}}}}^+(p)$, where ${p\ge 7}$ is an arbitrary prime number. Having implemented our algorithm, we prove the following.
Let $p$ be a prime number, ${11\le p \le 67}$, and let ${P\in X_{{\mathop{\mathrm{ns}}}}^+(p)({{\mathbb Q}})}$ be such that ${j(P)\in {{\mathbb Z}}}$. Then $P$ is a CM point (that is, $j(P)$ is one of the 13 numbers displayed in the second line of Table \[tacm\]).
One may conjecture that for any prime ${p\ge 11}$ the only integral points on $X_{{\mathop{\mathrm{ns}}}}^+(p)$ are the CM-points.
Plan of the article
-------------------
In Section \[sxg\] we recall basic definitions about modular curves. In particular, we remind the notions of the *nearest cusp* and the *$q$-parameter* at a given cusp, a basic tools in the calculus on modular curves.
In Section \[sbakergen\] we give a general informal overview on how Baker’s method applies to modular curves, highlighting both theoretical and numerical aspects.
In Sections \[ssiegel\] and \[squad\] we revise the theory of modular units, an indispensable tool in the Diophantine analysis of modular curves. In Section \[scpu\] we apply this general theory in the special case of the curve $X_{{\mathop{\mathrm{ns}}}}^+(p)$, constructing especially “economical” units on this curve.
In Section \[sprinrel\] we evaluate the unit constructed in Section \[scpu\] at an integral point $P$, and express the value as multiplicative combination of certain algebraic numbers: ${U(P)=\eta_0^{b_0}\eta_1^{b_1}\cdots\eta_r^{b_r}}$. We then express the exponents $b_k$ in terms of the $q$-parameter of $P$. These expression, while pretty trivial, will play fundamental role in the remaining part of the article.
In Section \[sbans\] we use Baker’s method to obtain a huge explicit bound for ${B=\max_k|b_k|}$. We follow [@BS13] with the modifications stemming from our present needs. In Section \[sred\] we show how this bound can be drastically reduced in practical situations. In the final section we show how to check which values of $b_k$ below the reduced bound indeed correspond to an integral point.
Notation and Conventions
------------------------
#### Modular functions
Throughout the article, the letter $j$ may have four different meaning, sometimes in the same equation, like in and : the modular invariant $j(\tau)$ on the Poincaré upper halfplane ${{\mathcal H}}$; the modular invariant $j(E)$ of an elliptic curve $E$; the “modular invariant” rational function on a modular curve; the sum of the familiar series ${j(q)=q^{-1}+ 744+ 196884q+\ldots}$ It should be always clear from the context which meaning of $j$ is used. A similar convention applies to other modular functions as well.
#### The $O_1(\cdot)$ notation
We shall use the notation $O_1(\cdot)$, which is a quantitative analogue of the familiar $O(\cdot)$. Precisely, ${A=O_1(B)}$ means that ${|A|\le B}$.
#### Absolute Values and Heights
Absolute values on number fields are normalized to extend standard absolute values on ${{\mathbb Q}}$: ${|p|_v=p^{-1}}$ if ${v\mid p<\infty}$ and ${|2013|_v=2013}$ if ${v\mid \infty}$. We denote by ${{{\mathop{\mathrm{h}}}}(\cdot)}$ the usual *absolute logarithmic height*: if ${\alpha \in \bar {{\mathbb Q}}}$ then $${{\mathop{\mathrm{h}}}}(\alpha) = [K:{{\mathbb Q}}]^{-1}\sum_{v\in M_K}[K_v:{{\mathbb Q}}_v]\log^+|\alpha|_v, \qquad \log^+=\max\{\log, 0\}.$$ where $K$ is a number field containing $\alpha$. If ${\alpha \in {{\mathcal O}}_K}$ then $${{\mathop{\mathrm{h}}}}(\alpha) = [K:{{\mathbb Q}}]^{-1}\sum_{\sigma:K\hookrightarrow {{\mathbb C}}}\log^+|\alpha^\sigma|,$$ the sum being over the complex embeddings of $K$.
Modular Curves, Fundamental Domains, $q$-Parameters {#sxg}
===================================================
Let $N$ be a positive integer. The modular curve $X(N)$ has a geometrically irreducible model over the cyclotomic field ${{\mathbb Q}}(\zeta_N)$, and the Galois group ${{{\mathop{\mathrm{Gal}}}}\bigl({{\mathbb Q}}(\zeta_N)(X(N)\bigr)\big/{{\mathbb Q}}(j)}$ is canonically isomorphic to ${{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})/\{\pm1\}$, with ${{\mathop{\mathrm{SL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})/\{\pm1\}$ being the group ${{{\mathop{\mathrm{Gal}}}}\bigl({{\mathbb Q}}(\zeta_N)(X(N)\bigr)\big/{{\mathbb Q}}(\zeta_N,j)}$, see [@La73 Chapter 6]. We write the Galois action of ${{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})$ on the field ${{\mathbb Q}}(\zeta_N)\bigl(X(N)\bigr)$ exponentially. In the following proposition we collect the properties of this action.
\[pgalomod\]
1. \[ifsig\] For ${f\in {{\mathbb Q}}(\zeta_N)\bigl(X(N)\bigr)}$ and ${\sigma \in {{\mathop{\mathrm{SL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})}$ we have $$f^\sigma= f\circ{{\tilde\sigma}},$$ where on the right we view $f$ as a $\Gamma(N)$-automorphic function on the extended Poincaré plane $\bar{{\mathcal H}}$, and ${{\tilde\sigma}}$ is a lifting of $\sigma$ to ${\Gamma(1)={{\mathop{\mathrm{SL}}}}_2({{\mathbb Z}})}$. Clearly, the result is independent of the choice of the lifting.
2. For ${\sigma \in {{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})}$ we have $$\zeta_N^\sigma=\zeta_N^{\det\sigma}.$$
3. \[igalqexp\] Recall that ${f\in {{\mathbb Q}}(\zeta_N)\bigl(X(N)\bigr)}$ has the “$q$-expansion” $$f=\sum_{k=k_0}^\infty a_kq^{k/N}\in{{\mathbb Q}}(\zeta_N)((q)).$$ Then for ${\sigma=(\begin{smallmatrix}1&0\\0&d\end{smallmatrix})}$ the $q$-expansion of $f^\sigma$ is $$f^\sigma=\sum_{k=k_0}^\infty a_k^\sigma q^{k/N}.$$
Let $G$ be a subgroup of ${{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})$ containing ${-I}$. We denote by $X_G$ the associated modular curve. It corresponds to the $G$-invariant subfield of the field ${{{\mathbb Q}}(\zeta_N)\bigl(X(N)\bigr)}$. The constant subfield of this field is ${{{\mathbb Q}}(\zeta_N)^{\det G}}$, where ${\det:{{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})\to({{\mathbb Z}}/N{{\mathbb Z}})^\times}$ is the determinant, and we identify $({{\mathbb Z}}/N{{\mathbb Z}})^\times$ with the Galois group ${{\mathop{\mathrm{Gal}}}}({{\mathbb Q}}(\zeta_N)/{{\mathbb Q}})$. In particular, if ${\det G=({{\mathbb Z}}/N{{\mathbb Z}})^\times}$ then the constant subfield is ${{\mathbb Q}}$ and the corresponding modular curve $X_G$ is defined (that is, has a geometrically irreducible model) over ${{\mathbb Q}}$.
For a subgroup $H$ of $({{\mathbb Z}}/N{{\mathbb Z}})^\times$ put $$\label{egh}
G_H=\{g\in G: \det g\in H\}.$$ In particular, ${G_{({{\mathbb Z}}/N{{\mathbb Z}})^\times}=G}$ and ${G_1=G\cap{{\mathop{\mathrm{SL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})}$. If $H$ is contained in $\det G$, then the subfield of ${{\mathbb Q}}(\zeta_N\bigl(X(N)\bigr)$ stabilized by $G_H$ is $K(X_G)$, where ${K={{\mathbb Q}}(\zeta_N)^H}$.
\[rorbits\] Let $M_N$ be the subset of the abelian group $({{\mathbb Z}}/N{{\mathbb Z}})^2$ consisting of the elements of exact order $N$. Then the set of cusps of the modular curve $X_G$ stays in natural one-to-one correspondence with the set ${G_1\backslash M_N}$ of orbits of the natural (left) action of $G_1$ on $M_N$ [@BI11 Lemma 2.3]. Formally, we do not need this property of cusps in the present article, but it provides a nice “visual” presentation of the cusps; we shall use it in Section \[scpu\].
The cusps are defined over the cyclotomic field ${{\mathbb Q}}(\zeta_N)$. Identifying the groups ${{{\mathop{\mathrm{Gal}}}}\bigl({{\mathbb Q}}(\zeta_N)/{{\mathbb Q}}\bigr)}$ and ${({{\mathbb Z}}/N{{\mathbb Z}})^\times}$, the natural left action of ${({{\mathbb Z}}/N{{\mathbb Z}})^\times}$ on the set ${G_1\backslash M_N}$ coincides with the Galois action on the cusps. Hence, if $H$ is a subgroup of $({{\mathbb Z}}/N{{\mathbb Z}})^\times$ then the set of $H$-orbits of cusps stands in a one-to-one correspondence with (left) $G_H$-orbits on $M_N$.
Optimal System of Representatives {#ssopt}
---------------------------------
Let $\Gamma$ be the subgroup of ${\Gamma(1)={{\mathop{\mathrm{SL}}}}_2({{\mathbb Z}})}$ obtained by lifting $G_1$. Then the set of complex points $X_G({{\mathbb C}})$ is analytically isomorphic to ${\Gamma\backslash\bar{{\mathcal H}}}$, where ${\bar{{\mathcal H}}={{\mathcal H}}\cup{{\mathbb Q}}\cup\{i\infty\}}$ is the extended Poincaré plane.
Let $\Sigma$ be a system of representatives of the right cosets $\Gamma\backslash\Gamma(1)$. We say that $\Sigma$ is an *optimal* system of representatives if it has the following property: given ${\sigma_1,\sigma_2\in \Sigma}$ such that $\sigma_1(i\infty)$ and $\sigma_2(i\infty)$ are $\Gamma$-equivalent, we have ${\sigma_1(i\infty)=\sigma_2(i\infty)}$. An optimal system of representatives always exists. Indeed, start from any $\Sigma$. The $\Gamma$-equivalence of $\sigma_1(i\infty)$ and $\sigma_2(i\infty)$ defines an equivalence relation on $\Sigma$. Fix an equivalence class ${\sigma_1, \ldots,\sigma_m}$. Then there exist ${\gamma_2,\ldots, \gamma_m\in \Gamma}$ such that ${\sigma_1(i\infty)=\gamma_k\circ\sigma_k(i\infty)}$ for ${k=2, \ldots, m}$. Replacing ${\sigma_2, \ldots, \sigma_m}$ by ${\gamma_2\circ\sigma_2, \ldots, \gamma_m\circ\sigma_m}$, and doing a similar operation for every other equivalence class, we obtain an optimal system of representatives.
Moreover, the argument of the previous paragraph shows the following. Let $\Sigma'$ be a subset of $\Gamma(1)$ with the property that for any two elements ${\sigma_1',\sigma_2'\in \Sigma'}$ the points $\sigma_1(i\infty)$ and $\sigma_2(i\infty)$ are *not* $\Gamma$-equivalent. Then $\Sigma'$ can be completed to an optimal system of representatives of $\Gamma\backslash\Gamma(1)$.
We fix, once and for all, an optimal system of representatives $\Sigma$. The function ${\tau\mapsto q(\tau)=q^{2\pi i\tau}}$ is an analytic function on ${{\mathcal H}}$, which vanishes at $i\infty$; it will be called the *$q$-parameter*. For every cusp $c$ of our curve $X_G$ we define the *$q$-parameter at $c$* as follows. Let ${\sigma \in \Sigma}$ be such that $\sigma(i\infty)$ represents $c$. Then the $q$-parameter at $c$ is defined by ${q_c= q\circ\sigma^{-1}}$. Since $\Sigma$ is optimal, $q_c$ depends only on $\Sigma$, but not on the particular choice of $\sigma$. The function $q_c$ is analytic on ${{\mathcal H}}$ and vanishes at $\sigma(i\infty)$.
The Fundamental Domain and the $q$-Parameters {#ssfund}
---------------------------------------------
We denote by $D$ the familiar fundamental domain of the modular group ${\Gamma(1)={{\mathop{\mathrm{SL}}}}_2({{\mathbb Z}})}$: the hyperbolic triangular with vertices ${e^{i\pi/3},e^{2i\pi/3},i\infty}$, and with the geodesics ${(i,e^{2i\pi/3}]}$ and ${[e^{2i\pi/3},i\infty]}$ excluded. Then the set $$\Delta=\bigcup_{\sigma\in \Sigma}\sigma D$$ is a fundamental domain for $\Gamma$. This means that there is a bijection between the set $\Delta$ and the set $Y_G(p)({{\mathbb C}})$ of non-cuspidal complex points of $X_G$. Thus, to every non-cuspidal ${P\in X_G({{\mathbb C}})}$ we uniquely associate ${\tau=\tau(P)\in \Delta}$.
Further, there is a natural projection ${\Delta\to D}$, coinciding with $\sigma^{-1}$ on every $\sigma D$. The image of $\tau(P)$ under this projection will be called $\tau_0(P)$. For a non-cuspidal complex point $P$ we have $$\begin{gathered}
|\tau_0(P)|\ge 1, \qquad {{\mathop{\mathrm{Im}}}}\tau_0(P)\ge \sqrt3/2 \nonumber \\
\label{ejptau}
j(P)=j(\tau(P))=j(\tau_0(P)) \end{gathered}$$ For every cusp $c$ we define the $\Omega_c$ in $X_G({{\mathbb C}})$ by $$\label{eomegac}
\Omega_c = \text{the image of}\left(\bigcup_{\sigma(i\infty)=c}\sigma D\right)\cup\{c\},$$ the union being over all ${\sigma \in \Sigma}$ representing the cusp $c$. The sets $\Omega_c$ are pairwise disjoint and cover $X_G({{\mathbb C}})$: $$\bigcup_{c}\Omega_c= X_G({{\mathbb C}}), \qquad \Omega_c\cap\Omega_{c'}=\varnothing \quad (c\ne c').$$ If ${P\in X_G({{\mathbb C}})}$ belongs to $\Omega_c$, we call $c$ the *nearest cusp* to $P$.
The $q$-parameter $q_c$ defines a holomorphic function on an open neighborhood of $\Omega_c$; this function is denoted by $q_c$. For ${P\in \Omega_c}$ we have ${q_c(P)=e^{2\pi i\tau_0(P)}}$ and $$\label{ejpq}
j(P)=j(q_c(P))$$ Since ${{{\mathop{\mathrm{Im}}}}\tau_0(P)\ge\sqrt3/2}$ for ${P\in \Omega_c}$, we have $$\label{eless}
|q_c(P)|\le e^{-\pi\sqrt3}<0.0044.$$
Denote by $e_c$ the ramification index at $c$ of the natural morphism ${X_G\to X(1)}$. Then $q_c^{1/e_c}$ can be viewed as a “local parameter” at $c$. This means the following: if ${u\in {{\mathbb C}}(X_G)}$ is a ${{\mathbb C}}$-rational function on $X_G$, then in a neighborhood of $c$ we have $$\label{elocalp}
\log|u(P)|= \frac{{{\mathop{\mathrm{Ord}}}}_cu}{e_c}\log|q_c(P)|+O(1).$$
The following property will be routinely used.
\[preal\] For a non-cusp point ${P\in X_G({{\mathbb C}})}$ the following two conditions are equivalent.
1. ${j(P)\in {{\mathbb R}}}$;
2. ${{{\mathop{\mathrm{Re}}}}(\tau_0(P))\in \{0,1/2\}}$ or ${|\tau_0(P)|=1}$.
More precisely: $$\begin{array}{lclcl}
j(P) \in [1728,+\infty) &\Longleftrightarrow& {{\mathop{\mathrm{Re}}}}(\tau_0(P))=0&\Longleftrightarrow&q_c(P)>0,\\
j(P) \in [-\infty,0) &\Longleftrightarrow& {{\mathop{\mathrm{Re}}}}(\tau_0(P))=1/2&\Longleftrightarrow& q_c(P) <0,\\
j(P) \in [0;1728] &\Longleftrightarrow& |\tau_0(P)|=1,
\end{array}$$ where $c$ is the nearest cusp to $P$.
We shall also need an approximate formula for the the $j$-invariant. Write the familiar expansion as[^2] $$j(q)=q^{-1}+c_0+c_1q+c_2q^2+\ldots,$$ with ${c_0=744}$, ${c_1=196884}$ etc. For a positive integer $N$ write $$j_N(q)= q^{-1}+ \sum_{n=0}^N c_nq^n.$$
For ${P\in \Omega_c}$ we have $$\label{eestj}
j(P) = j_N(q_c(P)) +R_N, \qquad |R_N|\le j(e^{-\pi\sqrt3})-j_N(e^{-\pi\sqrt3})$$ for any positive integer $N$.
#### Proof
Since $j$ is $\Gamma(1)$-invariant, we may assume that $c$ is the cusp at infinity and ${q_c(P)=q(P)}$. Since the coefficients $c_n$ are known to be positive and ${|q(P)|\le e^{-\pi\sqrt3}}$, we have $$|j(P) - j_N(q_c(P))|\le \sum_{n=N+1}^\infty c_n|q(P)|^n \le \sum_{n=N+1}^\infty c_n\bigl|e^{-\pi\sqrt3}\bigr|^n= j(e^{-\pi\sqrt3})-j_N(e^{-\pi\sqrt3}),$$ proving . [[$\square$]{}]{}
Integral Points and Baker’s Method {#sbakergen}
==================================
In this section we give a general overview of Baker’s method applied to modular curves. For more details, see [@Bi02].
Let $N$ and $G$ be as in Section \[sxg\], let $K$ be a number field containing ${{{\mathbb Q}}(\zeta_N)^{\det G}}$ and ${{\mathcal O}}_K$ the ring of integers of $K$. We define the set of integral points $$X_G({{\mathcal O}}_K)=\{P\in X_G(K): j(P)\in {{\mathcal O}}_K\}.$$ We want to bound the height ${{\mathop{\mathrm{h}}}}(j(P))$ for ${P\in X_G({{\mathcal O}}_K)}$. We show how to do this under the assumption $$\label{e3}
\nu_\infty(G)\ge 3,$$ where $\nu_\infty(G)$ denotes the number of cusps of $X_G$.
A *modular unit* is a rational function (defined over $\bar K$) on $X_G$ with no zeros and no poles outside the cusps. Equivalently, ${u\in \bar K(X_G)}$ is a modular unit if both $u$ and $u^{-1}$ are integral over the ring ${{{\mathbb Q}}[j]}$. Principal divisors of modular units form a subgroup in the group of degree $0$ divisors supported on the cusps. The latter is a free abelian group of rank ${\nu_\infty(G)-1}$, so the group of principal divisors of modular units must be of rank not exceeding ${\nu_\infty(G)-1}$. It is of fundamental importance for us that it is of the maximal possible rank; this is sometimes called the “Manin-Drinfeld theorem”.
\[tmandr\] The principal divisors of modular units form a free abelian group of rank ${\nu_\infty(G)-1}$.
See [@La78 Chapter 4, Theorem 2.1]. Here is an immediate consequence.
\[cmandr\] Assume that ${\nu_\infty(G)\ge 3}$. Then for any cusp $c$ there exists a non-constant modular unit $u$ such that ${u(c)=1}$.
If ${j(P)\in {{\mathcal O}}_K}$ then ${{{\mathop{\mathrm{h}}}}(j(P))= [K:{{\mathbb Q}}]^{-1}\sum_{\sigma:K\hookrightarrow {{\mathbb C}}}\log^+|j(P)^\sigma|}$, the sum being over the complex embeddings of $K$. For some embedding $\sigma$ we have ${{{\mathop{\mathrm{h}}}}(j(P)) \le \log|j(P)^\sigma|}$. We fix this embedding from now on and view $K$ as a subfield of ${{\mathbb C}}$. Thus, we have to bound $|j(P)|$ from above.
The point $P$ belongs to one of the sets $\Omega_c$, defined in , and the corresponding $c$ is the “nearest cusp” to $P$. Now, since ${\nu_\infty(G)\ge 3}$, we may use Corollary \[cmandr\] and find a non-constant modular unit $u$ with ${u(c)=1}$. The rational function $u$ is defined over the number field $K(\zeta_N)$.
If ${u(P)=1}$ then it is easy to bound $P$ as one of the zeros of the rational function ${u-1}$. From now on we assume that ${u(P)\ne 1}$. Since ${u(c)=1}$, we have $$u(P)= 1+ O(|q_c(P)|^{1/{e_c}}).$$ (Here and below in this section, the constant implied by the $O(\cdot)$-notation, as well as by the Vinogradov notation “$\ll$” and “$\gg$”, may depend on $N$ and $K$, but not on $P$.) Thus, $u(P)$ is a complex algebraic number, distinct from $1$ but “close” to $1$ if $q_c(P)$ is small.
Since both $u$ and $u^{-1}$ are integral over ${{\mathbb Q}}[j]$, that there exist non-zero ${A, B\in {{\mathbb Z}}}$, which can be easily determined explicitly, such that $Au$ and $Bu^{-1}$ are integral over ${{\mathbb Z}}[j]$. Since ${j(P)\in {{\mathcal O}}_K}$, both ${Au(P)}$ and $Bu(P)^{-1}$ belong to ${{\mathcal O}}_{K(\zeta_N)}$. It follows that there are only finitely many possibilities for the principal ideal $(u(P))$ (viewed as a fractional ideal in the field $K(\zeta_N)$). In other words, we have ${u(P)=\eta_0\eta}$, where $\eta_0$ belongs to a finite subset of $K$ (that can be explicitly determined), and $\eta$ is a Dirichlet unit of $K$. Fixing a base $\eta_1, \ldots, \eta_r$ of the group of Dirichlet units of $K(\zeta_N)$, we obtain ${u(P)=\eta_0\eta_1^{b_1}\cdots \eta_r^{b_r}}$, where ${b_1,\ldots, b_r}$ are rational integers depending of $P$. We obtain the inequality $$\label{eforb}
\left|\eta_0\eta_1^{b_1}\cdots \eta_r^{b_r} -1\right|\ll q_c(P)^{1/{e_c}}.$$ It is easy to show that ${B\ll {{\mathop{\mathrm{h}}}}(\eta)}$, see [@Bi02 bottom of page 77]. It follows that ${B\ll {{\mathop{\mathrm{h}}}}(u(P))+1}$. On the other hand, the general property of quasi-equivalence of heights on an algebraic curve implies that ${{{\mathop{\mathrm{h}}}}(u(P)) \ll {{\mathop{\mathrm{h}}}}(j(P))+1}$. It follows that $$\label{equasi}
B\ll {{\mathop{\mathrm{h}}}}(j(P)) \le \log|j(P)| = \log|q_c(P)^{-1}| +O(1).$$
On the other hand, one can bound the left-hand side of from below using the so-called *Baker’s inequality*. We state it in a full detail in Section \[sbans\]. Here we just remark that Baker’s inequality implies that either the left-hand side of is $0$ (in which case ${u(P)=\beta_c}$ and ${{\mathop{\mathrm{h}}}}(j(P))$ is bounded), or it is bounded from below by ${\exp(-\kappa\log B)}$, where ${B=\max\{|b_1|, \ldots ,|b_r|, 3\}}$ and $\kappa$ is a positive effective constant depending on ${\eta_0, \eta_1, \ldots, \eta_r}$ but independent of $B$. Combining this with , we obtain ${\log|q_c(P)^{-1}| \ll \log B}$. Together with this bounds $|q_c(P)|$ from below, which implies a bound for $|j(P)|$ from above.
In a similar fashion one can study $S$-integral points on $X_G$: the new ingredients to be added are the $p$-adic version of Baker’s inequality, due to Yu [@Yu07], and the $p$-adic analogue of the notion of the “nearest cusp”, see [@BP11 Section 3]. To make all this explicit, one needs to construct modular units explicitly. The standard tool for this are *Siegel functions*, see Section \[ssiegel\] below. One also needs explicit version for various statements above like the quasi-equivalence of heights, etc. All this is a part of a forthcoming Ph.D. thesis of Sha [@Sh13; @Sh14].
In the present work, we are interested in a somewhat different task: not just bound the heights of integral points, but determine them completely. We restrict ourselves to the case ${K={{\mathbb Q}}}$ and ${N=p}$ a prime number. In this case the most interesting class of modular curves for which integral points are unknown is $X_{{\mathop{\mathrm{ns}}}}^+(p)$, when the group $G$ is the normalizer of a non-split Cartan subgroup of ${{\mathop{\mathrm{GL}}}}_2({{\mathbb F}}_p)$.
The principal point here is that bounding the height of integral points, even explicitly in all parameters, is not sufficient for the actual calculation of the points. The problem is that the bounds obtained by Baker’s method are very high and not suitable for computational purposes.
Fortunately, one can reduce Baker’s bound using the technique of numerical Diophantine approximations. This reduction is described in detail in [@BH96; @BH99; @Ha00] in the context of the Diophantine equation of Thue. Recall that this is the equation of the form ${f(x,y)=A}$, where the ${f(x,y)\in {{\mathbb Z}}[x,y]}$ is a ${{\mathbb Q}}$-irreducible form of degree ${n\ge 3}$, and $A$ is a non-zero integer. In [@BH98] the method was extended to the superelliptic Diophantine equations. Here we adapt this reduction method to the modular curves.
Several observations are to be made.
1. Usually, to perform the computations, one should know explicitly the algebraic data of the number field(s) involved (in the case of Thue equation, this is the field generated over ${{\mathbb Q}}$ by a root of $f(1,y)$). By the algebraic data we mean here the unit group (with explicit generators), the class group (again, for every class one should have an explicit ideal representing this class), and so on. Fortunately, in the special case of the curve $X_{{\mathop{\mathrm{ns}}}}^+(p)$ these tasks are radically simplified. First, the field we are going to deal with is the real cyclotomic field ${{{\mathbb Q}}(\zeta_p+\bar\zeta_p)}$ (or its subfield, see below) for which the unit group (or at least a full-rank subgroup of the latter, which is sufficient, see below) are given explicitly by the *circular units*. Second, the only ideal we are going to deal with is the one above $p$, which is principal and has an obvious explicit generator ${(\zeta_p-\bar\zeta_p)^2}$. This was already used in [@BHV01] for solving Thue equations ${\Phi_n(x,y)=p}$, where $\Phi_n(1,y)$ is the $n$-th real cyclotomic polynomial, and $p$ is a primer divisor of $n$.
2. To make the calculations more efficient, it is useful to replace the field ${{{\mathbb Q}}(\zeta_p+\bar\zeta_p)}$ by a smaller subfield, whenever possible. This was suggested in [@BH99] and was very efficiently exploited in [@BHV01].
3. Also, it is not necessary to have the full unit group; a full-rank subgroup would suffice, as explained in [@Ha00]. This was used in [@BHV01] as well. In the present work we use only full unit groups, but one should keep this opportunity in mind for further applications.
4. Adapting numerical methods developed for Thue equations to modular curves is not straightforward. In the Thue case one has formulas with very strong error estimates, typically $O(|x|^{-n})$, where $n$ is the degree of the equation; see, for instance [@BH96 Proposition 2.4.1]. This is quite good even for small solutions $x$. However, for modular curves of level $p$ we have, typically, errors $O(|j(P)|^{-1/p})$. For small solutions this error can be too large to deal with, and we have to use high order asymptotic expansions for the modular functions involved, which makes the things more complicated. See Subsection \[ssslow\].
Siegel Functions {#ssiegel}
================
In this section we recall the principal facts about Klein forms and Siegel functions. For more details the reader can consult [@KL81 Section 2.1] and [@KS10].
Klein Forms and Siegel Functions
--------------------------------
Let ${{{\mathbf a}}=(a_1,a_2)\in {{\mathbb Q}}^2}$ be such that ${{{\mathbf a}}\notin {{\mathbb Z}}^2}$. We denote by ${{\mathfrak k}}_{{\mathbf a}}(\tau)$ the Klein form associated to ${{\mathbf a}}$, which is a holomorphic function on the Poincaré plane ${{\mathcal H}}$. We collect some properties Klein forms in the proposition below.
\[pprokl\]
1. The Klein forms do not vanish on ${{\mathcal H}}$.
2. \[iklega1\] The Klein forms well behave under the action of $\Gamma(1)$: for ${\gamma=(\begin{smallmatrix}a&b\\c&d\end{smallmatrix})\in \Gamma(1)}$ we have $${{\mathfrak k}}_{{\mathbf a}}\circ\gamma(\tau)=(c\tau+d)^{-1}{{\mathfrak k}}_{{{\mathbf a}}\gamma}(\tau),$$ where ${\gamma(\tau)= \frac{a\tau+b}{c\tau+d}}$. In particular, with ${\gamma=-I}$ this gives $$\label{eklemin}
{{\mathfrak k}}_{-{{\mathbf a}}}=-{{\mathfrak k}}_{{\mathbf a}}.$$
3. \[iklemod1\] For ${{{\mathbf a}}=(a_1,a_2) \in {{\mathbb Q}}^2\setminus{{\mathbb Z}}^2}$ and ${{{\mathbf b}}=(b_1,b_2)\in {{\mathbb Z}}^2}$ we have $${{\mathfrak k}}_{{{\mathbf a}}+{{\mathbf b}}}= {\varepsilon}({{\mathbf a}},{{\mathbf b}}){{\mathfrak k}}_{{\mathbf a}}, \qquad {\varepsilon}({{\mathbf a}},{{\mathbf b}})= (-1)^{b_1b_2+b_1+b_2}e^{\pi i(a_1b_2-a_2b_1)}.$$ Notice that ${{\varepsilon}({{\mathbf a}},{{\mathbf b}})^{2N}=1}$, where $N$ is a denominator of ${{\mathbf a}}$ (a common denominator of $a_1$ and $a_2$).
4. Let $N$ be a denominator of ${{\mathbf a}}$. Then ${{\mathfrak k}}_{{\mathbf a}}$ is “nearly” $\Gamma(N)$-automorphic of weight $-1$. Precisely, for ${\gamma=(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}) \in \Gamma(N)}$ we have $${{\mathfrak k}}_{{\mathbf a}}\circ\gamma(\tau)={\varepsilon}'({{\mathbf a}},\gamma)(c\tau+d)^{-1}{{\mathfrak k}}_{{\mathbf a}}(\tau), \qquad {\varepsilon}'({{\mathbf a}},\gamma)^{2N}=1.$$
The following result is a consequence of the properties above.
\[pkle\] Let $N$ be a denominator of ${{\mathbf a}}$. Then ${{\mathfrak k}}_{{\mathbf a}}^{2N}$ depends only on the residue class of ${{\mathbf a}}$ modulo ${{\mathbb Z}}^2$, and is $\Gamma(N)$-automorphic of weight ${-2N}$.
Further, for ${{{\mathbf a}}=(a_1,a_2)\in {{\mathbb Q}}^2\smallsetminus {{\mathbb Z}}^2}$ we define the *Siegel function* $g_{{\mathbf a}}(\tau)$ by $$g_{{\mathbf a}}(\tau)= {{\mathfrak k}}_{{\mathbf a}}(\tau)\eta(\tau)^2,$$ where $\eta(\tau)$ is the Dedekind $\eta$-function.
As usual, write ${q=q(\tau)=e^{2\pi i\tau}}$. For a rational number $a$ we define ${q^a=e^{2\pi i a\tau}}$. Then the Siegel function $g_{{\mathbf a}}$ has the following infinite product presentation [@KL81 page 29]: $$\label{epga}
g_{{\mathbf a}}(\tau)= -q^{B_2(a_1)/2}e^{\pi ia_2(a_1-1)}\prod_{n=0}^\infty\left(
1-q^{n+a_1}e^{2\pi ia_2}\right)\left(1-q^{n+1-a_1}e^{-2\pi i a_2}\right),$$ where $B_2(T)$ is the second Bernoulli polynomial. Together with item \[iklemod1\] of Proposition \[pprokl\] this has the following consequence.
\[pella\] We have ${{{\mathop{\mathrm{Ord}}}}_qg_{{\mathbf a}}=\ell_{{\mathbf a}}}$, where ${\ell_{{\mathbf a}}=B_2(a_1-\lfloor a_1\rfloor)/2}$.
Here the $q$-order ${{\mathop{\mathrm{Ord}}}}_q$ is defined by ${\lim_{q\to0}q^{{{\mathop{\mathrm{Ord}}}}_qg_{{\mathbf a}}}g_{{\mathbf a}}(q)\ne0,\infty}$.
Since ${\eta(\tau)^{24}=\Delta(\tau)}$ is $\Gamma(1)$-automorphic of weight $12$, Proposition \[pkle\] implies the following.
\[t12N\] In the set-up of Proposition \[pkle\], the function $g_{{\mathbf a}}^{12N}$ depends only on the residue class of ${{\mathbf a}}$ modulo ${{\mathbb Z}}^2$, and is $\Gamma(N)$-automorphic of weight $0$.
It follows, in particular, that Siegel functions $g_{{\mathbf a}}$ are algebraic over the field ${{\mathbb C}}(j)$ (because so are $\Gamma(N)$-automorphic functions). In addition to this, $g_{{\mathbf a}}$ is holomorphic and does not vanish on the Poincaré plane ${{\mathcal H}}$ (because so are the Klein forms and the Dedekind $\eta$). It follows that both $g_{{\mathbf a}}$ and $g_{{\mathbf a}}^{-1}$ must be integral over the ring ${{\mathbb C}}[j]$. Actually, a stronger assertion holds (see, for instance, Proposition 2.2 from [@BP11a]).
\[psiu\] Let $N$ be the smallest denominator of ${{\mathbf a}}$ and $\zeta_N$ a primitive $N$-th root of unity. Then both $g_{{\mathbf a}}$ and ${\left(1-\zeta_N\right)g_{{\mathbf a}}^{-1}}$ are integral over ${{\mathbb Z}}[j]$.
Simplest Modular Units {#sssmu}
----------------------
Now let us fix a positive integer $N$. By Theorem \[t12N\], for ${{{\mathbf a}}\in N^{-1}{{\mathbb Z}}^2\smallsetminus{{\mathbb Z}}^2}$ the function $g_{{\mathbf a}}^{12N}$ defines a ${{\mathbb C}}$-rational function on the modular curve $X(N)$, to be denoted by $u_{{\mathbf a}}$. Moreover, $u_{{\mathbf a}}$ is well-defined when ${{\mathbf a}}$ is a non-zero element of the abelian group ${(N^{-1}{{\mathbb Z}}/{{\mathbb Z}})^2}$, which will be assumed until the end of the subsection. Identity implies that $u_{{\mathbf a}}=u_{-{{\mathbf a}}}$. The infinite product implies that the $q$-expansion of $u_{{\mathbf a}}$ has coefficients in the cyclotomic field ${{\mathbb Q}}(\zeta_N)$. It follows that ${u_{{\mathbf a}}\in {{\mathbb Q}}(\zeta_N)\bigl(X(N)\bigr)}$. Moreover, the Galois action of the group ${{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})$ on the field ${{{\mathbb Q}}(\zeta_N)\bigl(X(N)\bigr)}$ (see Section \[sxg\]) is compatible with the natural right action of on the set ${(N^{-1}{{\mathbb Z}}/{{\mathbb Z}})^2}$ in the following sense: for a non-zero ${{{\mathbf a}}\in (N^{-1}{{\mathbb Z}}/{{\mathbb Z}})^2}$ and ${\sigma \in {{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})}$ we have $$\label{egalua}
u_{{{\mathbf a}}\sigma}=u_{{\mathbf a}}^\sigma.$$ See [@BP11 Section 4.2] for more details.
The functions $u_{{\mathbf a}}$ give simplest explicit examples of the *modular units*, already mentioned in Section \[sbakergen\]: they have no zeros and no poles outside the cusps. It follows that their principal divisors generate a free abelian subgroup of rank at most ${\nu_\infty(N)-1}$, where $\nu_\infty(N)$ is the number of cusps of $X(N)$. It turns out that this rank is maximal possible, which provides an explicit form of the “Manin-Drinfeld theorem” (Theorem \[tmandr\]):
\[tmdr\] The principal divisors $(u_{{\mathbf a}})$ generate a free abelian group of rank ${\nu_\infty(N)-1}$.
For the proof see Theorem 3.1 in [@KL81 Chapter 2].
In fact, one can show (we shall not need this) that already the principal divisors $(u_{{\mathbf a}})$, where ${{\mathbf a}}$ runs through the set $M_N$, consisting of the elements of $(N^{-1}Z/{{\mathbb Z}})^2$ of exact order $N$, generate a free abelian group of rank ${\nu_\infty(N)-1}$. The number of such ${{\mathbf a}}$ is $2\nu_\infty(N)$. It follows that, besides the relations ${u_{{\mathbf a}}=u_{-{{\mathbf a}}}}$, there can exist exactly one relation between the the principal divisors $(u_{{\mathbf a}})$ with ${{{\mathbf a}}\in M_N}$. This relation is $$\sum_{{{\mathbf a}}\in M_N}(u_{{\mathbf a}})=0.$$ In fact, we have a more precise statement: $$\label{eprodua}
\prod_{{{\mathbf a}}\in M_N}u_{{\mathbf a}}= \pm\Phi_N(1)^{12N},$$ where $\Phi_N(t)$ is the $N$-th cyclotomic polynomial. In particular, if ${N=p}$ is a prime number, we obtain the following identity: $$\label{eproduap}
\prod_{{{\mathbf a}}\in M_p}u_{{\mathbf a}}= \pm p^{12p}.$$ (One can show that the sign is actually $+$.)
Let us prove . Since the set $M_N$ is stable with respect to ${{{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})}$, the left-hand side of is stable with respect to the Galois action over the field ${{\mathbb Q}}(X(1))$. Hence it is a unit on the curve $X(1)$, defined over ${{\mathbb Q}}$. Since $X(1)$ has only one cusp, it has no non-constant units. Hence the left-hand side of is a constant belonging to ${{\mathbb Q}}$.
To determine the value of this constant, we evaluate it at the cusp at infinity. The left-hand side of is a product of a root of unity and the terms of the type ${\bigl(1-e^{2\pi i a_2}q^{n+a_1}\bigr)^{12N}}$ and of the type ${\bigl(1-e^{2\pi i -a_2}q^{n+1-a_1}\bigr)^{12N}}$, where $n$ runs through non-negative integers, and $(a_1,a_2)$ runs through the lifting of the set $M_N$ to the unit square $[0,1)^2$. When we set ${q=0}$, all these terms become $1$ except the terms ${\bigl(1-e^{2\pi i a_2}q^{n+a_1}\bigr)^{12N}}$ with ${n=0}$ and ${a_1=0}$. Hence, up to a root of unity, the left-hand side of is $$\prod_{\genfrac{}{}{0pt}{}{a_2\in N^{-1}{{\mathbb Z}}/{{\mathbb Z}}}{\text{$a_2$ is of order~$N$}}} \bigl(1-e^{2\pi i a_2}\bigr)^{12N}= \prod_{\genfrac{}{}{0pt}{}{0\le k <N}{(k,N)=1}}\bigl(1-e^{2\pi i k/N}\bigr)^{12N}= \Phi_N(1)^{12N}.$$ Since the only roots of unity in ${{\mathbb Q}}$ are $\pm1$, this proves .
Asymptotic Expansions {#ssasymp}
---------------------
In this subsection we obtain several types of asymptotic expansions for the Siegel function. Our main tool will be the infinite product presentation . Recall (see Proposition \[pella\]) that ${{{\mathop{\mathrm{Ord}}}}_qg_{{\mathbf a}}=\ell_{{\mathbf a}}}$, where ${\ell_{{\mathbf a}}=B_2(a_1-\lfloor a_1\rfloor)/2}$. Clearly, ${|\ell_{{\mathbf a}}|\le1/12}$. We also put $$\label{egammaa}
\gamma_{{\mathbf a}}=
\begin{cases}
e^{\pi ia_2(a_1-1)}, & a_1\ne 0,\\
e^{\pi ia_2(a_1-1)}(1-e^{2\pi i a_2}), &a_1=0.
\end{cases}$$ We start from a purely formal statement, by estimating the coefficients of a fractional power series representing the logarithm of (properly normalized) $g_{{\mathbf a}}$. In the sequel $v$ is an absolute value of $\bar{{\mathbb Q}}$, extending a standard absolute value of ${{\mathbb Q}}$.
\[pasympform\] Let $N$ be a denominator of ${{\mathbf a}}$. Then there exist ${\beta_1,\beta_2,\beta_3\ldots \in {{\mathbb Q}}(\zeta_N)}$ such that $$\log\frac{g_{{\mathbf a}}(q)}{\gamma_{{\mathbf a}}q^{\ell_{{\mathbf a}}}} = \sum_{k=1}^\infty\beta_kq^{k/N},$$ and $$\label{ebetak}
|\beta_k|_v \le
\begin{cases}
|k|_v^{-1}, & v\mid p<\infty,\\
2k/N+2, & v\mid \infty
\end{cases} \qquad (k=1,2,\ldots).$$
#### Proof
We may assume that ${0\le a_1<1}$. For a fixed non-negative integer $n$ (where we assume ${n>0}$ if ${a_1=0}$) write $$\log(1-e^{2\pi i a_2}q^{n+a_1})= \sum_{k=1}^\infty\alpha_kq^{k/N},$$ An immediate verification shows that ${\alpha_k\in {{\mathbb Q}}(\zeta_N)}$ and $$|\alpha_k|_v \le
\begin{cases}
|k|_v^{-1}, & \text{$v$ finite},\\
1, & \text{$v$ infinite}
\end{cases}
\qquad (k=1,2,\ldots).$$ Same estimates hold true for the coefficients of the $q$-series for ${\log(1-e^{2\pi i a_2}q^{n+1-a_1})}$.
Coefficients of at most $k/N+1$ series for ${\log(1-e^{2\pi i a_2}q^{n+a_1})}$ (those with ${0\le n\le k/N}$) may contribute to $\beta_k$, and the same is true for the series for ${\log(1-e^{2\pi i a_2}q^{n+1-a_1})}$. The result now follows by summation. [[$\square$]{}]{}
We can now replace the infinite sum by a finite and estimate the error term, but the estimate would be very poor when $q^{1/N}$ is close to $1$. To solve this problem, we take away one logarithmic term.
\[pasymplog\]
1. In the set-up of Proposition \[pasympform\] assume that ${0< a_1 <1}$ and put $$(a_1', a_2')=
\begin{cases}
(a_1,a_2), & 0< a_1</1/2,\\
(1-a_1, -a_2),& 1/2\le a_1 <1.
\end{cases}$$ Then with a suitable choice of the logarithms and for ${|q|<0.0044}$ we have $$\label{easymponlylog}
\log\frac{g_{{\mathbf a}}(q)}{\gamma_{{\mathbf a}}q^{\ell_{{\mathbf a}}}} = \log(1- q^{a_1'}e^{2\pi i a_2'})+ O_1(1.2|q|^{1/2}).$$ Further, there exist ${\beta_1',\beta_2',\beta_3'\ldots \in {{\mathbb Q}}(\zeta_N)}$ such that the following holds. Let $\nu$ be a non-negative integer. Then with a suitable choice of the logarithms and for ${|q|<0.0044}$ we have $$\label{easymponelog}
\log\frac{g_{{\mathbf a}}(q)}{\gamma_{{\mathbf a}}q^{\ell_{{\mathbf a}}}} = \log(1- q^{a_1'}e^{2\pi i a_2'})+\sum_{k=1}^\nu\beta_k'q^{k/N} + O_1\bigl((2.2\nu/N+3.1) |q|^{(\nu+1)/N}\bigr).$$
2. When ${a_1=0}$ we have $$\log\frac{g_{{\mathbf a}}(q)}{\gamma_{{\mathbf a}}q^{\ell_{{\mathbf a}}}} = O_1(2.02|q|).$$ Also, an analog of holds true with ${\beta_k'=\beta_k}$ and without the additional logarithmic term on the right: $$\log\frac{g_{{\mathbf a}}(q)}{\gamma_{{\mathbf a}}q^{\ell_{{\mathbf a}}}} = \sum_{k=1}^\nu\beta_kq^{k/N} + O_1\bigl((2.2\nu/N+3.1) |q|^{(\nu+1)/N}\bigr).$$
(Estimates similar to hold for the numbers $\beta'_k$ as well but we shall not need them.)
#### Proof
For ${|z|\le r<1}$ and non-negative integer $m$ we have $$\label{elogappr}
\log(1-z) = -\sum_{k=1}^m\frac{z^k}k+ O_1\left(\frac{|z|^{m+1}}{1-r}\right).$$ (This estimate is very rough but sufficient for us.)
Assume, for instance, that ${0<a_1<1/2}$, so that ${(a_1',a_2')=(a_1,a_2)}$. Let $n$ be a positive integer. Then we have $$\label{easymplogs}
\begin{aligned}
\log \frac{g_{{\mathbf a}}(q)}{\gamma_{{\mathbf a}}q^{\ell_{{\mathbf a}}}}&= \sum_{k=0}^{n-1}\left(\log(1-q^{k+a_1}e^{2\pi ia_2})+\log(1-q^{k+1-a_1}e^{-2\pi i a_2})\right)
+O_1\bigl(2.02|q|^{n+a_1}\bigr).
\end{aligned}$$ To prove we have to estimate the sum $$\sum_{k=n+1}^\infty\left(\left|\log(1-q^{k+a_1}e^{2\pi ia_2})\right|+\left|\log(1-q^{k+1-a_1}e^{-2\pi i a_2})\right|\right).$$ Applying with ${m=0}$ and ${r=0.0044}$ to each term of the latter sum, we estimate the sum as $$1.0045\frac{|q|^{n+a_1}+|q|^{n+1-a_1}}{1-|q|}\le 1.01(|q|^{n+a_1}+|q|^{n+1-a_1})\le 2.02|q|^{n+a_1},$$ because ${n+1-a_1\ge n+a_1}$. This proves .
Now to establish we take ${n=1}$. We obtain $$\log \frac{g_{{\mathbf a}}(q)}{\gamma_{{\mathbf a}}q^{\ell_{{\mathbf a}}}}= \log(1-q^{a_1}e^{2\pi ia_2})+\log(1-q^{1-a_1}e^{-2\pi i a_2})
+O_1\bigl(2.02|q|\bigr).$$ Since ${a_1<1/2}$, we have ${|q^{1-a_1}|\le |q|^{1/2}\le 0.067}$. Applying with ${m=0}$ and ${r=0.067}$, we obtain . $$\begin{aligned}
\log \frac{g_{{\mathbf a}}(q)}{\gamma_{{\mathbf a}}q^{\ell_{{\mathbf a}}}}&= \log(1-q^{a_1}e^{2\pi ia_2})
+O_1(1.072|q|^{1/2}+ 2.02|q|)\\
&= \log(1-q^{a_1}e^{2\pi ia_2})
+O_1(1.2|q|^{1/2}),\end{aligned}$$ proving .
To prove we define $n$ as the smallest integer such that ${n+a_1> \nu/N}$. Now, for ${k\ge 1}$ we have ${|q^{k+a_1}|\le |q|\le 0.0044}$, and for ${k\ge 0}$ we have ${|q^{k+1-a_1}|\le |q|^{1/2}\le 0.067}$. Applying with ${r=0.067}$ and with suitable $m$ to each logarithmic term of the right-hand side of except the term ${\log(1-q^{a_1}e^{2\pi i a_2})}$, we obtain with the error term ${\bigl(1.08(2n-1)+ 2.02)q^{(\nu+1)/N}}$. Since ${n\le \nu/N+1}$, the latter quantity is bounded by ${(2.2\nu/N+3.1)q^{(\nu+1)/N}}$, as wanted.
In a similar way one treats the case ${1/2\le a_1<1}$, but now the term ${\log(1-q^{1-a_1}e^{-2\pi i a_2})}$ must be excluded.
In the case ${a_1=0}$ the proofs are similar and simpler, and we omit them. [[$\square$]{}]{}
When $|q|^{1/N}$ is small enough, the extra term can be omitted as well.
\[casympnolog\] In the set-up of Proposition \[pasympform\] assume that ${|q|\le 2^{-N}}$. Then $$\begin{aligned}
\log\frac{g_{{\mathbf a}}(q)}{\gamma_{{\mathbf a}}q^{\ell_{{\mathbf a}}}} &= O_1(3.2|q|^{1/N}) ,\\
\label{easympnolog}
\log\frac{g_{{\mathbf a}}(q)}{\gamma_{{\mathbf a}}q^{\ell_{{\mathbf a}}}} &= \sum_{k=1}^\nu\beta_kq^{k/N} + O_1\bigl((2.2\nu/N+5.1) q^{(\nu+1)/N}\bigr) . \end{aligned}$$
#### Proof
We may assume that ${0< a_1< 1}$ and combine or with . [[$\square$]{}]{}
General Modular Units {#squad}
=====================
In this section we review and complement some of the results of Kubert and Lang [@KL81]. Our purpose is to construct “economical” modular units on the curve $X_G$.
The “naive” way to do is as follows. Let $G$ be a subgroup of ${{{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})}$ and $H$ a subgroup of $\det G$, which itself is a subgroup in $({{\mathbb Z}}/N{{\mathbb Z}})^\times$, viewed as the Galois group of the cyclotomic field ${{\mathbb Q}}(\zeta_N)$. Then $H$ left-acts naturally on the set of the cusps of $X_G$. Denote by $\nu_\infty(G)$ the number of cusps and by $\nu_\infty(G, H)$ the number $H$-orbits of cusps.
On the other hand, the group $G_H$, defined in , right-acts on the set $(N^{-1}{{\mathbb Z}}/{{\mathbb Z}})^2$. If ${{{\mathcal O}}\subset (N^{-1}{{\mathbb Z}}/{{\mathbb Z}})^2}$ is a non-zero orbit of this action, then $$\label{euo}
\prod_{{{\mathbf a}}\in {{\mathcal O}}}u_{{\mathbf a}}$$ is a rational function on the curve $X_G$ defined over the field ${{{\mathbb Q}}(\zeta_N)^H}$.
It is not difficult to deduce from Theorem \[tmdr\] that the principal divisors defined by products , where ${{\mathcal O}}$ runs the non-zero $G_H$-orbits, generate a free abelian group whose rank is ${\nu_\infty(G, H)-1}$.
Product can be written as $$\label{ega12N}
\prod_{{{\mathbf a}}\in {{\mathcal O}}}g_{{\tilde{{\mathbf a}}}}^{12N},$$ where ${{{\tilde{{\mathbf a}}}}\in N^{-1}{{\mathbb Z}}^2}$ is a lifting of ${{{\mathbf a}}\in (N^{-1}{{\mathbb Z}}/{{\mathbb Z}})^2}$. It turns out that in many interesting cases the exponents $12N$ can be considerably reduced, which is important for numerical purposes. This is the principal goal of this section.
Quadratic Relations {#ssquad}
-------------------
Let $N$ be a positive integer. We have the natural group isomorphism ${(N^{-1}{{\mathbb Z}}/{{\mathbb Z}})^2\cong ({{\mathbb Z}}/N{{\mathbb Z}})^2}$, and, with some abuse of speech, we identify the two groups. In particular, for ${{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}$ we have the corresponding element in ${(N^{-1}{{\mathbb Z}}/{{\mathbb Z}})^2}$, and for this latter we may fix a lifting in ${N^{-1}{{\mathbb Z}}^2}$, which will be called a lifting of ${{\mathbf a}}$ to ${N^{-1}{{\mathbb Z}}^2}$. By a lifting of a set ${A\subset ({{\mathbb Z}}/N{{\mathbb Z}})^2}$ we mean a mapping ${A\to N^{-1}{{\mathbb Z}}^2}$ such that for every ${{{\mathbf a}}\in A}$ its image ${{{\tilde{{\mathbf a}}}}\in N^{-1}{{\mathbb Z}}^2}$ is a lifting of ${{\mathbf a}}$ in the sense defined above.
Our principal tool will be the following result of Kubert and Lang [@KL81], see Theorem 5.2 in Chapter 3.
\[tquad\]
To every non-zero ${{{\mathbf a}}=(a_1,a_2) \in ({{\mathbb Z}}/N{{\mathbb Z}})^2}$ we associate an integer $m({{\mathbf a}})$. Fix a lifting ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}}$ of the set of non-zero elements of ${({{\mathbb Z}}/N{{\mathbb Z}})^2}$. Put $$\label{esumma}
\Lambda=\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}m({{\mathbf a}}).$$
1. Assume that $N$ is odd. Then $$\label{eprodkama}
\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}{{\mathfrak k}}_{{\tilde{{\mathbf a}}}}^{m({{\mathbf a}})}$$ is $\Gamma(N)$-automorphic (of level $-\Lambda$) if and only if $$\label{equadN}
\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}m({{\mathbf a}})a_1^2=
\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}m({{\mathbf a}})a_2^2=
\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}m({{\mathbf a}})a_1a_2= 0.$$
2. \[iprodgama\] Assume that ${\gcd(N,6)=1}$. Then the function $$\label{eprodgama}
\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}g_{{\tilde{{\mathbf a}}}}^{m({{\mathbf a}})}$$ is $\Gamma(N)$-automorphic (of level $0$) if and only if holds and ${12\mid \Lambda}$.
\[rquad\]
1. Kubert and Lang call “quadratic relations” (modulo $N$).
2. One may notice that $$\label{ewithdelta}
\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}g_{{\tilde{{\mathbf a}}}}^{m({{\mathbf a}})}=\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}{{\mathfrak k}}_{{\tilde{{\mathbf a}}}}^{m({{\mathbf a}})} \cdot \Delta^{\Lambda/12},$$ where ${\Delta=\eta^{24}}$.
3. The assumption ${\gcd(N,6)=1}$ is purely technical: in a slightly modified form the statement holds true when $N$ is divisible by $2$ and/or by $3$. However, assuming that ${\gcd(N,6)=1}$ will not hurt us, since we shall apply Theorem \[tquad\] only when $N$ is prime and ${N\ge 7}$.
4. \[izetan\] Theorem \[tquad\] implies that product defines a function ${f\in {{\mathbb C}}\bigl(X(N)\bigr)}$. By considering the $q$-expansion, as in Subsection \[sssmu\], we conclude that in fact ${f\in {{\mathbb Q}}(\zeta_N)\bigl(X(N)\bigr)}$.
Contrary to product , product may depend on the choice of the lifting ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}}$. Proposition \[pprokl\]:\[iklemod1\] implies that if we choose a different lifting ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}'}$ then and will be multiplied by a $2N$-th root of unity. Though this is pretty trivial, we state this as a proposition for further reference.
\[ptriv\] For every non-zero ${{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}$ pick an integer $m({{\mathbf a}})$ and fix **two** liftings ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}}$ and ${{{\mathbf a}}\mapsto {{\tilde{{\mathbf a}}}}'}$ of the set of non-zero elements of ${({{\mathbb Z}}/N{{\mathbb Z}})^2}$. Then there exists a $2N$-th root of unity ${\varepsilon}$ such that $$\label{etrivkle}
\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}{{\mathfrak k}}_{{{\tilde{{\mathbf a}}}}'}^{m({{\mathbf a}})} ={\varepsilon}\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}{{\mathfrak k}}_{{\tilde{{\mathbf a}}}}^{m({{\mathbf a}})}.$$ If, in addition, ${12\mid \Lambda}$, where $\Lambda$ defined in , then $$\label{etrivg}
\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}g_{{{\tilde{{\mathbf a}}}}'}^{m({{\mathbf a}})} ={\varepsilon}\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}g_{{\tilde{{\mathbf a}}}}^{m({{\mathbf a}})}.$$ If every $2\mid m({{\mathbf a}})$ for every ${{\mathbf a}}$ then $$\label{etriveps}
{\varepsilon}^N=1.$$
#### Proof
Statements and follow from Proposition \[pprokl\]:\[iklemod1\], and follows from and . [[$\square$]{}]{}
Galois Action
-------------
As we mentioned in Subsection \[sssmu\], the Galois action by the group ${{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})$ on the “simplest” modular units ${u_{{\mathbf a}}=g_{{\mathbf a}}^{12N}}$ is very easy to describe: it is given by relation . We want to obtain a similar result for “general” modular units .
\[psiegal\] Assume the set-up of item \[iprodgama\] of Theorem \[tquad\], so that $$f=\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}g_{{\tilde{{\mathbf a}}}}^{m({{\mathbf a}})}$$ defines a function in ${{\mathbb Q}}(\zeta_N)(X_G)$ (see item \[izetan\] in Remark \[rquad\]).
1. \[isl2\] Assume that ${\sigma \in {{\mathop{\mathrm{SL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})}$ and let ${{\tilde\sigma}}$ be a lifting of $\sigma$ to $\Gamma(1)$. Then $$\label{efsigma}
f^\sigma=\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}g_{{{\tilde{{\mathbf a}}}}{{\tilde\sigma}}}^{m({{\mathbf a}})}.$$
2. \[igl2\] Assume that ${\sigma \in {{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})}$. Then it has a lifting ${{{\tilde\sigma}}\in \mathrm M_2({{\mathbb Z}})}$ such that holds.
#### Proof
Item \[isl2\] is a consequence of Proposition \[pgalomod\]:\[ifsig\], Proposition \[pprokl\]:\[iklega1\] and . Indeed, write ${{{\tilde\sigma}}=(\begin{smallmatrix}a&b\\c&d\end{smallmatrix})}$, and recall that ${\Delta=\eta^{24}}$ is $\Gamma(1)$-automorphic of weight $12$. We obtain: $$\begin{aligned}
f^\sigma(\tau)&=f\circ{{\tilde\sigma}}(\tau)\\
&= \prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}\bigl({{\mathfrak k}}_{{\tilde{{\mathbf a}}}}\circ{{\tilde\sigma}}(\tau)\bigr)^{m({{\mathbf a}})} \cdot \bigl(\Delta\circ{{\tilde\sigma}}(\tau)\bigr)^{\Lambda/12}\\
&=(c\tau+d)^{-\Lambda}\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}{{\mathfrak k}}_{{{\tilde{{\mathbf a}}}}{{\tilde\sigma}}}(\tau)^{m({{\mathbf a}})}\cdot (c\tau+d)^{\Lambda} \Delta(\tau)^{\Lambda/12}\\
&=\prod_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in ({{\mathbb Z}}/N{{\mathbb Z}})^2}{{{\mathbf a}}\ne 0}}g_{{{\tilde{{\mathbf a}}}}{{\tilde\sigma}}}(\tau)^{m({{\mathbf a}})},\end{aligned}$$ as wanted.
In the proof of item \[igl2\] we may assume that $\sigma$ is of the form $(\begin{smallmatrix}1&0\\0&d\end{smallmatrix})$, because any ${\sigma\in {{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})}$ can be presented as $\sigma_1\sigma_2$ with ${\sigma_1 \in {{\mathop{\mathrm{SL}}}}_2({{\mathbb Z}}/N{{\mathbb Z}})}$ and $\sigma_2$ of this form. We lift ${\sigma=(\begin{smallmatrix}1&0\\0&d\end{smallmatrix})}$ as ${{{\tilde\sigma}}=(\begin{smallmatrix}1&0\\0&{{\tilde{d}}}\end{smallmatrix})}$, and the result follows immediately from Proposition \[pgalomod\]:\[igalqexp\] and infinite product . [[$\square$]{}]{}
“Economical” Modular Units on $X_G$
-----------------------------------
In this subsection, to avoid technicalities, we restrict to the prime level. Thus, let $p$ be a prime number, $G$ a subgroup in ${{\mathop{\mathrm{GL}}}}_2({{\mathbb F}}_p)$ and $H$ a subgroup in $\det G$. The group $G_H$, defined in , right-acts on the set ${M_p={{\mathbb F}}_p^2\smallsetminus \{0\}}$ (as in the previous subsection, we tacitly identify the sets ${{\mathbb F}}_p^2$ and ${(p^{-1}{{\mathbb Z}}/{{\mathbb Z}})^2}$). Let ${{{\mathcal O}}\subset M_p}$ be an orbit of this action, or, more generally, a $G_H$-invariant subset of $M_p$. We fix a lifting ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}}$ of the set ${{\mathcal O}}$ (as defined in the beginning of Subsection \[ssquad\]) and want to find an exponent $m$ such that $$\label{euoo}
\prod_{{{\mathbf a}}\in {{\mathcal O}}}g_{{\tilde{{\mathbf a}}}}^m$$ defines a function in $K(X_G)$, where ${K={{\mathbb Q}}(\zeta_p)^H}$. Clearly, ${m=12p}$ would do. It turns out that in some cases one can do much better, sometimes introducing a root of unity factor. We fix a $p$-th primitive root of unity and denote it by $\zeta_p$.
\[tecoun\] Let ${p\ge 5}$ be a prime number and ${G\ni-I}$ a semi-simple subgroup of ${{\mathop{\mathrm{GL}}}}_2({{\mathbb F}}_p)$ (with is equivalent to saying that $|G|$ is not divisible by $p$). Let $H$ be a subgroup of $\det G$ and ${{{\mathcal O}}\subset M_p}$ a $G_H$-invariant subset of $M_p$ satisfying $$\label{equadoo}
\sum_{{{\mathbf a}}\in {{\mathcal O}}}a_1^2= \sum_{{{\mathbf a}}\in {{\mathcal O}}}a_1a_2=\sum_{{{\mathbf a}}\in {{\mathcal O}}}a_2^2.$$ Let $m$ be an integer such that $$\label{e12moo}
2\mid m, \qquad 12\mid m|{{\mathcal O}}|.$$ Fix a lifting ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}}$ of the set ${{\mathcal O}}$ and denote by $f$ product . Then $f$ defines a function in ${{\mathbb Q}}(\zeta_p)(X_G)$ (denoted by $f$ as well). Further, there exists ${k\in {{\mathbb Z}}}$ (which is unique $\bmod p$ when ${H\ne 1}$) such that ${\zeta_p^kf\in K(X_G)}$, where ${K={{\mathbb Q}}(\zeta_p)^H}$.
The proof requires a lemma, which is the simplest special case of Kummer’s theory (see any textbook in algebra).
\[lkum\] Let $p$ be a prime number and $F$ a field of characteristic distinct from $p$. Let $\alpha$ be an element in the algebraic closure $\bar F$, and ${\zeta_p\in \bar F}$ a primitive $p$-th root of unity. Assume that ${\alpha^p\in F}$. Then either ${[F(\alpha):F]=p}$ or there exists ${k\in {{\mathbb Z}}}$ (which is unique $\bmod p$ when ${\zeta_p\notin F}$) such that ${\zeta_p^k\alpha \in F}$. In particular, if ${\zeta_p\in F}$ then either ${[F(\alpha):F]=p}$ or ${\alpha\in F}$.
#### Proof of Theorem \[tecoun\]
Theorem \[tquad\] (together with item \[izetan\] of Remark \[rquad\]) implies that $f$ defines a function in ${{\mathbb Q}}(\zeta_p)\bigl(X(p)\bigr)$. We want to study the Galois action of $G_H$ on $f$. Thus, fix ${\sigma \in G_H}$. Proposition \[psiegal\]:\[igl2\] implies that there exists a lifting ${{{\tilde\sigma}}\in \mathrm M_2({{\mathbb Z}})}$ such that $$\label{efsigoo}
f^\sigma= \prod_{{{\mathbf a}}\in {{\mathcal O}}} g_{{{\tilde{{\mathbf a}}}}{{\tilde\sigma}}}^m.$$ Since ${{\mathcal O}}$ is $G_H$-invariant, we have ${{{\mathcal O}}\sigma^{-1}={{\mathcal O}}}$. Consider a different lifting ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}'}$ of ${{\mathcal O}}$ defined by ${{{\tilde{{\mathbf a}}}}'= \widetilde{{{\mathbf a}}\sigma^{-1}}{{\tilde\sigma}}}$, where ${\widetilde{{{\mathbf a}}\sigma^{-1}}}$ is the lifting of ${{{\mathbf a}}\sigma^{-1}}$. Then can be re-written as $$f^\sigma= \prod_{{{\mathbf a}}\in {{\mathcal O}}} g_{{{\tilde{{\mathbf a}}}}'}^m.$$ Now Proposition \[ptriv\] implies that ${f^\sigma/f}$ is a $p$-th root of unity. We have proved that $f^p$ is invariant under the Galois action by $G_H$, which implies that ${f^p \in K(X_G)}$, the $G_H$-invariant subfield of ${{\mathbb Q}}(\zeta_p)\bigl(X(p)\bigr)$. Now Lemma \[lkum\] completes the proof. [[$\square$]{}]{}
There is an important special case when $f$ itself belongs to $K(X_G)$, without multiplication by a root of unity. Assume that $G_H$ contains $(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix})$. In this case ${{{\mathbf a}}=(a_1,a_2)}$ belongs to a $G_H$-orbit ${{\mathcal O}}$ if and only if its “complex conjugate” ${\bar{{\mathbf a}}=(a_1,-a_2)}$ does. We say that a lifting ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}}$ *respects the complex conjugation* if the following holds: if ${{{\mathbf a}}=(a_1,a_2)\in {{\mathcal O}}}$ is lifted to ${{{\tilde{{\mathbf a}}}}=({{\tilde{a}}}_1,{{\tilde{a}}}_2)}$, then the lifting of $\bar{{\mathbf a}}$ is $({{\tilde{a}}}_1,-{{\tilde{a}}}_2)$. This can be expressed briefly as ${\tilde{\bar{{\mathbf a}}}=\bar{{\tilde{{\mathbf a}}}}}$.
\[cecoun\] In the set-up of Theorem \[tecoun\] assume that ${(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix})\in G_H}$ and the lifting respects the complex conjugation. Then ${f\in K(X_G)}$.
#### Proof
The assumption ${(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix})\in G_H}$ implies that ${K\subseteq {{\mathbb Q}}(\zeta_p+\bar\zeta_p)}$. Further, since the lifting respects the complex conjugation, we have ${f^\iota=f}$, where ${\iota=(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix})}$. The subfield of ${{\mathbb Q}}(\zeta_p)(X_G)$ stabilized by $\iota$ is ${{{\mathbb Q}}(\zeta_p+\bar\zeta_p)(X_G)}$. Thus, ${f\in {{\mathbb Q}}(\zeta_p+\bar\zeta_p)(X_G)}$ and ${\zeta_p^kf\in K(X_G)}$ with ${K\subseteq {{\mathbb Q}}(\zeta_p+\bar\zeta_p)}$. It follows that ${\zeta_p^k\in {{\mathbb Q}}(\zeta_p+\bar\zeta_p)}$ which is only possible if ${\zeta_p^k=1}$. [[$\square$]{}]{}
An Example
----------
We conclude this section with an example. It will not be used in the sequel, but it gives a good illustration of how Theorem \[tecoun\] can be used.
We take as $G$ the diagonal subgroup of ${{\mathop{\mathrm{GL}}}}_2({{\mathbb F}}_p)$ and set ${H=\{1,-1\}}$, so that $$G_H=\bigl\{(\begin{smallmatrix}a&0\\0&d\end{smallmatrix}): ad=\pm1\bigr\}$$ and ${K={{\mathbb Q}}(\zeta_p+\bar\zeta_p)}$.
There are ${(p-1)/2}$ orbits, which are of the form ${\{{{\mathbf a}}: a_1a_2=\pm c\}}$ with ${c=1, \ldots, (p-1)/2}$. Quadratic relations are clearly satisfied, and to have it suffices to take $$m=
\begin{cases}
2, & p\equiv 1\bmod3,\\
6, & p\equiv -1\bmod3.
\end{cases}$$ Selecting a lifting respecting the complex conjugation, we obtain ${(p-1)/2}$ modular units in the field $K(X_G)$.
Cusp Points and Units on $X_{{\mathop{\mathrm{ns}}}}^+(p)$ {#scpu}
==========================================================
From now on we restrict to the case when ${N=p}$ is a prime number and $G$ is the normalizer of a non-split Cartan subgroup of ${{\mathop{\mathrm{GL}}}}_2({{\mathbb Z}}/p{{\mathbb Z}})$. A very detailed account of various properties of this curve (even for an arbitrary $N$) can be found in Sections 3 and 6 of Baran’s article [@Ba10].
We may and will assume that $$\label{eexpl}
G=\left\lbrace \begin{pmatrix} \alpha & \Xi\beta \\ \beta & \alpha \end{pmatrix} , \begin{pmatrix} \alpha & \Xi\beta \\ -\beta & -\alpha \end{pmatrix} : \alpha, \beta \in {{\mathbb F}}_p,\ (\alpha,\beta)\ne (0,0) \right\rbrace ,$$ where $\Xi$ is a quadratic non-residue modulo $p$, which will be fixed from now on. In particular, one can take ${\Xi=-1}$ if ${p\equiv 3\bmod4}$.
We fix, until the end of the article, a lifting ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}}$ of the set $M_p$ to $p^{-1}{{\mathbb Z}}^2$, which respects the complex conjugation (as defined before Corollary \[cecoun\]) and, in addition to this, has the following property: $$\label{e01}
\text{if ${{{\tilde{{\mathbf a}}}}=({{\tilde{a}}}_1,{{\tilde{a}}}_2)}$ is a lifting of ${{{\mathbf a}}\in M_p}$ then ${0\le {{\tilde{a}}}_1<1}$.}$$
Cusps {#sscusps}
-----
The curve ${X_G=X_{{\mathop{\mathrm{ns}}}}^+(p)}$ has ${(p-1)/2}$ cusps, defined over the real cyclotomic fields ${{{\mathbb Q}}(\zeta_p+\bar\zeta_p)}$, and the Galois group ${{{\mathop{\mathrm{Gal}}}}\bigl({{\mathbb Q}}(\zeta_p+\bar\zeta_p)/{{\mathbb Q}}\bigr)={{\mathbb F}}_p/\{\pm1\}}$ acts transitively on the cusps.
According to Remark \[rorbits\], the cusps stay in one-to-one correspondence with the the orbits of the left $G_1$-action on the set ${M_p={{\mathbb F}}_p^2\smallsetminus\{(0,0)\}}$. These orbits are the sets defined by ${x^2-\Xi y^2=\pm c}$, where $c$ runs through (representatives of) cosets ${{{\mathbb F}}_p^\times/\{\pm1\}}$, the cusp at infinity corresponding to ${c=1}$.
For every ${c\in{{\mathbb F}}_p^\times/\{\pm1\}}$ fix ${(a,b) \in {{\mathbb F}}_p^2}$ such that ${a^2-\Xi b^2=c^{-1}}$ and let $\sigma_c$ be a lifting to $\Gamma(1)$ of the matrix $\begin{pmatrix}ca&b\Xi\\cb&a\end{pmatrix}$. For ${c=1}$ we take ${(a,b)=(1,0)}$ and ${\sigma_1=I}$. Then the set ${\bigl\{\sigma_c(i\infty): c\in{{\mathbb F}}_p^\times/\{\pm1\}\bigr\}}$ is a full system of representatives of cusps on $\bar{{\mathcal H}}$. We can complete the set ${\bigl\{\sigma_c: c\in{{\mathbb F}}_p^\times/\{\pm1\}\bigr\}}$ to an optimal system of representatives of cosets of $\Gamma_{{\mathop{\mathrm{ns}}}}^+\backslash\Gamma(1)$, as explained in Subsection \[ssopt\].
In the sequel we fix a subgroup $H$ of ${{\mathbb F}}_p^\times$, containing ${-1}$ and put ${d=[{{\mathbb F}}_p^\times:H]}$. In particular, $$d= [K:{{\mathbb Q}}],$$ where ${K={{\mathbb Q}}(\zeta_p)^H}$. The group $H$ acts on the set of cusps by Galois conjugation, and this action has exactly $d$ orbits, each of them being defined over $K$ as a set. The Galois group ${{{\mathop{\mathrm{Gal}}}}(K/{{\mathbb Q}})={{\mathbb F}}_p^\times/H}$ acts on the set of $H$-orbits transitively. These $H$-orbits of cusps are in one-to-one correspondence with the sets defined by ${x^2-\Xi y^2\in cH}$, with $cH$ running through the cosets ${{\mathbb F}}_p^\times/H$.
Units {#ssuoo}
-----
Besides the left action, the group $G_H$ acts on the set $M_p$ on the right, There are again $d$ orbits of this action, and they are defined by ${\Xi x^2- y^2\in cH}$. These orbits will be used to define modular units in $K(X_G)$. Recall that we fixed a lifting ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}}$ of $M_p$ to ${p^{-1}{{\mathbb Z}}^2}$, respecting the complex conjugation.
\[tu\] Let ${{\mathcal O}}$ be right $G_H$-orbit on $M_p$. Pick a lifting ${{{\mathbf a}}\mapsto{{\tilde{{\mathbf a}}}}}$ of ${{\mathcal O}}$ to ${p^{-1}{{\mathbb Z}}^2}$. Put $$\label{edefm}
m=
\begin{cases}
2, & 3\mid (p+1)|H|, \\
6, & \text{otherwise}.
\end{cases}$$ Then the product $$\label{efum}
u_{{\mathcal O}}=\prod_{{{\mathbf a}}\in {{\mathcal O}}}g_{{\tilde{{\mathbf a}}}}^m$$ is well-defined (depends only on the orbit ${{\mathcal O}}$ but not on the particular lifting) and defines a function in $K(X_G)$.
We deduce this theorem from Theorem \[tecoun\] (more precisely, from Corollary \[cecoun\]), using some elementary lemmas about finite fields. We thank Julia Baoulina for useful explanations and for the proof of Lemma \[lba\] below.
\[lln\] Let ${P(x_1, \ldots, x_n)\in {{\mathbb F}}[x_1, \ldots, x_n]}$ be a polynomial over a finite field ${{{\mathbb F}}={{\mathbb F}}_q}$ of degree ${\deg P< n(q-1)}$. Then $$\sum_{{{\mathbf b}}\in {{\mathbb F}}^n}P({{\mathbf b}}) =0.$$
#### Proof
This is Lemma 6.4 in [@LN97]. [[$\square$]{}]{}
\[lba\] Let ${{\mathbb F}}$ be a finite field of odd characteristic and having more than $3$ elements. Further, let ${f(x,y), g(x,y)\in {{\mathbb F}}[x,y]}$ be quadratic forms over ${{\mathbb F}}$. Then for ${c \in {{\mathbb F}}^\times}$ we have $$\sum_{\genfrac{}{}{0pt}{}{a,b\in {{\mathbb F}}}{g(a,b)=\pm c}}f(a,b)=0.$$ where the sum is over the couples ${(a,b)\in {{\mathbb F}}^2}$ such that ${g(a,b)=\pm c}$.
#### Proof
Write ${q=|{{\mathbb F}}|}$, so that ${{{\mathbb F}}={{\mathbb F}}_q}$. Then $$\sum_{\genfrac{}{}{0pt}{}{a,b\in {{\mathbb F}}}{g(a,b)=\pm c}}f(a,b)= \sum_{a,b\in {{\mathbb F}}} f(a,b)(2-(g(a,b)-c)^{q-1}-(g(a,b)+c)^{q-1}).$$ We have $$f(x,y)(2-(g(x,y)-c)^{q-1}-(g(x,y)+c)^{q-1})= -2f(x,y)g(x,y)^{q-1}+\text{terms of degree $<2(q-1)$},$$ and Lemma \[lln\] implies that the wanted sum is equal to ${-2}$ times ${\sum_{a,b\in {{\mathbb F}}}f(x,y)g(x,y)^{q-1}}$ The latter sum is $$\sum_{\genfrac{}{}{0pt}{}{a,b\in {{\mathbb F}}}{g(a,b)\ne0}}f(a,b),$$ which, again by Lemma \[lln\] and by the assumption ${q>3}$ is equal to $-1$ times $$\sum_{\genfrac{}{}{0pt}{}{a,b\in {{\mathbb F}}}{g(a,b)=0}}f(a,b).$$ If the quadratic form $g(x,y)$ is anisotropic over ${{\mathbb F}}$ then the latter sum consists only of the term $f(0,0)$ and there is nothing to prove. And if it is isotropic then, after a variable change, we may assume that ${g(x,y)=xy}$. Writing ${f(x,y)=\alpha x^2+\beta xy+\gamma y^2}$, the latter sum becomes ${(\alpha+\gamma)\sum_{a\in {{\mathbb F}}}a^2}$. Lemma \[lln\] implies that ${\sum_{a\in {{\mathbb F}}}a^2=0}$ when ${{\mathbb F}}$ has more than $3$ elements. This completes the proof. [[$\square$]{}]{}
#### Proof of Theorem \[tu\]
Recall that the orbit ${{\mathcal O}}$ consists of ${(x,y)\in {{\mathbb F}}_p^2}$ satisfying ${\Xi x^2-y^2\in cH}$ with some ${c\in {{\mathbb F}}_p^\times}$. Since ${H\ni -1}$, Lemma \[lba\] implies that the quadratic relations hold true. Further, for each ${c\in{{\mathbb F}}_p^\times}$ there is exactly ${p+1}$ elements of ${{\mathbb F}}_{p^2}$ of norm $c$, which implies that our orbit ${{\mathcal O}}$ has exactly ${(p+1)|H|}$ elements, and with our choice of $m$ the divisibility conditions hold true as well. Corollary \[cecoun\] now implies that ${u\in K(X_G)}$.
Finally, $u_{{\mathcal O}}$ does not depend on the lifting. Indeed, if we choose two different liftings respecting the complex conjugation and obtain the products, say, $u$ and $u'$, then ${u/u'}$ is a $p$-th root of unity by Proposition \[ptriv\]. On the other hand, ${u,u'\in K(X_G)}$, which implies that ${u/u'\in K}$, a totally real field. Hence ${u=u'}$. The theorem is proved. [[$\square$]{}]{}
Asymptotics
-----------
Wi fix a right $G_H$-orbit ${{\mathcal O}}$. Using the results of Subsection \[ssasymp\], we may obtain two types of asymptotic expansions for the unit $u_{{\mathcal O}}$ defined in Subsection \[ssuoo\]. Let $c$ be a cusp of $X_G$. We define the set $\Omega_c$ and the $q$-parameter $q_c$ as in Subsection \[ssfund\], and with respect to the optimal system of representatives defined in Subsection \[sscusps\]: ${q_c(\tau) = e^{2\pi i\sigma_c^{-1}(\tau) }}$. Put $$\label{egamco}
\gamma_c=\gamma_{c,{{\mathcal O}}}= \prod_{{{\mathbf a}}\in {{\mathcal O}}\sigma_c}\gamma_{{\mathbf a}}^m,$$ where $\gamma_{{\mathbf a}}$ is defined in .
It follows from the definition of $\gamma_{{\mathbf a}}$ that ${{{\mathop{\mathrm{h}}}}(\gamma_{{\mathbf a}})=0}$ if ${a_1\ne 0}$ and ${{{\mathop{\mathrm{h}}}}(\gamma_{{\mathbf a}})\le \log 2}$ if ${a_1=0}$. Hence $$\label{ehgam}
{{\mathop{\mathrm{h}}}}(\gamma_{c,{{\mathcal O}}})\le (\log 2) \bigl|\{{{\mathbf a}}\in {{\mathcal O}}: a_1=0\}\bigr|\le 2|H|\log 2.$$
\[pasympuoo\] Let $c$ be cusp of $X_G$. Then for ${k=1,2,\ldots}$ there exist algebraic numbers ${\beta_{k,c},\beta_{k,c}'\in {{\mathbb Q}}(\zeta_p)}$ such that for any absolute value $v$ of ${{\mathbb Q}}(\zeta_p)$ we have[^3] $$\label{ebetakc}
|\beta_{k,c}|_v \le
\begin{cases}
|k|_v^{-1}, & v\mid p<\infty,\\
2m(p+1)|H|(k/N+1), & v\mid \infty
\end{cases} \qquad (k=1,2,\ldots)$$ and the following holds. Let $\nu$ be a non-negative integer. Then for ${P\in \Omega_c}$ we have, with a suitable choice of logarithms, $$\begin{aligned}
\log \frac{u_{{\mathcal O}}(P)}{q_c^{{{\mathop{\mathrm{Ord}}}}_cu_{{\mathcal O}}/p}\gamma_c}&=
m\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_c}{0<{{\tilde{a}}}_1<1/2}}\log (1-q_c^{{{\tilde{a}}}_1}e^{2\pi i {{\tilde{a}}}_2})+
m\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_c}{1/2\le {{\tilde{a}}}_1<1}}\log (1-q_c^{1-{{\tilde{a}}}_1}e^{-2\pi i {{\tilde{a}}}_2})
\\& \hphantom{=} +
O_1\left(1.2m(p+1)|H||q_c|^{1/2}\right)\\
\log \frac{u_{{\mathcal O}}(P)}{q_c^{{{\mathop{\mathrm{Ord}}}}_cu_{{\mathcal O}}/p}\gamma_c}&=
m\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_c}{0<{{\tilde{a}}}_1<1/2}}\log (1-q_c^{{{\tilde{a}}}_1}e^{2\pi i {{\tilde{a}}}_2})+
m\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_c}{1/2\le {{\tilde{a}}}_1<1}}\log (1-q_c^{1-{{\tilde{a}}}_1}e^{-2\pi i {{\tilde{a}}}_2})
\\& \hphantom{=} +
\sum_{k=1}^\nu \beta_{k,c}'q_c^{k/p}+
O_1\left( m(p+1)|H|\left(2.2\nu/p+3.1\right)|q_c|^{(\nu+1)/p}\right)
$$ where here and below we write ${q_c=q_c(P)}$. If, in addition, ${|q_c(P)|\le 2^{-p}}$ then $$\begin{aligned}
\label{esmall}
\log \frac{u_{{\mathcal O}}(P)}{q_c^{{{\mathop{\mathrm{Ord}}}}_cu_{{\mathcal O}}/p}\gamma_c}&=
O_1\left( 3.2m(p+1)|H||q_c|^{1/p}\right),\\
\label{esmaller}
\log \frac{u_{{\mathcal O}}(P)}{q_c^{{{\mathop{\mathrm{Ord}}}}_cu_{{\mathcal O}}/p}\gamma_c}&=
\sum_{k=1}^\nu \beta_{k,c}q_c^{k/p}+
O_1\left( m(p+1)|H|\left(2.2\nu/p+5.1\right)|q_c|^{(\nu+1)/p}\right).
$$
#### Proof
In the case ${c=c_\infty}$ this is an immediate consequence of the results of Subsection \[ssasymp\]. In particular, the error terms from and and the archimedean part of the estimate should be multiplied by ${m(p+1)|H|}$, because $u_{{\mathcal O}}$ is a product of exactly $m|{{\mathcal O}}|$ Siegel functions, and ${|{{\mathcal O}}|=(p+1)|H|}$. The general case is treated similarly, using the variable change ${\tau\mapsto\sigma_c\tau}$. [[$\square$]{}]{}
Taking the real parts, we obtain the following consequence.
\[casympuoo\] In the set-up of Proposition \[pasympuoo\] we have $$\begin{aligned}
\log |u_{{\mathcal O}}(P)|&= \frac{{{\mathop{\mathrm{Ord}}}}_cu_{{\mathcal O}}}p \log|q_c|+ \log|\gamma_c|
\\& \hphantom{=} +
m\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_c}{0<{{\tilde{a}}}_1<1/2}}\log |1-q_c^{{{\tilde{a}}}_1}e^{2\pi i {{\tilde{a}}}_2}|+
m\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_c}{1/2\le {{\tilde{a}}}_1<1}}\log |1-q_c^{1-{{\tilde{a}}}_1}e^{-2\pi i {{\tilde{a}}}_2}|
\\& \hphantom{=} +
O_1\left(1.2m(p+1)|H||q_c|^{1/2}\right)
\end{aligned}$$ If ${|q_c(P)|\le 2^{-p}}$ then we also have $$\begin{aligned}
\log |u_{{\mathcal O}}(P)|&= \frac{{{\mathop{\mathrm{Ord}}}}_cu_{{\mathcal O}}}p \log|q_c|+
O_1\left(3.2m(p+1)|H||q_c|^{1/p}\right).
\end{aligned}$$ If ${q_c(P)\in {{\mathbb R}}}$ and $\nu$ is a positive integer then we also have $$\begin{aligned}
\log |u_{{\mathcal O}}(P)|&= \frac{{{\mathop{\mathrm{Ord}}}}_cu_{{\mathcal O}}}p \log|q_c|+ \log|\gamma_c|
\\& \hphantom{=} +
m\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_c}{0<{{\tilde{a}}}_1<1/2}}\log |1-q_c^{{{\tilde{a}}}_1}e^{2\pi i {{\tilde{a}}}_2}|+
m\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_c}{1/2\le {{\tilde{a}}}_1<1}}\log |1-q_c^{1-{{\tilde{a}}}_1}e^{-2\pi i {{\tilde{a}}}_2}|
\\& \hphantom{=} +
\sum_{k=1}^\nu {{\mathop{\mathrm{Re}}}}(\beta_{k,c}')q_c^{k/p}+
O_1\left( m(p+1)|H|\left(2.2\nu/p+3.1\right)|q_c|^{(\nu+1)/p}\right).
\end{aligned}$$
We complete this subsection by estimating the orders of $u_{{\mathcal O}}$ at the cusps, and its degree as a rational function on $X_G$. Clearly, $${{\mathop{\mathrm{Ord}}}}_cu_{{\mathcal O}}=pm\sum_{{{\mathbf a}}\in {{\mathcal O}}\sigma_c}\ell_{{\mathbf a}},$$ where $\ell_{{\mathbf a}}$ is defined in Proposition \[pella\]. Since ${|\ell_{{\mathbf a}}|\le 1/12}$, this has the following consequence, to be used in Section \[sbans\].
\[pordeg\] For any cusp $c$ we have ${{{\mathop{\mathrm{Ord}}}}_cu_{{\mathcal O}}\le \frac1{12}mp(p+1)|H|}$. The degree of $u_{{\mathcal O}}$ as a rational function on $X_G$ does not exceed ${\frac1{48}mp(p^2-1)|H|}$.
Galois Action on the Units {#ssgalou}
--------------------------
Consider first the case of general algebraic curves. The proof of the following proposition is a standard exercise in Galois theory.
Let $K/k$ be a finite Galois extension of fields of characteristic $0$, and let $X$ be a projective curve defined (that is, having a geometrically irreducible model) over $k$. Then the extension $K(X)/k(X)$ is Galois, and the restriction map $${{\mathop{\mathrm{Gal}}}}\bigl(K(X)/k(X)\bigr)\to {{\mathop{\mathrm{Gal}}}}(K/k), \quad \sigma\mapsto\sigma\vert_K$$ defines isomorphism of Galois groups. Further, for ${P\in X(k)}$ and ${f\in K(X)}$ we have ${f(P)\in K}$, and given ${\sigma \in {{\mathop{\mathrm{Gal}}}}\bigl(K(X)/k(X)\bigr)= {{\mathop{\mathrm{Gal}}}}(K/k)}$, we have ${f^\sigma(P)=f(P)^\sigma}$.
In our case the group $${{\mathop{\mathrm{Gal}}}}\bigl(K(X_G)/{{\mathbb Q}}(X_G)\bigr)={{\mathop{\mathrm{Gal}}}}(K/{{\mathbb Q}})=G/G_H={{\mathbb F}}_p^\times/H$$ acts transitively and faithfully on the right $G_H$-orbits, and this action agrees with the Galois action: for ${\sigma \in {{\mathop{\mathrm{Gal}}}}(K/{{\mathbb Q}})={{\mathbb F}}_p^\times/H}$ we have ${u_{{\mathcal O}}^\sigma=u_{{{\mathcal O}}\sigma}}$. Fixing an orbit ${{\mathcal O}}$ and putting ${U=u_{{\mathcal O}}}$, we obtain the following.
For ${P\in X_G({{\mathbb Q}})}$ we have ${U(P)\in K}$ and ${U^\sigma(P)=U(P)^\sigma}$ for ${\sigma \in {{\mathop{\mathrm{Gal}}}}(K/{{\mathbb Q}})}$.
Since distinct orbits are disjoint, Theorem \[tmdr\] and the discussion thereafter have the following consequence (recall that ${d=[K:{{\mathbb Q}}]=[{{\mathbb F}}_p^\times:H]}$).
\[pmdrns\] The $d$ principal divisors $(U^\sigma)$, where ${\sigma \in {{\mathop{\mathrm{Gal}}}}(K/{{\mathbb Q}})}$, generate an abelian group of rank ${d-1}$, the only relation being ${\sum_\sigma(U^\sigma)=0}$. In particular, if ${d\ge 3}$ and ${\sigma \ne 1}$ then $U$ and $U^\sigma$ are multiplicatively independent modulo the constants.
Finally, equation implies that $$\label{eprodusig}
\prod_{\sigma \in {{\mathop{\mathrm{Gal}}}}(K/{{\mathbb Q}})}U^\sigma = \pm p^m$$
The Principal Relation {#sprinrel}
======================
We retain the set-up of Section \[scpu\] and, in particular, of Subsection \[ssgalou\]:
- ${p\ge 5}$ is a prime number, $\zeta_p$ is a primitive $p$-th root of unity;
- ${{{\mathbf a}}\mapsto {{\tilde{{\mathbf a}}}}}$ is a lifting of the set ${M_p={{\mathbb F}}_p^2\smallsetminus\{(0,0)\}}$ which respects the complex conjugation and satisfies ;
- $G$ is the normalizer of a non-split Cartan subgroup of ${{\mathop{\mathrm{GL}}}}_2({{\mathbb F}}_p)$, realized as in ;
- $H$ is a subgroup of ${{\mathbb F}}_p^\times$, ${H\ni -1}$;
- ${m=2}$ or ${6}$ according to .
- ${K={{\mathbb Q}}(\zeta_p)^H}$, ${d=[K:{{\mathbb Q}}]=[{{\mathbb F}}_p^\times:H]}$;
- ${{\mathcal O}}$ is a fixed right $G_H$-orbit in ${M_p={{\mathbb F}}_p\smallsetminus\{0,0\}}$, ${U=u_{{\mathcal O}}}$ as defined in Theorem \[tu\].
We fix a system ${\eta_1, \ldots, \eta_{d-1}}$ of fundamental units of the field $K$. We also put $$\label{eeta0}
\eta_0={{\mathcal N}}_{{{\mathbb Q}}(\zeta_p)/K}(1-\zeta_p).$$ Clearly, $$\label{eheta0}
{{\mathop{\mathrm{h}}}}(\eta_0)\le |H|\log2.$$ Also, $\eta_0$ generates the prime ideal ${{\mathfrak p}}$ of $K$ above $p$; recall that ${{{\mathfrak p}}^d=(p)}$.
Recall that we call a point ${P\in X_G({{\mathbb Q}})}$ *integral* if ${j(P)\in {{\mathbb Z}}}$. Proposition \[psiu\] implies that for an integral point $P$ on $X_G$, the principal ideal $\bigl(U(P)\bigr)$ is an integral ideal of the field $K$, and, moreover, it is a power of ${{\mathfrak p}}$. Since ${{{\mathfrak p}}^\sigma={{\mathfrak p}}}$ for ${\sigma\in {{\mathop{\mathrm{Gal}}}}(K/{{\mathbb Q}})}$, relation implies that ${\bigl(U(P)\bigr)= {{\mathfrak p}}^m}$. Thus, we have $$\label{eup}
U(P) = \pm \eta_0^{b_0}\eta_1^{b_1}\cdots \eta_{d_1}^{b_{d-1}},$$ where ${b_0=m}$ and ${b_1, \cdots, b_{d-1}}$ are some rational integers depending on $P$.
The purpose of this section is to express the exponents $b_k$ in terms of the point $P$; more precisely, in terms of $q_c(P)$, where $c$ is the nearest cusp to $P$ (Subsection \[ssfund\]). This can be viewed as an analogue of equation (20) on page 378 of [@BH96].
For ${\sigma \in {{\mathop{\mathrm{Gal}}}}(K/{{\mathbb Q}})}$ we have $$U^\sigma(P) = \pm (\eta_0^\sigma)^{b_0}(\eta_1^\sigma)^{b_1}\cdots (\eta_{d_1}^\sigma)^{b_{d-1}}.$$ Fix an ordering on the elements of the Galois group: ${{{\mathop{\mathrm{Gal}}}}(K/{{\mathbb Q}})=\{\sigma_0={{\mathop{\mathrm{id}}}}, \sigma_1, \ldots, \sigma_{d-1}\}}$. Since the real algebraic numbers ${\eta_0,\eta_1, \ldots, \eta_{d-1}}$ are multiplicatively independent, the ${d\times d}$ real matrix $\bigl(\log |\eta_\ell^{\sigma_k}|\bigr)_{0\le k,\ell\le d-1}$ is non-singular. Let $\bigl(\alpha_{k\ell}\bigr)_{0\le k,\ell\le d-1}$ be the inverse matrix. Then $$\label{ebkprel}
b_k = \sum_{\ell=0}^{d-1}\alpha_{k\ell}\log|U^{\sigma_\ell}(P)| \qquad (k=0, 1,\ldots, d-1).$$
Now, combining with Corollary \[casympuoo\], we may express $b_k$ in terms of $P$. Let us introduce some notation. Let $c$ be a cusp of $X_G$, and $q_c$ is the corresponding $q$-parameter (with respect to the optimal system of representatives defined in Subsection \[sscusps\]). Define the following quantities: $$\label{edelvart}
\begin{gathered}
\delta_{c,k} = -\frac1p\sum_{\ell=0}^{d-1}\alpha_{k\ell}{{\mathop{\mathrm{Ord}}}}_cU^{\sigma_\ell},\qquad
\gamma_{c,\ell} = \prod_{{{\mathbf a}}\in {{\mathcal O}}\sigma_\ell\sigma_c}\gamma_{{\mathbf a}}^m, \qquad
\vartheta_{c,k}= \sum_{\ell=0}^{d-1}\alpha_{k\ell}\log|\gamma_{c,\ell}|,\\
\kappa=\max_k\sum_{\ell=0}^{d-1} |\alpha_{k\ell}| ,\qquad \Theta=\kappa m(p+1)|H| .
\end{gathered}$$ where $\gamma_{{\mathbf a}}$ is defined in .
\[rtriv\] It is easy to see that ${\delta_{c,0}=0}$ and at least one of the numbers ${\delta_{c,1}, \delta_{c,2}, \ldots,\delta_{c,d-1}}$ is non-zero. Indeed, we have $$\label{etriv}
\left(\begin{smallmatrix}{{\mathop{\mathrm{Ord}}}}_cU^{\sigma_0}\\{{\mathop{\mathrm{Ord}}}}_cU^{\sigma_1}\\ \vdots\\{{\mathop{\mathrm{Ord}}}}_cU^{\sigma_{d-1}}\end{smallmatrix}\right)=\bigl(\log |\eta_\ell^{\sigma_k}|\bigr)_{0\le k,\ell\le d-1}\left(\begin{smallmatrix}\delta_{c,0}\\\delta_{c,1}\\ \vdots\\\delta_{c,d-1}\end{smallmatrix}\right).$$ Multiplying both sides by the $d$-line $(1,\ldots,1)$ on the left, we obtain ${\delta_{c,0}=0}$. Further, since the column-vector on the left of is non-zero, neither is the column-vector on the right.
\[pformula20\] Let $P$ be an integral point on $X_G$ having $c$ as the nearest cusp (that is, ${P\in \Omega_c}$). Then for ${k=0, \ldots, d-1}$ we have $$\label{erelation1}
\begin{aligned}
b_k &= \delta_{c,k}\log|q_c^{-1}|+ \vartheta_{c,k} \\
&\hphantom{=} + m\sum_{\ell=0}^{d-1}\alpha_{k\ell} \left( \sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_{\ell}\sigma_c}{0<{{\tilde{a}}}_1<1/2}}\log|1-q_c^{{{\tilde{a}}}_1}e^{2\pi i{{\tilde{a}}}_2}|+\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_{\ell}\sigma_c}{1/2\le{{\tilde{a}}}_1<1}}\log|1-q_c^{1-{{\tilde{a}}}_1}e^{-2\pi i{{\tilde{a}}}_2}|\right) \\
&\hphantom{=} +O_1\bigl(1.2\Theta |q_c|^{1/2}\bigr),
\end{aligned}$$ where here and below we write $q_c$ for $q_c(P)$. If, in addition, ${|q_c(P)|\le 2^{-p}}$ then we also have $$\label{erelation2}
b_k= \delta_{c,k}\log|q_c^{-1}|+ \vartheta_{c,k} +O_1\bigl( 3.2\Theta |q_c|^{1/p}\bigr).$$
Further, let $\nu$ be a positive integer. Then there exist polynomials ${Q_{c,k}(t)\in{{\mathbb R}}[t]}$ of degree at most $\nu$ and satisfying ${Q_{c,k}(0)=0}$ such that the following holds. Assume that $$\label{enot11727}
j(P)\notin\{1,2,\ldots, 1727\}$$ Then for ${k=0, \ldots, d-1}$ we have $$\label{erelation3}
\begin{aligned}
b_k &= \delta_{c,k}\log|q_c^{-1}|+ \vartheta_{c,k} \\
&\hphantom{=} + m\sum_{\ell=0}^{d-1}\alpha_{k\ell} \left( \sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_{\ell}\sigma_c}{0<{{\tilde{a}}}_1<1/2}}|\log|1-q_c^{{{\tilde{a}}}_1}e^{2\pi i{{\tilde{a}}}_2}|+\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_{\ell}\sigma_c}{1/2\le{{\tilde{a}}}_1<1}}\log|1-q_c^{1-{{\tilde{a}}}_1}e^{-2\pi i{{\tilde{a}}}_2}|\right) \\
&\hphantom{=} + Q_{c,k}(q_c^{1/p}) + O_1\left(\Theta\left(2.2\nu/p+3.1\right)|q_c|^{(\nu+1)/p}\right).
\end{aligned}$$
(We omit the explicit formulas for the polynomials $Q_{c,k}$, which are similar to those for the real numbers $\delta_{c,k}$ and $\vartheta_{c,k}$.)
#### Proof
As follows from Proposition \[preal\], assumption implies that ${q_c(P)\in {{\mathbb R}}}$. Now combining and Corollary \[casympuoo\] we complete the proof. [[$\square$]{}]{}
In particular, we may bound $b_k$ in terms of $q_c(P)$. Put $$\label{ebb}
B=B(P)=\max\{|b_1|, \ldots, |b_{d-1}|\}$$
In the set-up of Proposition \[pformula20\] we have $$\label{ebq}
B\le \delta_{\max} \log|q_c^{-1}|+\vartheta_{\max}+ \Theta\log p .$$ where $$\label{edeltamax}
\delta_{\max}=\delta_{\max,c}=\max_k|\delta_{c,k}|, \quad \vartheta_{\max}=\vartheta_{\max,c}=\max_k|\vartheta_{c,k}|.$$ If, in addition, ${|q_c(P)|\le 2^{-p}}$ then we also have $$\begin{aligned}
\label{eboundb1}
B&\le \delta_{\max}\log|q_c^{-1}|+ \vartheta_{\max} +3.2\Theta |q_c|^{1/p}\\
\label{eboundb2}
&\le \delta_{\max}\log|q_c^{-1}|+ \vartheta_{\max} +1.6\Theta. \end{aligned}$$
#### Proof
For ${e^{-r}<|z|<1}$ we have $$\label{estup}
\bigl|\log|1-z|\bigr|\le \max\left\{\log2, \log\left|\frac1{\log |z|}\right|+\log\frac r{1+e^{-r}}\right\}.$$ Since ${|q_c^{{{\tilde{a}}}_1}|,|q_c^{1-{{\tilde{a}}}_1}| \le e^{-(\pi\sqrt3)/p}}$, applying with ${r=(\pi\sqrt3)/5}$ (recall that ${p\ge 5}$) gives $$\bigl|\log|1-q_c^{{{\tilde{a}}}_1}e^{2\pi i{{\tilde{a}}}_2}|\bigr|\le \max\left\{\log 2, \log\frac p{\pi\sqrt3}+0.5\right\}\le \log p-0.5,$$ and similarly for ${\bigl|\log|1-q_c^{1-{{\tilde{a}}}_1}e^{-2\pi i{{\tilde{a}}}_2}|\bigr|}$. Hence the sum of the logarithmic terms in the right-hand side of is bounded in absolute value by ${m\kappa|{{\mathcal O}}| (\log p-0.5)}$. This proves . Finally is immediate from , and is immediate from . [[$\square$]{}]{}
Baker’s Method on $X_{{\mathop{\mathrm{ns}}}}^+(p)$ {#sbans}
===================================================
In this section we bound the quantity $B(P)$ defined in using Baker’s method. We mainly follows [@BS13], with appropriate changes.
We shall use Baker’s inequality in the following form, due to Matveev [@Ma00 Corollary 2.3].
\[tmatveev\] Let $L$ be a number field of degree $\delta$ over ${{\mathbb Q}}$, embedded in ${{\mathbb C}}$. Let ${\alpha_{1}, \dots, \alpha_{n}}$ be non-zero elements of $L$, and ${b_{1}, \dots, b_n}$ rational integers. We fix some values of logarithms ${\log\alpha_1, \ldots, \log\alpha_n}$ and put $$\Lambda=b_{1} \log{\alpha_{1}}+\cdots + b_{n} \log{\alpha_{n}}.$$ Further, let real numbers ${A_1, \ldots, A_n}$ satisfy $$\label{eaks}
A_k\ge \max \bigl\lbrace \delta{{\mathop{\mathrm{h}}}}(\alpha_k), |\log\alpha_k|, 0.16 \bigr\rbrace,$$ where ${{\mathop{\mathrm{h}}}}(\cdot)$ is the absolute logarithmic height. Finally, put $$\mho=A_{1}\cdots A_{n}, \quad C(n)= 40000\cdot 30^n n^{5.5}.$$ Then either ${\Lambda=0}$ or $$\label{eminor}
|\Lambda| > e^{-C(n) \delta^{2} \mho (1+\log \delta)(1+ \log B)}.$$
It will be more convenient for us to use the following consequence. By the principal values of $\log z$ and $\arg z$ we mean those satisfying $$-\pi <{{\mathop{\mathrm{Im}}}}\log z =\arg z \le \pi.$$
\[cmatveev\] In the set-up of Theorem \[tmatveev\] take the principal values of all logarithms and assume that $$A_k\ge \delta{{\mathop{\mathrm{h}}}}(\alpha_k)+\pi \qquad (k=1, \ldots, n).$$ Then we have either ${\alpha_1^{b_1}\cdots \alpha_n^{b_n}=1}$ or $$\label{ebakexp}
\bigl|\log(\alpha_1^{b_1}\cdots \alpha_n^{b_n})\bigr|\ge e^{-\pi C(n+1) \delta^{2} \mho (1+\log \delta)(1+\log n+\log B)},$$ again with the principal choice of the logarithm.
#### Proof
Notice first of all that ${\log|\alpha_k|\le \delta{{\mathop{\mathrm{h}}}}(\alpha_k)}$ and, since we have principal values of the logarithms, $$|\log\alpha_k|\le \delta{{\mathop{\mathrm{h}}}}(\alpha_k)+\pi.$$ Hence is satisfied.
Further, there exists ${b\in {{\mathbb Z}}}$ such that $$\log(\alpha_1^{b_1}\cdots \alpha_n^{b_n})= \Lambda-b\pi i.$$ We may assume that ${\bigl|\arg(\alpha_1^{b_1}\cdots \alpha_n^{b_n})\bigr|<\pi/2}$ (otherwise there is nothing to prove), and we have ${|\arg\alpha_k|\le \pi}$ by the assumption. It follows that ${|b|\le nB+1/2}$, and by integrality that ${|b|\le nB}$. Now put ${\alpha_{n+1}=-1}$, ${\log\alpha_{n+1}=\pi i}$, ${b_{n+1}=-b}$ and ${A_{n+1}=\pi}$ and apply Theorem \[tmatveev\] to the logarithmic form $$\Lambda'=\Lambda-b\pi i= b_1\log \alpha_1+\cdots+b_{n+1}\log \alpha_{n+1}.$$ We must replace $C(n)$ by ${C(n+1)}$, $\mho$ by $\pi\mho$ and $B$ by ${nB}$. We obtain $$|\Lambda'|\ge e^{-\pi C(n+1) d^{2} \mho (1+\log d)(1+\log n+\log B)},$$ as wanted. [[$\square$]{}]{}
Now we resume the set-up of Section \[sprinrel\] and will use Corollary \[cmatveev\] to bound $B$ from .
In the set-up of Proposition \[pformula20\] assume that $$d\ge 3$$ (and in particular ${p\ge 7}$). Define $$\begin{aligned}
\mho_1&= 10^8\delta_{\max}9^dd^6 p^{4d+2}{{\mathop{\mathrm{h}}}}(\eta_1)\cdots{{\mathop{\mathrm{h}}}}(\eta_{d-1}),\\
\mho_2&=\mho_1+\vartheta_{\max} +4\kappa p^3,\\
B_0&=2\mho_1\log\mho_1+2\mho_2, \end{aligned}$$ Then for ${B=B(P)=\max\{|b_0|, \ldots, |b_{d-1}|\}}$ we have ${B \le B_0}$.
By the famous result of Schinzel [@Sc74; @HS93] we have $$\label{eschin}
{{\mathop{\mathrm{h}}}}(\eta_k) \ge \frac12\log\frac{1+\sqrt5}2\ge0.24 \qquad (k=1, \ldots, d-1).$$ Since ${p\ge 7}$ this implies that $$\label{elower}
\mho_1\ge 10^8\delta_{\max}9^dd^6 p^{3d+3}.$$
#### Proof
We define the function ${W\in K(X_G)}$ as follows: $$W =
\begin{cases}
U^{{{\mathop{\mathrm{Ord}}}}_{c}U^{\sigma}} (U^{\sigma})^{-{{\mathop{\mathrm{Ord}}}}_{c}U}, & {{\mathop{\mathrm{Ord}}}}_{c}U \ne 0,\\
U,&{{\mathop{\mathrm{Ord}}}}_{c}U=0
\end{cases}$$ Then ${{{\mathop{\mathrm{Ord}}}}_{c}W=0}$ and $W$ is not a constant function by Proposition \[pmdrns\]. We have $$W(c)=
\begin{cases}
\gamma_{c,{{\mathcal O}}}^{{{\mathop{\mathrm{Ord}}}}_{c}U^{\sigma}} \gamma_{c,{{\mathcal O}}\sigma}^{-{{\mathop{\mathrm{Ord}}}}_{c}U}, & {{\mathop{\mathrm{Ord}}}}_{c}U \ne 0,\\
\gamma_{c,{{\mathcal O}}},&{{\mathop{\mathrm{Ord}}}}_{c}U=0
\end{cases}$$ where $\gamma_{c,{{\mathcal O}}}$ is defined in . Using and Proposition \[pordeg\] we may estimate the height of $W(c)$: $$\label{ehwc}
{{\mathop{\mathrm{h}}}}(W(c))\le \frac{\log 2}3mp(p+1)|H|^2.$$ We may assume that ${|q_c(P)|\le 1/2^p}$: otherwise and imply that $$B\le 2\delta_{\max} +\vartheta_{\max}+ \Theta\log p\le \mho_2\le B_0.$$ The rest of the proof splits into two cases, treated quite differently.
#### Case 1: ${W(P)\ne W(c)}$
This is the principal case, which requires use of Baker’s inequality. We have $$1\ne\frac{W(P)}{W(c)}=\alpha_0\alpha_1^{b_1}\cdots\alpha_{d-1}^{b_{d-1}},$$ where $$\begin{aligned}
\alpha_0
&=
\begin{cases}
W(C)^{-1}\eta_0^{m{{\mathop{\mathrm{Ord}}}}_{c}U^{\sigma}} (\eta_0^{\sigma})^{-m{{\mathop{\mathrm{Ord}}}}_{c}U}, & {{\mathop{\mathrm{Ord}}}}_{c}U \ne 0,\\
W(C)^{-1}\eta_0^m,&{{\mathop{\mathrm{Ord}}}}_{c}U=0,
\end{cases} \\
\alpha_k &=
\begin{cases}
\eta_k^{{{\mathop{\mathrm{Ord}}}}_{c}U^{\sigma}} (\eta_k^{\sigma})^{-{{\mathop{\mathrm{Ord}}}}_{c}U}, & {{\mathop{\mathrm{Ord}}}}_{c}U \ne 0,\\
\eta_k,&{{\mathop{\mathrm{Ord}}}}_{c}U=0,
\end{cases}
\qquad (k=1, \ldots, d-1). \end{aligned}$$ We shall apply Corollary \[cmatveev\] with ${L={{\mathbb Q}}(\zeta_p+\bar\zeta_p)}$ and ${\delta=(p-1)/2}$. Using , and Proposition \[pordeg\], we obtain the estimates $$\delta{{\mathop{\mathrm{h}}}}(\alpha_0)+\pi\le \frac{\log 2}3mp(p+1)\delta|H|^2 +\frac{\log2}{6}m^2p(p+1)\delta|H|^2+\pi \le 0.4p^5.$$ (recall that ${|H|\le (p-1)/3}$, ${p\ge 7}$ and ${m\le 6}$). Further, using Proposition \[pordeg\] and , we obtain $$\delta{{\mathop{\mathrm{h}}}}(\alpha_k)+\pi\le \frac16mp(p+1)\delta|H|{{\mathop{\mathrm{h}}}}(\eta_k)+\pi\le 0.3p^4{{\mathop{\mathrm{h}}}}(\eta_k)\qquad (k=1, \ldots, d-1)$$ Applying Corollary \[cmatveev\] with ${n=d}$ and ${\mho=A_0A_1\cdots A_{d-1}}$, where ${A_0=0.4p^5}$ and ${A_k=0.3p^4{{\mathop{\mathrm{h}}}}(\eta_k)}$ for ${k=1, \ldots, d-1}$, we obtain $$\label{eourbak}
\begin{gathered}
\left|\log\frac{W(P)}{W(c)}\right|=\left|\log\bigl(\alpha_0\alpha_1^{b_1}\cdots\alpha_{d-1}^{b_{d-1}}\bigr)\right|\ge e^{-\mho_0(1+\log d+\log B)},\\ \mho_0= 1.6\cdot10^6\cdot9^d(d+1)^{5.5}\bigl(1+\log (d+1)\bigr) p^{4d+1}{{\mathop{\mathrm{h}}}}(\eta_1)\cdots{{\mathop{\mathrm{h}}}}(\eta_{d-1}).
\end{gathered}$$ On the other hand, together with Proposition \[pordeg\] implies that $$\left|\log\frac{W(P)}{W(c)}\right|\le 0.6m^2p(p+1)^2|H|^2|q_c(P)|^{1/p} \le 2.4p^5|q_c(P)|^{1/p},$$ which, together with gives $$\log|q_c(P)^{-1}|\le p\mho_0(\log B+\log d+1)+6p\log p.$$ Combining this with , we obtain $$\begin{aligned}
B&\le p\delta_{\max}\mho_0\log B + p\delta_{\max}\mho_0(\log d+1) + 6\delta_{\max}p\log p+ \vartheta_{\max} +1.6\Theta \\
&\le \mho_1\log B+\mho_2. \end{aligned}$$ Now Lemma 2.3.3 from [@BH96] implies that ${B\le 2\mho_1\log\mho_1+2\mho_2}$, as wanted.
#### Case 2: ${W(P)=W(c)}$
In this case Baker’s inequality does not apply. Instead, we invoke an elementary argument using power series expansion of $W$.
For a moment we forget that we fixed $P$ and consider $P$ as a varying point in $\Omega_c$. Propositions \[pasympuoo\] and \[pordeg\] imply that for ${k=1,2,\ldots}$ there exist algebraic numbers ${\theta_k\in {{\mathbb Q}}(\zeta_p)}$ such that for any absolute value $v$ of ${{\mathbb Q}}(\zeta_p)$ we have $$\label{ethetak}
|\theta_k|_v \le
\begin{cases}
|k|_v^{-1}, & v\mid p<\infty,\\
\frac13m^2p(p+1)^2|H|^2(k/p+1), & v\mid \infty
\end{cases} \qquad (k=1,2,\ldots)$$ and the following holds. Let $\nu$ be a non-negative integer. Then for ${P\in \Omega_c}$ such that ${|q_c(P)|<1/2^p}$ we have, with a suitable choice of logarithms, $$\log \frac{W(P)}{W(c)}=
\sum_{k=1}^\nu \theta_kq_c(P)^{k/p}+
O_1\left( \frac16 m^2p(p+1)^2|H|^2\left(2.2\nu/p+5.1\right)|q_c(P)|^{(\nu+1)/p}\right).$$ Now specify $\nu$ to be the smallest $k$ such that ${\theta_k\ne 0}$. (Since the function $W$ is non-constant such $k$ always exist.) With this choice of $\nu$ the relation above would look like $$\label{elog=}
\log \frac{W(P)}{W(c)}=
\theta_\nu q_c(P)^{\nu/p}+
O_1\left( \frac16 m^2p(p+1)^2|H|^2\left(2.2\nu/p+5.1\right)|q_c(P)|^{(\nu+1)/p}\right).$$ Also, with ${k=\nu}$ implies that $${{\mathop{\mathrm{h}}}}(\theta_\nu) \le \log \left(\frac13m^2p(p+1)^2|H|^2(\nu/p+1)\right)+\log \nu,$$ Since ${\theta_\nu\ne 0}$ and ${\theta_\nu \in {{\mathbb Q}}(\zeta_p)}$, this implies that $$\label{ethetanu}
|\theta_\nu|\ge e^{-(p-1){{\mathop{\mathrm{h}}}}(\theta_\nu)} \ge \left(\frac13m^2p(p+1)^2|H|^2(\nu/p+1)\nu\right)^{-(p-1)}.$$
We now return to the initial set-up of the proof: $P$ is an integral point belonging to $\Omega_c$ and satisfying ${W(P)=W(c)}$. Then the logarithm on the left of is either $0$ or at least $2\pi$ in absolute value. We consider these cases separately.
#### Sub-case 2.1: the logarithm on the left of is $0$.
In this case we have $$\theta_\nu q_c(P)^{\nu/p}=
O_1\left( \frac16 m^2p(p+1)^2|H|^2\left(2.2\nu/p+5.1\right)|q_c(P)|^{(\nu+1)/p}\right).$$ Together with this implies $$\label{ewithnu}
|q_c(P)^{-1}|\le \left(\frac{m^2p(p+1)^2|H|^2\left(2.2\nu/p+5.1\right)}{6\theta_\nu}\right)^p \le \left(\frac12m^2p(p+1)^2|H|^2(\nu/p+1)\nu\right)^{p^2}$$ To complete the proof we need to bound $\nu$. We claim the following.
#### Claim
*There exists ${k\le\frac1{288}m^2p^2(p-1)(p+1)^2|H|^2}$ such that ${\theta_k\ne 0}$.*
We assume the claim for now, postponing the proof until later. By the claim, $$\nu\le\frac1{288}m^2p^2(p-1)(p+1)^2|H|^2,$$ which implies that $$|q_c(P)^{-1}|\le \left(\frac1{576}m^4p^3(p+1)^4(p-1)|H|^4\left(\frac1{288}m^2p^2(p-1)(p+1)^2|H|^2+1\right)\right)^{p^2} \le p^{19p^2}.$$ Together with this implies that $$B\le 19\delta_{\max}p^2\log p+ \vartheta_{\max} +1.6\Theta \le \mho_2\le B_0$$ by .
#### Sub-case 2.2: the logarithm on the left of is at least $2\pi$ in absolute value.
In this case a bound for $|q_c(P)^{-1}|$ much sharper than easily follows. We omit the details which are straightforward but tedious.
We are left with proving the Claim.
#### Proof of the Claim
Denote by $\Delta$ the degree of $W$ as a rational function on $X_G$. Then the function ${W/W(c)-1}$ is of degree $\Delta$ as well. It follows that ${{{\mathop{\mathrm{Ord}}}}_c(W/W(c)-1)\le \Delta}$.
Extend the additive valuation ${{\mathop{\mathrm{Ord}}}}_c$ from the field ${{\mathbb C}}(X_G)$ to the field of formal power series ${{\mathbb C}}((q_c^{1/p}))$. Then ${{{\mathop{\mathrm{Ord}}}}_c(q_c^{1/p})=1}$ and ${{{\mathop{\mathrm{Ord}}}}_c\log(W/W(c))={{\mathop{\mathrm{Ord}}}}_c(W/W(c)-1)\le \Delta}$. It follows that the series in $q_c^{1/p}$ representing $\log(W/W(c))$ has a non-zero coefficient with index ${k\le \Delta}$.
Proposition \[pordeg\] implies that ${\Delta\le\frac1{288}m^2p^2(p-1)(p+1)^2|H|^2}$. This proves the claim. [[$\square$]{}]{}
Reduction of Baker’s Bound {#sred}
==========================
In the previous section we bounded ${B=\max\{|b_1|, \ldots, |b_{d-1}|\}}$ by an explicitly computable number $B_0$. In practical situation $B_0$ can be very huge (around $10^{100}$ or so), so not suitable for direct enumeration of all possible vectors ${(b_1, \ldots, b_{d-1})}$. It turns out, however, that it can be drastically reduced, using the numerical Diophantine approximations technique introduced by Baker and Davenport [@BD69] and developed in [@BH96; @TW89] in the context of the Diophantine equation of Thue.
In turns out that the method of [@BH96], suitably adapted, works in the present situation as well. As in the previous section we fix a cusp $c$ and consider integral points ${P\in \Omega_c}$. We shall usually omit index $c$, writing as ${\delta_{c,k}=\delta_k}$ and ${\vartheta_{c,k}=\vartheta_{k}}$ the quantities defined in .
As we have seen in Remark \[rtriv\], at least one of the numbers ${\delta_1, \ldots, \delta_{d-1}}$ is non-zero. To simplify notation we will assume that ${\delta_1\ne 0}$.
It turns out to be more practical to obtain a reduced bound for ${\log|q_c(P)^{-1}|}$; due to the results of Section \[sprinrel\] this would imply reduced bounds for the exponents $b_k$. In this section we will assume that $$\label{eqcht}
|q_c(P)|\le \max\{\Theta, 2\}^{-p},$$ where $\Theta$ is defined in .
Put $$\delta=\frac{\delta_{{2}}}{\delta_{{1}}}, \quad \lambda= \frac{\delta_{{2}}\vartheta_{{1}}-\delta_{{1}} \vartheta_{{2}}}{\delta_{{1}}}$$ Relation implies that $$\label{eredula}
\left| b_{{2}}-\delta b_{{1}}+ \lambda \right| \le 3.2(1+|\delta|)\Theta |q_c(P)|^{1/p},
$$ To bound ${\log|q_c(P)^{-1}|}$ we proceed now as follows. We fix a real number ${T\ge 2}$ (in practice, we take ${T=10}$). Next, using continued fraction we find a “good” rational approximation of $\delta$; precisely, we find a non-negative integer ${r \le T B_{0}}$ such that $$\| r \delta \| \le (T B_{0})^{-1}$$ where $\|\cdot \|$ is the distance to the nearest integer. Now, if $r\lambda$ is not “very close” to the nearest integer (in practice if ${\| r \lambda \| \ge 2 T^{-1}}$) then we can bound ${|q_c(P)^{-1}|}$. Indeed, multiply both sides of by $r$. Since ${|b_1|\le B_0}$, the left-hand side of the resulting inequality would be $$\left| rb_{{2}}-r\delta b_{{1}}+ r\lambda \right| \ge \|r\lambda\|-B_0\|r\delta\| \ge \|r\Lambda\|-T^{-1},$$ and the right-hand side will be bounded from above by ${3.2(1+|\delta|)\Theta TB_0 |q_c(P)|^{1/p}}$. This gives the following upper bound for ${|q_c(P)^{-1}|}$: $$\label{enbq}
\log|q_c(P)^{-1}|\le p\log\frac{3.2(1+|\delta|)\Theta TB_0}{\|r\lambda\|-T^{-1}} =: \Xi_1.$$ In the case when ${\|r\lambda\|<2T^{-1}}$ we increase $T$ (say, replace it by $10T$) and restart.
Substituting into and using , we obtain a new bound for $b_1$: $$|b_1|\le |\delta_1|\Xi+|\vartheta_1|+3.2=:B_1.$$ Since $B_1$ depends logarithmically on $B_0$, it is expected to be much smaller than $B_0$, and in practice it is.
We then repeat the same procedure, but this time with $B_1$ instead of $B_0$, and obtain for $\log|q_c(P)^{-1}|$ and $|b_1|$ new reduced bounds $\Xi_2$ and $B_2$, and so on. In practice, after three-four iterations of this procedure we obtain bounds for $\log|q_c(P)^{-1}|$ and $|b_1|$ that can no longer be reduced. We call ${{\widehat\Xi}}$ this reduced bound for $\log|q_c(P)^{-1}|$. In practice ${{\widehat\Xi}}$ is of order about $10^3$.
To be precise, since we assumed , we must replace ${{\widehat\Xi}}$ by ${\max\{{{\widehat\Xi}}, p\log \Theta, p\log 2\}}$. But in all practical cases we had ${{{\widehat\Xi}}>p\log \Theta>p\log2}$ with large margins.
Final enumeration
=================
The reduced upper bound for ${\log|q_c(P)^{-1}|}$ obtained in the previous section allows one to bound the exponents ${b_1,\ldots, b_ {d-1}}$ by some reasonable quantities (of magnitude $10^5$ or so). Still, the number of possible vectors ${(b_1,\ldots,b_{d-1})}$ is excessively large, and they cannot be fully enumerated in reasonable time.
In the similar situation in [@BH96] it was suggested to express all the $b_k$ in terms of one of them, say, $b_1$. Then, for each possible value of $b_1$ one calculates the corresponding values of other $b_k$, and if one of these values is not integer, then the corresponding value of $b_1$ is impossible.
We apply the same methodology here, though in the present situation the technical aspect is much more involved. In the sequel we shall assume that ${q_c(P)\in {{\mathbb R}}}$, which, according to Proposition \[preal\], is equivalent to saying that ${j(P)\notin \{1,2,\ldots, 1727\}}$. In Subsection \[ssmissing\] we briefly discuss how we dispose of these missing $j$.
Like in Section \[sred\], we omit indice $c$ and write ${\delta_{c,k}=\delta_k}$ and ${\vartheta_{c,k}=\vartheta_{k}}$, etc. Recall that these quantities, as well as the quantity $\Theta$ used below are defined in .
Quick enumeration
-----------------
If $\log|q_c(P)^{-1}|$ is not too small then one can express ${b_2, \ldots, b_{d-1}}$ in terms of $b_1$ with high precision using relation , as it is done for $b_2$ in . For the reader’s convenience, we reproduce relation , which is our main tool in this subsection: for ${k=1, \ldots, d-1}$ and ${|q_c(P)|\le 1/2^p}$ we have $$\label{erelation2bis}
b_k =\delta_{k}\log|q_c(P)^{-1}|+ \vartheta_{k}+O_1(3.2\Theta |q_c(P)|^{1/p}).$$
\[pupseg\] Let ${{\mathsf{Y}}}$ be a real number satisfying ${p\log 2 \le {{\mathsf{Y}}}\le {{\widehat\Xi}}}$. Put ${\epsilon=3.2|\delta_1|^{-1}\Theta e^{-{{\mathsf{Y}}}/p}}$. In the set-up of Proposition \[pformula20\] assume that ${\log|q_c(P)^{-1}| \ge {{\mathsf{Y}}}}$. Then $$\label{ebigframe}
{{\mathsf{Y}}}- \epsilon\le \ell_1\le {{\widehat\Xi}}+ \epsilon, \qquad \text{where}\quad \ell_1=\frac{b_1-\vartheta_1}{\delta_1}.
$$ Further, for ${k=2, \ldots, d-1}$ we have $$\label{esmallframe}
\left|b_k -(\delta_k\ell_1+\vartheta_k\right| \le \left(1+\left|\frac{\delta_k}{\delta_1}\right|\right)\epsilon_1,$$ where ${\epsilon_1= 3.2\Theta e^{(\epsilon-\ell_1)/p}}$.
#### Proof
We write $q$ instead of $q_c(P)$. Assume that ${\log|q^{-1}|\ge {{\mathsf{Y}}}}$. Then ${{{\mathsf{Y}}}\le \log|q^{-1}|\le {{\widehat\Xi}}}$ and ${|q|^{1/p} \le e^{-{{\mathsf{Y}}}/p}}$. We obtain from with ${k=1}$ that ${\bigl|\ell_1-\log|q^{-1}|\bigr|\le \epsilon}$, which implies, in particular, . Furthermore, we have ${\log|q^{-1}|\ge \ell_1-\epsilon}$, which gives ${3.2|\Theta| |q|^{1/p} \le \epsilon_1}$. Again using with ${k=1}$, we obtain ${\bigl|\ell_1-\log|q^{-1}|\bigr|\le |\delta^{-1}|\epsilon_1}$. All this combined with relations for ${k=2, \ldots, d-1}$ gives . [[$\square$]{}]{}
In practice we initially set ${{{\mathsf{Y}}}=p\log(50|\delta_1|^{-1}\Theta)}$. We list integers $b_1$ satisfying in the descending order of $\ell_1$, and for each $b_1$ we verify whether each of the ${d-2}$ intervals $$\label{einter}
\left[\delta_k\ell_1+\vartheta_k - \left(1+\left|\frac{\delta_k}{\delta_1}\right|\right)\epsilon_1,\ \delta_k\ell_1+\vartheta_k + \left(1+\left|\frac{\delta_k}{\delta_1}\right|\right)\epsilon_1\right] \qquad (2\le k \le d-1),$$ contains an integer. If at least one of these intervals does not contain an integer we pass to the next $b_1$ in the list. Otherwise, we re-set ${{{\mathsf{Y}}}=\ell_1+\epsilon}$ and terminate the algorithm. At the output, we obtain a new upper bound ${{\mathsf{Y}}}$ for $\log|q_c(P)^{-1}|$.
The choice of $b_1$ as the “independent variable” is quite arbitrary; any $b_k$ with ${\delta_k\ne 0}$ would do. We believe that the optimal choice is the one with the smallest (in absolute value) non-zero $\delta_k$, because it minimizes the range of possible values for $b_k$.
Slow enumeration: the overview {#ssslow}
------------------------------
When ${\log|q_c(P)^{-1}|\le {{\mathsf{Y}}}}$, simple relations is no longer sufficient to express ${b_2, \ldots,b_{d-1}}$ in terms of $b_1$, and one has to use more complicated relations like . For the reader’s convenience, we reproduce below. Let $\nu$ be a positive integer and let $Q_k(t)$ be the polynomials $Q_{c,k}(t)$ (depending on $\nu$) defined in Proposition \[pformula20\]. For ${k=1, \ldots, d-1}$ define the functions of the real variable $t$ by $$\begin{aligned}
f_k(t) &= -p\delta_{k}\log|t|+ \vartheta_{c}+ Q_\nu(t)+\\
&\hphantom{=}
+ m\sum_{\ell=0}^{d-1}\alpha_{k\ell} \left( \sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_{\ell}\sigma_c}{0<{{\tilde{a}}}_1<1/2}}\log|1-t^{p{{\tilde{a}}}_1}e^{2\pi i{{\tilde{a}}}_2}|+\sum_{\genfrac{}{}{0pt}{}{{{\mathbf a}}\in {{\mathcal O}}\sigma_{\ell}\sigma_c}{1/2\le{{\tilde{a}}}_1<1}}\log|1-t^{p-p{{\tilde{a}}}_1}e^{-2\pi i{{\tilde{a}}}_2}|\right).
\end{aligned}$$ Then, setting ${t=q_c(P)^{1/p}}$, we have, for ${k=1, \ldots, d-1}$, the following: $$\begin{aligned}
\label{erelation3bis}
b_k &= f_k(t) +
O_1\left(\Theta\left(2.2\nu/p+3.1\right)|t|^{\nu+1}\right),\end{aligned}$$ where $\Theta$ is defined in . We have either ${q_c(P)\ge e^{-2\pi}}$ or ${q_c(P)\le -e^{-\pi\sqrt3}}$. Since ${\log |q_c(P)|\le {{\mathsf{Y}}}}$, we obtain $$\label{edomt}
t \in [-e^{-\pi\sqrt3/p}, -e^{-{{\mathsf{Y}}}/p}] \cup [e^{-{{\mathsf{Y}}}/p}, e^{-2\pi/p}].$$
The principal steps of the final enumeration procedure can be described as follows.
1. \[imono\] Split domain into intervals of monotonicity of the function $f_1$.
2. \[it\] Let $I$ be one of the intervals found on step \[imono\] and ${J=f_1(I)}$. Fix ${b_1\in J}$ and compute ${t\in I}$ such that ${f_1(t)=b_1}$. Since $f_1$ is monotonic on $I$, only one such $t$ may exist. (Since equality in is approximate, one should not miss the possible $b_1$ outside the interval $J$, but close to it; see more on this Subsection \[sscomput\].)
3. \[ibk\] The real number $f_k(t)$ must then be “very close” to the integer $b_k$, for ${k=2, \ldots, d-1}$. If this fails for at least one $k$, we discard the corresponding $b_1$.
4. \[ij\] If each of $f_k(t)$ is close to an integer, then we compute $j(t^p)$ (for this purpose, we might need to know $t$ with higher precision than on step \[ibk\], see Subsection \[sscomput\]). If it is approximately equal to an integer $j$, then we verify whether $j$ gives rise to an integral point on $X_{{\mathop{\mathrm{ns}}}}^+(p)$.
5. \[irep\] Steps \[it\]–\[ij\] should be repeated for all ${b_1\in J}$.
6. Step \[irep\] must be repeated for every interval of monotonicity of $f_1$.
In Subsection \[sscomput\] we add some computational details on these steps.
Slow enumeration: computational remarks {#sscomput}
---------------------------------------
Notice first of all that one should take care about the computational errors arising from the very fact that our numerical data is given approximately. This also concerns the reduction and the quick enumeration steps. This is a standard problem in the numerical analysis, and we do not speak on it here, but we had to take care of it in our software implementations.
#### The monotonicity intervals
To determine the monotonicity intervals of $f_1$, one should find the zeros of its derivative. This derivative is a rational function, and its zeros can be found using any of the available methods for numerical solution of polynomial equation.
We used Brent’s method, implemented in [PARI](PARI), which efficiently combines several known methods of numerical resolution of equation. In most of the cases $f_1$ was monotonic already on each of the two intervals forming domain , but in a few cases we had to split them into smaller intervals.
#### Resolving the equation ${f_1(t)=b_1}$ and computing $f_k(t)$
Here we give some clarifications on the steps \[it\] and \[ibk\] of the slow enumeration procedure. Thus, we assume that $t$ belongs to some interval ${I=[\alpha, \beta]}$ and that $f_1$ is monotonic on $I$. Let $$\label{eepsilonn}
\epsilon = \Theta\left(2.2\nu/p+3.1\right)e^{-(\nu+1)\pi\sqrt3/p}$$ be an upper bound for the error term in , and let ${b_1 \in [\alpha-\epsilon, \beta+\epsilon]}$. Using Brent’s method, we compute the solutions ${t\in I}$ of ${f_1(t)=b_1\pm \epsilon}$; call them $\tau^+$ and $\tau^-$. In the (rare) case when ${b_1-\epsilon\notin I}$ we replace ${b_1-\epsilon}$ by $\alpha$, and similarly for ${b_1+\epsilon}$. Now $b_k$ must be between ${f_k(\tau^-)}$ and ${f_k(\tau^+)}$ (except the very few cases when $f_k'$ vanishes between these points, and one has to be slightly more careful). For overwhelming majority of choices of $b_1$, at least one of the ${d-2}$ intervals $$\label{eintervals}
\bigl[\min\{f_k(\tau^-), f_k(\tau^+)\}-\epsilon, \ \max\{f_k(\tau^-), f_k(\tau^+)\}+\epsilon\bigr], \qquad k=2, \ldots, d-1$$ does not contain an integer, and the corresponding $b_1$ can be discarded. In the few cases when this fails, one may refine intervals , in one of the following ways.
- Replace $\epsilon$ by $$\epsilon_1 = \Theta\left(2.2\nu/p+3.1\right)(\min\{\tau^-, \tau^+\})^{-(\nu+1)}.$$ Then $\tau^-$ and $\tau^+$ will be replaced by certain $\tau_1^-$ and $\tau_1^+$, which lie much closer to each other than $\tau^-$ and $\tau^+$, and intervals will be modified accordingly.
- Increase $\nu$ and re-define functions $f_1, \ldots, f_{d-1}$ accordingly.
Acting like this, we managed to exclude almost all the false positives.
#### Computing $j$ and verifying whether it gives rise to an integral point
If everything above fails to discard $b_1$, then probably this $b_1$ indeed corresponds to an integral point. To verify this, we compute ${j=j(t^p)}$. For this we need knowing $t$ with much higher precision than in the previous steps of the algorithm. Therefore we have to increase $\nu$, re-define $f_1$ and re-calculate $t$ with higher precision, then find $j(t^p)$ and see whether it lies close to an integer within the computational error. In all cases when it did, $j$ was a rational CM $j$-invariant, that is, one of the $13$ numbers from the bottom line of Table \[tacm\] on page .
#### Selection of $\nu$
Selecting the parameter $\nu$ correctly is quite important for optimizing the calculations. When $\nu$ is chosen too small, then precision of would be insufficient; but choosing $\nu$ too high leads to too complicated expressions for $f_k(t)$ and makes the Brent’s method too slow. Our experimentation shows that the nearly optimal value for $\nu$ is the one making the error term in bounded by $10^{-10}$; that is, the quantity $\epsilon$ defined in should be about ${10^{-10}}$.
The “slow enumeration” step is the bottleneck of the method, it accounts for more than 90% of the computational time. There are several possible way to accelerate this step.
- Refining the error term in and . In particular, refining the error term in would lead to more efficient “quick reduction step”, and, as a result, would reduce the domain for $t$ in the slow reduction.
- Using different expressions for the functions $f_k$, with fewer logarithmic terms; this can be achieved by replacing the logarithmic terms involving higher powers of $t$ by their finite Taylor expansions, and merging these expansions with the polynomial $Q_k(t)$.
- Splitting the domain into smaller parts, and using on each a more adapted expression for this domain; for instances, for smaller $t$ we need smaller $\nu$ and fewer logarithmic terms.
We are experimenting with these approaches and will report on our experiments in subsequent publications.
The Case ${j(P)\in \{1,2,\ldots, 1727\}}$ {#ssmissing}
-----------------------------------------
Recall that to an elliptic curve $E/{{\mathbb Q}}$ and a prime number $p$ we associate a Galois representation ${\rho_{E,p}:{{\mathop{\mathrm{Gal}}}}_{{\mathbb Q}}\to {{\mathop{\mathrm{GL}}}}(E[p])\cong {{\mathop{\mathrm{GL}}}}_2({{\mathbb F}}_p)}$, which is defined by the natural action of the absolute Galois group $G_{{\mathbb Q}}$ on the torsion group $E[p]$. Points in $X_{{\mathop{\mathrm{ns}}}}(p)({{\mathbb Q}})$ correspond to the elliptic curves $E/{{\mathbb Q}}$ such that the image of $\rho_{E,p}$ is contained in the normalizer of a non-split Cartan subgroup of ${{\mathop{\mathrm{GL}}}}_2({{\mathbb F}}_p)$.
It is known that, if this latter property holds for some elliptic curve $E/{{\mathbb Q}}$ with ${j(E)\ne 0,1728}$, then it holds for any quadratic twist of $E$, that is, for any other elliptic curve $E'$ with ${j(E')=j(E)}$. Indeed, $E'$ is isomorphic to $E$ over some field $K$ of degree at most $2$. Denote by $\chi_K$ is the character of $G_{{\mathbb Q}}$ corresponding to $K$. Then ${\rho_{E',p}=\rho_{E,p}\chi_K}$. Hence if the image of $\rho_{E,p}$ is contained in the normalizer of a non-split Cartan subgroup, then so is the image of $\rho_{E',p}$.
Hence, if we fix ${j\in {{\mathbb Q}}}$, distinct from $0$ and $1728$, then, to verify whether $X_{{\mathop{\mathrm{ns}}}}(p)$ has a rational point $P$ with ${j(P)=j}$, it suffices to verify for at least one curve $E/{{\mathbb Q}}$ with ${j(E)=j}$ whether the image of $\rho_{E,p}$ is contained in the normalizer of a non-split Cartan subgroup. This can be easily accomplished with the [SAGE](SAGE) package [@sage], functions [EllipticCurve](EllipticCurve) and [image\_type](image_type). Using it, we found that $X_{{\mathop{\mathrm{ns}}}}(p)({{\mathbb Q}})$ has no points with $j$-invariants from the set $\{1, \ldots,1727\}$ for all $p$ we considered.
#### Acknowledgments
Yuri Bilu was supported by the *Agence National de la Recherche* project “Hamot” (ANR 2010 BLAN-0115-01) and by the ALGANT scholarship program. We thank Julia Baoulina, Andreas Enge, Pierre Parent and Sha Min for useful discussions and suggestions.
Our algorithms are implemented using the computer algebra systems [PARI/GP](PARI/GP) [@pari] and [SAGE](SAGE) [@sage].
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<http://pari.math.u-bordeaux.fr/>
<http://www.sagemath.org/>
[^1]: Supported by the *Agence National de la Recherche* project “Hamot” (ANR 2010 BLAN-0115-01) and by the ALGANT scholarship program.
[^2]: The coefficients $c_n$ cannot be confused with the cusps.
[^3]: Similar estimates hold for the numbers $\beta'_{k,c}$ as well, but we shall not need them.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate the single-point probability density function of the velocity in three-dimensional stationary and decaying homogeneous isotropic turbulence. To this end we apply the statistical framework of the Lundgren–Monin–Novikov hierarchy combined with conditional averaging, identifying the quantities that determine the shape of the probability density function. In this framework the conditional averages of the rate of energy dissipation, the velocity diffusion and the pressure gradient with respect to velocity play a key role. Direct numerical simulations of the Navier–Stokes equation are used to complement the theoretical results and assess deviations from Gaussianity.'
author:
- 'Michael Wilczek, Anton Daitche, Rudolf Friedrich'
title: 'On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from Gaussianity'
---
Introduction
============
The spatio-temporal complexity of turbulent flows demands a statistical description, which can be formulated in terms of probability density functions (PDFs) of the fluctuating turbulent quantities such as velocity or vorticity. The Lundgren–Monin–Novikov (LMN) hierarchy [@lundgren67pof; @monin67pmm; @novikov68sdp; @ulinich69spj] provides a statistical framework, which, starting from the basic equations of motion (i.e. the Navier–Stokes equation or the vorticity equation), yields the temporal evolution of the multi-point PDFs of the velocity or vorticity. As typical for a statistical theory of turbulence, this approach has to face the famous closure problem of turbulence. When deriving the evolution equation of the single-point PDF, this equation couples to the two-point PDF. Now determining the evolution of the two-point PDF involves the three-point PDF and so forth, ending up in an ever increasing number of evolution equations for the multi-point PDFs. This chain of equations may be truncated at a given level by introducing reasonable closure approximations, which express the $(n+1)$-point PDF in terms of the $n$-point PDF. However, these approximations have to be performed with great care as a too crude approximation results in serious defects. An alternative approach is provided by conditional averaging, which, for example, is taken as a starting point for modelling in the realm of the well-known PDF methods by [@pope00book]. [@novikov93jfr] used this method in the case of the vorticity equation to elucidate the statistical balance of vortex stretching and vorticity diffusion. When combined with the LMN hierarchy, a truncation at a given level is possible by introducing the conditional averages of the terms appearing in the equations of motion as unknown functions. These functions then may be modelled or estimated from experiments or direct numerical simulations as recently exemplified in [@wilczek09pre] in the case of the single-point vorticity PDF. The advantage of this approach lies in the possibility to interpret the unknown functions in terms of their physical meaning. For instance, it turns out that the single-point velocity PDF depends on the conditionally averaged rate of energy dissipation.
Ever since the experimental measurements by Townsend presented in [@batchelor53book] it is an ongoing discussion, whether the single-point velocity PDF displays a Gaussian shape. Arguments in favour of this hypothesis often involve the central limit theorem, taking the velocity field as a spatial ensemble of independently fluctuating random variables with finite variances. This argument turns out to be rather hand-waving as the assumption of statistical independence is obviously violated. Moreover, the same kind of argument should be applicable to the vorticity PDF. This PDF, however, is known to display extremely non-Gaussian tails, which are often associated with the existence of vortex structures in the flow. When applied with care, however, the central limit theorem together with proper assumptions on the energy spectrum has been shown to produce sub-Gaussian tails by [@jimenez97jfm].
While the assumption of Gaussianity at least seems to be a good approximation, experimental and numerical measurements indicate the possibility of sub-Gaussian tails [@vincent91jfm; @noullez97jfm; @gotoh02pof]. Theoretical support in this direction was, in the case of forced turbulence, given by [@falkovich97prl]. The approach, however, depends on the type of forcing applied to the flow. This point recently was raised in [@hosokawa08pre], where it was hypothesized that decaying turbulence might be profoundly different from forced turbulence. In this work, a closure for the LMN hierarchy was suggested leading to a Gaussian velocity PDF for decaying turbulence. Gaussian PDFs are also found as a result of the cross-independence hypothesis [@tatsumi04fdr] for both the single-point PDF and joint PDFs in the inertial range. In both works the pressure contributions are neglected on the level of the single-point PDF.
The present work aims at clarifying these points. We make use of the statistical framework of the LMN hierarchy combined with conditional averaging to identify the terms which determine the shape and evolution of the single-point velocity PDF. This approach gives concise insights how the different terms interplay to determine the functional form of the PDF. The theory applies to forced as well as decaying turbulence. The results presented in [@hosokawa08pre] can be obtained via a simple approximation, which neglects the statistical dependence of the pressure gradient and the energy dissipation on the velocity. Testing the theory against results from direct numerical simulation, however, indicates slightly non-Gaussian PDFs for both forced and decaying turbulence, which is related to the statistical correlations between the velocity and the different dynamical effects such as pressure gradient and energy dissipation.
The remainder of this article is structured as follows. After reviewing the LMN hierarchy and conditional averaging, we discuss the implications of statistical symmetries, which help to drastically simplify the description. We then derive explicit formulae for the homogeneous and stationary velocity PDF. The conditional averages appearing in these formulae have to obey certain constraints which will be derived and used to motivate a simple closure approximation. We then turn to the numerical results of stationary and decaying turbulence examining the different quantities arising in the theoretical description. To highlight the genuine properties of the velocity statistics, some comparisons with the single-point statistics of the vorticity are drawn before we conclude.
Theoretical Framework
=====================
The Lundgren–Monin–Novikov Hierarchy
------------------------------------
As an introduction into the statistical framework used in this work, we start with revisiting the derivation of the LMN hierarchy. For further details, we refer the reader to [@lundgren67pof; @monin67pmm; @novikov68sdp; @pope00book].
The Navier–Stokes equation for an incompressible fluid reads as $$\label{eq:navier-stokes}
\frac{\partial}{\partial t}{\boldsymbol}u({\boldsymbol}x,t)+{\boldsymbol}u({\boldsymbol}x,t)\cdot{\boldsymbol}{\nabla} {\boldsymbol}u({\boldsymbol}x,t)=-{\boldsymbol}{\nabla} p({\boldsymbol}x,t)+\nu \Delta {\boldsymbol}u({\boldsymbol}x,t)+{\boldsymbol}F({\boldsymbol}x,t) .$$ Here ${\boldsymbol}u$ denotes the velocity field, $p$ is the pressure, $\nu$ denotes the kinematic viscosity and ${\boldsymbol}F$ denotes a large-scale forcing applied to the fluid to produce a statistically stationary flow. In the case of ${\boldsymbol}F({\boldsymbol}x,t)={\boldsymbol}0$, turbulence decays due to dissipation and statistical stationarity cannot be maintained.
Let ${\boldsymbol}u({\boldsymbol}x,t)$ denote a realization of the velocity field and ${\boldsymbol}v$ denote the corresponding sample space variable. The fine-grained PDF of the velocity is then defined by $\hat f({\boldsymbol}v;{\boldsymbol}x,t)=\delta({\boldsymbol}u({\boldsymbol}x,t)-{\boldsymbol}v)$, from which the PDF can be obtained by ensemble-averaging, $$f({\boldsymbol}v;{\boldsymbol}x,t)=\langle \hat f({\boldsymbol}v;{\boldsymbol}x,t) \rangle=\langle \delta({\boldsymbol}u({\boldsymbol}x,t)-{\boldsymbol}v) \rangle .$$ Taking the derivative of the fine-grained PDF with respect to time yields $$\frac{\partial }{\partial t} \hat f({\boldsymbol}v;{\boldsymbol}x,t)=-{\boldsymbol}{\nabla}_{{\boldsymbol}v} \cdot \bigg[ \frac{\partial {\boldsymbol}u}{\partial t}({\boldsymbol}x,t) \, \hat f({\boldsymbol}v;{\boldsymbol}x,t) \bigg] .$$ Together with an analogous calculation for the advective term we obtain $$\begin{aligned}
\label{eq:finegrained}
\frac{\partial }{\partial t} \hat f+{\boldsymbol}{\nabla}\cdot \left[ {\boldsymbol}u \hat f \right] &= -{\boldsymbol}{\nabla}_{{\boldsymbol}v} \cdot \bigg[ \left( \frac{\partial {\boldsymbol}u}{\partial t}+{\boldsymbol}u\cdot{\boldsymbol}{\nabla} {\boldsymbol}u \right) \hat f \bigg] \nonumber\\
&= -{\boldsymbol}{\nabla}_{{\boldsymbol}v} \cdot \bigg[ \left( -{\boldsymbol}{\nabla} p+\nu \Delta {\boldsymbol}u + {\boldsymbol}F \right) \hat f \bigg] .\end{aligned}$$ Here, the Navier–Stokes equation has been used to replace the terms of the left-hand side with the terms of the right-hand side. Incompressibility (${\boldsymbol}{\nabla}\cdot{\boldsymbol}u=0$) was used to rearrange the advective term. In order to turn this into an equation for the PDF, we have to perform an ensemble average of . The averaging of the left-hand side is straightforward yielding $\partial_t f+{\boldsymbol}v \cdot {\boldsymbol}{\nabla} f$; however, the terms on the right-hand side cannot be expressed in terms of $f$ and ${\boldsymbol}v$ only. Now there are several options how to proceed. One possibility is to express these unclosed terms on the right-hand side with the help of the two-point PDF. We exemplify this for the pressure term.
The two-point PDF of the velocity can be written as $f_2({\boldsymbol}v_1,{\boldsymbol}v_2; {\boldsymbol}x_1,{\boldsymbol}x_2,t)=\langle \delta({\boldsymbol}u({\boldsymbol}x_1,t)-{\boldsymbol}v_1) \, \delta({\boldsymbol}u({\boldsymbol}x_2,t)-{\boldsymbol}v_2) \rangle$. Now the pressure term can be expressed in terms of the velocity according to $$\begin{aligned}
\big\langle \big( -{\boldsymbol}{\nabla}_{{\boldsymbol}x_1} p \big) \hat f \big\rangle
&= \bigg\langle -\frac{1}{4\pi}\int\! \mathrm{d}{\boldsymbol}x_2 \left({\boldsymbol}{\nabla}_{{\boldsymbol}x_1}\frac{{\boldsymbol}{\nabla}_{{\boldsymbol}x_2} \cdot [ {\boldsymbol}u({\boldsymbol}x_2,t)\cdot{\boldsymbol}{\nabla}_{{\boldsymbol}x_2}{\boldsymbol}u({\boldsymbol}x_2,t) ]}{|{\boldsymbol}x_2-{\boldsymbol}x_1|} \right) \, \delta({\boldsymbol}u({\boldsymbol}x_1,t)-{\boldsymbol}v_1) \bigg\rangle \displaybreak[0] \nonumber \\
&= \bigg\langle -\frac{1}{4\pi} \int\! \mathrm{d}{\boldsymbol}x_2 \left( {\boldsymbol}{\nabla}_{{\boldsymbol}x_1} \frac{\mathrm{Tr}\,[({\boldsymbol}{\nabla}_{{\boldsymbol}x_2}{\boldsymbol}{\nabla}_{{\boldsymbol}x_2}^{\mathrm T}) ({\boldsymbol}u({\boldsymbol}x_2,t) {\boldsymbol}u^{\mathrm T}({\boldsymbol}x_2,t))]}{|{\boldsymbol}x_2-{\boldsymbol}x_1|} \right) \, \times \nonumber \\
& \qquad \qquad \qquad \delta({\boldsymbol}u({\boldsymbol}x_1,t)-{\boldsymbol}v_1) \left( \int\! \mathrm{d}{\boldsymbol}v_2 \, \delta({\boldsymbol}u({\boldsymbol}x_2,t)-{\boldsymbol}v_2) \right) \bigg \rangle \nonumber \displaybreak[0]\\
&= -\frac{1}{4\pi} \int\! \mathrm{d}{\boldsymbol}x_2 \, \mathrm{d}{\boldsymbol}v_2 \left( {\boldsymbol}{\nabla}_{{\boldsymbol}x_1} \frac{1}{|{\boldsymbol}x_2-{\boldsymbol}x_1|} \right) \, \left( {\boldsymbol}v_2 \cdot {\boldsymbol}{\nabla}_{{\boldsymbol}x_2} \right)^2 \, f_2({\boldsymbol}v_1,{\boldsymbol}v_2; {\boldsymbol}x_1,{\boldsymbol}x_2,t) ,\end{aligned}$$ where we have assumed an infinite domain without further boundary conditions. Here, the identity , incompressibility and the sifting property of the delta function, ${\boldsymbol}u({\boldsymbol}x_2,t) \, \delta({\boldsymbol}u({\boldsymbol}x_2,t)-{\boldsymbol}v_2)={\boldsymbol}v_2 \, \delta({\boldsymbol}u({\boldsymbol}x_2,t)-{\boldsymbol}v_2)$, have been used. This manipulation explicitly shows how the single-point statistics couples to the two-point statistics. The remaining unclosed terms can be treated in a similar manner; for more details, we refer the reader to [@lundgren67pof]. Now determining the evolution equation for $f_2$ basically involves the same steps, leading to a coupling to $f_3$. This eventually leads to the aforementioned never-ending hierarchy of evolution equations for the multi-point velocity distributions.
Instead of establishing this chain of evolution equations, it is possible to truncate the hierarchy on a given level by introducing conditional averages according to e.g. $$\big\langle (-{\boldsymbol}{\nabla} p) \hat f \big\rangle=\big\langle (\, -{\boldsymbol}{\nabla} p({\boldsymbol}x,t) \,) \, \delta({\boldsymbol}u({\boldsymbol}x,t) - {\boldsymbol}v) \big\rangle=\big\langle -{\boldsymbol}{\nabla} p \big| {\boldsymbol}v \big\rangle f,$$ i.e. now the coupling to a higher-order PDF is replaced by introducing unknown functions in the form of conditionally averaged quantities, which may be modelled or obtained by experimental or numerical means. The conditional averages may be regarded as a measure for the correlation (in the sense of statistical dependence) of the different quantities on the single-point level. We will focus on this approach in the following. The temporal evolution of the velocity PDF then takes the general form $$\label{eq:pdfvel}
\frac{\partial}{\partial t}f+{\boldsymbol}v\cdot{\boldsymbol}{\nabla} f=-{\boldsymbol}{\nabla}_{{\boldsymbol}v} \cdot \left[ \big\langle -{\boldsymbol}{\nabla} p+\nu\Delta {\boldsymbol}u + {\boldsymbol}F \big| {\boldsymbol}v \big\rangle f \right].$$ To rewrite the advective term, here again incompressibility and the sifting property have been used. In this kinetic description for the evolution of the single-point velocity PDF, several unclosed terms appear: the conditionally averaged pressure gradient, the Laplacian of the velocity as well as the external forcing. We take this equation as a starting point for the following considerations and will now utilize statistical symmetries to simplify the description.
Statistical Symmetries
----------------------
### Homogeneity
For homogeneous flows, several simplifications arise. First, the advective term on the left-hand side of vanishes as homogeneity means that $f$ cannot depend on ${\boldsymbol}x$. This simplifies the kinetic equation resulting in $$\label{eq:pdfvelhomo}
\frac{\partial}{\partial t}f=-{\boldsymbol}{\nabla}_{{\boldsymbol}v} \cdot \left[ \big\langle -{\boldsymbol}{\nabla} p + \nu\Delta {\boldsymbol}u + {\boldsymbol}F \big| {\boldsymbol}v \big\rangle f \right].$$ Note that in the case of decaying turbulence, the evolution is determined by the conditionally averaged pressure gradient and the dissipative term only. Homogeneity allows to recast the latter. Calculating the Laplacian of $\hat f$ and averaging yields[^1] $$\label{eq:hom}
\frac{\partial^2}{\partial x_k^2} f = 0 =-\frac{\partial }{\partial v_i} \bigg \langle \frac{\partial^2 u_i}{\partial x_k^2} \bigg | {\boldsymbol}v \bigg \rangle f +\frac{\partial }{\partial v_i}\frac{\partial }{\partial v_j} \bigg \langle \frac{\partial u_i}{\partial x_k} \frac{\partial u_j}{\partial x_k} \bigg | {\boldsymbol}v \bigg \rangle f.$$ This is a multi-dimensional version of the homogeneity relation introduced by [@ching96pre]. With this expression, the kinetic equation takes the form $$\label{eq:pdfvelhomo2}
\frac{\partial}{\partial t}f=-\frac{\partial}{\partial v_i} \bigg \langle -\frac{\partial}{\partial x_i} p + F_i \bigg | {\boldsymbol}v \bigg \rangle f - \frac{\partial }{\partial v_i}\frac{\partial }{\partial v_j} \bigg \langle \nu \frac{\partial u_i}{\partial x_k} \frac{\partial u_j}{\partial x_k} \bigg | {\boldsymbol}v \bigg \rangle f ,$$ which has also been derived for turbulence with mean flow by [@pope00book]. This introduces the tensor $D_{ij}= \big \langle \nu (\partial u_i/\partial x_k) (\partial u_j/\partial x_k) \big | {\boldsymbol}v \big \rangle $, sometimes termed conditionally averaged dissipation tensor, which will be of central interest in the following.
Before we proceed to the simplifications that arise due to isotropy, we pause for a side remark as we would like to highlight the connection to moment equations which can be obtained directly from the PDF equation . The law of energy decay may serve as an example. In the decaying case, the only two terms in governing the evolution of the velocity PDF are the conditionally averaged pressure gradient and the conditional dissipation tensor. While the former takes the form of a drift induced by the non-local pressure contributions, the latter term may be interpreted as a diffusive term, however with a negative sign. This sign can be explained in a physically sound way as for a diffusion process an initially localized concentration spreads over time, eventually being dispersed over a large domain. Regarding the decay of the velocity field of a turbulent fluid, the opposite takes place. An initially broad distribution of velocity contracts as the velocity field dies away. When the fluid has come to rest, the PDF is localized sharply in probability space, $ \lim \limits_{ t \rightarrow \infty} f({\boldsymbol}v;t)=\delta({\boldsymbol}v)$, expressing that we have probability one finding a vanishing velocity.
To proceed, we multiply by ${\boldsymbol}v^2/2$ and integrate over ${\boldsymbol}v$, $$\frac{\partial}{\partial t} E_{kin}=\int \mathrm{d}{\boldsymbol}v \frac{{\boldsymbol}v^2}{2} \frac{\partial}{\partial t}f=-\int \mathrm{d}{\boldsymbol}v \frac{v_k v_k}{2} \left[ \frac{\partial }{\partial v_i} \bigg \langle -\frac{\partial}{\partial x_i} p \bigg | {\boldsymbol}v \bigg \rangle f + \frac{\partial }{\partial v_i}\frac{\partial }{\partial v_j} \bigg\langle \nu \frac{\partial u_i}{\partial x_k} \frac{\partial u_j}{\partial x_k} \bigg | {\boldsymbol}v \bigg\rangle f \right ].$$ The right-hand side may be integrated by parts. Assuming a sufficiently rapid decay of the PDF, one finds $$\begin{aligned}
\label{eq:endecay}
\frac{\partial}{\partial t} E_{kin} &= \int \mathrm{d}{\boldsymbol}v \, \bigg [ -\big\langle {\boldsymbol}u \cdot {\boldsymbol}{\nabla} p \big| {\boldsymbol}v \big\rangle f -\bigg\langle \nu \frac{\partial u_i}{\partial x_k} \frac{\partial u_i}{\partial x_k} \bigg | {\boldsymbol}v \bigg\rangle f\bigg] \nonumber\\
&= -\big\langle {\boldsymbol}u \cdot {\boldsymbol}{\nabla} p\big\rangle -\frac{1}{2} \big\langle \varepsilon + \nu \omega^2 \big\rangle \nonumber\\
&= - \big\langle \varepsilon \big\rangle .\end{aligned}$$ The validity of the second equality will become clear in the next section (see ). The last equality comes from the fact that the pressure-related average vanishes due to homogeneity and incompressibility and that the rate of energy dissipation and squared vorticity (multiplied by $\nu$) have the same spatial and hence also ensemble average. Equation is what one expects and also what is long known as the evolution equation of the kinetic energy. It is not surprising that the correct equation for the energy follows from the equations for the velocity PDF as those were derived directly from the Navier–Stokes equation without using any approximations.
### Isotropy
The conditional averages appearing in and are vector- and symmetric-tensor-valued functions of the velocity vector ${\boldsymbol}v$. Statistical isotropy, which is here assumed as invariance under rotation[^2], imposes further constraints on these functions e.g. $${\mathrm}{D}({\mathrm}{R}{\boldsymbol}v)={\mathrm}{R}{\mathrm}{D}({\boldsymbol}v){\mathrm}{R}^{\mathrm T}\qquad {\mathrm}{R}\in\mathrm{SO}(3),
\label{}$$ where ${\mathrm}{R}$ is a rotation matrix. In three dimensions it can be shown that, because of these constraints, isotropic vector- and symmetric-tensor-valued functions like $\langle{\boldsymbol}{\nabla} p\big |{\boldsymbol}v\rangle$ and ${\mathrm}{D}({\boldsymbol}v)$ take the form [@robertson40pps; @batchelor53book] $$\begin{aligned}
a_i({\boldsymbol}v) &= a(v) \, \frac{v_i}{v} \label{eq:isotropy_a},\\
B_{ij}({\boldsymbol}v) &= \mu(v) \, \delta_{ij} + \left[ \lambda(v)-\mu(v)\right] \frac{v_i v_j}{v^2} \label{eq:isotropy_B},\end{aligned}$$ where $a(v)=\hat{{\boldsymbol}v}\cdot{\boldsymbol}a({\boldsymbol}v)$ is the projection of ${\boldsymbol}a({\boldsymbol}v)$ onto the unit vector in the direction of the velocity and $\lambda(v)$, $\mu(v)$ are the eigenvalues of the matrix ${\mathrm}{B}$. Here $\lambda$ is the eigenvalue of the eigenvector ${\boldsymbol}v$, whereas $\mu$ is the eigenvalue of the (two-dimensional) eigenspace perpendicular to ${\boldsymbol}v$. Note that because $a$, $\lambda$ and $\mu$ are isotropic scalar-valued functions, they depend only on the absolute value of the velocity $v$. The same goes for the probability density $f({\boldsymbol}v)$, i.e. it is only a function of the absolute value $v$. This, however, may not be confused with the PDF of the absolute value of the velocity, in the following denoted by $\tilde f(v)$. However, there is a simple relation between those two, $$\label{eq:isopdf}
\tilde f(v) = 4 \pi v^2 f({\boldsymbol}v),$$ indicating that it suffices to determine the PDF of the absolute value in order to specify the PDF of the full vector.
From the consideration about isotropic vector-valued functions, we find $$\begin{aligned}
\left\langle -{\boldsymbol}{\nabla} p \big | {\boldsymbol}v \right\rangle = \Pi(v) \hat{{\boldsymbol}v},
&\qquad \Pi(v)=\left\langle - \hat{{\boldsymbol}u} \cdot {\boldsymbol}{\nabla} p \big | v \right\rangle \label{eq:iso-pressure},\\
\left\langle \nu \Delta {\boldsymbol}u \big | {\boldsymbol}v \right\rangle = \Lambda(v) \hat{{\boldsymbol}v},
&\qquad \Lambda(v)=\left\langle \nu \hat{{\boldsymbol}u} \cdot \Delta {\boldsymbol}u \big | v \right\rangle \label{eq:iso-laplace},\\
\left\langle {\boldsymbol}F \big | {\boldsymbol}v \right\rangle = \Phi(v) \hat{{\boldsymbol}v},
&\qquad \Phi(v)=\left\langle \hat{{\boldsymbol}u} \cdot {\boldsymbol}F \big | v \right\rangle \label{eq:iso-force},\end{aligned}$$ where $\hat{{\boldsymbol}u}$ and $\hat{{\boldsymbol}v}$ denote the unit vectors in the direction of ${\boldsymbol}u$ and ${\boldsymbol}v$. Note that it is sufficient to take the conditional average with respect to the magnitude of the velocity to obtain the functions $\Pi$, $\Lambda$ and $\Phi$. This considerably simplifies the numerical estimation of the conditional averages. Let us now turn to the conditional dissipation tensor, which takes the form $${\mathrm}{D}({\boldsymbol}v)=\big\langle \nu {\mathrm}{A}{\mathrm}{A}^{\mathrm T} \big| {\boldsymbol}v \big\rangle,$$ where $A_{ij}=\partial u_i/\partial x_j$ is the velocity gradient tensor. ${\mathrm}{A}$ may be decomposed into symmetric and antisymmetric parts according to ${\mathrm}{A}={\mathrm}{S}+{\mathrm}{W}$, where ${\mathrm}{S}=\frac{1}{2}({\mathrm}{A}+{\mathrm}{A}^{\mathrm T})$ and ${\mathrm}{W}=\frac{1}{2}({\mathrm}{A}-{\mathrm}{A}^{\mathrm T})$. These two tensors characterize the local rate of stretching and the rate of rotation of the fluid. In the case of statistical isotropy, ${\mathrm}{D}$ is determined by its eigenvalues (see ), which can be obtained using the relations $$\begin{aligned}
\mathrm{Tr}({\mathrm}{D}) & =\lambda(v) + 2\mu(v),\\
\hat{{\boldsymbol}v}{\mathrm}{D}\hat{{\boldsymbol}v} &=\lambda(v) .\end{aligned}$$ The trace of ${\mathrm}{D}$ is determined by the conditional averages of the local rate of energy dissipation $\varepsilon=2\nu \mathrm{Tr}({\mathrm}{S}^2)$ and the squared vorticity $\omega^2$: $$\label{eq:trace}
\mathrm{Tr}({\mathrm}{D})=\nu \left\langle \mathrm{Tr}({\mathrm}{S}^2)- \mathrm{Tr}({\mathrm}{W}^2) \big | v \right\rangle =\frac{1}{2}\left\langle \varepsilon+\nu \omega^2 \big | v\right\rangle .$$ The second scalar quantity needed to determine ${\mathrm}{D}$ is $$\label{eq:con2}
\hat{{\boldsymbol}v}{\mathrm}{D}\hat{{\boldsymbol}v}=\left\langle \nu \, \hat{{\boldsymbol}u}{\mathrm}{AA}^{\mathrm T}\hat{{\boldsymbol}u}\big | v\right\rangle =\left\langle \nu({\mathrm}{A}^{\mathrm T}\hat{{\boldsymbol}u})^2\big | v\right\rangle .$$ This rather formally looking quantity has a simple physical interpretation. As ${\mathrm}{A}$ may be decomposed in symmetric and antisymmetric parts, we write ${\mathrm}{A}^{\mathrm T} \hat{{\boldsymbol}u }=({\mathrm}{S}-{\mathrm}{W})\hat{{\boldsymbol}u }$. The last term may also be written as ${\mathrm}{W}\hat{{\boldsymbol}u}=\frac{1}{2}{\boldsymbol}\omega \times \hat{{\boldsymbol}u}$ due to the relation $W_{ij}=-\frac{1}{2}\epsilon_{ijk}\omega_k$. Hence, the conditional average appearing in involves the absolute value of the difference between the rate of stretching in the direction of the velocity vector and rate of rotation of the unit vector $\hat{{\boldsymbol}u}$. Summing up, the conditional dissipation tensor ${\mathrm}{D}$ in isotropic turbulence has the form $$\begin{aligned}
D_{ij}({\boldsymbol}v) &= \mu(v) \, \delta_{ij} + \left[ \lambda(v)-\mu(v)\right] \frac{v_i v_j}{v^2},\label{eq:D-lambda-mu}\\
\mu(v) &= \frac{1}{4}\left\langle \varepsilon+\nu \omega^2 \big | v\right\rangle-\frac{1}{2}\left\langle \nu({\mathrm}{A}^{\mathrm T}\hat{{\boldsymbol}u})^2\big | v\right\rangle, \label{eq:mu_relation} \\
\lambda(v) &= \left\langle \nu({\mathrm}{A}^{\mathrm T}\hat{{\boldsymbol}u})^2\big | v\right\rangle \label{eq:lambda_relation} .\end{aligned}$$ These relations can now be used to simplify the structure of the PDF equations –. It turns out that by exploiting statistical isotropy, we have to deal with an effectively one-dimensional problem. Take for example $$\begin{aligned}
\frac{\partial}{\partial v_i} \big\langle \nu \Delta u_i \big| {\boldsymbol}v \big\rangle f({\boldsymbol}v) &= \frac{\partial}{\partial v_i} \Lambda(v) \frac{v_i}{4\pi v^3} \tilde f(v) \nonumber\\
&= \frac{1}{4\pi v^2} \frac{\partial}{\partial v} \Lambda(v) \tilde f(v),\end{aligned}$$ which makes clear that this term depends on $v$ only. The same applies to terms involving ${\mathrm}{D}$, a short calculation yields $$\begin{aligned}
\frac{\partial}{\partial v_i}\frac{\partial}{\partial v_j} D_{ij}({\boldsymbol}v)f({\boldsymbol}v)=\frac{1}{4\pi v^2} \bigg[ \frac{\partial^2}{ \partial v^2} \lambda(v) \tilde f(v)-\frac{\partial}{ \partial v} \frac{2}{v} \mu(v) \tilde f(v) \bigg].\end{aligned}$$ This leads to the isotropic form of –: $$\begin{aligned}
\frac{\partial}{\partial t} \tilde f&=-\frac{\partial}{\partial v} \left( \Pi+\Lambda+\Phi \right) \tilde f\label{eq:pdfveliso},\\
0&=-\frac{\partial}{\partial v} \left( \Lambda+\frac{2\mu}{v} \right) \tilde f + \frac{\partial^2}{\partial v^2} \lambda \tilde f\label{eq:pdfhomoiso},\\
\frac{\partial}{\partial t} \tilde f&=-\frac{\partial}{\partial v} \left( \Pi+\Phi-\frac{2\mu}{v} \right) \tilde f - \frac{\partial^2}{\partial v^2} \lambda \tilde f\label{eq:pdfvelhomoiso}.\end{aligned}$$ We note in passing that these equations can also be derived without the assumption of statistical isotropy and therefore correctly describe $\tilde{f}(v)$ also in statistically anisotropic turbulence. However, in that case they are not equivalent to – and $f({\boldsymbol}v)$ is not fully determined by $\tilde{f}(v)$. Furthermore, the relations – and are not valid in that case and do not allow a simple interpretation of the quantities $\Pi$, $\Lambda$, $\Phi$, $\lambda$ and $\mu$.
Homogeneous and Stationary PDFs
-------------------------------
Because now having reduced the problem to a one-dimensional, is easily integrated yielding the velocity PDF in homogeneous turbulence, $$\label{eq:homosol}
\tilde f(v;t)=\frac{{\cal N}}{\lambda(v,t)} \exp \int_{v_0}^v \mathrm{d}v' \, \frac{ \Lambda(v',t)+\frac{2}{v'}\mu(v',t)}{\lambda(v',t)},$$ showing that it can be expressed as a function of the conditional averages involving the velocity diffusion, dissipation, enstrophy and the stretching and turning of the velocity vector. Here ${\cal N}$ denotes a normalization constant, which depends on $v_0$. Note that by only imposing isotropy and homogeneity, the conditional averages will be a function of time. In the following, we will refer to this solution as the ‘homogeneous solution’.
In the case of forced turbulence, it is possible to maintain a statistically stationary flow. This immediately implies $\partial_tf=0$ and from follows $$\label{eq:probcurrent}
\Pi(v)+\Lambda(v)+\Phi(v)=0 ,$$ i.e. the pressure gradient term, the viscous term and the term stemming from the external forcing identically cancel. This also means that in the case of stationary, homogeneous and isotropic turbulence the probability current on the right-hand side of identically vanishes. Note that this is not obvious and is due to the fact that the problem becomes effectively one-dimensional by virtue of isotropy. In the case of the turbulent vorticity, this balance was discussed in [@novikov93jfr; @novikov94mpl] and more recently in [@wilczek09pre]. In the stationary case, and are equivalent due to the conditional balance . The stationary solution of reads as $$\label{eq:statsol}
\tilde f(v)=\frac{{\cal N}}{\lambda(v)} \exp \int_{v_0}^v \mathrm{d}v' \, \frac{ -\Pi(v')-\Phi(v')+\frac{2}{v'}\mu(v')}{\lambda(v')},$$ and can be either obtained by solving or from by utilizing the conditional balance and omitting the time dependence. We emphasize here that introducing the second derivatives with the help of the homogeneity relation is essential to obtain a unique stationary solution which cannot be obtained solely from .
We now have formally identified the quantities which determine the shape of the single-point velocity PDF, however, the explicit functional form of these quantities is unknown. Further input from the numerical or experimental side is needed to specify the PDF, which will be presented below, after discussing some more theoretical points. Note that the equations and relations derived so far are exact as no approximations have been made.
Constraints on the Functional Form of the Conditional Averages {#sec:Constraints}
--------------------------------------------------------------
Now that the theoretical framework is set up, we are faced with the closure problem of turbulence in terms of the unknown conditional averages $\lambda$, $\mu$, $\Lambda$, $\Pi$ and $\Phi$. In this section, we present constraints on the functional form of these quantities, which follow from elementary statistical relations and statistical isotropy. These constraints are useful to narrow down the possible functional forms of the conditional averages.
We start out with the pressure gradient. As already mentioned, its averaged projection on the velocity vector has to vanish because of incompressibility and homogeneity. This in turn gives an integral constraint on $\Pi(v)$[^3]: $$0 = \left \langle {\boldsymbol}{u}\cdot{\boldsymbol}{\nabla} p \right \rangle = \int_{0}^{\infty} \mathrm{d}v \, \left \langle {\boldsymbol}{u}\cdot{\boldsymbol}{\nabla} p \big | v \right \rangle \tilde f(v) = -\int_{0}^{\infty} \mathrm{d}v \, v\,\Pi(v) \tilde f(v).
\label{pressure-int-constraint}$$ With an analogous argumentation one also finds constraints for $\Lambda$, $\lambda$ and $\mu$, $$\begin{aligned}
-\left\langle \varepsilon \right\rangle &= \left\langle \nu {\boldsymbol}u \cdot \Delta {\boldsymbol}u\right\rangle= \int_{0}^{\infty} \mathrm{d}v \, v\,\Lambda(v) \tilde f(v) \label{Lambda-int-constraint}, \\
\left\langle \varepsilon \right\rangle &= \left\langle \mathrm{Tr}({\mathrm}{D}) \right\rangle= \int_{0}^{\infty} \mathrm{d}v \, [\lambda(v)+2\mu(v)] \tilde f(v) \label{lambda-mu-int-constraint},\end{aligned}$$ and in the stationary case also for the forcing $$\left\langle \varepsilon \right\rangle = \left\langle {\boldsymbol}u \cdot {\boldsymbol}F\right\rangle= \int_{0}^{\infty} \mathrm{d}v \, v\,\Phi(v) \tilde f(v).
\label{Phi-int-constraint}$$
A second class of constraints can be deduced from statistical isotropy, which states that the conditional averages are invariant under rotations e.g. $$\left\langle {\boldsymbol}{\nabla} p \big| {\mathrm}{R}{\boldsymbol}v\right\rangle={\mathrm}{R}\left\langle {\boldsymbol}{\nabla} p \big| {\boldsymbol}v\right\rangle,\qquad {\mathrm}{R}\in\mathrm{SO}(3).$$ From this relation, it follows that the *constant* vector $\left\langle {\boldsymbol}{\nabla} p \big| {\boldsymbol}v={\boldsymbol}{0}\right\rangle$ is invariant under rotations. However, this is possible only for the zero vector, thus $$\begin{aligned}
\left\langle {\boldsymbol}{\nabla} p \big| {\boldsymbol}v={\boldsymbol}{0}\right\rangle={\boldsymbol}{0} \quad \Rightarrow & \quad \Pi(0)=\lim_{v\rightarrow0}\hat{{\boldsymbol}v}\cdot\left\langle -{\boldsymbol}{\nabla} p \big| {\boldsymbol}v\right\rangle=0 .\end{aligned}$$ Note that the limit $\lim_{v\rightarrow0}\hat{{\boldsymbol}v}$ is not defined; however, the boundedness of $\hat{{\boldsymbol}v}$ is enough for the above result. In summary, we find $$\begin{aligned}
\left\langle {\boldsymbol}{\nabla} p \big| {\boldsymbol}v={\boldsymbol}{0}\right\rangle={\boldsymbol}{0} \quad \Rightarrow & \quad \Pi(0)=0 \label{pressure-zero-constraint}\\
\left\langle \nu \Delta {\boldsymbol}u \big| {\boldsymbol}v={\boldsymbol}{0}\right\rangle={\boldsymbol}{0} \quad \Rightarrow & \quad \Lambda(0)=0 \label{Lambda-zero-constraint}\\
\left\langle {\boldsymbol}F \big| {\boldsymbol}v={\boldsymbol}{0}\right\rangle={\boldsymbol}{0} \quad \Rightarrow & \quad \Phi(0)=0 \label{force-zero-constraint} .\end{aligned}$$ This type of argument can also be applied to tensor-valued isotropic functions. In three dimensions the only constant isotropic tensor of rank two is $\delta_{ij}$, which leads to constraints on the eigenvalues $$D_{ij}({\boldsymbol}v={\boldsymbol}{0}) \sim \delta_{ij}\quad\Rightarrow\quad\lambda(0)=\mu(0).
\label{lambda-eq-mu-at-zero}$$ We can go even further and use this type of argument to obtain constraints on the derivatives at ${\boldsymbol}{v}={\boldsymbol}0$. For this, first note that the isotropic forms of vector- and tensor-valued functions and are also invariant under reflections. Therefore, all the isotropic functions considered here are actually invariant under transformations in $\mathrm{O}(3)$. This means that in our case, the assumption of invariance under $\mathrm{SO}(3)$ yields invariance under $\mathrm{O}(3)$. Now, it can be easily checked that if a tensor-valued function is invariant under some transformations, then so are also its derivatives. Therefore, the $n$-th derivative of $D({\boldsymbol}{v})$ at ${\boldsymbol}{v}={\boldsymbol}0$ is a constant $(n+2)$-th-order tensor, invariant under $\mathrm{O}(3)$. For $n$ being odd, the derivative has to vanish, because the only constant tensor of odd order which is invariant under reflections is the zero tensor and it follows that $$\left. \frac{\partial}{\partial v_{i_1}}\cdots\frac{\partial}{\partial v_{i_n}} D_{ij}\right|_{{\boldsymbol}{v}=0}=0
\quad \Rightarrow \quad
\left.\frac{\mathrm{d}^n\lambda}{\mathrm{d}v^n}\right|_{v=0}=
\left.\frac{\mathrm{d}^n\mu}{\mathrm{d}v^n}\right|_{v=0}=0,$$ where $n$ is odd. With the very same argumentation applied to vectors (i.e. tensors of rank one) one also finds $$\begin{aligned}
\left.\frac{\mathrm{d}^n\Pi}{\mathrm{d}v^n}\right|_{v=0}=\left.\frac{\mathrm{d}^n\Lambda}{\mathrm{d}v^n}\right|_{v=0}=\left.\frac{\mathrm{d}^n\Phi}{\mathrm{d}v^n}\right|_{v=0}=0,
\label{derivative-general}\end{aligned}$$ where $n$ is even.
This shows that in isotropic turbulence, the series expansion of $\lambda$ and $\mu$ contains only even powers, whereas that of $\Pi$, $\Lambda$ and $\Phi$ contains only odd. These constraints have also to be kept in mind, when approximating the above functions with polynomials.
A Simple Analytical Closure {#sec:simple-closure}
---------------------------
We now try to find the most simple functional forms of the conditional averages which still fulfil the constraints described in the previous section. By ‘most simple‘ we mean the lowest-order polynomial. For $\Lambda$, a constant is not allowed because of and . We choose a linear function and determine the pre-factor through . We obtain $$\Lambda(v)=-\frac{\langle\varepsilon\rangle}{3\sigma^2}v$$ where $\sigma=\sqrt{\langle {\boldsymbol}{u}^2\rangle/3}$ is the standard deviation of the velocity. In the stationary case, we further find $\Phi(v)=\langle\varepsilon\rangle/(3\sigma^2)v$. For $\lambda$ and $\mu$, the most simple choice which is consistent with and is a constant: $$\lambda(v)=\mu(v)=\frac{\langle\varepsilon\rangle}{3}.$$ Inserting this ansatz for $\lambda$, $\mu$, $\Lambda$ into the homogeneous solution yields $$\tilde{f}(v;t)=\sqrt{\frac{2}{\pi}}\frac{v^2}{\sigma(t)^3}\exp\left( -\frac{1}{2} \frac{v^2}{\sigma(t)^2} \right),
\label{gaussian-angleintegrated}$$ which is the angle-integrated Gaussian distribution, i.e. the velocity *vector* is distributed according to a Gaussian $$f({\boldsymbol}{v};t)=\frac{1}{(2\pi\sigma(t)^{2})^{3/2}}\exp\left( -\frac{1}{2}\frac{{\boldsymbol}{v}^2}{\sigma(t)^2} \right),
\label{gaussian}$$ with standard deviation $\sigma(t)$. This solution is valid for decaying as well as stationary turbulence, where $\sigma(t)$ is a monotonously decaying function in the first and a constant in the latter case. The above solution rests upon the homogeneity relation , which in the stationary case is equivalent to the evolution equation . However, for decaying turbulence, this evolution equation gives another possibility to obtain the PDF which we demonstrate in the following. For this, we choose the most simple functional form for the pressure gradient which is consistent with and , namely $\Pi(v)=0$. This ansatz can be also viewed as an approximation which neglects the statistical dependence of the pressure gradient and the velocity. Furthermore, $\lambda$ and $\mu$ are chosen as above, representing statistical independence of the dissipation tensor and the velocity. With these assumptions, the evolution equation simplifies to $$\label{eq:simpleapprox}
\frac{\partial}{\partial t}f=-\frac{\langle\varepsilon\rangle(t)}{3}\Delta_{{\boldsymbol}v}f.$$ This is exactly the equation derived in [@hosokawa08pre] by neglecting pressure contributions and approximating the coupling of the dissipative term to the two-point PDF. By a change of variables [@hosokawa08pre], $$\tau(t)=\frac{1}{3}E_{kin}(t)=\frac{1}{3}\left[ E_{kin}(t_0) - \int^t_{t_0} \mathrm{d}t' \, \langle \varepsilon \rangle(t') \right]=\frac{1}{2}\sigma(t)^2
\label{tau-energy-sigma}$$ the equation takes the form of the heat equation $$\frac{\partial}{\partial \tau}f=\Delta_{{\boldsymbol}v}f.$$ The change of sign indicates that the process runs backwards in time. As stated before, when the fluid has come to rest, we find $f({\boldsymbol}v)=\delta({\boldsymbol}v)$, which now may serve as an initial condition. The solution reads as $$f({\boldsymbol}v;t)=\frac{1}{(4\pi\tau(t))^{3/2}}\exp\left( -\frac{{\boldsymbol}v^2}{4\tau(t)} \right) ,$$ i.e. we have a Gaussian PDF, whose time evolution is solely determined by the mean rate of energy dissipation. Note that this solution is consistent with as $\tau(t)=\sigma(t)^2/2$. Equation highlights the connection between $\langle\varepsilon\rangle(t)$ and $\sigma(t)$ for decaying turbulence; because the energy dissipation reduces the kinetic energy of the flow, the PDF becomes narrower and narrower as a function of time.
The results presented in this section should not be understood as a justification for the Gaussianity of the velocity as they have been obtained by using purely mathematical arguments, which do not account for any physical properties of the velocity field. For example, the argumentation using the homogeneous solution (presented at the beginning of this section) can also be applied to the vorticity field, which is known to exhibit a highly non-Gaussian distribution. The validity of the above results depends on how well the simple choices for the conditional averages agree with their functional forms measured in turbulence. For this comparison, we refer to the next sections, where the numerical results will be presented.
With the results of this section, the Gaussian distribution can be viewed as the most simple solution of the equations describing the velocity PDF, which is consistent with the constraints presented above. In a sense it is a zeroth-order approximation to the real velocity PDF. Deviations of the conditional averages from their most simple forms lead to deviations from the Gaussian distribution. In the following sections, we will see that this zeroth-order approximation roughly describes the velocity PDF, but completely fails for the vorticity PDF.
DNS results
===========
Some comments on the numerical simulations
------------------------------------------
$N^3$ $R_{\lambda}$ $L$ $T$ $u_{rms}$ $\nu$ $\left\langle \varepsilon \right\rangle$ $ k_{max}\eta$
-- --------- --------------- ------ ------ ----------- ------------ ------------------------------------------ ---------------- -- --
$512^3$ 112 1.55 2.86 0.543 $10^{-3} $ 0.103 2.03
: Major simulation parameters. Number of grid points $N^3$, Reynolds number based on the Taylor micro-scale $R_{\lambda}$, integral length scale $L$, large-eddy turnover time $T$, root-mean-square velocity $u_{rms}$, kinematic viscosity $\nu$, average rate of energy dissipation $\left\langle \varepsilon \right\rangle$. Here $k_{max}\eta$ characterizes the spatial resolution of the smallest scales, where $\eta$ is the Kolmogorov length scale and $k_{max}=0.8 N/2$ is the highest resolved wavenumber (the factor 0.8 is due to the aliasing filter).[]{data-label="tab:simpara"}
Before presenting the numerical results, some comments on the numerical scheme are in order. The presented results are generated with a standard, dealiased Fourier-pseudo-spectral code [@canuto87book; @hou2007jcp] for the vorticity equation. The integration domain is a triply periodic box of box length $2\pi$. The time-stepping scheme is a third-order Runge–Kutta scheme [@shu88jcp]. For the statistically stationary simulations, a large-scale forcing has to be applied to the flow. Here, care has to be taken in order to fulfil the statistical symmetries we make use of in our theoretical framework. After numerous tests, we chose a large-scale forcing which conserves the energy of the flow. This forcing has been found to deliver accurate results concerning the statistical symmetries. However, the results presented here do not depend on the exact form of the forcing, which has been verified using other forcing methods (e.g. the common method which holds the amplitudes of the Fourier coefficients in a band constant).
The numerical results presented in the following are obtained from a simulation whose simulation parameters are listed in table \[tab:simpara\]. Care has been taken to produce a well-resolved simulation with an integral length scale which is not too large compared with the box length of $2\pi$. We have found that the statistical symmetries, especially isotropy, are not valid for a snapshot of the velocity field at a single point in time, but can be obtained through time- or ensemble-averaging, respectively. This is due to long-range correlation of the velocity and is not as pronounced for the vorticity, a short-range correlated quantity. To ensure the validity of the statistical symmetries, the averages for the stationary case have been obtained through spatial and temporal averaging over more than 150 large-eddy turnover times, whereas for decaying turbulence a spatial and an ensemble average over 12 realizations has been applied.
It may be doubted that the results obtained from a single numerical set-up are of universal nature, for example there may be influences of the imposed (periodic) boundary conditions. This critique, of course, also applies to any experimental result, so that a compilation of data from many different numerical and experimental set-ups would be useful to discuss the issue on a more general level. However, the results presented in the following will make a good point that there are deviations from the Gaussian shape in both stationary and decaying turbulence.
Stationary Turbulence {#sec:stationary-turbulence}
---------------------
The homogeneous and stationary solutions and of the PDF equations allow a detailed analysis of the connection between the functional shape of the PDF and the statistical correlations of the different dynamical quantities, represented by the conditional averages $\Pi$, $\Lambda$, $\Phi$, $\lambda$ and $\mu$. This analysis is presented in this section for the case of stationary turbulence.
![The conditional averages $\Pi(v)$, $\Lambda(v)$, $\Phi(v)$ and the PDF $\tilde f(v)$. The $v$-axis is scaled with the standard deviation of the velocity $\sigma=u_{rms}=\sqrt{\left\langle {\boldsymbol}u^2/3\right\rangle}$. The conditional averages show an explicit $v$-dependence indicating statistical correlations.[]{data-label="fig:PILAPHI"}](PILAPHI){width="65.00000%"}
To start with, we consider the diffusive term, shown in figure \[fig:PILAPHI\]. It exhibits a negative correlation with the velocity, indicating that a fluid element will on average be decelerated due to viscous forces. The pressure term, however, displays an interesting zero-crossing. This means that a fluid particle will on average be decelerated for low values of velocity and accelerated for high values of velocity. If the pressure term is non-vanishing, it *has* to exhibit this zero-crossing, because the integral constraint can be fulfilled only through a zero-crossing or vanishing of $\Pi(v)$. Our numerical investigations show that the pressure contributions to the single-point velocity PDF cannot be neglected, in contrast to the assumptions of some recent theories [@tatsumi04fdr; @hosokawa08pre]. This is especially true for high velocities, where we see a strong $v$-dependence of $\Pi$ and hence a clear statistical dependence of the pressure gradient and the absolute value of the velocity.
The conditionally averaged forcing term in figure \[fig:PILAPHI\] was computed based on the conditional balance . This is because the forcing in our numerical scheme is implicitly applied to the vorticity field and is not available as an additive field that we can average conditionally. However, some careful tests have been performed to ensure that this balance actually holds. An interesting feature of $\Phi$ is that it seems to saturate for large values of $v$ in contrast to $\Pi$ and $\Lambda$, suggesting that the forcing plays a minor role for the statistics of high velocities. In [@falkovich97prl] it has been argued that the tails of the velocity PDF are related to those of the forcing. Our numerical results, however, suggest that the main contribution comes from the dynamical effects of the pressure gradient and the velocity diffusion. Furthermore, the saturation of $\Phi$ suggests a simple relation between $\Pi$ and $\Lambda$ for high velocities, namely an approximately constant offset. This shows that for high velocities, the statistical dependences of the pressure gradient and the velocity diffusion on the velocity are strongly related. A natural question concerning the conditionally averaged forcing is whether its form depends on the type of forcing. We have tested two other common forcing methods (these methods manipulate the vorticity in a Fourier band by holding either the Fourier coefficients or their amplitudes constant), finding that the general shape of $\Phi$ does not depend on the forcing method.
![The conditional averages $\mu(v)$, $\lambda(v)$, $\left\langle \varepsilon |v \right\rangle$, $\left\langle \nu \omega^2 |v \right\rangle$ and the PDF $\tilde f(v)$. The conditional averages show an explicit $v$-dependence indicating statistical correlations.[]{data-label="fig:mulaepsz"}](mulaepsz){width="65.00000%"}
Now let us proceed to investigate the conditional energy dissipation tensor, its eigenvalues are shown in figure [\[fig:mulaepsz\]]{} together with the PDF $\tilde f(v)$. Both eigenvalues have a similar dependence on $v$, they are almost constant for small and medium velocities, but display a strong $v$-dependence for high velocities. This shows that the occurrence of high velocities is statistically correlated with strong dissipation and enstrophy.
![Comparison of the PDF $\tilde f(v)$ with the evaluation of . The reconstructed PDF collapses with the directly estimated PDF; therefore, the latter is marked by black dots to indicate its presence. The excellent agreement demonstrates the consistency of the theoretical results. A comparison with an angle-integrated Gaussian shows that, although being close to Gaussian, significant deviations exist.[]{data-label="fig:recpdf"}](reconstructed_vs_measured){width="65.00000%"}
The conditional averages shown in figures \[fig:PILAPHI\] and \[fig:mulaepsz\] have been tested to obey the constraints presented in §\[sec:Constraints\]. The validity of the constraints at $v=0$ can be seen in the figures, whereas the integral constraints have been verified numerically. As a test for consistency, the conditional averages are inserted into to calculate the homogeneous PDF, and the result is presented in figure \[fig:recpdf\]. An excellent agreement with the PDF directly estimated from the numerical data is found. The same results can be obtained using as it is equivalent to in the stationary case. A comparison with an angle-integrated Gaussian shows that, although the PDF is similar to Gaussian, significant deviations occur. Especially notable is the sub-Gaussian tail, which has also been found in experiments and other numerical simulations [@vincent91jfm; @noullez97jfm; @gotoh02pof].
It is well known for a long time that the velocity PDF is similar to a Gaussian distribution. However, this fact (coming from experiments and simulations) is *a priori* not clear at all. One can certainly arrive at a Gaussian distribution by neglecting certain terms or assuming the most simple forms for the conditional averages (see §\[sec:simple-closure\]). However, as can be clearly seen in figures \[fig:PILAPHI\] and \[fig:mulaepsz\], those assumptions are violated. Furthermore, some of the arguments can also be applied to the vorticity which is known to exhibit a highly non-Gaussian PDF. This shows that one has to be very careful with the utilization of such simple arguments. Therefore, here we rather use the measured conditional averages to understand the similarity of the velocity PDF to the Gaussian distribution. The question sought after is, how it is possible to find strong statistical correlations for the various conditional averages and the velocity, but still only moderate deviations from Gaussianity for the PDF.
As already stated, there are clear deviations of the conditional averages from the simple forms suggested in §\[sec:simple-closure\]. We see that for small and medium velocities, the assumption of linear $\Lambda$ (or equivalently of linear $\Phi$ and vanishing $\Pi$) and constant $\lambda,\mu$ can be seen as a rough approximation. Because of this, the PDF stays close to a Gaussian in this range of velocities. However, for high velocities we see a strong $v$-dependence of $\lambda$, $\mu$ and a functional form of $\Lambda$ which is clearly not linear. Still, only moderate deviations from a Gaussian tail occur. To understand this, let us examine again the relation between the PDF and the conditional averages: $$\tilde f(v)=\frac{{\cal N}}{\lambda(v)} \exp \int_{v_0}^v \mathrm{d}v' \, \frac{ \Lambda(v')+\frac{2}{v'}\mu(v')}{\lambda(v')}.
\label{eq:homsol-again}$$ We see that only the quotients $\Lambda/\lambda$ and $\mu/\lambda$ enter the exponential function. As $-\Lambda$ and $\lambda$ both are bended upwards for high velocities, their quotient might still be linear, i.e. the deviations from the simple forms of §\[sec:simple-closure\] might compensate each other. Indeed, this is roughly true as can be seen in figure \[fig:integrands-analyzed\], which also shows that $\lambda$ and $\mu$ are roughly equal. Furthermore, a comparison with the case of vorticity is shown, which will be discussed later. Assuming for the moment a linear $\Lambda/\lambda$ and $\mu/\lambda=1$ yields a Gaussian shape for the exponential factor in . In this case, the second factor, $1/\lambda$, is then responsible for (slight) modifications of the Gaussian shape. Therefore, it appears that the compensation of $\Lambda$ and $\lambda$ to yield an approximately linear quotient $\Lambda/\lambda$ is crucial for the similarity of the velocity PDF to the Gaussian distribution. For example, if one artificially sets $\lambda=\mu=\langle\varepsilon\rangle/3$ as in §\[sec:simple-closure\], but still uses the measured and highly nonlinear $\Lambda$ for the reconstruction, one obtains a PDF which decays much faster than a Gaussian.
To discuss the issue of the sub-Gaussian tails, we need to take into account the deviations of the quotients from a linear and constant function. A detailed analysis shows that the deviations from Gaussianity cannot be solely attributed to the pre-factor $1/\lambda$, but are also due to the aforementioned deviations of the quotients. What makes the issue even more complicated is that the relative amount of contribution to the deviations by the two factors in seems to vary with the Reynolds number, as has been checked with a number of simulations not presented here. This shows that the detailed properties of the PDF of the velocity are due to a subtle interplay of the statistical dependences of dynamical quantities such as the dissipation, the velocity diffusion or the pressure gradient.
### Approximating the Dissipation Tensor
In view of the complicated statistical dependences, it is natural to ask for approximations which simplify the matter but still describe the velocity PDF reasonably well e.g. maintain the sub-Gaussianity of the velocity PDF. It turns out that the conditional dissipation tensor allows simplifications which contain some interesting physical insights. It appears from figure \[fig:mulaepsz\] that the eigenvalues of ${\mathrm}{D}$ have a similar functional form, suggesting the approximation $\lambda(v)\approx\mu(v)$. Thus, all eigenvalues of ${\mathrm}{D}$ are approximately equal, which basically states that the conditional dissipation tensor does not contain any directional information and is proportional to the identity matrix $$D_{ij}({\boldsymbol}v)\approx\frac{1}{3}\mathrm{Tr}({\mathrm}{D})\,\delta_{ij}=\frac{1}{6}\langle \varepsilon + \nu \omega^2 |v \rangle \, \delta_{ij}.$$ In this approximation, ${\mathrm}{D}$ depends only on the magnitude of the velocity, i.e. the dissipation tensor $\nu{\mathrm}{AA}^{\mathrm T}$ and the *direction* of the velocity vector are uncorrelated. Note that this property is not a consequence of the statistical isotropy, but rather a special property of the velocity in turbulent flows (this will become more clear when we examine the dissipation tensor of the vorticity).
The equality of the eigenvalues may be motivated with a simple scale-separation argument. The velocity itself is known to be a rather smooth vector field varying quite slowly in space. Compared to that, the gradients vary on a much shorter scale (see figure \[fig:vortvel\]). The numerical results suggest that quantities like, for example, the dissipation depend on the absolute value $v$. If we now assume that the dependence on $v$ is crucial, but due to scale separation no correlation between the direction of ${\boldsymbol}u$ and the velocity gradient tensor exists, we may consider a projection on $\hat{{\boldsymbol}u}$ as a projection on a random direction. This immediately gives $$\langle \nu ({\mathrm}{A}^{\mathrm T}\hat{{\boldsymbol}u})^2 | v \rangle=\langle \nu \hat{{\boldsymbol}u}{\mathrm}{AA}^{\mathrm T}\hat{{\boldsymbol}u} | v \rangle=\frac{1}{6}\langle \varepsilon + \nu \omega^2 | v \rangle,$$ i.e. averaging the projection of a tensor on a random direction yields one-third of the conditionally averaged trace of the tensor. This in turn yields $\lambda(v)=\mu(v)$, which is suggested by the numerical results as an approximation.
As the trace is the central quantity, one may wonder if further simplifications are possible. To this end, we consider the average proportionality of enstrophy and dissipation in homogeneous turbulence, $\langle \varepsilon \rangle=\langle \nu \omega^2 \rangle $, which comes due to $\langle \Delta p \rangle=0$. It is now straightforward to ask whether this balance also holds in a stronger sense, when the conditional average is considered. The numerical evaluation of these terms is also presented in figure \[fig:mulaepsz\], supporting that this stronger equality actually holds. As a consequence, the approximation $\frac{1}{2}\langle \varepsilon + \nu \omega^2 | v \rangle \approx \langle \varepsilon | v \rangle$ seems reasonable. Taking this all together, the conditional dissipation tensor can be approximated as $$D_{ij}({\boldsymbol}v) \approx \frac{1}{3} \langle \varepsilon | v \rangle \, \delta_{ij},$$ depending only on the conditionally averaged rate of kinetic energy dissipation.
It may be hypothesized that the approximate relations so far only hold asymptotically with e.g. increasing Reynolds number. By comparing simulations with different Reynolds numbers (up to $Re_\lambda\approx200$), we were not able to clearly verify (or falsify) this statement; hence, simulations with significantly higher Reynolds numbers are needed to answer this question.
In the light of these simplifications, it is interesting to reconsider the functional form of the velocity PDF. Equations and together with $\lambda(v)\approx\mu(v)\approx \langle \varepsilon | v \rangle/3$ instantly yield $$\begin{aligned}
\label{eq:simplepdf}
f({\boldsymbol}v)&=\frac{{\cal N}}{\langle \varepsilon | v \rangle} \exp \int_{v_0}^v \mathrm{d} v' \, 3 \, \frac{\langle \nu \hat{{\boldsymbol}u}\cdot \Delta {\boldsymbol}u | v' \rangle}{\langle \varepsilon | v' \rangle} \nonumber \\
&=\frac{{\cal N}}{\langle \varepsilon | v \rangle} \exp \int_{v_0}^v \mathrm{d} v' \, 3 \, \frac{\langle \hat{{\boldsymbol}u}\cdot {\boldsymbol}{\nabla} p-\hat{{\boldsymbol}u}\cdot {\boldsymbol}{F} | v' \rangle}{\langle \varepsilon | v' \rangle} ,\end{aligned}$$ indicating that the shape of the velocity PDF may be solely determined by the velocity diffusion and the rate of energy dissipation. As the diffusive term balances the sum of the pressure and the forcing term, the PDF is equivalently determined by how energy is injected into the system, redistributed by non-local pressure effects and finally dissipated.
![Comparison of the PDF $\tilde f(v)$ with the angle-integrated PDF of the approximation .[]{data-label="fig:approximations"}](reconstructed_vs_approx){width="65.00000%"}
Figure \[fig:approximations\] compares this approximation with the measured PDF and a Gaussian. We see that the above formula for the velocity PDF holds quite well for the core of the PDF, but leads to deviations in the tail. This (in addition to figure \[fig:mulaepsz\]) shows that the assumption of absent correlation between the dissipation tensor and the direction of the velocity becomes less valid for high velocities. Nevertheless, this approximation describes the velocity significantly better than the Gaussian distribution.
We note here that the closeness of the velocity PDF to a Gaussian can be attributed to the approximate validity of $$\frac{\langle \nu \hat{{\boldsymbol}u} \cdot \Delta {\boldsymbol}u | v \rangle}{\langle \varepsilon | v \rangle} \approx-\frac{v}{3\sigma^2}.
\label{eq:linear-quotient-approx}$$ It describes the relation of the dissipation and the velocity diffusion, which is actually the dynamical effect responsible for the dissipation. It is the theoretical framework presented in the previous sections and in particular the formula which allows to relate the observation of closely Gaussian velocity PDF to the non-obvious relation above.
Decaying Turbulence
-------------------
![Velocity PDF $\tilde f(v;t)$ for decaying turbulence at time instants separated by $0.35T$, $T$ being the large-eddy turnover time of the initial condition. The insets show the PDFs normalized to unit variance, indicating the existence of a self-similar regime.[]{data-label="fig:velpdftemp"}](decay_pdfs2){width="65.00000%"}
![Comparison of $\tilde f(v)$ (directly estimated and reconstructed from ) together with an angle-integrated Gaussian. A deviation from the Gaussian shape is also found in the decaying case. The structure of the conditional averages is similar to the case of stationary turbulence. The averages and the PDFs are computed from an ensemble for a decay time of $1.4T$.[]{data-label="fig:recpdfdecay"}](PILAPHI_2000 "fig:"){width="50.00000%"}![Comparison of $\tilde f(v)$ (directly estimated and reconstructed from ) together with an angle-integrated Gaussian. A deviation from the Gaussian shape is also found in the decaying case. The structure of the conditional averages is similar to the case of stationary turbulence. The averages and the PDFs are computed from an ensemble for a decay time of $1.4T$.[]{data-label="fig:recpdfdecay"}](mulaepsz_2000 "fig:"){width="50.00000%"}
![Sum of the pressure and diffusive terms for decaying turbulence together with the analytical relation in straight lines. Slight deviations are visible after $0.7T$, which are negligible later on.[]{data-label="fig:selfsim"}](selfsim){width="65.00000%"}
![Kinetic energy for decaying turbulence as a function of time. After a transient the energy decays algebraically with an exponent close to $1.4$. The inset shows the rate of energy dissipation, which also exhibits an approximately algebraic decay with an exponent close to $2.4$.[]{data-label="fig:decayenergy"}](decay_energy){width="65.00000%"}
We now study the shape and evolution of the velocity PDF in the case of decaying turbulence. The motivation for this is two fold. First, it has recently been hypothesized by [@hosokawa08pre] that decaying turbulence might be profoundly different from forced, stationary turbulence. As mentioned in §\[sec:simple-closure\], Hosokawa found Gaussian solutions for decaying turbulence by approximate closure assumptions. Second, in the work by [@falkovich97prl], the argument for sub-Gaussian tails of the velocity PDF relies strongly on the statistics of the external forcing. This effect, of course, is absent for decaying turbulence.
Figure \[fig:velpdftemp\] shows the velocity PDF for decaying turbulence $\tilde f(v;t)$ for different points in time. The results were obtained by averaging over an ensemble of 12 independent simulations to increase the statistical quality. The variance of the PDFs decreases as a function of time as the kinetic energy of the system is dissipated. However, when rescaled to unit variance, the PDFs collapse, indicating a self-similar regime. Hence, once the PDF is specified at the beginning of this regime, it is determined subsequently according to $$\label{eq:selfsimilarpdf}
\tilde f(v;t)= \frac{\sigma(t_0)}{\sigma(t)} \tilde f \left( \frac{\sigma(t_0)}{\sigma(t)} v; t_0 \right),$$ where $\sigma(t)$ may be related to the kinetic energy or the rate of energy dissipation according to . Turning to a closer investigation of the functional form of the PDF in this regime still reveals deviations from Gaussianity, as can be seen in figure \[fig:recpdfdecay\]. Here, also the conditional averages of the pressure term, the diffusive term and the terms related to the conditional dissipation tensor are shown. They all show a functional form coinciding with the observations in stationary turbulence, indicating that the statistical correlations and the shape of the PDF do not fundamentally differ for decaying turbulence. As a check for consistency, the PDF is computed from the conditional averages by , which again performs very well. On the basis of these results, we can conclude that sub-Gaussian velocity PDFs are also found for decaying turbulence, which can be tracked down to the interplay of statistical correlations already discussed in the stationary case.
It is interesting to study this self-similar decay more deeply in the framework of the PDF equation . For decaying turbulence, it is particularly useful to employ the method of characteristics. Let $V(t,v_0)$ denote a characteristic curve which starts from $v_0$, i.e. $V(t_0,v_0)=v_0$. The method of characteristics then yields the following equations: $$\begin{aligned}
\label{eq:characteristics}
\frac{\mathrm{d}}{\mathrm{d} t} V(t,v_0) &= \left[ \Pi(v,t)+\Lambda(v,t) \right]_{v=V(t,v_0)}, \\
\frac{\mathrm{d}}{\mathrm{d} t} \tilde f(V(t,v_0);t) &= \left[ -\frac{\partial}{\partial v} \big( \Pi(v,t)+\Lambda(v,t) \big) \right]_{v=V(t,v_0)} \, \tilde f(V(t,v_0);t) ,\end{aligned}$$ of which the latter is easily integrated. We obtain the evolution of $\tilde f$ along the characteristic curves: $$\tilde f(V(t,v_0);t) = \tilde f(v_0 ; t_0) \, \exp \left[ -\int_{t_0}^t\mathrm{d}t' \, \left[ \frac{\partial}{\partial v} \big( \Pi(v,t')+\Lambda(v,t') \big)\right]_{v=V(t',v_0)} \right] .$$ Of course, we are more interested in the temporal evolution of $f(v;t)$ instead of $f(V(t,v_0);t)$. This mapping can be achieved with the inverse function of $V(t,v_0)$, which is defined by $V^{-1}(t,V(t,v_0)) = v_0$. Thus, the temporal evolution of the PDF of the magnitude of velocity is given by $\tilde f(v;t) = \tilde f(V(t,V^{-1}(t,v));t)$, such that we obtain $$\tilde f(v;t) = \tilde f(V^{-1}(t,v) ; t_0) \, \exp \left[ -\int_{t_0}^t\mathrm{d}t' \, \left[ \frac{\partial}{\partial v'} \big( \Pi(v',t')+\Lambda(v',t') \big)\right]_{v'=V(t',V^{-1}(t,v))} \right] .$$ The interesting fact about this result now is that we obtain self-similar solutions of the form if and only if the characteristic curves are of the form $$V(t,v_0) = \frac{\sigma(t)}{\sigma(t_0)} v_0,$$ which due to gives rise to the relation $$\label{eq:selfsim}
\Pi(v,t)+\Lambda(v,t) = v \, \frac{\mathrm{d}}{\mathrm{d} t}\ln\left[ \frac{\sigma(t)}{\sigma(t_0)} \right] = -\frac{1}{2} \frac{\langle \varepsilon \rangle(t)}{E_{kin}(t)} \, v .$$ This means that we only obtain self-similar solutions if the sum of the conditional averages related to the pressure gradient and the Laplacian of the velocity is a linear function of $v$, with a negative slope proportional to the ratio of the rate of energy dissipation and the kinetic energy. Note that this non-trivial relation is a direct consequence of the observation of a self-similar regime of the velocity PDF, no matter whether it is Gaussian or not. The relation is also checked with our numerical results with some examples shown in figure \[fig:selfsim\]. Apart from the beginning of the decay phase, the analytical relation is in very good agreement with measured conditional averages.
To characterize the statistics of decaying turbulence further, we study the temporal evolution of the kinetic energy and the rate of energy dissipation. The decay rate of energy is a central question in turbulence research, which still remains a point of debate. We refer the reader to [@karman38prs; @kolmogorov41dan; @batchelor48prs; @saffman67pof; @george92pfa] for a detailed account of the theoretical aspects of this question as well as some classical experimental results. While a power-law decay of the kinetic energy is widely accepted, the prediction for the precise numeric value of the decay exponent varies. Detailed numerical investigations on this have been presented recently in [@ishida06jfm; @perot11aip]. Apart from the dependence of the decay exponent on some fundamental theoretical issues concerning the Loitsyansky integral, it has to be assumed that it depends on properties of the initial conditions [@george92pfa; @ishida06jfm]. As we have not investigated a larger class of initial conditions, the following results shall characterize the simulations rather than making a point for or against some of the cited theories.
The kinetic energy for a single run is shown in figure \[fig:decayenergy\]. After a short transient period of about $1T$ ($T$ is the large-eddy turnover time of the initial condition), the kinetic energy decays algebraically with an exponent close to $1.4$, i.e. $E_{kin}(t) \sim t^{-1.4}$, which is in good agreement with the results presented in [@ishida06jfm] and close to the Kolmogorov prediction of $E_{kin}(t) \sim t^{-10/7}$. Accordingly, the kinetic energy dissipation decays with an exponent close to $2.4$ in this regime, as shown in the inset of figure \[fig:decayenergy\]. The self-similar decay together with an algebraic evolution of the variance indicates a particularly simple evolution in time. It is also interesting to note that the slight deviations from linearity of the conditional acceleration observed in figure \[fig:selfsim\] lie within the short initial transient. It seems as if the turbulence needs a short relaxation time to switch from the forced regime to the decaying regime.
Comparison with the Vorticity {#vorticity-section}
-----------------------------
![Volume rendering of the magnitude of the velocity field (left top), the trace of the dissipation tensor $\nu\mathrm{Tr}({\mathrm}{AA}^{\mathrm T})=\frac{1}{2}(\varepsilon+\nu \omega^2)$ (also known as pseudo-dissipation, left bottom), the magnitude of the vorticity field (right top) and the trace of the enstrophy dissipation tensor $\nu\mathrm{Tr}\left[ {\boldsymbol}{\nabla} {\boldsymbol}{\omega} ({\boldsymbol}{\nabla} {\boldsymbol}{\omega})^{\mathrm T} \right]$ (right bottom). The colours blue, green, yellow, red in that order indicate increasing amplitude. The vorticity and both dissipation tensors tend to organize into entangled small-scale structures forming a complicated global structure. The velocity appears more unstructured but displays long-range correlations. The visualizations have been produced with VAPOR (`www.vapor.ucar.edu`).[]{data-label="fig:vortvel"}](velocity "fig:"){width="50.00000%"}![Volume rendering of the magnitude of the velocity field (left top), the trace of the dissipation tensor $\nu\mathrm{Tr}({\mathrm}{AA}^{\mathrm T})=\frac{1}{2}(\varepsilon+\nu \omega^2)$ (also known as pseudo-dissipation, left bottom), the magnitude of the vorticity field (right top) and the trace of the enstrophy dissipation tensor $\nu\mathrm{Tr}\left[ {\boldsymbol}{\nabla} {\boldsymbol}{\omega} ({\boldsymbol}{\nabla} {\boldsymbol}{\omega})^{\mathrm T} \right]$ (right bottom). The colours blue, green, yellow, red in that order indicate increasing amplitude. The vorticity and both dissipation tensors tend to organize into entangled small-scale structures forming a complicated global structure. The velocity appears more unstructured but displays long-range correlations. The visualizations have been produced with VAPOR (`www.vapor.ucar.edu`).[]{data-label="fig:vortvel"}](vorticity "fig:"){width="50.00000%"} ![Volume rendering of the magnitude of the velocity field (left top), the trace of the dissipation tensor $\nu\mathrm{Tr}({\mathrm}{AA}^{\mathrm T})=\frac{1}{2}(\varepsilon+\nu \omega^2)$ (also known as pseudo-dissipation, left bottom), the magnitude of the vorticity field (right top) and the trace of the enstrophy dissipation tensor $\nu\mathrm{Tr}\left[ {\boldsymbol}{\nabla} {\boldsymbol}{\omega} ({\boldsymbol}{\nabla} {\boldsymbol}{\omega})^{\mathrm T} \right]$ (right bottom). The colours blue, green, yellow, red in that order indicate increasing amplitude. The vorticity and both dissipation tensors tend to organize into entangled small-scale structures forming a complicated global structure. The velocity appears more unstructured but displays long-range correlations. The visualizations have been produced with VAPOR (`www.vapor.ucar.edu`).[]{data-label="fig:vortvel"}](TrD_u "fig:"){width="50.00000%"}![Volume rendering of the magnitude of the velocity field (left top), the trace of the dissipation tensor $\nu\mathrm{Tr}({\mathrm}{AA}^{\mathrm T})=\frac{1}{2}(\varepsilon+\nu \omega^2)$ (also known as pseudo-dissipation, left bottom), the magnitude of the vorticity field (right top) and the trace of the enstrophy dissipation tensor $\nu\mathrm{Tr}\left[ {\boldsymbol}{\nabla} {\boldsymbol}{\omega} ({\boldsymbol}{\nabla} {\boldsymbol}{\omega})^{\mathrm T} \right]$ (right bottom). The colours blue, green, yellow, red in that order indicate increasing amplitude. The vorticity and both dissipation tensors tend to organize into entangled small-scale structures forming a complicated global structure. The velocity appears more unstructured but displays long-range correlations. The visualizations have been produced with VAPOR (`www.vapor.ucar.edu`).[]{data-label="fig:vortvel"}](TrD_w "fig:"){width="50.00000%"}
To demonstrate that some of the results are unique features of the velocity statistics, we draw a comparison with the statistics of the turbulent vorticity, which has been studied more extensively in [@wilczek09pre]. A first glance at snapshots of the fields reveals a fundamental difference between velocity fields and vorticity fields as shown in figure \[fig:vortvel\]. While the vorticity tends to be organized in small filamentary structures, the velocity seems to be more structureless and smeared out. Recall that the velocity may be calculated from the vorticity with the help of the Biot–Savart law, which may be interpreted as some kind of a smoothing filter. On the level of the single-point statistics, the most striking difference is that the vorticity displays a highly non-Gaussian PDF with pronounced tails, which is closely related to the coherent structures present in the flow. Moreover, as for example reasoned in [@tennekes83book], in the case of the turbulent vorticity the average enstrophy production and enstrophy dissipation tend to cancel, which means that the external forcing has a negligible effect on the statistics of the vorticity at high Reynolds numbers.
The evolution equation for the vorticity PDF $f_{\Omega}({\boldsymbol}\Omega; {\boldsymbol}x,t)$ can be derived analogously to the PDF equation for the velocity. For homogeneous turbulence, it takes the form $$\frac{\partial}{\partial t}f_{\Omega} = - {\boldsymbol}{\nabla}_{{\boldsymbol}\Omega}\cdot\left[ \left\langle {\mathrm}{S} {\boldsymbol}\omega + \nu \Delta {\boldsymbol}\omega + {\boldsymbol}{\nabla} \times {\boldsymbol}F\, \big | {\boldsymbol}\Omega \right\rangle f_{\Omega} \right] \label{eq:evolution_liouville_omega}.$$ Exploiting homogeneity, the diffusive term may also be rearranged, such that the PDF equation takes the form $$\frac{\partial}{\partial t}f_{\Omega} = -{\boldsymbol}{\nabla}_{{\boldsymbol}\Omega}\cdot \left[ \left\langle {\mathrm}{S} {\boldsymbol}\omega + {\boldsymbol}{\nabla} \times {\boldsymbol}F\, \big| {\boldsymbol}\Omega \right\rangle f_{\Omega} \right]
-\frac{\partial^2}{\partial \Omega_i \partial \Omega_j} \left\langle \nu \frac{\partial \omega_i}{\partial x_k} \frac{\partial \omega_j}{\partial x_k} \bigg| {\boldsymbol}\Omega \right\rangle \! f_{\Omega} .$$ Hereby the conditional enstrophy dissipation tensor enters. Because of statistical isotropy the quantities in these equations take the form $$\begin{aligned}
\left\langle \nu \Delta {\boldsymbol}\omega \big | {\boldsymbol}\Omega \right\rangle &= \Lambda_\Omega(\Omega) \hat{{\boldsymbol}\Omega},
\qquad \Lambda_\Omega(\Omega)=\left\langle \nu \hat{{\boldsymbol}\omega} \cdot \Delta {\boldsymbol}\omega \big | \Omega \right\rangle,\\
\left\langle {\boldsymbol}{\nabla} \times {\boldsymbol}F \big | {\boldsymbol}\Omega \right\rangle &= \Phi_\Omega(\Omega) \hat{{\boldsymbol}\Omega},
\qquad \Phi_\Omega(\Omega)=\left\langle \hat{{\boldsymbol}\omega} \cdot ({\boldsymbol}{\nabla} \times {\boldsymbol}F) \big | \Omega \right\rangle,\end{aligned}$$ $$\left\langle S_{ij} \big| {\boldsymbol}\Omega \right\rangle = \frac{1}{2} \Sigma(\Omega) \left(3 \frac{\Omega_i \Omega_j}{\Omega^2} - \delta_{ij} \right),\qquad \Sigma(\Omega) = \left\langle \hat{{\boldsymbol}\omega} {\mathrm}{S} \hat{{\boldsymbol}\omega} \big| \Omega \right\rangle,$$ $$D_{\Omega;ij}({\boldsymbol}\Omega)=\left\langle \nu \frac{\partial \omega_i}{\partial x_k} \frac{\partial \omega_j}{\partial x_k} \bigg | {\boldsymbol}\Omega \right \rangle=\mu_\Omega(\Omega)\delta_{ij}+\left[ \lambda_\Omega(\Omega)-\mu_\Omega(\Omega)\right]\frac{\Omega_i \Omega_j}{\Omega^2},$$ where $\hat{{\boldsymbol}\Omega}$ and $\hat{{\boldsymbol}\omega}$ are the unit vectors belonging to ${\boldsymbol}\Omega$ and ${\boldsymbol}\omega$. The rate-of-strain tensor may, due to incompressibility, be characterized by a single scalar function $\Sigma(\Omega)$ related to the enstrophy production. The conditionally averaged vortex stretching term can then be expressed as $$\left\langle {\mathrm}{S} {\boldsymbol}\omega \big | {\boldsymbol}\Omega \right\rangle = \left\langle {\mathrm}{S} \big | {\boldsymbol}\Omega \right\rangle {\boldsymbol}\Omega=\Sigma(\Omega)\Omega\,\hat{{\boldsymbol}\Omega} .$$ The eigenvalues of ${\mathrm}{D}_{\Omega}$ can be computed using relations analogous to and . The isotropic forms of the evolution equations and the homogeneity relation (analogous to –) are $$\begin{aligned}
\frac{\partial}{\partial t} \tilde f_\Omega&=-\frac{\partial}{\partial \Omega} \left( \Sigma\,\Omega+\Lambda_\Omega+\Phi_\Omega \right) \tilde f_\Omega\label{eq:pdfveliso_omega}\\
0&=-\frac{\partial}{\partial \Omega} \left( \Lambda_\Omega+\frac{2\mu_\Omega}{\Omega} \right) \tilde f_\Omega + \frac{\partial^2}{\partial \Omega^2} \lambda_\Omega \tilde f_\Omega\label{eq:pdfhomoiso_omega}\\
\frac{\partial}{\partial t} \tilde f_\Omega&=-\frac{\partial}{\partial \Omega} \left( \Sigma\,\Omega+\Phi_\Omega-\frac{2\mu_\Omega}{\Omega} \right) \tilde f_\Omega - \frac{\partial^2}{\partial \Omega^2} \lambda_\Omega \tilde f_\Omega\label{eq:pdfvelhomoiso_omega} .\end{aligned}$$ Solving yields a relation between the PDF and the conditional averages $$\label{eq:homosol_omega}
\tilde f_\Omega(\Omega;t)=\frac{{\cal N}}{\lambda_\Omega(\Omega,t)} \exp \int_{\Omega_0}^\Omega \mathrm{d}\Omega' \, \frac{ \Lambda_\Omega(\Omega',t)+\frac{2}{\Omega'}\mu_\Omega(\Omega',t)}{\lambda_\Omega(\Omega',t)}.$$ The reconstruction of the PDF using this formula again works perfectly well, as shown in figure \[fig:everything\_about\_omega\]. For stationary turbulence, the probability current in has to vanish and therefore also the conditional average in . However, it turns out that the effect of the external forcing may be neglected here, as can be seen from figure \[fig:everything\_about\_omega\]. Therefore, the conditionally averaged vortex stretching term and the diffusive term tend to cancel almost identically. This means that the balance between enstrophy production and dissipation holds even under conditional averaging, which has already been shown in [@novikov93jfr; @novikov94mpl; @wilczek09pre]. As a consequence, we may regard the enstrophy production as an internal process, which is decoupled from the external forcing. This interpretation is consistent with the common assumption that the small-scale properties of turbulence become independent of the forcing mechanism given a sufficiently high Reynolds number. As demonstrated in the preceding paragraphs, the forcing term may not be neglected when determining the stationary PDF of the velocity.
![The PDF of the vorticity and the conditional averages which determine its shape. Note that the forcing term vanishes and therefore $\Lambda_\Omega$ and $\Sigma \, \Omega$ cancel. The abscissa is normalized by the standard deviation of the vorticity, whereas the ordinate is normalized using the average of $\varepsilon_\Omega=\nu/2\mathrm{Tr}\left( \left[ {\boldsymbol}{\nabla} {\boldsymbol}\omega+({\boldsymbol}{\nabla} {\boldsymbol}\omega)^{\mathrm T}\right] ^2\right)$ and the standard deviation of the vorticity $\sigma_\Omega$. Note that not $\lambda_\Omega$ but $\lambda_\Omega/4$ is shown in the right figure.[]{data-label="fig:everything_about_omega"}](PILAPHI_omega "fig:"){width="50.00000%"}![The PDF of the vorticity and the conditional averages which determine its shape. Note that the forcing term vanishes and therefore $\Lambda_\Omega$ and $\Sigma \, \Omega$ cancel. The abscissa is normalized by the standard deviation of the vorticity, whereas the ordinate is normalized using the average of $\varepsilon_\Omega=\nu/2\mathrm{Tr}\left( \left[ {\boldsymbol}{\nabla} {\boldsymbol}\omega+({\boldsymbol}{\nabla} {\boldsymbol}\omega)^{\mathrm T}\right] ^2\right)$ and the standard deviation of the vorticity $\sigma_\Omega$. Note that not $\lambda_\Omega$ but $\lambda_\Omega/4$ is shown in the right figure.[]{data-label="fig:everything_about_omega"}](mulaepsz_omega "fig:"){width="50.00000%"}
The conditional averages are shown in figure \[fig:everything\_about\_omega\] together with the vorticity PDF. They display a strong dependence on $\Omega$, especially the eigenvalues which do not show a plateau for small values of $\Omega$ like in the case of velocity. An important observation here is that, in contrast to the velocity statistics $\lambda_\Omega\not\approx\mu_\Omega$, but rather $\lambda_\Omega\approx4\mu_\Omega$. This shows that the tensor ${\mathrm}{D}_\Omega$ depends on the direction of ${\boldsymbol}\Omega$, i.e. the enstrophy dissipation tensor $\nu{\boldsymbol}{\nabla}{\boldsymbol}{\omega}\left( {\boldsymbol}{\nabla}{\boldsymbol}{\omega} \right)^{\mathrm T}$ and the direction of the vorticity $\hat{{\boldsymbol}\Omega}$ are obviously correlated, in contrast to an almost vanishing correlation in the case of velocity. One may hypothesize that the simple scale-separation argument, which works in the case of the conditional dissipation tensor, breaks down for the conditional enstrophy dissipation tensor, because there is no clear scale separation between the vorticity and its gradients. This can be observed in figure \[fig:vortvel\], where a snapshot of the velocity, vorticity and the traces of the corresponding dissipation tensors are shown. It appears that the fields of the vorticity and both dissipation tensors consist of structures of similar size, whereas the velocity field consists of fairly large patches. Furthermore, the structures[^4] of vorticity and the dissipation tensors are oriented in almost the same direction, but no obvious correlation of the *orientation* of the structures of the velocity and the other fields is visible. However, the *amplitudes* of the fields are correlated. Therefore it is understandable that ${\mathrm}{D}$ and ${\mathrm}{D}_\Omega$ both depend on the magnitude of the corresponding variable, but the dependence on the direction of the variable ($\hat{{\boldsymbol}v}$ and $\hat{{\boldsymbol}\Omega}$ respectively) is weak in the case of velocity and strong in the case of vorticity.
![The quotients $\Lambda/\lambda$ and $\mu/\lambda$ appearing in and for the case of velocity (left) and vorticity (right).[]{data-label="fig:integrands-analyzed"}](integrand_analyzed "fig:"){width="50.00000%"}![The quotients $\Lambda/\lambda$ and $\mu/\lambda$ appearing in and for the case of velocity (left) and vorticity (right).[]{data-label="fig:integrands-analyzed"}](integrand_analyzed_omega "fig:"){width="50.00000%"}
We have already noted that the arguments in §\[sec:simple-closure\], which lead to a Gaussian PDF, could, in principle, also be applied to the vorticity. Figure \[fig:everything\_about\_omega\] shows a comparison of $\tilde{f}_\Omega$ with an angle-integrated Gaussian. This exemplifies that such types of arguments can fail completely and therefore cannot be used as proofs of Gaussianity. To understand the reasons for a strongly non-Gaussian vorticity PDF we again examine the quotients $\Lambda_\Omega/\lambda_\Omega$ and $\mu_\Omega/\lambda_\Omega$, depicted in figure \[fig:integrands-analyzed\]. We see that $\Lambda_\Omega/\lambda_\Omega$ is clearly not linear and has a complicated dependence on $\Omega$, showing that there is a complicated relation between vorticity diffusion and the enstrophy dissipation tensor. The quotient of the eigenvalues also shows a dependence on $\Omega$, corresponding to the fact that $\mu_\Omega\!\nsim\!\lambda_\Omega$. A detailed analysis shows that the complicated form of the vorticity PDF depends crucially on the non-trivial form of *both* quotients, i.e. it cannot be attributed to one of them. Comparing the quotients for the case of velocity and vorticity, we see that there is much more non-trivial behaviour in the vorticity case. This leads to a highly non-Gaussian vorticity PDF, whereas the velocity PDF displays only moderate, but still significant, deviations from a Gaussian distribution.
In this section, we have examined the difference between the velocity and vorticity PDFs, using the conditional averages which determine these PDFs. We have seen that the approximate relations $\lambda\approx\mu$ and $\Lambda/\lambda\sim v$, which have been found for the velocity, are clearly violated in the case of vorticity. Overall, we can say that the conditional averages or the statistical dependences which determine the shape of the vorticity PDF are significantly more complicated for the vorticity and hence yield a highly non-Gaussian PDF.
Summary and Conclusions
=======================
To sum up, we studied the statistics of the single-point velocity PDF within the framework of the LMN hierarchy. Combined with conditional averaging and the use of statistical symmetries, this framework provides a clear-cut identification of the quantities which determine the details of the velocity PDF. The theory identifies the conditional diffusion of velocity and the conditional dissipation tensor in the homogeneous case as the central quantities of interest. In the case of stationary turbulence, the diffusive term is equivalently represented by the conditional pressure gradient and the external forcing. Exact expressions for the velocity PDF are presented in terms of these functions, demonstrating that the closure problem of turbulence may be expressed in terms of *a priori* unknown statistical correlations of different dynamical contributions directly related to the Navier–Stokes equation.
In an analytical treatment, we suggest simple closure approximations that comply with the functional constraints on the conditional averages leading to Gaussian PDFs for both stationary and decaying turbulence. It turns out that the presented closure approximations contain the results recently derived in [@hosokawa08pre], explaining which correlations have to be neglected to yield Gaussian statistics.
DNS simulations of stationary and decaying turbulence are then used to study deviations from Gaussianity. It turns out that the velocity PDF displays slight deviations from a Gaussian distribution with sub-Gaussian tails. A detailed investigation of the terms arising in the theoretical framework allows to investigate why only moderate deviations from Gaussianity occur in spite of pronounced correlations. Unlike suggested in recent theories, we show that the pressure contributions may not be neglected. Furthermore, the correlations of the velocity with the external forcing do not seem to be the main contributor to the deviations from Gaussianity. The conditional energy dissipation tensor turns out to display strong correlations for high values of the velocity, which contribute to the deviations. The numerical results additionally suggest that this tensor is approximately diagonal with identical eigenvalues related to the conditional rate of energy dissipation. The investigation of decaying turbulence gives similar results, indicating that forced and decaying turbulence do not differ in a fundamental way. Interestingly, a self-similar range with an algebraic decay of the kinetic energy has been observed, which considerably simplifies the description of the evolution of the PDF during the decay phase and implies a simple relation for the conditionally averaged right-hand side of the Navier–Stokes equation.
To highlight the genuine features of the velocity statistics, some comparisons to the vorticity PDF are drawn. The general outcome here is that the vorticity is stronger correlated with the terms arising in the vorticity equation that determine the local dynamics such as enstrophy production and dissipation. The external forcing has a negligible effect as has been found in various other studies. The conditional enstrophy dissipation tensor cannot be assumed to be diagonal, in contrast to the conditional energy dissipation tensor, indicating the presence of non-negligible directional correlations. This fact may be related to the presence of coherent structures in the case of the vorticity and their absence in the case of the velocity; however, a concise description of this fact is still lacking.
In conclusion, the approach presented above gives a comprehensive characterization of the shape and evolution of the single-point velocity PDF. Although the introduction of conditional averages does not solve, but rather reformulates the closure problem of turbulence, valuable insights have been obtained highlighting the statistical correlations of the different dynamical influences and the velocity. Regarding a statistical theory of turbulence, it is interesting to see how PDFs may be expressed in terms of these influences. A theoretical derivation of the functional shape of these quantities remains a challenging task for the future.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank O. Kamps, J. Lülff and F. Jenko for valuable discussions and careful reading of the manuscript as well as the Editor and one of the Referees for constructive input. Computational resources were granted within the project h0963 at the LRZ Munich.
[29]{} natexlab\#1[\#1]{}
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[^1]: Throughout the paper the Einstein summation convention is assumed.
[^2]: Invariance under reflections is not necessary for the results presented in this paper.
[^3]: We omit the $t$-dependence to simplify the presentation. However, we do not assume stationarity unless explicitly mentioned.
[^4]: The vorticity field consists of tube-like structures, whereas the pseudo-dissipation field $\nu\mathrm{Tr}({\mathrm}{AA}^{\mathrm T})=\frac{1}{2}(\varepsilon+\nu\omega^2)$ consists of structures, which are a superposition of sheet-like and tube-like structures.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
A group of order $p^n$ ($p$ prime) has an indecomposable polynomial invariant of degree at least $p^{n-1}$ if and only if the group has a cyclic subgroup of index at most $p$ or it is isomorphic to the elementary abelian group of order 8 or the Heisenberg group of order 27.\
[*Keywords:* ]{} polynomial invariants, degree bounds, zero-sum sequences
author:
- |
Kálmán Cziszter [^1]\
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences\
Reáltanoda u. 13 – 15, 1053 Budapest, Hungary
title: 'On the Noether number of $p$-groups'
---
Introduction
============
Let $G$ be a finite group and $V$ a $G$-module over a field $\operatorname{\mathbb{F}}$ of characteristic not dividing the group order $|G|$. The Noether-number $\beta(G,V)$ is the maximal degree in a minimal generating set of the ring of polynomial invariants $\mathbb F[V]^G$. It is known that $\beta(G,V) \le |G|$ (see [@noether], [@fogarty], [@fleishmann]). Even more, it was observed that $\beta(G) := \sup_{V}(G,V)$ (where $V$ runs over all $G$-modules over the base field $\operatorname{\mathbb{F}}$) is typically much less than $|G|$. For an algebraically closed base field of characteristic zero it was proved in [@schmid] that $\beta(G) = |G|$ holds only if $G$ is cyclic. Then it turned out that $\beta(G) \le \frac 3 4 |G|$ for any non-cyclic group $G$ (see [@DH] and [@sezer]). Moreover $\beta(G) \ge \frac 1 2 |G|$ holds if and only if $G$ has a cyclic subgroup of index at most two, with the exception of four particular groups of small order (see [@CzD:3 Theorem 1.1]). Recently some asymptotic extensions of this result were given in [@HP]. Our goal in the present article is to establish the following strengthening of this kind of results for the class of $p$-groups:
\[fo2\] If $G$ is a finite $p$-group for a prime $p$ and the characteristic of the base field $\operatorname{\mathbb{F}}$ is zero or greater than $p$ then the inequality $$\begin{aligned}
\label{fotul}\beta(G) \ge \frac 1 p |G| \end{aligned}$$ holds if and only if $G$ has a cyclic subgroup of index at most $p$ or $G$ is the elementary abelian group $C_2\times C_2 \times C_2$ or the Heisenberg group of order 27.
The proof of Theorem \[fo2\] will be reduced to the study of a single critical case, the Heisenberg group $H_p$, which is the extraspecial group of order $p^3$ and exponent $p$ for an odd prime $p$. We prove about this the following result:
\[beta\_H\_p\] For any prime $p \ge 5$ and base field $\operatorname{\mathbb{F}}$ of characteristic $0$ or greater than $p$ we have $ \beta(H_p) < p^2$.
The paper is organised as follows. Section \[Sec:zerosum\] contains some technical results on zero-sum sequences over abelian groups that will be needed later. In Section \[Sec:reduction\] we reduce the proof of Theorem \[fo2\] to that of Theorem \[beta\_H\_p\]. Then in Section \[Sec:invariant\] we explain the main invariant theoretic idea behind the proof of Theorem \[beta\_H\_p\] which is also applicable in a more general setting. The proof itself of Theorem \[beta\_H\_p\] will then be carried out in full detail in Section \[Sec:3c\]. Finally, Section \[Sec:4\] completes our argument by showing that for the case $p=3$ we have $\beta(H_3)= 9$ in any non-modular characteristic.
Some preliminaries on zero-sum sequences {#Sec:zerosum}
========================================
We follow here in our notations and terminology the usage fixed in [@CzDG]. Let $A$ be an abelian group noted additively. By a sequence $S$ over a subset $A_0 \subseteq A$ we mean a multiset of elements of $A_0$. They form a free commutative monoid with respect to concatenation, denoted by $S\cdot T$, and unit element the empty sequence $\emptyset$; this has to be distinguished from $0$, the zero element of $A$. The sequence $ a\cdot a \cdots a$, obtained by the $k$-fold repetition of an element $a \in A$, is denoted by $a^{[k]}$; this has to be distinguished from the product $ka \in A$. The multiplicity of an element $a\in A$ in a sequence $S$ is denoted by $\mathsf v_a(S)$. We also write $a \in S$ to indicate that $\mathsf v_a(S) >0$. We say that $T$ is a subsequence of $S$, and write $T\mid S$, if there is a sequence $R$ such that $S = T \cdot R$. In this case we also write $R = S \cdot T^{[-1]}$. The *length* of a sequence, denoted by $|S|$, can be expressed as $\sum_{a \in A} \mathsf v_a(S)$, whereas the *sum* of a sequence $S = a_1 \cdots a_n$ is $\sigma(S) := a_1 + \ldots + a_n \in A$ and by convention we set $\sigma(\emptyset) = 0$. We say that $S$ is a *zero-sum sequence* if $\sigma(S) =0$.
The relevance of zero-sum sequences for our topic is due to the fact that for an abelian group $A$ the Noether number $\beta(A)$ coincides with the Davenport constant $\operatorname{\mathsf{D}}(A)$, which is defined as the maximal length of a zero-sum sequence over $A$ not containing any non-empty, proper zero-sum subsequence (see e.g. [@CzDG Chapter 5]). Its value for $p$-groups is given by the following formula [@bible Theorem 5.5.9]: $$\begin{aligned}
\label{olson_p}
\operatorname{\mathsf{D}}(C_{p^{n_1}} \times \dots \times C_{p^{n_r}} )= \sum_{i=1}^r (p^{n_i} -1) +1.\end{aligned}$$
A variant of this notion is the $k$th Davenport constant $\mathsf D_k(A)$ defined for any $k \ge 1$ as the maximal length of a zero-sum sequence $S$ that cannot be factored as the concatenation $S = S_1 \cdots S_{k+1}$ of non-empty zero-sum sequences $S_i$ over $A$. Its numerical value is much less known (for some recent results see [@freeze-schmid]); we shall only need the fact that according to [@bible Theorem 6.1.5.2]: $$\begin{aligned}
\label{halter_koch}
\operatorname{\mathsf{D}}_{k} (C_p \times C_p) &= kp +p-1.\end{aligned}$$
The following consequence of the definition of $\mathsf D_k(A)$ will also be used:
\[HK\] Any sequence $S$ over an abelian group $A$ of length at least $\mathsf D_k(A)$ factors as $S=S_1 \cdots S_k\cdot R$ with some non-empty zero-sum sequences $S_i$.
We define for any sequence $S$ over $A$ the set of all partial sums of $S$ as $\Sigma(S) := \{ \sigma(T): \emptyset \neq T \mid S \} $. If $0 \not\in \Sigma(S)$ then $S$ is called *zero-sum free*. The next result could also be deduced from the Cauchy-Davenport theorem (see [@bible Corollary 5.2.8.1]) but we provide here an elementary proof for the reader’s convenience:
\[CD\] Let $p$ be a prime. Then for any sequence $S$ over $C_p \setminus \{ 0 \}$ we have $|\Sigma(S)|\ge \min \{p, |S|\}$.
We use induction on the length of $S$. For $|S| = 0$ the claim is trivial. Otherwise consider a sequence $S\cdot a$ where the claim holds for $S$. We have $\Sigma(S \cdot a) = \Sigma(S) \cup \{ a\} \cup (a + \Sigma(S))$, where $a+ \Sigma(S) := \{a + s : s \in \Sigma(S) \}$. Then either $|\Sigma(S\cdot a)| \ge |\Sigma(S)| +1$, or else $a \in \Sigma(S)$ and $a+\Sigma(S) = \Sigma(S)$, that is when $\Sigma(S)$ is a subgroup of $C_p$ containing $a$. But since $C_p$ has only two subgroups and by assumption $\Sigma(S) \ni a \neq 0$, this means that $\Sigma(S) = C_p$.
\[zerosumfree\] A sequence $S$ over $C_p$ ($p$ prime) of length $|S| =p-1$ is zero-sum free if and only if $S= a^{[p-1]}$ for some $a \in C_p \setminus \{0\}$.
\[eta\] Let $p$ be a prime and $S$ be a sequence over $C_p \times C_p$ of length $|S|\ge 3p-2$. Then $S$ has a zero-sum subsequence $X \mid S$ of length $p$ or $2p$.
We close this section with a technical result. Its motivation and relevance will become apparent through its application in the proof of Proposition \[amorf\]. For any function $\pi$ defined on $A$ and any sequence $S$ over $A$ we will write $\pi(S)$ for the sequence obtained from $S$ by applying $\pi$ element-wise.
\[nullak\] Let $S$ be a sequence over $C_p$ of length $|S| \ge p^2-1$. If we have $\mathsf v_0(S)\ge p+1$ then $S= S_1 \cdots S_{\ell} \cdot R$, where each $S_i$ is a non-empty zero-sum sequence and $\ell \ge 2p-1$.
Let $\ell$ denote the maximal integer such that $S =S _1 \cdots S_{\ell} \cdot R$ for some non-empty zero-sum sequences $S_i$. Then each $S_i$ is irreducible, hence $|S_i| \le p$ and $R$ is zero-sum free, hence $|R|\le p-1$. Assuming that $\ell \le 2p-2 $ we get $$p^2-1 \le |S| \le \mathsf v_0 (S) + (\ell - \mathsf v_0(S))p + p-1 \le (p-1) (2p+1 - \mathsf v_0(S))$$ whence $\mathsf v_0(S) \le p$ follows, in contradiction with our assumption.
\[separ\] Let $A= C_p \times C_p$ for some prime $p \ge 5$ and $\pi:A \to C_p$ the projection onto the first component. If $S$ is a sequence over $A$ with $|S| \ge p^2-1$ and $\mathsf v_0(\pi(S)) \le p$ then for any given subsequence $T \mid S $ of length $|T| \le p-1$ there is a a factorisation $S = S_1 \cdots S_{p-1}\cdot R$, where each $S_i$ is a non-empty zero-sum sequence over $A$, while $T \mid S \cdot (S_1 \cdot S_2)^{[-1]}$ and $\Sigma(\pi(S_1)) = C_p$.
Let $S^* \mid S \cdot T^{[-1]}$ be the maximal subsequence such that $0 \not\in \pi(S^*)$. Then by assumption $|S^*| \ge |S| -2p +1\ge 3p$, as $p \ge 5$, so there is a zero-sum subsequence $X \mid S^*$ of length $p$ or $2p$ by Lemma \[eta\]. We have two cases:
\(i) If $|X| = 2p$ then $X =S_1\cdot S_2$ for some non-empty zero-sum sequences $S_1, S_2$ such that $|S_1| \ge p$ and $|S_2| \le p$, as $\mathsf D(C_p \times C_p) =2p-1$ by .
\(ii) If $|X| = p$ then we can take $S_1 := X$. Then we have $|S \cdot (S_1\cdot T)^{[-1]}| \ge |S| -2p+1 \ge 3p$, so again by Lemma \[eta\] we find a non-empty zero-sum sequence $S_2 \mid S \cdot (S_1\cdot T)^{[-1]}$ of length $|S_2| \le p$ as above.
In both cases $T \mid S \cdot (S_1 \cdot S_2)^{[-1]}$ and $|S_1 \cdot S_2| \le 2p$ by construction. Consequently $|S \cdot (S_1 \cdot S_2)^{[-1]}| \ge |S| -2p \ge p^2-2p-1 = \mathsf D_{p-3}(C_p\times C_p)$, hence by Lemma \[HK\] we have a factorisation $S \cdot (S_1 \cdot S_2)^{[-1]} = S_3 \cdots S_{p-1} \cdot R$ with non-empty zero-sum sequences $S_i$ for each $i \ge 3$. Finally, in both cases we had $|S_1| \ge p$ and $0 \not \in \pi(S_1)$, hence $|\Sigma(\pi(S_1))| = p$ by Lemma \[CD\].
Reduction of Theorem \[fo2\] to Theorem \[beta\_H\_p\] {#Sec:reduction}
======================================================
Our main tool here will be the $k$th Noether number $\beta_k(G,V)$ which is defined for any $k \ge 1$ as the greatest integer $d$ such that some invariant of degree $d$ exists which is not contained in the ideal of $\operatorname{\mathbb{F}}[V]^G$ generated by the products of at least $k+1$ invariants of positive degree. This notion was introduced in [@CzD:1 Section 2] with the goal of estimating the ordinary Noether number from information on its composition factors. This was made possible by [@CzD:3 Lemma 1.4] according to which for any normal subgroup $N \triangleleft G$ we have: $$\begin{aligned}
\label{reduction}
\beta(G,V) &\le \beta_{\beta(G/N)}(N,V). \end{aligned}$$ As observed in [@CzDG Chapter 5], if $A$ is an abelian group then $\beta_k(A)$ coincides with $\operatorname{\mathsf{D}}_k(A)$, so that we can use in the applications of .
The “if” part follows from [@schmid Proposition 5.1] which states that $\beta(C) \le \beta(G)$ for any subgroup $C \le G$. So if $C$ is cyclic of index at most $p$ then $ \beta(G) \ge \beta(C)= |C|=|G|/[G:C] \ge \frac 1 p |G|$. Moreover $\beta(C_2^3) =4$ by and $\beta(H_3) \ge 9$ by Proposition \[H3\_felso\] below.
The “only if” part for $p=2$ follows from [@CzD:3 Theorem 1.1] so for the rest we may assume that $p \ge 3$. Let $G$ be a group of order $p^n$ for which holds. If $G$ is non-cyclic then it has a normal subgroup $N \cong C_p \times C_p$ by [@berk Lemma 1.4]. We claim that $G/N$ must be cyclic. For otherwise by applying [@berk Lemma 1.4] to the factor group $G/N$ we find a subgroup $K$ such that $N \triangleleft K \triangleleft G$ and $K/N \cong C_p \times C_p$. But then we get using and that $$\beta(K) \le \beta_{\beta(C_p \times C_p)}(C_p \times C_p) = p(2p-1)+p-1 = 2p^2-1 < p^3=\frac 1 p |K|.$$ As $\beta(G)/|G| \le \beta(K)/|K| $ by [@CzD:3 Lemma 1.2] we get a contradiction with .
Now let $g \in G$ be such that $gN$ generates $G/N \cong C_{p^{n-2}}$. Then $g^{p^{n-2}} \in N$ has order $p$ or $1$. In the first case ${\langle}g {\rangle}$ has index $p$ in $G$ and we are done. In the other case ${\langle}g {\rangle}\cap N = \{1\}$ hence $G \cong N \rtimes {\langle}g {\rangle}$. If $g$ acts trivially on $N$ then $G$ contains a subgroup $H \cong C_p \times C_p \times C_p$ for which we have $\beta(H) = 3p -2$ by hence $\beta(G)/|G| \le \beta(H)/|H| < 3/p^2 \le 1/p$, as $p\ge 3$, a contradiction. This shows that $g$ must act non-trivially on $C_p \times C_p$. It is well known that $\operatorname{Aut}(C_p \times C_p) = \operatorname{GL}(2,p)$ has order $ (p^2-1)(p^2-p)$, so its Sylow $p$-subgroup must have order $p$ and it is isomorphic to $C_p$. Therefore $g^p$ must act trivially on $N$, so if $n \ge 4$ then $g^p \neq 1$ and the subgroup $\langle N, g^p\rangle $ is isomorphic to $ C_p \times C_p \times C_p$, but this was excluded before. The only case which remains open is that $n = 3$ and $G \cong (C_p \times C_p) \rtimes C_p$, where the factor group $C_p$ acts non-trivially on $C_p \times C_p$. This is the Heisenberg group denoted by $H_{p}$. By Theorem \[beta\_H\_p\] we have $\beta(H_p) < p^2$ for all $p>3$ under our assumption on the characteristic of the base field $\operatorname{\mathbb{F}}$. So among the Heisenberg groups the inequality can only hold for $H_3$.
The precise value of the Noether number is already known for all the $p$-groups which satisfy according to Theorem \[fo2\]. As the Theorem states, equality holds in for $C_2^3$ and $H_3$. For the rest, the groups of order $p^n$ which have a cyclic subgroup of index $p$ were classified by Burnside (see e.g. [@berk Theorem 1.2]) as follows:
\(i) if $G$ is abelian, then either $G$ is cyclic with $\beta(G) = p^n$ or $G = C_{p^{n-1}} \times C_p$ in which case it has $\beta(G) = p^{n-1} +p -1$ by
\(ii) if $G$ is non-abelian and $p > 2$ then $G$ is isomorphic to the modular group $M_{p^n} \cong C_{p^{n-1}} \rtimes C_p$. We have $\beta(M_{p^n}) = p^{n-1} + p-1$ by [@CzD:2 Remark 10.4].
\(iii) if $G$ is non-abelian and $p=2$ then $G$ is the dihedral group $D_{2^n}$ or the semi-dihedral group $SD_{2^n}$ or the generalised quaternion group $Q_{2^n}$. We have $\beta(Q_{2^n}) = 2^{n-1} +2$ and $\beta(D_{2^n}) = \beta(SD_{2^n}) = 2^{n-1}+1$ by [@CzD:2 Theorem 10.3].
Altogether these results imply that for any non-cyclic $p$-group $G$ we have $$\begin{aligned}
\beta(G) \le \frac 1 p |G| + p \end{aligned}$$ and this inequality is sharp only for the case $p= 2$.
The notion of the Davenport constant $\mathsf D(G)$, originally defined only for abelian groups as in Section \[Sec:zerosum\], was extended to any finite group $G$ in [@GeGryn; @Gryn]. For the conjectural connection between the Noether number and this generalisation of the Davenport constant see [@CzDG Section 5.1] and [@CzDSz].
Invariant theoretic lemmas {#Sec:invariant}
==========================
Let us fix here some notations related to invariant rings. For any vector space $V$ over a field $\operatorname{\mathbb{F}}$ we denote its coordinate ring by $\operatorname{\mathbb{F}}[V]$. We say that a group $G$ has a left action on $V$, or that $V$ is a $G$-module, if a group homomorphism $\rho: G \to \operatorname{GL}(V)$ is given and we abbreviate $\rho(g)(v)$ by writing $g \cdot v$ for any $g \in G$ and $v \in V$. By setting $f^g(v) := f(g\cdot v)$ for any $f \in \operatorname{\mathbb{F}}[V]$ we obtain a right action of $G$ on $\operatorname{\mathbb{F}}[V]$. The ring of polynomial invariants is defined as $\operatorname{\mathbb{F}}[V]^G := \{f \in \operatorname{\mathbb{F}}[V]: f^g =f \; \text{ for all } g \in G \}$. If the ring $\operatorname{\mathbb{F}}[V]^N$ is already known for some normal subgroup $N \triangleleft G$ then $\operatorname{\mathbb{F}}[V]^{G}$ as a vector space is spanned by its elements of the form $\tau_N^G(m)$, where $m$ runs over the set of all monomials and $\tau_N^G: \operatorname{\mathbb{F}}[V]^N \to \operatorname{\mathbb{F}}[V]^{G}$ is the $\operatorname{\mathbb{F}}[V]^G$-module epimorphism defined as $$\tau_N^G(m) = \frac 1 {|G/N|} \sum_{g \in G } m^g$$ (see e.g. [@NeuselSmith Chapter 2.2]). When $N$ is trivial this definition amounts to the Reynolds operator $\tau := \tau_{\{ 1\}}^G$. Given any character $\chi \in \widehat{G}:=\operatorname{Hom}(G, \operatorname{\mathbb{F}}^{\times})$ the set $\operatorname{\mathbb{F}}[V]^{G,\chi} := \{ f\in \operatorname{\mathbb{F}}[V] : f^g = \chi(g) f\}$ constitutes the $\operatorname{\mathbb{F}}[V]^G$-module of $G$-semi-invariants of weight $\chi$. If the restriction of $\chi$ to $N$ is trivial, i.e. when $\chi \in \widehat{G/N}$, then these semi-invariants can be obtained by the projection map $\tau_{\chi}: \operatorname{\mathbb{F}}[V]^N \to \operatorname{\mathbb{F}}[V]^{G,\chi}$ defined with the analogous formula $$\tau_{\chi}(u) = \frac 1 {|G/N|}\sum_{g \in G/N} \chi^{-1}(g) u^g.$$
$\operatorname{\mathbb{F}}[V]$ and $\operatorname{\mathbb{F}}[V]^G$ are graded rings: $\operatorname{\mathbb{F}}[V]_d$ denotes for any $d \ge 0$ the vector space of degree $d$ homogeneous polynomials and $\operatorname{\mathbb{F}}[V]^G_d = \operatorname{\mathbb{F}}[V]^G \cap \operatorname{\mathbb{F}}[V]_d$. The set $\operatorname{\mathbb{F}}[V]^G_+ := \bigoplus_{d \ge 1} \operatorname{\mathbb{F}}[V]^G_d$ is a maximal ideal in $\operatorname{\mathbb{F}}[V]^G$, while $\operatorname{\mathbb{F}}[V]^G_+\operatorname{\mathbb{F}}[V]$, the ideal of $\operatorname{\mathbb{F}}[V]$ generated by all $G$-invariant polynomials of positive degree, is the so called *Hilbert-ideal*. This ideal will be our main object of interest since, as observed in [@CzD:1 Section 3], the graded factor ring $\operatorname{\mathbb{F}}[V]/\operatorname{\mathbb{F}}[V]^G_+\operatorname{\mathbb{F}}[V]$ is finite dimensional and its top degree, denoted by $b(G,V)$, yields an upper bound on the Noether number by an easy argument using the Reynolds operator: $$\begin{aligned}
\label{b_beta}
\beta(G,V) \le b(G,V) +1.\end{aligned}$$
It is well known that $\beta(G,V)$ is unchanged when we extend the base field so we will assume throughout this paper that $\mathbb{F}$ is algebraically closed.
\[b+1\] Let $G$ be a finite group with a normal subgroup $N$ such that $G/N$ is abelian. Let $W$ be a $G$-module over $\operatorname{\mathbb{F}}$ and assume that $|G| \in \operatorname{\mathbb{F}}^{\times}$. Then $(\operatorname{\mathbb{F}}[W]_+^N)^k \subseteq {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ for any $k \ge \mathsf D(G/N)$.
$\operatorname{\mathbb{F}}[W]^N$ regarded as a $G/N$-module has the direct sum decomposition $\bigoplus_{\chi \in \widehat{G/N}} \operatorname{\mathbb{F}}[W]^{G,\chi}$. (Here we used both our assumptions on $\operatorname{\mathbb{F}}$.) This means that any element $u \in \operatorname{\mathbb{F}}[W]^N_+$ can be written as a sum $u = \sum_{\chi \in \widehat{G/N}} \tau_{\chi}(u)$. Now for any $k \ge 1$ and $u_1,\ldots,u_{k} \in \operatorname{\mathbb{F}}[W]_+^N$ we have $$\begin{aligned}
\label{trukk:1}
\prod_{i=1}^{k}u_i =
\prod _{i=1}^{k} \left(\sum_{\chi \in \widehat{G/N}} \tau_{\chi}(u_i) \right) =
\sum_{\chi_1,\ldots,\chi_{k} \in \widehat{G/N}} \tau_{\chi_1}(u_1) \cdots \tau_{\chi_{k}}(u_{k}).\end{aligned}$$ The term $\tau_{\chi_1}(u_1) \cdots \tau_{\chi_{k}}(u_{k})$ belongs to the ideal ${\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ whenever the sequence $(\chi_1,\ldots,\chi_{k})$ over $\widehat{G/N} \cong G/N$ contains a non-empty zero-sum subsequence. But this holds for every term on the right of as $k \ge \mathsf D (G/N)$.
\[trukk\] If in Lemma \[b+1\] the factor group $G/N \cong C_p$ is cyclic of prime order then for any $g\in G/N$ and any elements $u_1,\ldots,u_{p-1} \in \operatorname{\mathbb{F}}[W]^N_+$ we have the relation: $$\begin{aligned}
\label{trukk:2} u_1 \cdots u_{p-1} - u_1^{g}u_2^{-g}u_3 \cdots u_{p-1} \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}.\end{aligned}$$
Observe that in with $k = p-1$ the weight sequence $(\chi_1,\ldots,\chi_{p-1})$ over $\widehat{C}_p$ is zero-sum free if and only if $\chi_1=\ldots=\chi_{p-1}$ and $\chi_1$ is non-trivial (by Lemma \[zerosumfree\]). As a result we get: $$\begin{aligned}
u_1\cdots u_{p-1} \in \sum_{\chi \in \widehat{C}_p \setminus \{ 1 \}} \tau_{\chi}(u_1) \cdots \tau_{\chi}(u_{p-1})
+ {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}.\end{aligned}$$ Replacing here $u_1$ and $u_2$ with $u_1^g$ and $u_2^{-g}$, respectively, and observing that by definition we have $\tau_{\chi}(u^g) = \chi(g)\tau(u)$ for any $u\in \operatorname{\mathbb{F}}[W]^N$ we infer that $u_1^g u_2^{-g} u_3\cdots u_{p-1}$ must belong to the same residue class modulo the ideal ${\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ to which $u_1\cdots u_{p-1}$ does belong. This proves our claim.
The Heisenberg group $H_p$ {#Sec:3c}
==========================
The Heisenberg group $H_p = {\langle}a, b {\rangle}$ can be defined by the presentation: $$\begin{aligned}
\label{H_rel}
a^p=b^p=c^p =1 \quad
[a,b] =c \quad
[a,c] =[b,c] =1\end{aligned}$$ where $[a,b]$ denotes the commutator $a^{-1} b^{-1} ab$. The subgroups $A:= {\langle}a,c {\rangle}$ and $B:= {\langle}b,c {\rangle}$ are normal and isomorphic to $C_p \times C_p$. The Frattini-subgroup, the center and the derived subgroup of $H_p$ all coincide with ${\langle}c{\rangle}$, so that $H_p$ is extraspecial. In particular $H_p / {\langle}c {\rangle}$ is also isomorphic to $C_p \times C_p$. Taking into account only the subgroup structure of $H_p$ the best upper bound that we can give about its Noether number by means of and is the following: $$\begin{aligned}
\label{apriori}
\beta(H_p) \le \beta_{\beta(C_p)}(C_p \times C_p) = p^2 + p -1. \end{aligned}$$ Our goal in this section will be to enhance this estimate by analysing more closely the invariant rings of $H_p$.
Let $\operatorname{\mathbb{F}}$ be an algebraically closed field with $\operatorname{char}(\operatorname{\mathbb{F}}) \neq p$, so that there is a primitive $p$-th root of unity $\omega \in \operatorname{\mathbb{F}}$ that will be regarded as fixed throughout this paper. The irreducible $H_p$-modules over $\operatorname{\mathbb{F}}$ are then of two types:
\(i) Composing any group homomorphism $\rho \in \operatorname{Hom}(C_p \times C_p, \operatorname{\mathbb{F}}^{\times} )$ with the canonic surjection $H_p \to H_p/{\langle}c{\rangle}\cong C_p \times C_p$ yields $p^2$ non-isomorphic $1$-dimensional irreducible representations of $H_p$.
\(ii) For each primitive $p$-th root of unity $\omega^i \in \operatorname{\mathbb{F}}$, where $i=1,\ldots,p-1$, take the induced representation $V_{\omega^i} := \operatorname{Ind}_A^{H_p}{\langle}v {\rangle}$, where ${\langle}v {\rangle}$ is a $1$-dimensional left $A$-module such that $a\cdot v =v$ and $c \cdot v = \omega^i v$. In the basis $\{v, b \cdot v, ..., {b^{p-1}} \cdot v \}$ this representation is then given in terms of matrices in the following form, with $I_p$ the $p \times p$ identity matrix: $$\begin{aligned}
\label{H_irred}
a \mapsto \left(\begin{array}{cccc}1 & & & \\ & \omega^i & & \\ & & \ddots & \\ & & & \omega^{i(p-1)}\end{array}\right)
\quad
b \mapsto \left(\begin{array}{cccc} 0& \cdots & \cdots & 1 \\1 & & & \vdots \\ & \ddots & & \vdots\\ & & 1 & 0\end{array}\right)
\quad
c \mapsto \omega^i I_p.\end{aligned}$$ Each $V_{\omega^i}$ is irreducible by Mackey’s criterion (see e.g. [@serre]) and for $\omega^i \neq \omega^{i'}$ it is easily seen (e.g. from the matrix corresponding to $c$) that $V_{\omega^i}$ and $V_{\omega^{i'}}$ are non-isomorphic as $G$-modules.
Adding the squares of the dimensions of the above irreducible $H_p$-modules we get $p^2\cdot 1 + (p-1)p^2 =p^3= |H_p|$, so that no other irreducible $H_p$-modules exist. As a result an arbitrary $H_p$-module $W$ over $\operatorname{\mathbb{F}}$ has the canonic direct sum decomposition $$\begin{aligned}
\label{direct}
W = U \oplus V_1 \oplus \ldots \oplus V_{p-1} \end{aligned}$$ where $U$ consists only of $1$-dimensional irreducible representations of $H_p$ with ${\langle}c {\rangle}$ in their kernel, while each $V_i$ is an isotypic $H_p$-module consisting of the direct sum of $n_i \ge 0$ isomorphic copies of the irreducible representation $V_{\omega^i}$: $$\begin{aligned}
\label{isotypic}
V_i = \underbrace{V_{\omega^i} \oplus \ldots \oplus V_{\omega^i}}_ {n_i \text{ times}}.\end{aligned}$$
Next we recall how does the action of $G$ on $W$ extend to the coordinate ring $\operatorname{\mathbb{F}}[W]$. When speaking of a coordinate ring $\operatorname{\mathbb{F}}[V_{\omega^i} ] = \operatorname{\mathbb{F}}[x_{i,0},...,x_{i,p-1}]$ we always tacitly assume that the variables $x_{i,k}$ form a dual basis of the basis used at . By our convention from Section \[Sec:invariant\], $H_p$ acts from the right on the variables, i.e. $x^g(v) = x(g \cdot v)$ for all $g\in H_p$, so we can rewrite as: $$\begin{aligned}
\label{action}
x_{i,k}^b &= x_{i,(k-1) \mathrm{mod } \, p} \qquad \qquad
x_{i,k}^a = \omega^{ik} x_{i,k} \qquad \qquad
x_{i,k}^c = \omega^{i} x_{i,k} .\end{aligned}$$ (Here, by some abuse of notation, we identified the integers $k=0,1,\ldots,p-1$ occurring as indexes with the modulo $p$ residue classes they represent.) This shows that the action of the subgroup $A$ on a variable $x_{i,k}$ is completely determined by the modulo $p$ residue classes of the exponents $ik$ and $i$ of $\omega$ in ; we will call $\phi(x_{i,k}) := (ik , i) \in {\mathbb{Z}}/ p {\mathbb{Z}}\times {\mathbb{Z}}/ p {\mathbb{Z}}$ the *weight* of the variable $x_{i,k}$. We shall also refer to the projections $\phi_a(x_{i,k})= ik $ and $\phi_c(x_{i,k}) =i$. With this notation it is immediate from that for any $n \in {\mathbb{Z}}$ and $x=x_{i,k}$ $$\begin{aligned}
\label{action2}
\phi_a(x^{b^n}) = \phi_a(x) - n \, \phi_c(x)
\quad \text{ and } \quad
\phi_c(x^{b^n}) = \phi_c(x)\end{aligned}$$ where the subtraction and multiplication with $n$ is understood in ${\mathbb{Z}}/ p {\mathbb{Z}}$. This implies the observation, which will be used frequently later on, that for any variable $x$ with $\phi_c(x) \neq 0$ and any arbitrarily given $w \in {\mathbb{Z}}/ p {\mathbb{Z}}$ there is always an element $g \in {\langle}b {\rangle}$ such that $\phi_a(x^g) = w$. Our discussion also shows that for a variable $y \in \operatorname{\mathbb{F}}[W]$ we have $\phi_c(y) = 0$ if and only if $y \in \operatorname{\mathbb{F}}[U]$, and otherwise the value $\phi_c(y) =i $ determines the isotypic $H_p$-module $V_i$ such that $y \in \operatorname{\mathbb{F}}[V_i]$.
Any monomial $u\in \operatorname{\mathbb{F}}[W]$ is an $A$-eigenvector, too, hence we can associate a weight $\phi(u) := (j,i) \in {\mathbb{Z}}/ p {\mathbb{Z}}\times {\mathbb{Z}}/ p {\mathbb{Z}}$ to it so that $u^a = \omega^ju$ and $u^c = \omega^i u$. Obviously then $\phi(uv) = \phi(u) + \phi(v)$ for any monomials $u,v$. If $u = y_1 \cdots y_n$ for some variables $y_i \in \operatorname{\mathbb{F}}[W]$, with repetitions allowed, then we can form the sequence $\Phi(u) := \phi(y_1) \cdots \phi(y_n)$ over $A$, which will be called the *weight sequence* of $u$. Obviously $\phi(u) = \sigma(\Phi(u)) = \phi(y_1) + \cdots+\phi(y_n)$ with the notations of Section \[Sec:zerosum\]. Observe that a monomial $u$ is $A$-invariant if and only if $\phi(u) =0$, that is if $\Phi(u)$ is a zero-sum sequence over $A$. Finally, we set $\Phi_a(u) := (\phi_a(y_1), \ldots, \phi_a(y_n)) $ and $\Phi_c(u) := (\phi_c(y_1), \ldots, \phi_c(y_n)) $.
We call two monomials $u,v\in \operatorname{\mathbb{F}}[W]$ *homologous*, denoted by $u \sim v$, if $\deg(u) = \deg(v) = d$ and $u= \prod_{n=1}^d y_n$ while $v = \prod_{n=1}^d y_n^{g_n}$ for some variables $y_n \in \operatorname{\mathbb{F}}[W]$ (with repetitions allowed) and group elements $g_n \in \langle b \rangle$.
Observe that a monomial $v$ obtained from a monomial $u$ by repeated applications of will be homologous to it in the above sense.
\[amorf\] Let $p \ge 5$. If $u \in \operatorname{\mathbb{F}}[W]$ is a monomial with $\deg(u) \ge p^2- 1$, $\mathsf v_0(\Phi_c(u)) \le p$ and $v \mid u$ is a monomial such that $\deg(v) \le p $ and $0 \not\in \Phi_c(v)$ then for any homologous monomial $v' \sim v$ there is a homologous monomial $u' \sim u$ such that $v' \mid u'$ and $u'-u \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$.
We use induction on the degree $d:= \deg(v) = \deg(v')$. If $d =0$ then $v=v'=1$, so we are done by taking $u'=u$. Suppose now that the claim holds for some $d \le p-1$. It suffices to prove that for any given divisor $xv \mid u$, where $x$ is a variable, $\deg(v) = d$, $0 \not\in \Phi_c(xv)$, and for any $v'\sim v$ and $g \in \langle b \rangle$ a monomial $u'' \sim u$ exists such that $x^gv' \mid u''$ and $u'' - u \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$.
By the inductive hypothesis we already have a monomial $u' \sim u$ such that $v' \mid u'$ and $u'-u \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$. As $u'/v' \sim u/v$ and $x$ divides $ u/v$ there is a $t \in \langle b \rangle$ such that $x^t$ divides $u'/v'$. By applying Proposition \[separ\] to the weight sequences $S := \Phi(u')$, $T := \Phi( v')$ we obtain a factorisation $u' =u_1 \cdots u_{p-1}u_p$ such that $u_i \in \operatorname{\mathbb{F}}[W]_+^A$ for all $i=1,\ldots, p-1$, $u_p \in \operatorname{\mathbb{F}}[W]$, $v'$ divides $u'/u_1u_2$ and $\Sigma(\Phi_c(u_1)) = {\mathbb{Z}}/ p {\mathbb{Z}}$. We have two cases:
i\) If $x^t \mid u_1$ (or similarly if $x^t \mid u_2$) then take $u'' := u_1^{-t+g}u_2^{t-g}u_3 \cdots u_{p-1}u_p$. We have $x^gv' \mid u''$ and $u'' \sim u' \sim u$, while $u''-u' \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ by Lemma \[trukk\].
ii\) Otherwise $x^t \mid u_k$ for some $k>2$. By our assumption on $\Sigma(\Phi_c(u_1))$ there is a divisor $w \mid u_1$ with $\phi_c(w) = -\phi_c(x^t) $. As $\phi_c(x^t) = \phi_c(x) \neq 0$ there is an $h \in \langle b \rangle$ for which $\phi_a(w^h) = -\phi_a(x^t)$. Then for $\hat u := u_1^hu_2^{-h} u_3 \cdots u_{p-1}r$ we have $\hat u \sim u$ and $\hat u -u \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ by Lemma \[trukk\]. Take the factorisation $\hat u = \hat u_1 \cdots \hat u_p$ where $\hat u_1 = w^hx^t$, $\hat u_2 = u_2^{-h}$, $\hat u_k = (u_k/x^t) (u_1^h/w^h)$ and $\hat u_i = u_i$ for the rest. By construction $\hat u _i \in \operatorname{\mathbb{F}}[W]^A_+$ for all $i \le p-1$, $v'$ divides $\hat u / \hat u_1 \hat u_2$ and $x^t \mid \hat u_1$, so this factorisation of $\hat u$ falls under case i) and we are done.
We need some further notations. The decomposition induces an isomorphism $\operatorname{\mathbb{F}}[W] \cong \operatorname{\mathbb{F}}[U] \otimes \operatorname{\mathbb{F}}[V_1] \otimes \ldots \otimes \operatorname{\mathbb{F}}[V_{p-1}] $ which in turn yields for any monomial $m \in \operatorname{\mathbb{F}}[W]$ a factorisation $m=m_0 m_1 \cdots m_{p-1}$ such that $m_0 \in \operatorname{\mathbb{F}}[U]$ and $m_i \in \operatorname{\mathbb{F}}[V_i]$ for all $i$. Then for each $i$ the decomposition gives the identifications $\operatorname{\mathbb{F}}[V_i] = \bigotimes_{j=1}^{n_i} \operatorname{\mathbb{F}}[V_{\omega^i}]=\operatorname{\mathbb{F}}[x_{i,k}^{(j)}: k=0,\ldots,p-1; j=1,\ldots,n_i]$, where we set $x_{i,k}^{(j)} := 1 \otimes \cdots \otimes x_{i,k}\otimes \cdots \otimes 1$, i.e. the variable $x_{i,k}$ introduced at is placed in the $j$th tensor factor. So for any monomial $m_i \in \operatorname{\mathbb{F}}[V_i]$ we have a factorisation $m_i = m_i^{(1)} \cdots m_i^{(n_i)}$ where each monomial $m_i^{(j)}$ depends only on the set of variables $\{x_{i,k}^{(j)}$: $k = 0,1,\ldots, p-1\}$. Observe finally that two monomials $u,v \in \operatorname{\mathbb{F}}[V_1 \oplus \ldots \oplus V_{p-1}]$ are homologous, $u \sim v$, if and only if $\deg(u_{i}^{(j)}) = \deg(v_{i}^{(j)})$ for all $i=1,\ldots,p-1$ and $j=1,\ldots, n_i$.
We shall also need the polarisation operators defined for any polynomial $f \in \operatorname{\mathbb{F}}[W]$ by the formula $$\begin{aligned}
\Delta_i^{s,t}(f) := \sum_{k=0}^{p-1} x^{(t)} _{i,k}\partial^{(s)}_{i,k}f\end{aligned}$$ where $\partial^{(s)}_{i,k}$ denotes partial derivation with respect to the variable $x_{i,k}^{(s)}$. All polarisation operations $\Delta:=\Delta^{s,t}_i$ are degree preserving, $\deg(\Delta(f) ) = \deg(f)$, and $G$-equivariant, i.e. $\Delta(f^g) = \Delta(f)^g$. Therefore by the Leibniz rule $$\label{Leibniz}
\begin{aligned}
\Delta({\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}) & \subseteq {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}\quad \text{ and } \\ \Delta({\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}) & \subseteq {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}.
\end{aligned}$$
\[majdnem\_tiszta\] Let $p \ge 5$ and assume that $\mathrm{char}(\operatorname{\mathbb{F}})$ is $0$ or greater than $p$. If a monomial $m \in \operatorname{\mathbb{F}}[W]$ has $\deg(m) \ge p^2-1$ then $m \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$.
Consider the factorisation $m= m_0 m_1 \cdots m_{p-1} $ derived from as described above. Observe that for the weight sequence $S = \Phi(m)$ we have $\mathsf v_0(\Phi_c(m)) = \deg(m_0)$. So if $\deg(m_0) \ge p+1$ then $m \in (\operatorname{\mathbb{F}}[W]^{\langle c \rangle}_+)^{2p-1}\operatorname{\mathbb{F}}[W]$ by Lemma \[nullak\] and we are done, as $\mathsf D(G/\langle c \rangle) = \mathsf D (C_p \times C_p) = 2p-1$ by hence $(\operatorname{\mathbb{F}}[W]^{\langle c \rangle}_+)^{2p-1} \subseteq {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ by Lemma \[b+1\].
It remains that $\deg(m_0) \le p$. Then we must have $\deg(m_i) \ge p$ for some $i\ge 1$, say $i=1$, as otherwise $\deg(m) \le \deg(m_0) +(p-1)^2 \le p^2-p+1$ would follow. Take the factorisation $m_1 = m_{1}^{(1)} \cdots m_{1}^{(n_1)}$ corresponding to the direct decomposition . We proceed by induction on $\mu(m) := \max_{j=1}^{n_1} \deg(m_{1}^{(j)})$.
Assume first that $\mu(m) \ge p$. This means that $\deg(m_{1}^{(j)}) \ge p$ for some $j$, say $j=1$. Now let $v$ be an arbitrary divisor of $ m_{1}^{(1)}$ with degree $\deg(v) = p$ and let $v' = \prod_{g \in \langle b \rangle} x^g$ for some variable $x \in \operatorname{\mathbb{F}}[V_{\omega^1}^{(1)}]$. Then $v'$ is $b$-invariant by construction. Moreover by we have $\phi_c(v') = p\phi_c(x) =0$ and $\phi_a(v') = p\phi_a(x) - (1+2+\cdots+p-1)\phi_c(x) =0$, and consequently $v'$ is $G$-invariant. Now as $v' \sim v$, we can find by Proposition \[amorf\] a monomial $m' \sim m$ such that $m-m' \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ and $v' \mid m'$. But then $m' \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ and we are done for this case.
Now let $\mu(m) < p$. As $\deg(m_1)\ge p$, we can take a divisor $v \mid m_1$ such that $v = v^{(i)} v^{(j)}$ for some indices $i \neq j \leq n_1$ where we have $\deg(v^{(i)}) = \mu(m)$ and $\deg(v^{(j)} ) =1$. Then the monomial $v' := (x^{(i)}_{1,1} )^{\mu(m)}x_{1,1}^{(j)}$ is homologous with this $v$ and consequently, by Proposition \[amorf\], a monomial $m' \in \operatorname{\mathbb{F}}[W]$ exists such that $m-m' \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ and $v' \mid m'$. Our claim will now follow by proving that $m' \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$.
To this end observe that for the monomial $\tilde m := x_{1,1}^{(i)}m'/x_{1,1}^{(j)}$ we have $\mu(\tilde m) = \mu(m) +1$, hence by the induction hypothesis $\tilde m \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ already holds. Moreover $ \Delta_1^{i,j}(\tilde m) = (\mu(m)+1) m'$ by construction, hence $(\mu(m) +1)m ' \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]}$ by and we are finished because by our assumption on $\operatorname{\mathbb{F}}$ we are allowed to divide by $\mu(m)+1 \le p < \mathrm{char} (\operatorname{\mathbb{F}})$.
From Proposition \[majdnem\_tiszta\] we see that $\operatorname{\mathbb{F}}[W]$ as a module over $\operatorname{\mathbb{F}}[W]^G$ is generated by elements of degree at most $p^2 -2$. Equivalently, for the top degree in the factor ring $\operatorname{\mathbb{F}}[W]/\operatorname{\mathbb{F}}[W]_+^G\operatorname{\mathbb{F}}[W]$ we have the estimate $b(G,W) \le p^2-2$, whence by we conclude that $\beta(G,W)\le p^2-1$.
The case p=3 {#Sec:4}
============
\[H3\_also\] Consider $V=V_{\omega}$ for a primitive third root of unity $\omega \in \operatorname{\mathbb{F}}$ as given by . Then $\beta(H_3, V)\ge 9$.
Let $\operatorname{\mathbb{F}}[V] = \operatorname{\mathbb{F}}[x,y,z]$ with the variables conforming our conventions. $\operatorname{\mathbb{F}}[V]^{H_3}$ is spanned by the elements $\tau(m):= \tau_A^{H_3}(m)=\frac 1 3(m+m^b + m^{b^2})$ where $m$ is any $A$-invariant monomial. An easy argument shows that $xyz, x^3, y^3, z^3$ are the only irreducible $A$-invariant monomials. Then by enumerating all $A$-invariant monomials of degree at most $8$ we see that they have degree $3$ or $6$ so that for $d \le 8$ we have $\operatorname{\mathbb{F}}[V]^{H_3}_d = R_d$, where $ R := \operatorname{\mathbb{F}}[xyz, \tau(x^3), \tau(x^3y^3)]$. Now if we assume that $\beta(H_3,V) \le 8$ then $\operatorname{\mathbb{F}}[V]^{H_3} = R$ follows. Observe however that all the generators of $R$ are symmetric polynomials, so that $R \subseteq \operatorname{\mathbb{F}}[V]^{S_3}$. On the other hand $\tau(x^6y^3) \in \operatorname{\mathbb{F}}[V]^{H_3}$ is not a symmetric polynomial, whence $\tau(x^6y^3) \not\in R.$ This is a contradiction which proves that $\beta(H_3,V ) \ge 9$.
The upper bound on $\beta(H_3)$ will be obtained by an argument very similar to Propositions \[separ\], \[amorf\] and \[majdnem\_tiszta\], but since there are many different details, too, we preferred to give a self-contained treatment of this case here:
\[H3\_felso\] If $\mathrm{char}(\operatorname{\mathbb{F}}) \neq 3$ then $\beta(H_3) \le 9$.
Suppose that $\beta(H_3, W) \ge 10$ holds for a $H_3$-module $W$. Then there is a monomial $m \in \operatorname{\mathbb{F}}[W]^A$ with $\deg(m) \ge 10$ such that $m \not\in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$ (as otherwise for any $d \ge 10$ the space $\operatorname{\mathbb{F}}[W]^G_d$ spanned by the elements $\tau(m)$ would be contained in $(\operatorname{\mathbb{F}}[W]_+^G)^2$). Let $S = \Phi_c(m)$, identify ${\langle}c {\rangle}$ with $\mathbb Z / 3 \mathbb Z$ and let $d_i = \mathsf v_i(S)$ for $i\in \mathbb Z/ 3 \mathbb Z =\{0,1,2\}$. Recall that we have the factorisation $m=m_0m_1m_2$ corresponding to the direct decomposition $W = U \oplus V_1 \oplus V_2$, so that $\deg(m_i) = d_i $ for $i=0,1,2$. We may assume by symmetry that $d_1 \ge d_2$.
[**A.**]{} [*We claim that $d_1 \ge 5$.*]{}
$S$ is a zero-sum sequence over $\mathbb Z / 3 \mathbb Z$ and this is only possible if $d_1 - d_2 \equiv 0 \mod{3}$. So let $d_1 - d_2 = 3k$ for some integer $k \ge 0$. Denoting by $\ell(S)$ the maximum number of non-empty zero-sum sequences into which $S$ can be factored, we have $\ell(S) = d_0 +d_2 +k \le 5$, as otherwise by Lemma \[b+1\] applied with $N = \langle c \rangle$ we get $m \in (\operatorname{\mathbb{F}}[W]_+^{\langle c\rangle})^6 \subseteq {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$, since $H_3/N \cong C_3 \times C_3$ and $\operatorname{\mathsf{D}}(C_3^2) = 5$ by . On the other hand $ |S| = d_0 +d_1+d_2 \ge 10$. Subtracting from this inequality the previous one yields $ d_1 -k \ge 5$, whence the claim.
[**B.**]{} [*For any $w \mid m_1$ with $\deg(w) \le 2$ there is a factorisation $m=u_1u_2u_3$ with $u_i \in \operatorname{\mathbb{F}}[W]^A_+$ such that $w \mid u_3$ and $y \mid u_1$ for some variable $y \mid m_1$.* ]{}
As $\deg(m/w) \ge 8 = \operatorname{\mathsf{D}}_2(C_3^2)$ there is a factorisation $m/w= u_1u_2r$ with $u_1,u_2 \in \operatorname{\mathbb{F}}[W]^A_+$. Setting $u_3 = rw$ enforces $u_3 \in \operatorname{\mathbb{F}}[W]^A_+$. Here $\deg(u_3) \le \mathsf D(A)=5$ as otherwise $u_3 \in (\operatorname{\mathbb{F}}[W]^A_+)^2$ and $m \in (\operatorname{\mathbb{F}}[W]^A_+)^4 \subseteq {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$ by Lemma \[b+1\], a contradiction. Therefore we cannot have $m_1 \mid u_3$, for then by [**A.**]{} we have $5 \le \deg(m_1) \le \deg(u_3) \le 5$, so that $m_1 = u_3$ and $\Phi_c(m_1) = 1^{[5]}$, contradicting the assumption that $\Phi(u_3)$ is a zero-sum sequence over $A$. As a result there is a variable $y \mid m_1$ not dividing $u_3$, whence the claim.
[**C.**]{} [*For any divisor $v \mid m_1$ with $\deg(v) \le 3$ and any monomial $v' \sim v$ there is a monomial $m' \in \operatorname{\mathbb{F}}[W]$ such that $v' \mid m'$ and $m -m' \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$.*]{}
Let $v= xw$ and $v'=x^gw'$ where $\deg(w) \le 2$, $w' \sim w$ and $g\in \langle b \rangle$. By induction on $ \deg(v) $ assume that we already have a monomial $m'' \sim m$ such that $m''-m \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$ and $x^tw' \mid m''$ for some $t \in \langle b \rangle$. According to [**B.**]{} there are factorisations $m'' = u_1u_2u_3$ with $u_i \in \operatorname{\mathbb{F}}[W]^A_+$ such that $w'\mid u_3$ and $y \mid u_1$ for some variable $y \in \operatorname{\mathbb{F}}[V_1]$. We have two cases: i) If we can take $y = x^t$ in one these factorisations then for $m' := u_1^{-t+g}u_2^{t-g} u_3 \sim m''$ we have $m'-m'' \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$ by Lemma \[trukk\] so we are done as $v' \mid m'$. ii) Otherwise necessarily $x^tw' \mid u_3$ and $y \neq x^t$. Still however, there is an $h \in \langle b \rangle$ such that $\phi(y^h) = \phi(x^t)$ hence for $\tilde m := u_1^hu_2^{-h}u_3 \sim m''$ we have $m-\tilde m \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$ by Lemma \[trukk\] and we obtain a factorisation $\tilde m = \tilde u_1 \tilde u_2 \tilde u_3$ falling under case i) by setting $\tilde u_1 = x^tu_1^h/y^h$, $\tilde u_2 = u_2^{-h}$, $\tilde u_3 = y^hu_3/x^t$, so we are done again.
[**D.**]{} Now we proceed as in the proof of Proposition \[majdnem\_tiszta\]. For the sake of simplicity from now on we rename our variables so that $\operatorname{\mathbb{F}}[V_1] = \bigotimes_{i=1}^{n_1} \operatorname{\mathbb{F}}[V_{\omega}]= \operatorname{\mathbb{F}}[x_i,y_i,z_i: i=1,\ldots, n_1]$. Moreover we abbreviate $\Delta_1^{s,t}$ as $\Delta^{s,t}$.
1\) If we have $\deg(m_{1}^{(i)}) \ge 3$ for some $1 \le i \le n_1$ then we can apply [**C.**]{} with $v' := x_iy_iz_i \in \operatorname{\mathbb{F}}[W]_+^G$, concluding that $m \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$, a contradiction.
2\) Otherwise if $\deg(m_{1}^{(i)}) =2$ for some $i$ then still there is a $j \neq i$ such that $ \deg(m_{1}^{(j)}) \ge 1$. After an application of [**C.**]{} we may assume that $m$ is divisible by $ x_i^2x_j$. But then $ m = \frac 1 3 \Delta^{j,i} (\tilde m)$ for the monomial $\tilde m := m x_i/x_j$ which falls under case 1) hence $m \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$ by , a contradiction.
3\) Finally, if $\deg(m_1^{(1)}) = \ldots = \deg(m_1^{(n_1)}) =1$ then after an application of [**C.**]{} we may assume that $x_1y_2z_3 \mid m$. Now consider the relation: $$\begin{aligned}
\label{polar}
\Delta^{1,2} (x_1y_1z_3) + \Delta^{2,3} (x_1 y_2 z_2) + \Delta^{3,1} (x_3 y_2 z_3) = 3 x_1y_2z_3 + \tau(x_3y_2z_1)\end{aligned}$$ After multiplying with $m' := m / x_1y_2z_3 $ we get on the left hand side $\Delta^{1,2} ( my_1/y_2) + \Delta^{2,3} (m z_2/z_3 ) + \Delta^{3,1} ( mx_3/x_1) \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$ by , as all the three monomials occurring here fall under case 2), and on the right hand side $\tau(x_3y_2z_1)m' \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$, whence $3m = 3x_1y_2z_3m' \in {\operatorname{\mathbb{F}}[W]^G_+\operatorname{\mathbb{F}}[W]_+}$ follows. This contradiction completes our proof.
Now comparing Proposition \[H3\_also\] and \[H3\_felso\] immediately gives:
\[H3\] If $\mathrm{char}(\operatorname{\mathbb{F}}) \neq 3$ then $\beta(H_3) = 9$.
It would be interesting to know if Theorem \[beta\_H\_p\] also extends to the whole non-modular case, i.e. for any field $\operatorname{\mathbb{F}}$ whose characteristic does not divide $|G|$, just as it is the case for $p=3$ by the above result.
Acknowledgements {#acknowledgements .unnumbered}
================
The author is grateful to Mátyás Domokos for many valuable comments on the manuscript of this paper. He also thanks the anonymous referee for many suggestions to improve the presentation of this material.
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[^1]: Partially supported by the National Research, Development and Innovation Office (NKFIH) grants PD113138, ERC HU 15 118286, K115799 and K119934.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper outlines a geometric interpretation of flows generated by the collisionless Boltzmann equation, focusing in particular on the coarse-grained approach towards a time-independent equilibrium. The starting point is the recognition that the collisionless Boltzmann equation is a noncanonical Hamiltonian system with the distribution function $f$ as the fundamental dynamical variable, the mean field energy ${\cal H}[f]$ playing the role of the Hamiltonian and the natural arena of physics being ${\Gamma}$, the infinite-dimensional phase space of distribution functions. Every time-independent equilibrium $f_{0}$ is an energy extremal with respect to all perturbations ${\delta f}$ that preserve the constraints (Casimirs) associated with Liouville’s Theorem. If the extremal is a local energy minimum, $f_{0}$ must be linearly stable but, if it corresponds instead to a saddle point, $f_{0}$ may be unstable. If an initial $f(t=0)$ is sufficiently close to some linearly stable lower energy $f_{0}$, its evolution can be visualised as involving linear phase space oscillations about $f_{0}$ which, in many cases, would be expected to exhibit linear Landau damping. If instead $f(0)$ is far from any stable extremal, the flow will be more complicated but, in general, one might anticipate that the evolution can be visualised as involving nonlinear oscillations about some lower energy $f_{0}$. In this picture, the coarse-grained approach towards equilibrium usually termed violent relaxation is interpreted as nonlinear Landau damping. Evolution of a generic initial $f(0)$ involves a coherent initial excitation ${\delta f}(0){\;}{\equiv}{\;}f(0)-f_{0}$, not necessarily small, being converted into incoherent motion associated with nonlinear oscillations about some $f_{0}$ which, in general, will exhibit destructive interference. This picture allows for distinctions between regular and chaotic “orbits” in ${\Gamma}$: Stable extremals $f_{0}$ all have vanishing Lyapunov exponents, even though “orbits” oscillating about $f_{0}$ may well correspond to chaotic trajectories with one or more positive Lyapunov exponents.'
---
=3ex =4ex
**Violent Relaxation, Phase Mixing, and**
.1in
**Gravitational Landau Damping**
.2in
**Henry E. Kandrup**
.15in
Department of Astronomy and Department of Physics and
.05in
Institute for Fundamental Theory, University of Florida
.05in
Gainesville, FL 32611 USA
.2in
**1. Introduction and Motivation**
.1in
The problem addressed in this paper is how to visualise flows generated by the collisionless Boltzmann equation ([*CBE*]{}), i.e., the gravitational analogue of the electrostatic Vlasov equation from plasma physics.
It is generally accepted that many physical problems arising in galactic dynamics and cosmology can be modeled in terms of the [*CBE*]{}, perhaps allowing also for low amplitude discreteness effects, modeled as friction and noise through the formulation of a Fokker-Planck equation, or for a coupling to a dissipative fluid described, e.g., by the Navier-Stokes equation. Astronomers recognise that an evolution described completely by the [*CBE*]{} is special because of the constraints associated with Liouville’s Theorem, and that, at some level, the flow must be Hamiltonian, which precludes the possibility of any pointwise approach towards a time-independent equilibrium: in the absence of dissipation, one can only speak meaningfully of a coarse-grained approach towards equilibrium. However, there does not seem to be a clear sense of exactly how one ought to visualise a flow governed by the [*CBE*]{} or of what sort of coarse-graining one ought to implement in order to identify an approach towards equilibrium.
The conventional wisdom of galactic dynamics (cf. Binney and Tremaine 1987), as articulated, e.g., by Maoz (1991), draws sharp distinctions between different aspects of the evolution, speaking separately of phase mixing, (linear) Landau damping, and violent relaxation. However, such distinctions, even if useful in addressing specific physical effects, are arguably [*ad hoc*]{} and, as such, may obscure the overall character of the flow. Plasma physicists (cf. van Kampen 1955, Case 1959) are well acquainted with the fact that, appropriately interpreted, linear Landau damping [*is*]{} a phase mixing associated with the evolution of a wave packet constructed from a continuous set of normal modes. Moreover, even though conventional wisdom makes a sharp distinction between violent relaxation and phase mixing/Landau damping, one can argue that, as is implicit in Lynden-Bell’s (1967) original paper on violent relaxation, it too is a phase mixing process.
The objective here is to present a coherent mathematical description of an evolution described by the [*CBE*]{} that manifests explicitly the Hamiltonian character of the flow. This entails a synthesis and extension of existing work in both plasma physics and galactic dynamics (cf. Morrison 1980, Morrison and Eliezur 1986, Kandrup 1989, 1998 and numerous references cited therein) which, in the context of galactic dynamics, has proven useful in understanding problems related to both linear and global stability, as well as stability in the presence of weak dissipation (cf. Kandrup 1991a,b, Perez and Aly 1996, Perez, Alimi, Aly, and Scholl 1996). Section 2 describes the precise sense in which the [*CBE*]{} is an infinite-dimensional Hamiltonian system, identifying the natural phase space, exhibiting the noncanonical Hamiltonian structure, and then speculating on the possible meaning of regular versus chaotic flows.
Section 3 turns to the problem of linear stability for time-independent equilibria. This is addressed both in the context of the full noncanonical Hamiltonian dynamics and in terms of a simpler canonical Hamiltonian structure associated with the tangent dynamics, i.e., identifying explicitly a set of canonically conjugate variables in terms of which to analyse linear perturbations. One immediate by-product of this discussion is a simple explanation (cf. Habib, Kandrup, and Yip 1986) of linear Landau damping which manifests explicitly that it is in fact a phase mixing process: Even though a perturbation cannot “die away” in any pointwise sense, one may expect a coarse-grained approach towards equilibrium in which observables like the density perturbation ${\delta}{\rho}$ eventually decay to zero.
Section 4 generalises the preceding to the case of nonlinear stability, allowing for perturbations ${\delta f}$ away from some equilibrium $f_{0}$ which are not necessarily small. The intuition derived from that problem is then used to motivate one possible way in which to visualise the flow associated with a generic initial $f(t=0)$. The obvious point is that a generic initial $f(0)$ can be viewed as a (possibly strongly nonlinear) perturbation of [*some*]{} equilibrium $f_{0}$, the form of which, however, need not be known explicitly. To the extent that this interpretation is accepted, those aspects of the flow typically denoted violent relaxation should be viewed as nonlinear Landau damping/phase mixing (cf. Kandrup 1998). Section 5 concludes by describing the mathematical issues which must be resolved to make the preceding discussion rigorous and complete.
A simple mechanical model, which can help in visualising the basic ideas described in this paper, is the following: Consider a point particle moving in some complicated, many-dimensional potential $V({\bf r})$ which is characterised generically by multiple extremal points but which, being bounded from below, will have a (in general nondegenerate) global minimum. If one chooses initial data corresponding to a configuration space point ${\bf r}$ close to but slightly above some local minimum ${\bf r}_{0}$ and a velocity ${\bf v}$ whose magnitude is very small, the subsequent evolution will involve linear oscillations about ${\bf r}_{0}$, whether or not that point corresponds to a global minimum. The trajectory of the point particle thus corresponds to a regular orbit in what appears locally as a harmonic potential. If the initial deviation from the extremal point becomes somewhat larger, because $|{\bf r}-{\bf r}_{0}|$ and/or $|{\bf v}|$ is bigger, one would still anticipate oscillations around ${\bf r}_{0}$, but these will now become nonlinear and the particle trajectory may well correspond to a chaotic orbit. Suppose, however, that ${\bf r}_{0}$ is [*not*]{} the global minimum. In this case, one would expect that, for initial data sufficiently far from ${\bf r}_{0}$, the particle will have left the “basin of attraction” associated with the local minimum and will instead (generically) exhibit strongly nonlinear oscillations about the global minimum (it could of course oscillate around a different nonglobal minimum!). In the absence of dissipation, there is no pointwise sense in which the particle evolves towards the global minimum. However, the nonlinear oscillations in different directions will in general interfere destructively, so that any initial coherence between motions in different directions will eventually be lost (at least for times short compared with the Poincaré recurrence time). It is this loss of coherence which, for the [*CBE*]{}, gives rise to (linear or nonlinear) Landau damping. .2in
**2. The Noncanonical Hamiltonian Formulation**
.1in If one considers the Liouville equation appropriate for a collection of noninteracting particles evolving in a fixed potential ${\Phi}({\bf x})$, the natural phase space is the six-dimensional phase space associated with the canonical pair $({\bf x},{\bf v})$. If, however, one considers the full [*CBE*]{}, allowing for a self-consistent potential ${\Phi}[f({\bf x},{\bf v})]$ determined by the free-streaming particles, this is no longer so. In this case, the fundamental dynamical variable is the distribution function itself, and the natural phase space ${\Gamma}$ is the infinite-dimensional phase space of distribution functions. In general, it is not easy to identify conjugate coordinates and momenta in this phase space so as to rewrite the [*CBE*]{} in the form of Hamilton’s equations. However, one can still capture the Hamiltonian character at a formal algebraic level through the identification of an appropriate cosymplectic structure (cf. Arnold 1989).[^1]
In this context, manifesting the Hamiltonian character of the flow entails identifying a Lie bracket $[ \, . \, , \, . \, ]$, defined on pairs of phase space functionals ${\cal A}[f]$ and ${\cal B}[f]$, and a Hamiltonian functional ${\cal H}[f]$, so chosen that the [*CBE*]{} $${{\partial}f\over {\partial}t}+v{\cdot}{{\partial}f\over {\partial}{\bf x}}
-{\nabla}{\Phi}{\cdot}{{\partial}f\over {\partial}{\bf v}}= 0, \eqno(1)$$ with ${\Phi}({\bf x},t)$ the self-consistent potential satisfying $${\nabla}^{2}{\Phi}=4{\pi}G{\rho}{\;}{\equiv}{\;}\int\,d^{3}v\,f ,\eqno(2)$$ can be written in the form $${{\partial}f\over {\partial}t}+ {\bigl [}{\cal H},f {\bigr ]} = 0. \eqno(3)$$
The Hamiltonian ${\cal H}$ may be taken as $${\cal H}[f]={1\over 2}\,\int\,d{\Gamma}\,v^{2}\,f({\bf x},{\bf v}) -
{G\over 2}\,\int\,d{\Gamma}\,\int\,d{\Gamma}'\,
{f({\bf x},{\bf v})f({\bf x}',{\bf v}')\over |{\bf x}-{\bf x}'|}, \eqno(4)$$ with $d{\Gamma}{\;}{\equiv}{\;}d^{3}xd^{3}v$, which corresponds to the obvious mean field energy, as identified, e.g., by Lynden-Bell and Sanitt (1969). The bracket is then chosen to satisfy (Morrison 1980) $$[{\cal A},{\cal B}]=\int\,d{\Gamma}\,f\,{\Bigl\{}
{{\delta}{\cal A}\over {\delta}f},{{\delta}{\cal B}\over {\delta}f}{\Bigr\}},
\eqno(5)$$ where $\{g,h \}$ denotes the ordinary Poisson bracket acting on functions $g({\bf x},{\bf v})$ and $h({\bf x},{\bf v})$, and ${\delta}/{\delta}f$ denotes a functional derivative. It is straightforward to show that the operation defined by eq. (5) is a skew symmetric, bilinear form, satisfying the Jacobi identity $$[g,[h,k]] + [h,[k,g]] + [k,[g,h]] = 0, \eqno(6)$$ so that it defines a bona fide Lie bracket. However for this bracket one verifies immediately that eq. (3) reduces to the [*CBE*]{} in the form $${{\partial}f\over {\partial}t}-\{E,f\} = 0, \eqno(7)$$ where $E$ represents the energy of a unit mass test particle, i.e., $$E={1\over 2}v^{2}+{\Phi}({\bf x},t). \eqno(8)$$
A flow governed by the [*CBE*]{} is strongly constrained by Liouville’s Theorem, which implies the existence of an infinite number of conserved quantities, the so-called Casimirs ${\cal C}[f]$. Specificially, the flow has the property that, for any function ${\chi}(f)$, the value of the phase space integral $$C[f]=\int\,d{\Gamma}\,{\chi}(f) \eqno(9)$$ is invariant under time translation, i.e., $dC/dt=0$. The simplest case corresponds to the choice ${\chi}=f$, which leads to conservation of number (or mass): $${d\over dt}\,\int\;d{\Gamma}\,f{\;}{\equiv}{\;}0. \eqno(10)$$ By analogy with finite-dimensional systems, where Noether’s Theorem relates conserved quantities to continuous symmetries, these Casimirs reflect internal symmetries in the infinite-dimensional phase space ${\Gamma}$ (Morrison and Eliezur 1986).
The Casimirs play an important role in analysing the stability of equilibrium solutions $f_{0}$, where one must restrict attention to perturbations ${\delta}f$ that satisfy ${\delta}C{\;}{\equiv}{\;}0$ for all possible choices of ${\chi}$. As first noted by Bartholomew (1971), this demand implies that any allowed perturbation ${\delta}f$ is related to $f_0$ by a canonical transformation induced by some generating function $g$, i.e., $$f{\;}{\equiv}{\;}f_{0}+{\delta}f={\rm exp}(\{g,\, .\,\})f_{0}. \eqno(11)$$
In addition to the Casimirs, there is also at least one other conserved quantity, namely the mean field energy ${\cal H}[f]$. Specifically, it follows from the [*CBE*]{} that $d{\cal H}/dt{\;}{\equiv}{\;}0$. If one considers initial data $f(0)$ characterised by a high degree of symmetry, other conserved quantities may also exist. For example, if the initial data correspond to a potential ${\Phi}$ which is spherically symmetric, it follows that the numerical value of the angular momentum $${\bf J}{\;}{\equiv}{\;}
\int\,d^{3}xd^{3}v\,f\,{\bf x}{\times}{\bf v} \eqno(12)$$ is necessarily conserved. However, these conserved quantities, if they exist, are on a different footing from the Casimirs since they reflect symmetries in the particle phase space, rather than internal symmetries associated with the infinite-dimensional phase space of distribution functions.
Because of the infinite number of constraints associated with the Casimirs, the evolution of $f$ is reduced to a lower (but presumably still infinite-) dimensional phase space hypersurface, say ${\gamma}$. One might naively believe that, in the same sense as, e.g., for the Kortweg-de Vries equation (cf. Arnold 1989), the flow associated with the [*CBE*]{} is integrable. In point of fact, however, this is almost certainly not so (cf. Morrison 1987), the important point being that the Casimirs associated with the [*CBE*]{} are all “ultralocal” quantities which do not involve derivatives of $f$.
At the present time, there is no universally accepted notion of what precisely one should mean by chaos in an infinite-dimensional Hamiltonian system. However, one obvious tact entails comparing initially nearby flows and asking whether, for some given $f(t=0)$, there exist perturbations ${\delta}f(t=0)$ which grow exponentially. This leads naturally[^2] to the notion of a functional Lyapunov exponent which, at least formally, can be defined by analogy with the definition of an ordinary Lyapunov exponent in a finite-dimensional system (cf. Lichtenberg and Lieberman 1992). Specifically, given the introduction of an appropriate norm $|| \; ||$, one can write $${\chi}=\lim_{t\to\infty}\lim_{{\delta}f(0)\to 0} {1 \over t}
\;{||{\delta}f(t)|| \over ||{\delta}f(0) || }. \eqno(13)$$ For finite dimensional systems one knows that, independent of the choice of norm, the analogue of eq. (13) will, for a generic phase space perturbation ${\delta}z$, converge towards the largest Lyapunov exponent. Much less is known about the infinite-dimensional case. For specificity, it thus seems reasonable to choose $|| \; ||$ as corresponding to a (possibly weighted) $L^{2}$ norm defined in the phase space of distribution functions, i.e., $$||{\delta}f|| {\;}{\equiv}{\;} \int \, d{\Gamma} \,M({\bf x},{\bf v})\,
|{\delta}f({\bf x},{\bf v})|^{2} , \eqno(14)$$ where $M$ denotes a specified function of ${\bf x}$ and ${\bf v}$. This is, e.g., the type of norm that has been used in proving theorems about linear stability. .2in
**3. Linear Stability and Gravitational Landau Damping**
.1in The key fact underlying the interpretation of flows described by the [*CBE*]{} and, especially, the problem of stability, is that every time-independent equilibrium $f_{0}$ is an energy extremal with respect to “symplectic” perturbations ${\delta}f$ of the form (11) which preserve the numerical values of every Casimir. This implies that, if one restricts attention to the reduced phase space ${\gamma}$ obtained by freezing the value of each Casimir at its equilibrium value $C[f_{0}]$, every equilibrium $f_{0}$ corresponds to an isolated fixed point: To lowest order, the quantity ${\delta}{\cal H}{\;}{\equiv}{\;}0$ for any symplectic ${\delta}f$. As explained below, if $f_{0}$ is a local energy minimum, so that, to next leading order, ${\delta}{\cal H}{\;}{\ge}{\;}0$, $f_{0}$ must be linearly stable. Alternatively, if $f_{0}$ corresponds to a saddle point, so that ${\cal H}$ increases for some perturbations but decreases for others, linear stability is no longer guaranteed, although one cannot necessarily infer that $f_{0}$ must be linearly unstable.
The proof that, to lowest order, ${\delta}{\cal H}$ vanishes for any perturbation of the form (11) and the computation of ${\delta}{\cal H}$ to higher order are straightforward if one expands (11) perturbatively to infer that $${\delta}f=\{g,f_{0}\} + {1\over 2} \{g, \{g,f_{0}\} \} +. \, . \, . {\;}
{\equiv}{\;}{\delta}^{(1)}f + {\delta}^{(2)}f + . \, . \, . \, . \eqno(15)$$ It is easy to see that, for any ${\delta}^{(1)}f$, the first variation ${\delta}^{(1)}{\cal H}$ becomes $${\delta}^{(1)}{\cal H}=\int \, d{\Gamma}\;{\Bigl(}
{1\over 2}v^{2}-G\int\,d{\Gamma}'\, {f_{0}'\over |{\bf x}-{\bf x}'|} {\Bigr)}
{\delta}^{(1)}f = \int\,d{\Gamma}E_{0}{\delta}^{(1)}f , \eqno(16)$$ where $E_{0}$ is the particle energy associated with $f_{0}$. However, by combining eqs. (15) and (16) and then integrating by parts, one finds that $${\delta}^{(1)}{\cal H}=\int\,E_{0}\,\{g,f_{0}\} = -\int\,d{\Gamma}\,
g\{E_{0},f_{0}\} {\;}{\equiv}{\;} 0 , \eqno(17)$$ where (cf. eq. 7) the final equality follows from fact that $f_{0}$ is time-independent. Extending this calculation to one higher order shows that the second variation $${\delta}^{(2)}{\cal H}=-{1\over 2}\,\int\,d{\Gamma}\,
\{g,f_{0}\}\, \{ g,E_{0} \} - {G\over 2}\,\int\,d{\Gamma} \int\,d{\Gamma}'
{ \{g,f_{0}\} \, \{ g',f_{0}' \} \over |{\bf x}-{\bf x}'|} . \eqno(18)$$
To help visualise what is going on, and to understand why linear stability follows if ${\delta}^{(2)}{\cal H}$ is positive for all symplectic perturbations of the form (11), suppose that, in ordinary three-dimensional space, the $x$-$y$ plane corresponds to a hypersurface in the reduced ${\gamma}$-space of distribution functions. One can then “warp” this plane into a curved two-dimensional surface by assigning to each $x$-$y$ pair a coordinate $z$ which corresponds to the numerical value assumed by the energy ${\cal H}$. On this warped surface, the equilibrium points correspond to those pairs $(x_{0},y_{0})$ which are extremal in $z$, so that any infinitesimally displaced point $(x_{0}+{\delta}x,y_{0}+{\delta}y)$ assumes a new value $z+{\delta}z$.
If the equilibrium point is a local energy minimum, any infinitesimal displacement on the surface necessarily increases the value of $z$, so that, in the neighbourhood of $(x_{0},y_{0})$, the surface has the geometry of an upward opening paraboloid. Any perturbation comes with positive energy and corresponds to bounded motion on the paraboloid. Thus the equilibrium is linearly stable. In principle, the same conclusion also obtains if the extremal point is a local maximum, although one can show that, for realistic equilibria, ${\delta}^{(2)}{\cal H}$ is never strictly negative. If, however, the equilibrium corresponds to a saddle point, so that $z$ increases in some directions but decreases in others, the situation becomes more complicated. In this case, the linearised dynamics implies that it is possible to combine a very large negative energy perturbation in one direction with a very large positive energy perturbation in another to generate a total perturbation with vanishing energy. In itself, this does not guarantee a linear instability, but the simple geometric argument for stability that holds for a local minimum is no longer applicable.[^3]
That saddle points need not imply linear instability may seem surprising at first glance. However, the following two-dimensional example makes clear exactly what can go wrong: $$H={1\over 2}{\Bigl(}v_{1}^{2}+{\omega}_{1}^{2}x_{1}^{2}{\Bigr)}-
{1\over 2}{\Bigl(}v_{2}^{2}+{\omega}_{2}^{2}x_{2}^{2}{\Bigr)}. \eqno(19)$$ Here $x_{1}=v_{1}=x_{2}=v_{2}=0$ is a time-independent extremal point in the phase space which corresponds to a saddle but, nevertheless, the equilibrium is clearly stable. This model may seem somewhat contrived but, as discussed in Section V of Kandrup and Morrison (1993), such stable saddle points are not uncommon in various infinite-dimensional Hamiltonian systems.
The preceding argument for stability or lack thereof may seem somewhat unusual because it is formulated abstractly in phase space, without the introduction of conjugate coordinates and momenta. One might therefore hope that, by identifying an appropriate set of conjugate variables, a more intuitive proof could be derived. In certain cases, this is in fact possible. One knows that, when formulated in the full ${\Gamma}$-space, the dynamics cannot be decomposed completely into canonical variables because of the existence of the Casimirs, which correspond to null vectors of the cosymplectic structure. If, however, one passes to the reduced ${\gamma}$ space, where the values of all the Casimirs are frozen, one might expect that, at least locally, conjugate variables do exist. Indeed, for finite-dimensional systems it follows from Darboux’s Theorem (cf. Arnold 1989) that, if the cosymplectic structure has vanishing determinant, i.e., if there are no null eigenvectors, it is always possible to find a set of canonically conjugate variables, at least locally (see Section V of Kandrup and Morrison \[1993\] for a detailed discussion of this point).
One setting in which such a canonical formulation is possible is for the special case of linear perturbations of an equilibrium $f_{0}$ which is a function only of the one-particle energy $E$, i.e., $f_{0}=f_{0}(E)$, and for which the partial derivative $F_{E}{\;}{\equiv}{\;}{\partial}f/{\partial}E$ is strictly negative. Physically the latter restriction implies that the system does not exhibit a population inversion; mathematically it ensures that division by $F_{E}$ is well defined. The basic idea, due originally to Antonov (1960), is to split the linearised perturbation ${\delta}f$ into two pieces, ${\delta}f_{+}$ and ${\delta}f_{-}$, respectively even and odd under a velocity inversion ${\bf v}\to -{\bf v}$, and to view the single linearised perturbation equation for ${\delta}f$ as a coupled system for ${\delta}f_{\pm}$.
When linearised about some equilibrium $f_{0}$, the [*CBE*]{} reduces to $${\partial}_{t}{\delta}f - \{E,{\delta}f\} - \{ {\Phi}[{\delta}f],f_{0} \} =0,
\eqno(20)$$ where $E$ is the particle energy associated with $f_{0}$ and ${\Phi}[{\delta}f]$ denotes the gravitational potential “sourced” (cf. eq. 2) by the perturbation ${\delta}f$. If one observes that $E$ is an even function of ${\bf v}$, that the Poisson bracket is odd under velocity inversion, and that ${\Phi}[{\delta}f_{-}]$ vanishes identically, it is clear that eq. (20) is equivalent to the coupled system $${\partial}_{t}{\delta}f_{+} - \{E,{\delta}f_{-} \} =0$$ and $${\partial}_{t}{\delta}f_{-} - \{E,{\delta}f_{+} \} -
\{ {\Phi}[{\delta}f_{+}],f_{0} \} =0 .\eqno(21)$$ However, if one differentiates the second of these relations with respect to $t$, and uses the first to eliminate ${\partial}_{t}{\delta}f_{+}$, it follows that $${\partial}_{t}^{2}{\delta}f_{-} = \{ E, \{ E,{\delta}f_{-} \} \} +
\{ {\Phi}[ \{E,{\delta}f_{-}\}] , f_{0} \} {\;}{\equiv}{\;}F_{E}\,
{\cal A}{\delta}f_{-}, \eqno(22)$$ where ${\cal A}$ denotes a linear operator. One can then show that, given the identification of ${\delta}f_{-}$ and ${\partial}_{t}{\delta}f_{-}$ as conjugate variables, the equation $$(-F_{E})^{-1}{\partial}_{t}^{2}{\delta}f_{-} = -{\cal A}{\delta}f_{-}
\eqno(23)$$ can be derived from the Hamiltonian .05in $$\hskip -2.95in {\widehat{\cal H}}={1\over 2}\int\,{d{\Gamma}\over (-F_{E})}
({\partial}_{t}{\delta}f_{-})^{2} +
{1\over 2}\int\,d{\Gamma}\,{\delta}f_{-}{\cal A}{\delta}f_{-}$$ $$={1\over 2}\int\,{d{\Gamma}\over (-F_{E})}
({\partial}_{t}{\delta}f_{-})^{2} +
{1\over 2}\int\,{d{\Gamma}\over (-F_{E})}
\{E,{\delta}f_{-}\}^{2} - {G\over 2}\,\int\,d{\Gamma}\,\int\,d{\Gamma}'
{ \{E,{\delta}f_{-} \} \, \{E',{\delta}f_{-}' \}\over |{\bf x}-{\bf x}'|}.
\eqno(24)$$ The connection between $\widehat{\cal H}$ and the energy ${\delta}^{(2)}{\cal H}$ associated with a small symplectic perturbation is discussed in Kandrup (1989). In particular, one can show that ${\widehat{\cal H}}>0$ for all ${\delta}f_{-}$ if and only if ${\delta}^{(2)}{\cal H}>0$ for all symplectic perturbations.
The fact that ${\cal A}$ is a symmetric (i.e., hermitian) operator facilitates a proof that the equilibrium $f_{0}(E)$ is linearly stable if and only if ${\widehat{\cal H}}$ (and ${\delta}^{(2)}{\cal H}$) is positive. Specifically, a simple energy argument (cf. Laval, Mercier, and Pellat 1965) implies that the magnitude of ${\delta}f_{-}$, and hence ${\delta}f$, is bounded in time if ${\cal A}$ is a positive operator, so that ${\delta}^{(2)}{\cal H}>0$, whereas the possibility of perturbations with $\int\,d{\Gamma}{\delta}f_{-}{\cal A}{\delta}f_{-}<0$ implies the existence of solutions that grow exponentially. This is easy to understand in the language of normal modes. Since ${\cal A}$ is symmetric, it is clear that all solutions ${\delta}f_{-}{\;}{\propto}{\;}{\rm exp}(st)$ have $s^{2}$ real, so that the evolution is either purely oscillatory or purely exponential. If ${\cal A}$ is positive, $s^{2}$ must be negative, so that the modes are purely oscillatory. If, however, ${\cal A}$ is not a positive operator, there exist modes with $s^{2}>0$, which implies an exponential instability.[^4]
This sort of normal mode expansion facilitates a simple geometric picture of an infinite-dimensional configuration space of perturbations ${\delta}f_{-}$ which is (locally) embeddable in the reduced ${\gamma}$-space. The equilibrium $f_{0}$, which is necessarily an extremal point of the full Hamiltonian ${\cal H}$, satisfies ${\delta}f_{-}{\;}{\equiv}{\;}{\partial}_{t}{\delta}f_{-}{\;}{\equiv}{\;}0$. An arbitrary initial perturbation entails a kinetic energy ${\cal K}=
\int\,d{\Gamma}(-F_{E})^{-1}({\partial}_{t}{\delta}f_{-})^{2}$ which is necessarily positive and a potential energy ${\cal W}=
\int\,d{\Gamma}{\delta}f_{-}{\cal A}{\delta}f_{-}$ whose sign depends on the properties of ${\cal A}$. If ${\cal A}$ is a positive operator, the evolution in configuration space involves a particle with “mass” $(-F_{E})^{-1}$ moving in an infinite-dimensional harmonic potential which corresponds to an upwards opening paraboloid. Linear stability is therefore assured. If, however, ${\cal A}$ is not always positive, ${\delta}f_{-}{\;}{\equiv}{\;}0$ corresponds to a saddle point, rather than a local minimum, and the flow is linearly unstable.
In visualising all of this, there is the strong temptation to think of the normal modes as being discrete, i.e., corresponding to honest square integrable eigenfunctions rather than singular eigendistributions. This, however, is not necessarily justified.
Assuming completeness, one can always view any linear perturbation of an equilibrium $f_{0}(E)$ with $F_{E}<0$ as a superposition of normal modes, writing ${\delta}f$ as a formal sum $${\delta}f({\bf x},{\bf v},t)=\sum_{\sigma}\;
A_{\sigma}g_{\sigma}({\bf x},{\bf v}){\rm exp}(i{\sigma}t) , \eqno(25)$$ where $g_{\sigma}$ labels the eigenvector, ${\sigma}$ is the corresponding frequency, which is necessarily real, and $A_{\sigma}$ is an expansion coefficient.[^5] Modulo largely unimportant technical details, the modes then divide into two types, namely: (1) a countable set of discrete frequencies belonging to the point spectrum, for which the corresponding eigenvectors are well-behaved (e.g., square-integrable) eigenfunctions; and (2) a continuous set of frequencies belonging to the continuous spectrum, for which the eigenvectors are singular eigendistributions.
The distinction between these two types of modes is extremely important (cf. Habib, Kandrup, and Yip 1986). Because true eigenfunctions are nonsingular, they can in principle be triggered individually, i.e., one can choose a reasonable initial ${\delta}f$ which populates only a single discrete mode. By contrast, because eigendistributions are singular, one cannot sample a single continuous mode. Rather, any smooth ${\delta}f$ sampling the continuous spectrum must really be constructed as a wavepacket comprised of a continuous set of modes. The important point then is that, when evolved into the future, such a wavepacket implies a damping of coarse-grained observables like the density ${\rho}$. In other words, if the modes are continuous there is a precise sense in which the perturbation “dies away” and the system exhibits a coarse-grained approach towards the original equilibrium $f_{0}$.
The physics here is analogous to what arises in ordinary quantum mechanics. If, in that setting, one considers a physical observable like angular momentum with a discrete spectrum, one can construct well behaved eigenstates which, when evolved into the future, maintain their coherence for all time: the only effect of the evolution is a coherently oscillating phase. If, however, one considers an observable like position or linear momentum, where the spectrum is continuous, this is no longer so. In this case, a normalisable initial state must be constructed from a continuous set of singular eigendistributions, so that the best one can do is build a localised (e.g., minimum uncertainty) wavepacket. However, when evolved into the future such a wavepacket will necessarily spread because different eigendistributions have different phase velocities.
It is this loss of coherence associated with the spreading of a wavepacket that corresponds to (linear) Landau damping. In the context of plasma physics, Landau damping was derived originally (Landau 1946) in a very different way, through the introduction of a Fourier-Laplace transform and an analysis of poles in the complex plane. However, at least for the electrostatic Vlasov equation (cf. Case 1959), i.e., the electrostatic analogue of the [*CBE*]{}, the mathematical equivalence of these two pictures of Landau damping is well understood. The physics underlying their equivalence is discussed in Kandrup (1998).
For the special case of perturbations of an homogeneous neutral plasma characterised by an isotropic distribution of velocities that is everywhere nonvanishing, the modes can be computed explicitly (cf. Case 1959), and one finds generically that $$g_{\sigma}({\bf x},{\bf v})={\rm exp}(i{\bf k}{\cdot}{\bf x})\;
g_{\sigma}({\bf v}) , \eqno(26)$$ where $g_{\sigma}({\bf v})$ is a singular eigendistribution involving a Dirac delta. In this setting, an examination of the perturbation associated with a given ${\bf k}$-vector at a fixed phase space point $({\bf x}_{0},{\bf v}_{0})$ yields no evidence of damping away. Rather, one finds persistent oscillations ${\propto}{\;}{\rm exp}[i{\bf k}{\cdot}({\bf x}_{0}-{\bf v}_{0}t)].$ This is simply a manifestation of the fact that, without the introduction of some coarse-graining, one cannot speak of the system returning to equilibrium. If, however, a coarse-graining is implemented by integrating over any finite range of velocities, one discovers that the resulting $\int\,d^{3}v\,{\delta}f({\bf x},{\bf v},t)$ will in fact damp away.
The obvious question, therefore, is: will perturbations ${\delta}f$ of a generic equilibrium solution to the [*CBE*]{} correspond to discrete modes, continuous modes, or a combination of both? Unfortunately, this is a difficult question to answer. It appears impossible to calculate the modes explicitly for realistic equilibria, and a formal analysis is also difficult because the operator ${\cal T}$ entering into the linearised equation ${\partial}_{t}{\delta}f={\cal T}{\delta}f$ is not elliptic and involves a singular integral kernel. However, the normal modes can, and have, been computed for a variety of nontrivial equilibrium solutions to the corresponding electrostatic Vlasov equation (cf. van Kampen 1955, Case 1959), and the results derived thereby would seem suggestive.
Perhaps the most important result derived for the Vlasov equation is that discrete modes are seemingly the exception, rather than the norm, arising only if the equilibrium in question manifests nontrivial boundary conditions, e.g., the existence of a maximum speed $v_{m}$ such that $f({\bf v}){\;}{\equiv}{\;}0$ for $|{\bf v}|>v_{m}$. In particular, one can prove that the modes are always purely continuous if $f_{0}$ is an analytic function of ${\bf v}$ in the complex plane.[^6] The best known example of a nonempty point spectrum is the case of so-called van Kampen (1955) modes, which arise precisely in those configurations where there is maximum velocity. In the usual interpretation (cf. Stix 1962), Landau damping is understood as resulting from a resonance between “particles” (the unperturbed $f_{0}$) propagating with velocity ${\bf v}$ and a “wave” (the perturbation ${\delta}f$) that propagates with phase velocity ${\bf c}$. Discrete van Kampen modes correspond to perturbations which propagate with a phase velocity ${\bf c}$ for which $f_{0}({\bf c})=0$, so that no resonance is possible.
By analogy, one might therefore conjecture (cf. Habib, Kandrup, and Yip 1986) that, for the gravitational [*CBE*]{}, most perturbations will in fact correspond to continuous modes that damp away, but that some perturbations, especially longer wavelength disturbances that probe the phase space boundaries of the system, could in fact correspond to discrete modes. In this connection, it is interesting to note that there do in fact exist exact time-dependent solutions to the [*CBE*]{}, seemingly appropriate for a system like a galaxy, that exhibit finite amplitude undamped oscillations about some time-independent $f_{0}$ (cf. Louis and Gerhart 1988, Sridhar 1989). The interesting point, then is that in all these models the time-independent $f_{0}$ contains phase space “holes,” i.e., regions in the middle of the occupied phase space region where $f_{0}\to 0$. Whether these sorts of solutions are generic, and whether they could arise from reasonable initial conditions, is at the present unclear.
Finally, it should be noted that, in point of fact, one can in principle get (at least temporary) phase mixing or loss of coherence even for the much simpler case of a finite set of discrete modes. For example, if one considers the function $x(t)= \sum_{p=10}^{29}\rm{cos}(0.1pt)$ over the finite interval $0<t<1000$, one infers a rapid damping of the initial coherent excitation with $x=20$ to a much smaller value oscillating about $x=0$ with typical amplitude $|x|{\;}{\sim}{\;}1$. If, however, the evolution is tracked for a somewhat longer time one finds that the initial coherence is regained. An infinite set of continuous modes differs from this toy model in two important ways, namely (1) the recurrence time is infinitely long and (2) it is impossible to consider a smooth initial excitation that does not damp.
.2in
**4. Nonlinear Stability and Global Evolution**
.1in Suppose, once again, that attention is focused on some linearly stable equilibrium $f_{0}(E)$ with $F_{E}<0$, but that one is now interested in the effects of larger perturbations ${\delta}f$, i.e., the problem of nonlinear stability. To the extent that the normal modes of the linear problem remain complete, one can still envision evolution in terms of these modes, the important point, however, being that, because of nonlinearities, the modes will now interact. This is, e.g., the basis for the standard quasilinear analyses implemented in plasma physics, which allow for the effects of the quadratic term ${\nabla}{\Phi}{\cdot}({\partial}{\delta}f/{\partial}{\bf v})$ which is ignored when considering linear perturbations.
Mode-mode couplings are important in that they facilitate the transfer of energy between different modes, which makes the physics more complicated. However, one might still anticipate that, if the modes are continuous, Landau damping can and will occur. Because of the interactions between modes, the simple model of a dispersing quantum mechanical wavepacket is no longer directly applicable, but the basic phenomenon of loss of coherence is robust. Indeed, there are many examples in nonlinear dynamics of flows satisfying nonlinear evolution equations where phase mixing occurs. It thus seems reasonable to suppose that, when considering the nonlinear evolution of some perturbation ${\delta}f$, one will encounter nonlinear Landau damping. For the case of an electrostatic plasma, nonlinear Landau damping is a well known, and reasonably well understood, phenomenon (cf. Davidson 1972 and references cited therein). Indeed, there are simple geometries where the nonlinear evolution can be computed explicitly in the context of a systematic perturbation expansion, thus facilitating analytic formulae for exactly how this phenomenon works (cf. Montgomery 1963).
Mode-mode couplings can also lead to another important possibility, namely the onset of chaos. Because $f_{0}$ is a local energy minimum, one knows that any infinitesimal perturbation ${\delta}f$ will simply oscillate, each eigenvector corresponding to motion in a “direction” in configuration space that is orthogonal to the motion of all the other eigenvectors. This implies that, for the fixed point $f_{0}$, the Lyapunov exponents, which were defined in eq. (13) as probing the average linear instability of the orbit generated from some initial $f(0)$, must all vanish identically. One might anticipate further that, when evolved into the future, other phase space points sufficiently close to $f_{0}$ will also correspond to regular orbits with vanishing Lyapunov exponents. Thus, e.g., for finite-dimensional systems one knows that there is a regular phase space region of finite measure surrounding every stable periodic orbit. However, for sufficiently large ${\delta}f$, where mode-mode couplings become significant and the motion cannot be well approximated by orthogonal harmonic oscillations, one might anticipate that many, if not all, perturbations will evolve chaotically. If true, this would suggest that a “typical” perturbation with ${\delta}{\cal H}=
{\cal H}[f_{0}+{\delta}f]-{\cal H}[f_{0}]$ will evolve ergodically on (some subset of) the constant energy hypersurface in the ${\gamma}$-space with energy ${\cal H}[f_{0}+{\delta}f]$.
This idea of the onset and development of chaos is an infinite-dimensional generalisation of what is typically found when considering the motion of a point mass in a multi-dimensional nonlinear potential which has only one extremal point, a global minimum.[^7] Low energy orbits sufficiently close to the pit of the potential move in what is essentially a harmonic potential, so that their motion is regular. If, however, the energy is raised one finds generically that, unless the motions in different directions remain completely decoupled, there is an onset of global stochasticity which leads, for sufficiently high energies, to well developed chaotic regions.
This configuration space description is not appropriate when considering generic equilibria, where the energy $\widehat{\cal H}$ associated with a small perturbation cannot be written easily as a functional of conjugate variables, and there is no guarantee that $\widehat{\cal H}$ can be written as a simple sum of kinetic and potential contributions, ${\cal K}$ and ${\cal W}$. Modulo technical details, one might expect that canonical phase space coordinates do exist, at least in principle, but the energy $\widehat{\cal H}$ associated with the tangent dynamics could in general be an arbitrary quadratic functional $\widehat{\cal H}[q,p]$ of the conjugate variables $q$ and $p$. Moreover, even for the simple model of an equilibrium $f_{0}(E)$ with $F_{E}<0$, it may not be possible to extend the canonical description to allow for arbitrarily large perturbations ${\delta}f$. One really needs to return to a full phase space description.
As discussed in Section 3, if for some equilibrium $f_{0}$ the second variation ${\delta}^{(2)}{\cal H}$ is positive for all ${\delta}f$, a linearised perturbation corresponds in phase space to stable motion on an upwards opening infinite-dimensional paraboloid. As long as this surface remains convex, one would anticipate that stability will persist and, as such, one would expect intuitively that the equilibrium could remain nonlinearly stable even for small but finite ${\delta}f$. The normal modes of the linearised problem become coupled, but the geometric argument for stability should remain valid. In particular, one can presumably visualise the evolution of ${\delta}f$ as involving nonlinear [*phase space*]{} oscillations about the equilibrium point $f_{0}$.
If, however, $f_{0}$ corresponds to a stable saddle, one might suppose that even the smallest nonlinearities could trigger an instability (cf. Moser 1968, Morrison 1987). Thus, e.g., for the simple toy model of two stable oscillators described by eq. (19), it is possible to trigger an instability by introducing even very tiny mode-mode couplings which allow energy to be transferred between modes. Indeed, as noted by Cherry (1925), if the two frequencies are in an appropriate resonance, e.g., ${\omega}_{2}^{2}=2{\omega}_{1}^{2}$, the introduction of a simple cubic coupling implies that initial data arbitrarily close to $x_{1}=v_{1}=x_{2}=v_{2}=0$ can lead to solutions in which $x_{1}$, $x_{2}$, $v_{1}$, and $v_{2}$ all diverge in a finite time. If true, this expectation about saddle points would suggest that, even though they can be linearly stable, they cannot represent reasonable candidate equilibria in terms of which to model real astronomical objects.
If a linearly stable $f_{0}$ corresponds to a unique extremal point in the ${\gamma}$-space, the surface which near $f_{0}$ is a paraboloid will remain upwards opening even if ${\delta}f$ is very large, so that stability should persist for arbitrarily large perturbations. In other words, one would expect that the equilibrium $f_{0}$ is globally stable: In this case, any phase space deformation ${\delta}f$ increases the energy, and the evolution of an initial ${\delta}f(0)$ will involve nonlinear phase space oscillations around the unique stable fixed point.
If, however, there exist multiple extremal points in the ${\gamma}$-space, each corresponding to a local energy minimum, the situation is much more complicated. In this case, one would anticipate that, for sufficiently large ${\delta}f$, the distribution function can actually be transferred from the “basin of attraction” of one equilibrium $f_{0}$ to the “basin” of some other $f_{1}$. In other words, the evolution of ${\delta}f(0)$ could yield oscillations around $f_{1}$, rather than $f_{0}$. By suitably fine-tuning the perturbation, one can in principle displace the system from any one basin to any other. However, by analogy with the behaviour observed in finite-dimensional systems, one might expect generically that, if the perturbation is sufficiently large, its motion can be interpreted as involving nonlinear phase space oscillations about the global energy minimum. To the extent that this is true, one would anticipate that a sufficiently large perturbation will tend generically to push $f$ into the “basin of attraction” of the equilibrium $f_{0}$ that corresponds to a global energy minimum.
If one considers an initial perturbation ${\delta}f(0)$ that is sufficiently large, the subsequent evolution will in general be almost completely unrelated to the initial equilibrium $f_{0}$ and, as such, the way in which one visualises the evolution ${\delta}f(0)$ is really no different from the way in which one can, and arguably should, envision the evolution of a generic $f(0)$. In other words, the physical picture described above can be used equally well to visualise generic flows associated with the initial value problem, the only difference being that, in general, one may know nothing at all about what time-independent equilibria $f_{0}$ actually exist.
Specification of an initial $f(0)$ fixes the values of all the Casimirs for all times, thus determining ${\gamma}$, the reduced infinite-dimensional phase space which constitutes the natural arena of physics. This $f(0)$ also fixes the numerical value of the conserved energy ${\cal H}$ and, as such, determines the constant energy hypersurface in the ${\gamma}$-space to which the flow is necessarily restricted. By analogy with finite-dimensional Hamiltonian systems (cf. Kandrup and Mahon 1994) one might expect that, when evolved into the future, $f(0)$ will exhibit a coarse-grained approach towards an invariant measure on this hypersurface, i.e., a suitably defined microcanonical distribution. If the flow associated with $f(0)$ is chaotic, one might anticipate an approach towards this invariant measure that is exponential in time. If, alternatively, the flow is regular, one might instead expect a power law approach. However, in either case one might anticipate an approach towards a “phase-mixed” invariant measure. In this context, the crucial question is then: to what extent can this invariant measure be interpreted as corresponding to a distribution function $f$ executing phase space oscillations about one or more equilibrium solutions $f_{0}$?
It is easy to see that, in the ${\gamma}$-space, there must exist one or more extremal points with ${\delta}^{(1)}{\cal H}=0$, these corresponding to equilibrium solutions $f_{0}$ for which all the Casimirs share the same values as the Casimirs associated with $f(0)$. Indeed, one knows that, for sufficiently smooth initial data, the [*CBE*]{} admits global existence (cf. Pfaffelmoser 1992, Schaeffer 1991), so that ${\delta}f$ cannot diverge and, presumably, the Hamiltonian is bounded from below. However, this implies that there must exist at least one $f_{0}$, namely the global energy minimum (although in principle the global minimum could be degenerate). The question therefore becomes: in the basin of which $f_{0}$ (or $f_{0}$’s) does the flow reside?
In principle, the evolved distribution function $f$ could execute phase space oscillations about any $f_{0}$ with lower energy, which one presumably depending on the initial $f(0)$. However, one might conjecture that, if the initial $f(0)$ is sufficiently far from any equilibrium $f_{0}$, it will execute oscillations around the global minimum $f_{0}$. The initial $f(0)$ cannot exhibit a pointwise approach towards this, or any other, $f_{0}$. However, one might expect that, in general, the initial deviation ${\delta}f(0)=f(0)-f_{0}$ will exhibit nonlinear Landau damping so that, in terms of observables like the density ${\rho}$, ${\delta}f$ does indeed “die away,” and one can speak of a coarse-grained approach towards the equilibrium $f_{0}$. .2in
**5. Conclusions and Unanswered Questions**
.1in The aim of this paper is to suggest a potentially fruitful way in which to visualise flows described by the [*CBE*]{} and, in particular, the expected coarse-grained approach towards an equilibrium. No claim is made regarding mathematical rigor, and it is not clear that all the details are completely correct. However, the viewpoint developed here does have the advantage that it incorporates what [*is*]{} known rigorously about the [*CBE*]{}, and that it provides a framework in terms of which to pose precise, well defined questions. In this context, there are at least three basic questions which, if answered satisfactorily, would yield important insights into the physical properties of a flow generated by the [*CBE*]{}: .02in
1\. Will generic initial conditions exhibit effective Landau damping, thus allowing one to speak of an efficient coarse-grained evolution towards some equilibrium $f_{0}$? In the context of linear Landau damping, the answer to this question depends on the spectral properties of the linearised evolution equation. If the modes are all continuous, every initial perturbation will eventually phase mix away, so that physical observables like the density will damp to zero. If, however, some of the modes are discrete, it is possible to construct initial perturbations that do not damp away. At the present time, it is not clear whether, for realistic galactic models, the spectrum is purely continuous, although the investigation of various toy models is currently underway (Lynden-Bell 1997, private communication).
To the extent that $N$-body simulations are reliable and that, for sufficiently large $N$, they capture the same physics as the [*CBE*]{}, the fact that most initial conditions yield an efficient approach towards some statistical equilibrium can be interpreted as evidence that nonlinear Landau damping is in general very effective. However, there [*do*]{} exist toy models like one-dimensional gravity where one ends up with undamped oscillations. For example, the evolution of counterstreaming initial conditions in one-dimensional systems (either gravitational or electrostatic) can lead to a final state which corresponds seemingly to a distribution function $f$ exhibiting finite amplitude undamped oscillations about a (near-) equilibrium $f_{0}$ (cf. Mineau, Feix, and Rouet 1990). This toy model actually corroborates the physical intuition described in this paper in the sense that, as one would expect, the phase space contains a large “hole,” i.e., a region where $f_{0} \to 0$. Whether or not analogous results obtain for two- and three-dimensional systems is as yet unclear, although the problem is currently under investigation (Habib, Kandrup, Pogorelov, and Ryne, work in progress). .02in
2\. Are functional Lyapunov exponents the “right” way in which to identify chaos in infinite-dimensional systems and, assuming that they are, will a generic flow associated with the [*CBE*]{} be chaotic? Given this definition, will standard results from finite-dimensional chaos remain at least approximately valid? Although not proven for generic finite-dimensional systems, there is the physical expectation that, when evolved into the future, a chaotic initial condition will evolve towards an invariant distribution on a time scale that is related somehow to the spectrum of Lyapunov exponents. This implies however that, at asymptotically late times, one can visualise the flow as densely filling a chaotic phase space region of finite measure. Assuming, however, that this is true, the Ergodic Theorem provides important information about the statistical properties of the flow, implying the equivalence of time and phase space averages (cf. Lichtenberg and Lieberman 1992).
One other point about chaos in the [*CBE*]{} should be stressed: The definition proposed in this paper is, at least superficially, completely decoupled from the (also interesting) question of whether individual orbits in a self-consistent potential generated from the [*CBE*]{} are, or are not, chaotic. This latter question refers to the behaviour of nearby trajectories in the six-dimensional particle phase space. The “natural” definition of chaos for the [*CBE*]{} should presumably reflect properties of the flow in the infinite-dimensional phase space of distribution functions. .02in
3\. For a specified initial $f(0)$, towards which equilibrium $f_{0}$ will the system evolve? Given $f(0)$, one can compute the numerical value of all possible Casimirs, thus identifying explicitly the ${\gamma}$-space to which the evolution is restricted. The obvious problem, then, is to identify all time-independent equilibria $f_{0}$ in ${\gamma}$ and to determine which initial conditions correspond to which equilibria. Although unquestionably difficult, this is a problem that is both well defined mathematically and well motivated physically. Finding all equilibria is equivalent mathematically to finding all extremal points in ${\gamma}$. However, to the extent that one chooses to visualise the flow as involving oscillations in the ${\gamma}$-space, there is no question physically but that the extremal points define “basins of attraction” associated with the oscillations.
The basic points described in this paper are easily summarised:
1\. The [*CBE*]{} is a Hamiltonian system, albeit an unusual one. The fundamental dynamical variable is the distribution function $f$, not the particle ${\bf x}$’s and ${\bf v}$’s; and it is not always possible (at least easily) to identify canonically conjugate variables.
2\. Because the [*CBE*]{} is Hamiltonian, there can be no pointwise approach towards equilibrim. The best for which one can hope is a coarse-grained approach towards equilibrium.
3\. Even though the phase space ${\gamma}$ associated with the dynamics is infinite-dimensional, one might expect that much of one’s intuition from finite-dimensional systems remains valid. In particular, one might anticipate an asymptotic approach towards an invariant measure, and one might hope to make meaningful distinctions between regular and chaotic flows.
4\. The phenomenon normally designated as linear Landau damping can be interpreted as a phase mixing of a continuous set of normal modes. Whether a small initial perturbation will always eventually Landau damp/phase mix away depends on whether the normal modes for the linearised perturbation equation are discrete or continuous.
5\. To the extent that one’s ordinary intuition about finite-dimensional phase spaces remains approximately valid, the evolution of generic initial data should be interpreted as involving nonlinear (phase space) oscillations about one or more energy extremals, which correspond to time-independent equilibria $f_{0}$. The phenomenon of violent relaxation should thus be interpreted as nonlinear phase mixing/Landau damping which, if efficient, will facilitate a coarse-grained approach towards equilibrium. .2in
**Acknowledgments**
.1in
I am pleased to acknowledge useful discussions, collaborations, and correspondence with Salman Habib, Donald Lynden-Bell, Bruce Miller, Phil Morrison, and Daniel Pfenniger. I am grateful to Barbara Eckstein for comments on the exposition. Work on this manuscript began while I was a visitor at the Observatoire de Marseille, where I was supported by the [*C.N.R.S.*]{}, and continued during a visit to the Aspen Center for Physics. Limited financial support was provided by the National Science Foundation grant PHY92-03333.
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[^1]: One example of a noncanonical Hamiltonian system, well known to astronomers, is rigid body rotations described by the standard Euler equations (cf. Landau and Lifshitz 1960). Specifically, as described and generalised, e.g., in Kandrup (1990) and Kandrup and Morrison (1993), the Euler equations constitute a Hamiltonian system, formulated in the three-dimensional phase space coordinatised by the three components of angular momentum $J_{i}$, ($i=1,2,3$), with the Hamiltonian $H[J_{i}]=\sum_{i=1}^{3}J_{i}^{2}/2I_{i}$ (the analogue of eq. 4) defined in terms of the principal moments of inertia $I_{i}$, and the Lie bracket (the analogue of eq. 5) given as the natural bracket associated with the three-dimensional rotation group, i.e., $$[a,b]=\sum_{i,j,k}\,{\epsilon}_{ijk}J_{k}
{\bigl(}{{\partial}a\over {\partial}J_{i}}{\bigr)}
{\bigl(}{{\partial}b\over {\partial}J_{j}}{\bigr)}$$ for functions $a(J_{i})$ and $b(J_{i})$. As for the [*CBE*]{}, there is also a Casimir (the analogue of eq. 9), namely $C[J_{i}]=\sum_{i=1}^{3}J_{i}^{2}$, which restricts motion to the two-dimensional constant $C$ surface in the three-dimensional phase space.
Astronomers are also acquainted with infinite-dimensional Hamiltonian systems, at least those realisable in canonical coordinates, one simple example being the scalar wave equation ${\partial}^{2}_{t}{\Psi}-{\nabla}^{2}{\Psi}=0$, which derives from the Hamiltonian $${\cal H}={1\over 2}\,
\int\,d^{3}x{\Bigl(}{\Pi}^{2}({\bf x})+|{\nabla}{\Psi}({\bf x}|^{2}{\Bigr)},$$ where ${\Psi}$ and ${\Pi}$ are canonically conjugate.
[^2]: I thank Bruce Miller and Klaus Dietz for suggesting this point to me.
[^3]: Strictly speaking, the application of this finite-dimensional argument to an infinite-dimensional Hamiltonian system requires that the reduced phase space ${\gamma}$ be endowed with a metric, so that one knows what is meant by distance between points. In practice, this can be done by introducing an appropriate $L^2$ norm, which provides the natural extension of the Euclidean notion of distance to an infinite-dimensional space. In this context, a proof of stability entails showing that $||{\delta}f(t)||$ remains bounded for all times.
[^4]: In point of fact, one anticipates that, for this simple case, ${\cal A}$ is guaranteed to be positive. It is believed (cf. Binney and Tremaine 1987) that any $f_{0}$ depending only on $E$ corresponds to a spherically symmetric configuration; but assuming that the mass density ${\rho}$ associated with $f_{0}$ is spherical one can prove that ${\cal A}$ is indeed positive (cf. Kandrup 1989).
[^5]: Strictly speaking, this sum must be interpreted (cf. Riesz and Nagy 1955) as a Stiltjes integral.
[^6]: The validity of Landau’s original derivation of exponential damping actually relies on the implicit assumption that $f_{0}$ is analytic. If it is not, his manipulations of contours and evaluation of poles cannot be justified.
[^7]: Even order truncations of the Toda potential (cf. Kandrup and Mahon 1994) provide a simple two-dimensional example.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Based on the Log-Periodic Power Law (LPPL) methodology, with the universal preferred scaling factor $\lambda \approx 2$, the negative bubble on the oil market in 2014-2016 has been detected. Over the same period a positive bubble on the so called commodity currencies expressed in terms of the US dollar appears to take place with the oscillation pattern which largely is mirror reflected relative to oil price oscillation pattern. This documents recent strong anti-correlation between the dynamics of the oil price and of the USD. A related forecast made at the time of FENS 2015 conference (beginning of November) turned out to be quite satisfactory. These findings provide also further indication that such a log-periodically accelerating down-trend signals termination of the corresponding decreases.'
address: |
$^1$ Complex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences, PL–31-342 Kraków, Poland\
$^2$ Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology, PL–31-155 Kraków, Poland
author:
- 'Marcin Watorek$^{1}$[^1], Stanis[ł]{}aw Drożdż$^{1,2}$, Pawe[ł]{} Oświȩcimka$^{1}$'
title: |
World Financial 2014-2016 Market Bubbles:\
Oil Negative - US Dollar Positive
---
=by -1
Introduction
============
The concept of financial log-periodicity [@Fei96; @Sorr96; @sornette1998; @V98; @D1; @C7] often termed as Log-Periodic Power-Law (LPPL) model, has widely been used for detecting bubbles and subsequent crashes already for almost two decades. In spite of rising some controversies [@F11; @Fei1; @Br1], many successful attempts to describe [@A1; @B11; @C11; @L1; @F1; @G1; @J2; @J1; @K2; @K1; @K3; @W6; @Sorr16] and even to detect bubbles and their subsequent bursts by using this technique [@M1; @D21; @S1; @H1] have been reported. One of the most spectacular such examples is ex-ante exceptionally precise prediction of Brent Crude Oil bubble bursting time in early July 2008, delivered three months ahead as described in ref. [@X1] and also on Wojciech Bia[ł]{}ek blog [@Bi5]. Crucial in this connection was application of the universal preferred scaling factor $\lambda \approx 2$ [@D1; @C7; @B11] and decomposition of the entire oil-price development into long-term trend and a local super-bubble - general concept introduced in ref. [@C7] - here operating on the oil price in the first half of 2008 and violently terminating on July 11th, 2008, precisely as predicted. In longer terms the prediction also was that after this super-bubble burst the oil price will return to the longer-term still increasing trend with its ultimate termination in the second half of 2010. A minimally updated variant of the original prediction for this long-term oil development scenario as Figure 5 in ref. [@O1] was presented during FENS 4 conference in May 2009. Exactly this same scenario with the actual oil price course up to the beginning of 2014 is shown in Figure \[fig:Brent\] of the present contribution. Clearly, there is lot of truth even in this long-term forecast. As predicted, the oil price after recovery from the 2008 super-bubble burst went up sharply until the turn of 2010/2011 and this was the end of this long-term increasing trend, indeed. The following decline was probably at least partly delayed and slowed down by the Arab Spring in the years 2010 - 2013 [@AS; @AS1; @AS2]. The real decrease on the oil market started in mid 2014 and within less than 2 years it dropped by $75\%$ from 106\$ to 26\$ per barrel. Usually such a downward trend is associated with the decelerating log-periodic oscillations but in contrast to most of the previous cases [@B11; @H2; @Z1; @Z2; @Z3] this phase on the oil market appears to be dominated by the accelerating log-periodic oscillations. Simultaneously and in parallel a positive bubble on the so called commodity currencies expressed in terms of the US dollar (USD), exceptionally strongly anti-correlated with the oil price, has developed. This last period of the oil market dynamics is the main subject of the present contribution.
13.0cm
LPPL model for bubbles {#se1}
======================
The concept of financial log-periodicity is based on the assumption that the financial dynamics is governed by phenomena analogous to criticality in the statistical physics sense. In its conventional form criticality implies a scale invariance which, for a properly defined function $F(x)$ characterizing the system, means that: $$F(\lambda x) = \gamma F(x).
\label{eq:F}$$ A constant $\gamma$ in this equation reflects how the properties of the system change when it is rescaled by a factor $\lambda$. The general solution of Eq. (\[eq:F\]) reads: $$F(x) = x^{\alpha} P({\ln (x) / \ln(\lambda)}),
\label{eq:logper}$$ where the first term represents a standard power-law as it is characteristic to continuous scale-invariance with the critical exponent $\alpha = \ln(\gamma)/\ln(\lambda)$ and $P$ denotes a periodic function of period one. This general solution can be interpreted in terms of discrete scale invariance. Due to the second term the continuous dominating scaling acquires a correction that is periodic in $\ln(x)$. It is then meaningful to define $x = \vert t - t_c \vert$, where $t$ denotes the ordinary time labeling the original price time series. This variable $x$ represents a distance to the critical point $t_c$. The resulting spacings between the corresponding consecutive repeatable structures at $x_n$ (i.e., minima or maxima) of the log-periodic oscillations seen in the linear scale follow a geometric contraction according to the relation $\lambda= {x_{n+1} - x_n \over x_{n+2} - x_{n+1}}$. The time points $t_c$ thus correspond to the accumulation of such oscillations and, in the context of the financial dynamics such points indicate a reversal of the trend. One possible representation of periodic function $P$ is the first term of its Fourier expansion: $$P(\ln(x)/\ln(\lambda)) = A + B \cos ({\omega \over 2\pi} \ln(x) + \phi).
\label{eq:pfo}$$ This implies that $\omega = 2\pi / \ln(\lambda)$[@C7].
Negative bubble {#se2}
===============
One possible mechanism that gives rise to such log-periodic structures is positive feedback. This phenomenon leading to an increasing amplitude of the price momentum can also occur in a downward price regime and, as a result, a faster than exponential downward acceleration can take place. In a positive bubble, the positive feedback results from over optimistic expectations of future returns leading to self fulfilling but transient unsustainable price appreciations. In a negative bubble, the positive feedback reflects the rampant pessimism fueled by short positions leading investors to run away from the market which spirals downwards also in a self fulfilling process. The symmetry between positive and negative bubbles is obvious for currencies. If a currency A strongly appreciates against another currency B following a faster than exponential trajectory, the value of currency B expressed in currency A will correspondingly fall faster than exponentially in a downward spiral. In this example, the negative bubble is simply obtained by taking the inverse of the price [@N2].
An alternative related mechanism could be the herding behavior between hedge funds or investors which leads to extreme short positioning building up in the futures market. This regime is unstable and almost anything could trigger short squeeze which leads to rapid price growth. It was precisely this situation that existed in the oil market by the end of 2015 [@W1].
Adjusting procedure {#se3}
===================
In the time domain the Eq. \[eq:pfo\] can be rewritten as: $$\label{eq:lppl}
p(t)=A+B(t_c-t)^m+C(t_c-t)^m\cos(\omega\ln(t_c-t)-\phi).$$
This log-periodic power law (LPPL) model is described by 3 linear parameters ($A, B, C$) and 4 nonlinear parameters ($m, \omega, t_c, \phi$). These parameters are subject to the following constrains as proposed by Sornette [@B1]: $0 < m <1$, $6 \leq\omega\leq 13$, $B < 0$, $|C|<1$, $t \le t_c$.
To fit LPPL function Eq. \[eq:lppl\] to empirical data we use procedure proposed by Filimonov and Sornette [@C1], which reduces adjustment to just three nonlinear parameters: $t_c, m, \omega$. The key idea of this method is to decrease the number of nonlinear parameters and simultaneously to eliminate the interdependence between the phase $\phi$ and the angular log-frequency $\omega$. This one achieves by expanding the cosine term the formula (\[eq:lppl\]) as follows: $$\label{eq:lppl_new}
p(t) = A + B(t_c - t)^m + C_1(t_c - t)^m \cos(\omega \ln(t_c -t))+ C_2(t_c - t)^m \sin(\omega \ln(t_c -t)).$$
As seen from Eq.\[eq:lppl\_new\], the LPPL function has now only 3 nonlinear ($t_c, \omega, m$) and 4 linear ($A, B,C_1, C_2$) parameters, and the two new parameters $C_1$ and $C_2$ contain formerly the phase $\phi$. Based on previous evidence [@C7; @D1; @B11; @K3] we are using a constant scaling factor $\lambda \approx 2$, which further reduces the estimation problem ($\omega = 2\pi / \ln(\lambda)$).
In order to fit the LPPL function we select the initial parameters $t_c, m, \omega$. We then calculate linear parameters $A, B, C_1, C_2$ by ordinary least squares method and then minimize the cost function using nonlinear least squares method. All possible values of start-up parameters: $m \in[0.1, 0.9]$ with step 0.05 and $t_c\in[t+1, t+0.1\*n]$ (where $n$ is the length of time series) with step 5 were tested. To get more robust results we carried out the analysis on empirical data with moving starting point with the step of 5 trading days in a shrinking time window $[t_1, t_2]$. In our work $t_1$ is changing from 12.06.2014 to 10.07.2014, $t_2$ is fixed on 12.02.2016. The lowest sum of squared residuals (SSR) points to the best fit within each time window. In fitting process getting a stable value of $t_c$ is essential, therefore we compare the SSR’s from each time window by evaluating the mean squared error (MSE). The lowest MSE determines the best fit. In order to further illustrate the stability of the adjusting procedure we present the standard deviation for $t_c$ obtained from all fits with different $t_1$ (std($T_c$) in trading days).
Oil versus currency markets {#se4}
===========================
Already a visual chart inspection indicates that in around the end of 2015 the commodity currencies expressed in terms of the US dollar and the oil price develop similar patterns [@W5]. In order to quantify this we calculate the Pearson correlation coefficients from the time series representing the price changes of the currencies and of the Crude Light Oil (CL) in the period June 2014 - March 2016. The results are presented in Table \[Corr14\].
CL CLdiff
----- --------- ---------
AUD -0.9530 -0.3187
BRL -0.8897 -0.2485
CAD -0.9412 -0.5472
CLP -0.9039 -0.2086
GBP -0.9235 -0.2168
MXN -0.9221 -0.3911
NOK -0.9746 -0.3570
RUB -0.9717 -0.2542
: \[Corr14\] Pearson correlation coefficients of the oil (CL) vs 8 commodity currencies (Australian dollar, Brazilian real, Canadian dollar, Chilean peso, Pound sterling, Mexican peso, Norwegian krone, Russian ruble) in the period 01.06.2014-18.03.2016. 1st column - correlation coefficient calculated from the price time series, 2nd column - correlation coefficient calculated from the corresponding return (CLdiff) time series. Above results clearly show high correlations between commodity currencies vs USD and oil.
All these coefficients, even the ones calculated from the returns, are large and negative which reflects the fact that these currencies are anti-correlated with the oil price changes.
A highly coordinated behaviour of all these currencies expressed in USD can be seen from Figure \[fig:Commodtycurr\] where they all - in order to make their dynamics directly comparable - are standardized (scaled to have standard deviation 1 and centered to have mean 2). Already visually their oscillatory behaviour quite convincingly follows the same pattern of the log-periodic contractions. For this reason we construct a basket by summing up with equal weight all the considered commodity currencies, i.e. AUD, BRL, CAD, CLP, GBP, MXN, NOK, RUB. The LPPL best fit is performed on this basket and displayed in Figure \[fig:currbasket\]. The resulting critical time $t_c$=07.03.2016 and as such it was determined already in the beginning of November at the time of FENS 8 Conference. Interestingly, an independent fit performed at the same time to the inverse of the oil price changes, also shown in Figure \[fig:currbasket\] (standardized in the same way as currencies) points to exactly the same $t_c$. This reflects a highly correlated dynamics of the corresponding time series. This correlation somewhat weakened about five weeks before $t_c$ when the USD reached maximum against the entire basket of all these eight commodity currencies. The inverse oil price reached its highest level some three weeks before this date and started a systematic drawdown. Such a somewhat earlier than $t_c$ burst of the bubble determined by LPPL does not contradict applicability of this methodology and in fact is consistent with the concept of criticality that stays behind LPPL. The closer to $t_c$ is the system the more susceptible it becomes to perturbations that may turn it down [@sornette1998].
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Not all the currencies in the above commodity basket were equally correlated regarding their way of approaching $t_c$. The highest correlation is observed in the USD expressed in terms of the Mexican peso and for this reason it is shown in a separate Figure \[fig:MXNfit\]. In this case the trend reversal took place only two weeks before the original prediction.
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Finally, using the same adjusting procedure as described in section \[se3\] directly to the Crude Light Oil prices, as displayed in Figure \[fig:CLfit\], results in essentially the same critical time $t_c$ as for the inverse oil price and as for the currencies basket. An uncommon feature of this 2014-2016 oil price draw-down is that it is accompanied with the accelerating log-periodic oscillations whose accumulation point signals the real trend reversal which in this case occurred indeed. It therefore belongs to the category of negative bubbles [@N2; @D5] as confronted with the anti-bubbles [@J2; @Z1; @Z2; @Z3].
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Summary {#se5}
=======
The downward trend on the world oil market has fully developed starting in mid-2014, thus about four months before the end of quantitative easing in the USA. At around the same time the US dollar started to strengthen. The development of both these markets appears to be describable within the Log-Periodic Power Law methodology with the universal preferred scaling factor $\lambda \approx 2$. A novel aspect of this oil price dynamics is presence of the log-periodically accelerating oscillations accompanying the draw-down phase of the market, therefore termed negative bubbles, contrary to the common scenario where the draw-downs are log-periodically decelerating and are called anti-bubbles. Furthermore, this oil negative bubble appears strongly (anti-)correlated in phase with the US dollar (positive) bubble against the major commodity currencies. Both these bubbles ended in mid-February, 3 weeks before their ultimate limit of termination as set by the critical time $t_c$=07.03.2016. After reaching the low, the Crude Light Oil price surged from 13-year low by $50\%$ in one month. It was the biggest 18-session jump in oil prices over 25 years[@bialek].
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[^1]: e-mail: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Cardiovascular disease (CVD) is the global leading cause of death. A strong risk factor for CVD events is the amount of coronary artery calcium (CAC). To meet demands of the increasing interest in quantification of CAC, i.e. coronary calcium scoring, especially as an unrequested finding for screening and research, automatic methods have been proposed. Current automatic calcium scoring methods are relatively computationally expensive and only provide scores for one type of CT. To address this, we propose a computationally efficient method that employs two ConvNets: the first performs registration to align the fields of view of input CTs and the second performs direct regression of the calcium score, thereby circumventing time-consuming intermediate CAC segmentation. Optional decision feedback provides insight in the regions that contributed to the calcium score. Experiments were performed using 903 cardiac CT and 1,687 chest CT scans. The method predicted calcium scores in less than 0.3s. Intra-class correlation coefficient between predicted and manual calcium scores was 0.98 for both cardiac and chest CT. The method showed almost perfect agreement between automatic and manual CVD risk categorization in both datasets, with a linearly weighted Cohen’s kappa of 0.95 in cardiac CT and 0.93 in chest CT. Performance is similar to that of state-of-the-art methods, but the proposed method is hundreds of times faster. By providing visual feedback, insight is given in the decision process, making it readily implementable in clinical and research settings.'
author:
- 'Bob D. de Vos, Jelmer M. Wolterink, Tim Leiner, Pim A. de Jong, Nikolas Lessmann, Ivana Išgum [^1]'
title: Direct Automatic Coronary Calcium Scoring in Cardiac and Chest CT
---
Calcium scoring, Cardiac CT, Chest CT, Deep Learning, Convolutional Neural Network, Atlas-Registration, Regression.
Introduction
============
Cardiovascular disease (CVD) is the global leading cause of death[@gbd2016]. To reduce the burden of cardiovascular disease the World Health Organization underlines the need for early detection and treatment of individuals with CVD or those who are at high cardiovascular risk due to the presence of one or more risk factors [@whofactsheet]. A strong and independent risk factor for CVD events, e.g. myocardial infarction, is the quantity of coronary artery calcium (CAC) [@yeboah2012; @hecht2015; @hecht2017]. Quantification of CAC, i.e. calcium scoring, is typically performed in dedicated non-contrast-enhanced ECG-synchronized cardiac CT scans[@hecht2015]. Alternatively, calcium scoring can be performed in other non-contrast-enhanced CTs that visualize the heart; e.g. in low-dose CT attenuation correction scans acquired in hybrid PET/CT and SPECT/CT [@einstein2010; @mylonas2012], or in radiation therapy planning CTs of breast cancer patients [@gernaat2016]. Furthermore, it has been shown that calcium scoring in lung screening low-dose chest CT scans is a predictor for all-cause mortality [@jacobs2010; @chiles2015]. In fact, in the National Lung Screening Trial (NLST) CVD was the leading cause of mortality [@nlst2011b]. Thus, CAC quantification, especially as an unrequested finding, has garnered much attention.
Clinically, calcium scoring is performed by experts who manually identify CAC in CT image slices. This is a tedious process of finding and selecting high density voxels in the coronary arteries—commonly defined as two or more connected voxels above 130Hounsfield Units (HU). In scans not dedicated to calcium scoring this can be particularly cumbersome because of high noise, low resolution, and motion artifacts. Subsequently, when lesions are identified, region growing is used to fully segment the calcified lesions. Finally, after all CAC lesions have been segmented, CAC is quantified using the Agatston score [@agatston1990]. The Agatston score takes into account the lesion area and the weighted maximum density of the lesion. This score can be used to stratify patients into risk categories [@rumberger1999].
The additional cost involved with manual calcium scoring makes the process prohibitive in settings where it is not the primary request. To simplify the task, qualitative stratification into CVD risk groups was proposed [@shemesh2010; @chiles2015]. Qualitative calcium scoring is faster and it demonstrates good inter-rater agreement. However, such an analysis still demands experts who closely inspect the scans. With the ever-increasing amount of scans and the increasing interest in calcium scoring, especially as an unrequested finding, the use of fully-automatic methods might be the preferred direction.
Several automatic methods have been introduced for calcium scoring in non-contrast-enhanced CT, ranging from rule-based approaches [@gonzalez2016; @xie2017], to the better performing conventional machine learning approaches[@isgum2012; @shahzad2013; @wolterink2015; @durlak2017] and recent deep learning approaches [@wolterink2015miccai; @wolterink2016; @lessmann2016; @lessmann2018]. The main difficulty in automatic calcium scoring is to differentiate CAC from other dense structures. Obviously, CAC exclusively resides in the walls of the coronary arteries, thus most of the automatic methods exploit this prior knowledge.
Išgum et al. [@isgum2012] introduced the first method for automatic calcium scoring in chest CT. CAC lesions were described with features and subsequently classified using a two-stage classification approach of k-nearest neighbor and support vector classification. Among texture, size, and shape features, highly important for CAC identification, were the location features. Location features were determined by registering an input image to an atlas image and by extracting the location features from a map of a priori spatial probabilities of CAC. The probability map was created from known CAC locations in 237 chest CTs that were registered to a single priorly chosen atlas image. Shahzad et al. [@shahzad2013] used a similar machine learning approach for calcium scoring in cardiac CT, but they employed pair-wise deformable image registration to ten atlases that encoded the coronary arteries. The atlases were made from 85 contrast enhanced CT angiography scans with annotated coronary arteries. The methods of Išgum et al. [@isgum2012] and Shahzad et al. [@shahzad2013] relied on feature selection methods to reduce dimensionality. Wolterink et al. [@wolterink2015] circumvented feature selection by using an extremely randomized trees classifier. Their method also depended on location features that were obtained by deformable image registration of ten atlases with encoded coronary arteries, but these were obtanied from non-contrast-enhanced CTs. Durlak et al. [@durlak2017] combined the principles of the aforedescribed methods: they employed a random forest and made an a priori probability map of coronary arteries locations, made from automatically extracted coronary arteries from cardiac CT angiography images. Instead of using time-consuming deformable image registration to align input images and atlas images, they achieved a speed-up by using affine registration. Similarly, other methods employed information from CTA to aid calcium scoring in cardiac CT. These methods were specifically designed for the coronary calcium score (orCaScore) challenge, and employed rule-based image analysis or conventional machine learning [@wolterink2016orcascore].
Most recently proposed methods employ deep learning methods for automatic calcium scoring, in particular convolutional neural networks (ConvNets). ConvNets are known for their automatic feature extracting capabilities and alleviate the need for handcrafting features. Wolterink et al. [@wolterink2016] used ConvNets to classify CAC in cardiac CT angiography scans. All voxels were classified using a pair of ConvNets. One ConvNet identified voxels likely to be CAC and discarded the majority of non-CAC-like voxels such as lung and fatty tissue. The other ConvNet more precisely discriminated between CAC and CAC-like negatives. In the method of Lessmann et al. [@lessmann2016] a single ConvNet was used that classified candidate CAC lesions in lung screening chest CTs. To simplify the classification tasks, both these deep learning methods used an additional ConvNet that localized the heart with a bounding box [@devos2017localization]. More recently, the method of Lessmann et al.[@lessmann2018] fully exploited the feature extraction capabilities of ConvNets without dedicated localization methods. They employed two sequential ConvNets to classify CAC as well as aortic valve, mitral valve, and aorta calcifications in chest CT. The first ConvNet identified candidate calcifications based on their location, and the second ConvNet refined the classification results by reducing false positive errors.
![In a typical automatic calcium scoring workflow, CAC is first identified and subsequently quantified. The proposed method uses ConvNet regression to quantify CAC in image slices directly.[]{data-label="fig:workflows"}](Workflow.pdf){width="\linewidth"}
While all aforementioned methods use different strategies, they all follow a workflow similar to current clinical calcium scoring: CAC is first identified and thereafter quantified. The automatic methods show high accuracy, but often at considerable computational cost. Employing these methods on large datasets would require dedicated servers. To alleviate computational cost, we propose a workflow that circumvents intermediate identification and that performs direct quantification (see Figure \[fig:workflows\]). Direct quantification has proven to be useful for atrial and ventricle volume quantification [@hussain2017; @zhen2017; @xue2018]. Furthermore, attempts are being made to use it for calcium scoring. In our preliminary study we presented a direct calcium scoring method that uses 2-D ConvNet regression [@devos2017rsna; @devos2017arxiv]. The method performs direct calcium scoring in extracted image slices from bounding boxes cropped around the heart. In a recently proposed method, Cano-Espinosa et al. used a 3-D regression ConvNet for direct calcium scoring in downsampled CT volumes also cropped around the heart. However, their method could not be used in 14% of the scans, because heart localization failed. Furthermore, previously proposed automatic calcium scoring methods are dedicated to either cardiac CT or chest CT. These methods required retraining for application in other types of CT [@gernaat2016; @isgum2017].
We present an automatic method that performs real-time direct calcium scoring in different types of non-contrast-enhanced CT. Unlike previous methods that focused on a single type of CT, the proposed method is able to perform calcium scoring directly in multiple types of CT by using an unsupervised deep learning atlas-registration method to align their fields of view (FOVs). For this we employ two ConvNets: one for atlas-registration and one for calcium scoring, as shown in Figure \[fig:pipeline\]. The atlas-registration ConvNet makes the FOV of input CT images alike using Deep Learning Image Registration (DLIR) [@devos2017registration; @devos2018media] further developed to facilitate atlas-registration. Subsequently, a calcium scoring ConvNet predicts the calcium score in image slices mimicking clinical calcium scoring with the Agatston score. When desired, decision feedback can be queried for every slice with a predicted calcium score. For this purpose, a visual attention heatmap accurately reveals the regions that contributed to the calcium score. The method provides robust and accurate predictions of calcium scores and it is computationally efficient, obtaining an Agatston score in less than 0.3s in cardiac and chest CT.
Data
====
This study included two datasets used in previous studies that presented automatic coronary calcium scoring in cardiac CT [@wolterink2015] and in chest CT [@lessmann2018]. To allow a direct comparison of methods, the original training, validation, and test set distributions were used.
Cardiac CT
----------
The set of 903 cardiac CT scans (age range: 18 to 88 years, 31% women) originates from a set of routinely acquired scans for clinical calcium scoring of the University Medical Center Utrecht, Utrecht, The Netherlands. The need for informed consent was waived by the local Medical Research Ethics Committee. Scans were acquired with a 256-detector row Philips Brilliance iCT scanner (tube voltage 120kVp, tube current 55mAs) during a single breath-hold, with ECG-triggering and without contrast enhancement. The images were reconstructed to 3mm slice thickness and slice increment with in-plane resolution ranging from 0.29mm to 0.49mm, depending on patient size. The dataset was divided into 237 scans for training, 136 scans for validation, and 530 scans were in the hold-out test set only used for final evaluation.
Chest CT
--------
The set of 1,687 chest CT scans (age range: 43 to 74 years, 39% women) originates from a set of 6,000 available baseline scans from the National Lung Screening Trial (NLST) [@nlst2011b]. All scans were acquired during inspiratory breath-hold without contrast enhancement. Scans were acquired in 31 different hospitals with 120 or 140kVp tube voltage and 30-160mAs tube current. Axial images slices were reconstructed with varying kernels, varying slice thickness (1.00-3.00mm), varying slice increments (0.63-3.00mm), and with varying in-plane resolutions (0.49-0.98mm per voxel). In our study, scans with less than 100 slices or slices thicker than 3.00mm were not considered, because they were not adequate for calcium scoring. Furthermore, the scans were resampled to 3.00mm slice thickness and 1.50mm slice increment to make the scans suitable for calcium scoring [@rutten2011]. The dataset was divided into 1,012 scans for training, 169 scans for validation, and 506 scans were in the hold-out test set only used for final evaluation.
-- ------------ ----- ---- ----- ----- -----
Training 120 14 33 29 41
Validation 68 14 28 15 11
Test 260 49 89 70 62
Training 272 76 207 205 252
Validation 39 14 46 30 40
Test 128 42 99 112 125
-- ------------ ----- ---- ----- ----- -----
: Number of scans per CVD risk category for training, validation, and test sets. CVD risk categorization is based on the total Agatston score per scan: : very low $<1$, : low $[1, 10)$, : moderate $[10, 100)$, : moderately high $[100, 400)$, : high $\geq400$[]{data-label="tab:riskcatsalldata"}
Reference standard {#sec:refstandard}
------------------
The reference standard was defined by experts who manually identified CAC lesions in the scans. CAC lesions were segmented following a standard procedure: region growing was used to select 26-connected voxels $\geq$130HU. In the chest CTs with low radiation dose this procedure could lead to faulty segmentations (i.e. leakage) because of excessive noise. In such cases annotations were manually corrected by voxel painting [@lessmann2018]. Agatston scores were calculated in each axial slice for training. Total Agatston scores for each scan were calculated for final evaluation. Additionally, each subject was assigned to one of five CVD risk categories [@rumberger1999] based on the Agatston score: very low: $<$1; low: \[1, 10), moderate: \[10, 100), high: \[100, 400), very high: $\geq$400. Table \[tab:riskcatsalldata\] provides an overview of the number of scans per risk category per dataset.
Methods
=======
The method employs two ConvNets in sequence (Figure \[fig:pipeline\]). The first ConvNet registers input CTs to an cardiac CT atlas-image. The second ConvNet performs calcium scoring. When desired, visual feedback can be queried for image slices with a score. For this purpose an attention heatmap reveals the regions that contributed to the calcium score.
![Schematics of the proposed method. Input CTs of varying FOV are first aligned using an atlas-registration ConvNet. Subsequently, a calcium scoring ConvNet is used for direct calcium scoring in image slices. Finally, decision feedback can be visualized when desired.[]{data-label="fig:pipeline"}](Pipeline_1.pdf){width="\linewidth"}
Atlas-registration strategy {#sec:method:registration}
---------------------------
An atlas-registration ConvNet ensures that all input images have a similar FOV and resemble a cardiac CT. The ConvNet is trained with a modified version of our framework for Deep Learning Image Registration (DLIR) [@devos2017registration]. The DLIR framework uses an end-to-end unsupervised approach that trains a ConvNet for image registration. Similar to a conventional intensity-based image registration framework it exploits optimization of an image similarity metric. Figure \[fig:dlirframework\] shows the schematics of training an atlas-registration ConvNet using the atlas image as a static fixed image. The task of the ConvNet is to analyze moving images and predict the transformation parameters that warp the moving images to the atlas-image. Image similarity between the atlas and the warped image, is used for backpropagation during training. By optimizing image similarity (e.g. minimizing negative cross correlation) with gradient descent, the atlas-registration ConvNet learns the registration task in an unsupervised manner. After training, the ConvNet can register unseen moving images in one shot.
A cardiac atlas-image is created using an iterative inter-subject registration strategy [@jongen2004]. With this strategy an initial atlas image is made by averaging multiple images. The atlas image is iteratively refined by registering the individual images to the atlas. Subsequently, the final atlas image is used to train the atlas-registration ConvNets for cardiac and chest CT alignment used for subsequent calcium scoring.
![DLIR framework used to train a registration ConvNet. During a forward pass (indicated by the thick blue arrow) the registration ConvNet analyzes moving images and outputs transformation parameters. The transformation parameters are used by the interpolator to warp the moving image. During a backward pass (indicated by the thick red arrow) an image similarity loss (i.e. dissimilarity) is determined between the warped image and a fixed template image, and the resulting loss is backpropagated trough the ConvNet. The ConvNet is trained in multiple iterations of forward and backward passes, with mini-batch stochastic gradient descent. Once the ConvNet has been trained for registration it can take a moving image as its input and it can output registration parameters in one pass, thus non-iteratively.[]{data-label="fig:dlirframework"}](DLIR.pdf){width="\linewidth"}
Atlas-registration ConvNet training
-----------------------------------
For registration we propose a global 3-D rigid registration model with six degrees of freedom (shown in Figure \[fig:degrees\_of\_freedom\]). The model allows translations in any direction, but rotations are restricted to the axial ($z$) axis. Furthermore, scaling in the axial plane is isotropic and independent from scaling along the axial axis. These restrictions preserve the relation of reference Agatston scores that are defined on the original (unregistered) axial image slices. This facilitates training of the subsequent calcium scoring ConvNet.
(img) at (0,0) [![Rigid transformation model used for to train the registration ConvNet. The six degrees of freedom allow translation in any direction, rotation around the axial axis, and uniform scaling in the axial plane independent from scaling along the axial direction. By constraining the registration to the proposed transformation model, we can trivially exploit the model parameters for selection and warping of axial slices that are presented to the calcium scoring ConvNet.[]{data-label="fig:degrees_of_freedom"}](DegreesOfFreedom.pdf "fig:"){width=".35\linewidth"}]{}; at (32pt, 26pt) [$t_y$]{}; at (38pt,-14pt) [$t_x$]{}; at (-7pt,41pt) [$t_z$]{}; at (-32pt,-2pt) [$s_{xy}$]{}; at (10pt,-20pt) [$s_z$]{}; at (15pt,20pt) [$\theta_z$]{};
We use a computationally efficient ConvNet architecture that is listed in Table \[tab:convnetdesigns\]. For fast analysis, images are downsampled close to 3mm isotropic voxel dimensions; i.e. 6$\times$6$\times$1 downsampling for cardiac CT, and 6$\times$6$\times$2 downsampling for chest CT using average pooling. The ConvNet has three alternating layers of 3$\times$3$\times$3 convolutions and 2$\times$2$\times$2 average pooling and those are followed by two layers of 3$\times$3$\times$3 convolution. To facilitate a fixed output, global average pooling is applied before connection with two fully connected layers. The final output layer has six nodes, one for each transformation parameter. Throughout the network exponential linear units are used for activation, except in the output nodes. Three output nodes are unconstrained translation parameters ($t_x$, $t_y$, $t_z$), the rotation parameter ($\theta_z$) is constrained with a hyperbolic tangent between $-\pi$ and $\pi$, and the two scaling parameters ($s_{xy}$, $s_z$) are constrained with a hyperbolic tangent between $0.25$ and $4$ scaling factors. These output parameters are used to constitute the following 3-D transformation matrix: $$T_\textrm{3D} =
\setlength\arraycolsep{2pt}
\begin{bmatrix}
1 & 0 & 0 & t_x \\
0 & 1 & 0 & t_y \\
0 & 0 & 1 & t_z \\
0 & 0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
\cos\theta_z & -\sin\theta_z & 0 & 0 \\
\sin\theta_z & \phantom{-}\cos\theta_z & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
s_{xy} & 0 & 0 & 0 \\
0 & s_{xy} & 0 & 0 \\
0 & 0 & s_z & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}$$
Atlas-registration ConvNet inference
------------------------------------
We train an atlas-registration ConvNet for 3-D registration, but we use it for slice selection and 2-D warping. As a consequence, correspondence is guaranteed between warped axial slices and the per-slice calcium scores. Axial image slices are extracted from the original image from $t_z$ to $t_z + d_z/s_z$, where $d_z$ is the depth of the atlas image along the axial axis. These slices are resampled using bi-linear interpolation to a 256$\times$256 grid with the following 2-D transformation matrix: $$T_\textrm{2D} =
\begin{bmatrix*}
1 & 0 & t_x \\
0 & 1 & t_y \\
0 & 0 & 1 \\
\end{bmatrix*}
\begin{bmatrix*}
\cos\theta_z & -\sin\theta_z & 0 \\
\sin\theta_z & \phantom{-}\cos\theta_z & 0 \\
0 & 0 & 1 \\
\end{bmatrix*}
\begin{bmatrix*}
s_{xy} & 0 & 0 \\
0 & s_{xy} & 0 \\
0 & 0 & 1 \\
\end{bmatrix*}$$
**Atlas-Registration ConvNet** **Calcium Scoring ConvNet**
-------------------------------------- -----------------------------
512$\times$512$\times$N 3-D input 256$\times$256 2-D input
6$\times$6$\times${1,2} Avg. Pooling 224$\times$224 cropping
32\*3$\times$3$\times$3 Convolutions 32\*3$\times$3 Convolutions
2$\times$2$\times$2 Avg. Pooling 2$\times$2 Max Pooling
32\*3$\times$3$\times$3 Convolutions 32\*3$\times$3 Convolutions
2$\times$2$\times$2 Avg. Pooling 2$\times$2 Max Pooling
32\*3$\times$3$\times$3 Convolutions 32\*3$\times$3 Convolutions
2$\times$2$\times$2 Avg. Pooling 2$\times$2 Max Pooling
32\*3$\times$3$\times$3 Convolutions 32\*3$\times$3 Convolutions
32\*3$\times$3$\times$3 Convolutions 2$\times$2 Max Pooling
Global Avg. Pooling 32\*3$\times$3 Convolutions
2$\times$2 Max Pooling
32\*3$\times$3 Convolutions
2$\times$2 Max Pooling
64 Fully Connected Nodes 64 Fully Connected Nodes
64 Fully Connected Nodes 64 Fully Connected Nodes
6 Output Nodes 1 Output Node
: Efficient ConvNet architectures were used for atlas-registration as well as calcium scoring.[]{data-label="tab:convnetdesigns"}
Calcium scoring ConvNet {#sec:cacscoremethod}
-----------------------
The calcium scoring ConvNet employs direct regression to predict an Agatston score from input axial image slices. The choice of 2-D ConvNets, in favor of 3-D ConvNets, is based on the number of samples that are available for training. There are more image slices available than image volumes. Furthermore, [2-D]{} image analysis mimics clinical calculation of the Agatston calcium score that is performed in [2-D]{} axial slices: $$\textrm{Agatston Score} = {\sum_{S \in V}\sum_{l \in S}{A_l w_l\frac{i_S}{t_S}}}\,.$$ where $l$ is a 2-D CAC lesion in a slice $S$ of a CT volume $V$. $A_l$ is the area of the lesion. The weighted intensity $w_l$ is based on the maximum radio-density in HU of a 2-D lesion in the following manner: 1 = \[130, 200), 2 = \[200, 300), 3 = \[300, 400), and 4 = $\geq$400. The Agatston score is corrected when image slices are overlapping, thus when slice increment $i_S$ is not equal to slice thickness $t_S$ [@ohnesorg2002].
Agatston scores are dependent on the CAC lesion area. Given that input images have different voxels sizes, we chose to simplify the prediction task by determining a pseudo-Agatston score. This score is obtained by cancelling out the axial pixel dimensions, the slice increment, and the slice thickness of the original Agatston score. The resulting target is the product of the number of voxels in a lesion $n_l$, the predicted slice scaling factor $s_{xy}$, and the weighted intensity $w_l$: $$\textrm{Pseudo-Agatston Score} = \sum_{l \in S} n_l \cdot s_{xy} \cdot w_l\,.$$ The calcium scoring ConvNet uses an efficient architecture that is listed in Table \[tab:convnetdesigns\]. It analyzes random image croppings of 224$\times$224 pixels during training and center croppings during application. It has alternating layers of 3$\times$3 convolutions and $2\times2$ max pooling, followed by two fully connected layers, and an output layer of one node. Throughout the network batch normalization [@ioffe2015] is used and exponential linear units are used for activation[@clevert2016]. The final output node has a linear output to facilitate continuous prediction. However, given that clinically used CVD risk categories are exponentially increasing, the task of the calcium scoring ConvNet was modified to learn a log-transform of the pseudo-Agatston score: $$L = |\hat{y} - \ln(y + 1)|\,,$$ where $\hat{y}$ is the predicted score, and $y$ is the reference pseudo-Agatston score. The log-transform induces relatively high penalties for erroneous low calcium score predictions, and relatively low penalties for erroneous high calcium score predictions. Consequently, higher precision is forced for lower calcium burden, which is favorable for CVD risk stratification. During application of the calcium scoring ConvNet, the predicted outputs are converted to the original Agatston scores.
Decision feedback
-----------------
By employing regression of calcium scores, we circumvent time-consuming intermediate segmentation. On the other hand, it may be desirable to visualize regions in image slice that contributed to the calcium score. Inspired by the study of Zeiler and Fergus [@zeilerfergus2014], we provide such visualization by using a deConvNet. The deConvNet uses the same operations of filtering and pooling as a ConvNet, but in reverse order from output to input. The reverse operations map the activities back to the input pixel space, and it shows which input patterns originally contributed to the activations in the feature maps. To obtain a smooth visual attention heatmap, the deConvNet is applied until the third convolutional layer, by taking the absolute value per feature of this layer, and by summing these features along the feature map dimension to get 2-D matrix. Using third order interpolation we obtain a smooth map that can be superpositioned on the image slice as a heatmap. This resulting heatmap visualizes attention by highlighting the regions that contributed to the Agatston score.
Evaluation
==========
Automatically predicted per-subject Agatston scores were compared with manually determined reference scores. Evaluations were performed on the hold-out test sets which were not used during method development. Two-way mixed intra-class correlation coefficient (ICC) for absolute agreement was computed and Bland-Altman analysis was performed to evaluate bias between predicted and reference Agatston scores. In addition, for each subject, CVD risk category was determined based on the Agatston score as defined in section \[sec:refstandard\]. Agreement between predicted and reference CVD risk categories was determined using accuracy and Cohen’s linearly weighted kappa ($\kappa$).
Experiments and results
=======================
In this section we evaluate the atlas-registration ConvNet, the calcium scoring ConvNet, and the quality of decision feedback. In addition, we will evaluate whether the calcium scoring ConvNet requires to be trained on all data, or whether it can be trained on one dataset and applied to the other. Finally, we will compare state-of-the-art automatic calcium scoring methods with the proposed method. All experiments were performed with Theano [@theano2016], Lasagne [@lasagne2015], and OpenCV [@opencv] on an Intel Xeon E5-1620 3.60GHz CPU with an NVIDIA Titan X GPU.
Atlas-registration ConvNet
--------------------------
-- -- -- -- --
-- -- -- -- --
Figure \[fig:atlas:atlas1\] shows the initial atlas image that was created by aligning all cardiac training images using their geometric centroids. We chose the median dimensions and voxels sizes of all the cardiac training images define the atlas image space. The atlas can be iteratively refined, but given the constraints of the global registration model used here, only one update was sufficient. The final atlas image, shown in Figure \[fig:atlas:atlas2\], was used to train the atlas-registration ConvNets for cardiac and chest CT alignment. Thus, in total three ConvNet instances were trained: one to create an atlas image, one for cardiac CT alignment, and one for chest CT alignment. All ConvNets were trained in 15,000 iterations with mini-batches containing 32 randomly selected images. Training took about 40 hours per ConvNet. Adam [@kingma2014] was used with a learning rate of 0.001 for mini-batch gradient descent. To illustrate performance of the atlas-registration ConvNets, Figure \[fig:atlas\] shows images before and after registration. Figure \[fig:atlas:cardiac1\] shows the average image of the 530 cardiac CT images from the test set before registration and Figure \[fig:atlas:cardiac2\] shows these images after registration. Similarly, Figure \[fig:atlas:chest1\] shows an average image of the 506 chest CTs before registration and Figure \[fig:atlas:chest2\] shows these after registration. Note the similarity of the registered image with the refined atlas image shown in Figure \[fig:atlas:atlas2\].
Quantitative evaluation of registration results revealed that registration erroneously cropped CAC out of the selected slices. Between one and four image slices containing CAC were not selected in three cardiac CTs and three chest CTs. Upon closer inspection, two of the chest CTs had calcifications in the aortic arch and descending aorta incorrectly labeled as CAC in the reference, thereby affecting CVD risk categorization. Nevertheless, these annotations were left uncorrected in further analysis to facilitate a fair comparison with previously developed methods. The registration errors did not have an adverse effect on CVD risk categorization in the other cases.
Calcium scoring ConvNet {#sec:calciumscoringconvnet}
-----------------------
The calcium scoring ConvNet was trained in 150,000 iterations using Adam [@kingma2014]. Training took 21 hours with 100 image slices per mini-batch randomly selected from the registered image slices taken from the cardiac and chest CT training sets. High imbalance between the minority of slices with a calcium score and the majority of slices with zero calcium score prevented convergence during ConvNet training. To ensure convergence, the amount of image slices with CAC (Agatston score $>0$) and without CAC (Agatston score $=0$) were balanced during training. To prevent bias, training continued on the full imbalanced training set after 10,000 iterations. Additionally, we ensured stable convergence by decreasing the learning rate to 10% of its previous value every 50,000 iterations.
\
-- -- --------- -------- -------- -------- --------
**259** 0 1 0 0
9 **36** 4 0 0
2 3 **82** 2 0
0 1 2 **65** 2
0 0 0 11 **51**
-- -- --------- -------- -------- -------- --------
: Confusion matrices showing agreement in CVD risk categorization based on the total Agatston scores: : very low $<1$, : low $[1, 10)$, : moderate $[10, 100)$, : moderately high $[100, 400)$, : high $\geq400$. The method is evaluated separately on the test sets of cardiac CTs (left) and chest CTs (right). The corresponding linearly weighted $\kappa$ is shown below the confusion matrices.[]{data-label="fig:confusion:fullset"}
-- -- --------- -------- -------- -------- ---------
**118** 6 4 0 0
8 **29** 5 0 0
3 8 **85** 3 0
1 1 7 **99** 4
0 0 0 3 **122**
-- -- --------- -------- -------- -------- ---------
: Confusion matrices showing agreement in CVD risk categorization based on the total Agatston scores: : very low $<1$, : low $[1, 10)$, : moderate $[10, 100)$, : moderately high $[100, 400)$, : high $\geq400$. The method is evaluated separately on the test sets of cardiac CTs (left) and chest CTs (right). The corresponding linearly weighted $\kappa$ is shown below the confusion matrices.[]{data-label="fig:confusion:fullset"}
After training, the test sets were used to evaluate the calcium scoring ConvNet. Per-subject scores show high intraclass correlation coefficients (ICC); the ICC for cardiac CT and chest CT were both 0.98 with 95% confidence intervals of 0.98 to 0.99. Slight positive bias in cardiac and chest CT is visualized with the Bland-Altman plots shown in Figure \[fig:blandaltman\]. This was mainly caused by overestimations of the higher Agatston scores. However, this was not noticeable in CVD risk stratification. Table \[fig:confusion:fullset\] shows confusion matrices of predicted risk categories vs. the manual reference standard. In cardiac CT calcium scoring only four scans were two categories off, and in chest CT calcium scoring eight scans were two categories off. The scan that was three categories off was a scan with incorrectly annotated aorta calcium, as discussed in the previous section. Nonetheless, overall agreement was *almost perfect* [@mchugh2012] with Cohen’s linearly weighted $\kappa$s of 0.95 in cardiac CT and 0.93 in chest CT. Accuracy in CVD risk categorization was 0.93 for cardiac CT and 0.90 for chest CT. Because efficient network architectures are used, the method is able to achieve high speed when used on a single CPU core: within 5s a score for cardiac CT is obtained and within 11s a score for chest CT is obtained. When using a GPU, calcium scoring can be performed in real-time. Including image registration and image resampling, a calcium score for cardiac CT is obtained in less than 0.15s and for chest CT in less than 0.30s.
Decision feedback
-----------------
Decision feedback visualizes attention of the calcium scoring ConvNet. This feedback informs and end-user about the regions that contributed to the calcium score. Figure \[fig:feedback\] shows examples of such feedback. The feedback helps an expert to quickly navigate and evaluate the image slices containing CAC.
\
\
\
We propose visual feedback as an optional qualitative tool, but we have performed a quantitative analysis to provide insight in its accuracy. To obtain quantitative results we analyzed heatmaps for slices with predicted calcium scores. The heatmaps were warped to the original image spaces by using the inverse transformation matrices. The values of the heatmaps were scaled between 0 and 1 to mimic probability maps for CAC candidate voxels. CAC candidates were defined as high density 26-connected voxels with a volume between 1.5 and 1,500mm^3^[@wolterink2015]. For evaluation of these maps we performed precision-recall analysis (Figure \[fig:precisionrecall\]). We have defined an optimal threshold by selecting the maximum F1 (i.e. Dice) score on the validation set. Table \[tab:feedback\_evaluation\] shows the obtained scores using the selected threshold on the test sets. The results show that detection performance is very accurate on the validation set as well as the test set.
![Precision recall curve of CAC segmentation using the obtained visual feedback heatmaps. The analysis is performed on the validation set to obtain an optimal threshold for evaluation. Optimal F1 score was 0.81 at a threshold of 0.27. Final results for quantitative evaluation of visualization feedback are shown in Table \[tab:feedback\_evaluation\].[]{data-label="fig:precisionrecall"}](precision_recall.pdf){width=".8\linewidth"}
Additionally, decision feedback aided our analysis by clarifying incorrect calcium scores. Decision feedback revealed that the largest CVD miscategorizations were not caused by incorrect quantification but by incorrect recognition of CAC. Figure \[fig:incorrect\] shows six examples of the largest miscategorizations made by the calcium scoring ConvNet. The majority of errors were made in identification of calcifications near the coronary artery ostia. Calcifications near the ostia can be partly in the aorta and partly in the coronary artery. These calcifications are difficult to distinguish, especially when no information of neighboring slices is available.
Cardiac CT Chest CT
----------------- ------------ ----------
Precision 0.77 0.78
Recall 0.85 0.86
Accuracy 0.99 0.99
F1 (Dice) score 0.81 0.82
: Quantitative evaluation of visual feedback. Evaluation was performed segmenting CAC lesions with the visualization feedback. An optimal threshold was selected using precision recall analysis on the validation data shown in Figure \[fig:precisionrecall\]. Final results show that visualization by the heatmap is is as accurate on the validation as on the test set.[]{data-label="tab:feedback_evaluation"}
Influence of training data and registration
-------------------------------------------
For clinical application it would be useful to investigate whether the method needs training data from both datasets or if data from one set would suffice, and we investigated the influence of atlas-registration is required. Thus, we performed experiments using different combinations of training data with and without atlas-registration, as listed in Table \[tab:allexperiments\]. The calcium scoring ConvNets were trained with either cardiac CT images, chest CT images, or a combination thereof. To balance cardiac and chest CT data, a subset of chest CT images was created by taking images from 237 randomly selected subjects and by removing every other slice in the chest CT images. Additionally, the histograms shown in Figure \[fig:histograms\] provide insight in the distribution of calcium amount in the training data. Note that the chest CT subset has a very similar distribution compared to the cardiac CT training set.
![Histograms of per slice Agatston scores of the registered training datasets. Note that Agatston scores shown here are not corrected by factor $\frac{i_s}{t_s}$. Please see Section \[sec:cacscoremethod\] for application of this correction factor in the Agatston score.[]{data-label="fig:histograms"}](histogram.pdf){width="\columnwidth"}
The best performance was achieved using atlas-registration with a calcium scoring ConvNet trained on all cardiac and chest CT images. Lower scores are found when a calcium scoring ConvNet is only trained with cardiac CT or the subset of chest CTs. However, combining the two datasets increased the scores notably, giving a performance close to the ConvNet trained with all images. Furthermore, the results show that atlas-registration facilitated training on one type of data and high performance on the other: the ConvNet trained with the full set of chest CTs achieved a high performance on the cardiac CT test images that was very close to the best results.
-- -- --------------------------- ------- --------- -------------- ---------- ---------- ---------- ---------- ---------- ----------
Data CTs Slices Fraction CAC $\kappa$ Acc. ICC $\kappa$ Acc. ICC
Cardiac CT 237 10,468 10.4% 0.92 0.89 0.89 0.46 0.41 0.24
Chest CT 1,012 211,353 6.6% 0.48 0.59 0.24 0.91 0.86 0.93
Cardiac + Chest CT 1,239 221,821 6.7% 0.90 0.86 0.87 0.92 0.88 0.94
Cardiac CT 237 10,016 10.9% 0.92 0.88 0.97 0.86 0.79 0.90
Chest CT subset 237 11,716 14.8% 0.91 0.86 0.95 0.90 0.85 0.93
Cardiac + Chest CT subset 574 21,732 13.0% 0.94 0.92 **0.99** 0.91 0.88 0.97
Chest CT 1,012 100,379 13.8% 0.94 0.91 0.98 **0.93** 0.89 **0.98**
Cardiac + Chest CT 1,239 110,395 13.5% **0.95** **0.93** 0.98 **0.93** **0.90** **0.98**
-- -- --------------------------- ------- --------- -------------- ---------- ---------- ---------- ---------- ---------- ----------
Comparison with other methods
-----------------------------
Table \[tab:comparison\] shows a comparison with other state-of-the-art calcium scoring methods by Wolterink et al. [@wolterink2015] and Lessmann et al. [@lessmann2018] using the same datasets. The proposed method achieves similar performance compared to these methods, but it is hundreds of times faster. Even when ran on a single core of a CPU, the method achieves high speed. Additionally, we listed results from other direct calcium scoring methods by González et al. [@gonzalez2016] and Cano-Espinosa et al. [@cano2018] using chest CT data from the COPDGene study [@regan2010]. We provide similar performance metrics to give an indication, but please note that a direct comparison between these methods and ours was not possible.
[llc|cc|cc|cc|cc|cc]{} & & & & & &\
& Source & Number & ICC & $\rho$ & $\kappa$ & acc. & $\kappa$ & acc. & $\kappa$ & acc. & CPU & GPU\
\
Wolterink et al.[@wolterink2015] & UMCU & 530 & 0.96 & – & 0.95 & 0.91 & – & – & – & – & 20min & –\
Proposed method & UMCU & 530 & 0.97 & 0.99 & 0.95 & 0.93 & 0.95 & 0.96 & 0.94 & 0.93 & 5s & 0.15s\
\
Cano-Espinosa et al. [@cano2018] & COPDGene & 1,000 & – & 0.93 & – & – & – & – & 0.80 & 0.76 & – & –\
Lessmann et al. [@lessmann2018] & NLST & 506 & – & – & – & – & 0.91 & 0.91 & – & – & – & 7min\
Proposed method & NLST & 506 & 0.98 & 0.97 & 0.93 & 0.90 & 0.92 & 0.91 & 0.93 & 0.90 & 11s & 0.30s\
Performance on orCaScore data
-----------------------------
We evaluated our method on data from the orCaScore challenge [@wolterink2016orcascore]. This challenge provides data to evaluate a method for coronary calcium scoring. The data consists of non-contrast enhanced ECG-triggered cardiac CT acquired on CT scanners from four different vendors from four different hospitals. Training data is provided, but we evaluated our method on the test set of 40 patients without retraining. Table \[tab:orcascoreresult\] shows the obtained confusion matrix and lists the results of dedicated cardiac CT calcium scoring methods that competed in the challenge. Given that our method does not differentiate between location of CAC, we only provide total calcium scoring results.
-- -- ------- -------- ------- --------
**8** 0 0 0
0 **12** 0 0
0 0 **8** 0
0 0 1 **11**
-- -- ------- -------- ------- --------
: Results of the proposed method on orCaScore challenge data. Left: The confusion matrix shows agreement in CVD risk categorization based on the total Agatston scores: : $0$, : $[1, 100)$, : $[100, 300)$, :$>300$. The corresponding linearly weighted $\kappa$ is shown below the confusion matrix. Right: Comparison with other methods evaluated in the challenge [@wolterink2016orcascore].[]{data-label="tab:orcascoreresult"}
[c|ccc]{}\
Method & $\kappa$ & Acc. & ICC\
A[@shahzad2013] &0.88&0.85&0.97\
B[@wolterink2016orcascore] &0.98&0.98&0.99\
C[@wolterink2016orcascore] &0.96&0.95&0.98\
D[@wolterink2016orcascore] &0.80&0.80&0.60\
E[@wolterink2015] &1.00&1.00&0.99\
Ours & 0.98 & 0.98 & 0.98\
Per-artery calcium scores
-------------------------
Routine coronary artery calcium scoring is typically performed per artery. Currently, only total coronary calcium scores are reported and used for CVD risk prediction. For research purposes, per-artery calcium scores might provide interesting additional information. Hence, we evaluated performance of the proposed method for per-artery calcium scoring, i.e. scoring in the the LAD, LCX, and RCA. We chose to combine CAC scores in the LM and LAD, since it is difficult, if not impossible, to differentiate them in chest CT scans. The direct scoring ConvNet was adapted by changing the number of output nodes from one to to three. Similar to the experiment described in Section \[sec:calciumscoringconvnet\], training started with a balanced set of image slices with and without calcium scores for the first 10,000 iterations and continued with the full set of image slices thereafter. Additionally, each mini-batch had at least three image slices containing each type of arterial calcification. Risk categories are clinically not defined for per-artery calcium scores, but they are obtained for total calcium scores by summation of per-artery scores. The results are listed in Table \[tab:perarteryscores\].
------------ ------ ------ ------ ---------- ------ ------
LAD LCX RCA $\kappa$ Acc. ICC
Cardiac CT 0.93 0.88 0.97 0.94 0.91 0.97
Chest CT 0.91 0.80 0.98 0.92 0.88 0.96
------------ ------ ------ ------ ---------- ------ ------
: Intraclass correlation coefficient (ICC) for per-artery calcium scores. Since CVD risk categories are not defined for per-artery scores, CVD risk categorization was evaluated with linearly weighted $\kappa$ and accuracy (Acc.) on the total calcium scores obtained by summation.[]{data-label="tab:perarteryscores"}
Discussion
==========
We have presented a method for automatic coronary calcium scoring in cardiac CT and chest CT. The method uses an atlas-registration ConvNet to align FOVs making input images alike. The atlas-registration ConvNet is trained for 3-D registration, but its rigid model is constrained to enable 2-D slice selection and 2-D image warping. Selected and warped input image slices are presented to a calcium scoring ConvNet that directly predicts the Agatston score in these slices. The method circumvents time-consuming CAC segmentation. To provide decision feedback, a visual attention heatmap can be generated that shows the regions in an image contributing to the calcium score. The method achieves excellent agreement for calcium score prediction for CVD risk categorization compared to manual calcium scoring. The method achieves similar performance compared to state-of-the-art methods, but achieves it hundreds of times faster.
In preliminary experiments we found that only a small ConvNet architecture was able to learn direct calcium scoring. Large ConvNet architectures architectures were unstable and failed to converge during training. By limiting the degrees of freedom of a ConvNet, i.e. by using a small architecture, we were able to train a ConvNet that learned to differentiate coronary calcification from other types of calcification e.g. aorta calcification, pericardium calcification, and heart valve calcification.
To simplify the problem we extracted bounding boxes around the heart in our preliminary work [@devos2017rsna; @devos2017arxiv]. However, this was a supervised method that classified presence of the heart in image slices. In case of noisy images, consecutive image slices could have discontinuous predictions. Discontinuous predictions resulted in an incorrect bounding box extracting a partial heart. For atlas-registration used in our current work this is not an issue.
The atlas-registration ConvNets were highly successful in pre-alignment of input CTs, i.e. in slice selection and image warping. Only 4 out of 1,036 test images had slices containing CAC that were missed by erroneous slice selection. Erroneous slice selection was likely caused by incorrect focus of the atlas-registration ConvNet on high contrast areas like the diaphragm. A mask drawn around the heart might steer focus of the ConvNet and might increase registration performance. Alternatively, a simple adjustment could be made by padding slice selection with some slices. Nonetheless, the errors caused by registration had negligible impact on calcium scoring and did not affect CVD risk categorization. Calcium scoring is better with atlas-registration than without it. Moreover, registration allows training and application of direct calcium scoring on datasets with different FOVs.
In general accuracy of predicted Agatston scores was high. Although Bland-Altman analysis showed that the method underestimated subjects with high Agatston scores. In fact, this was by design, because the method estimates a log transformed Agatston score, which induces relatively low precision for higher scores, and high precision for lower scores. Because the clinically used CVD risk categories are based on exponentially increasing Agatston scores, it is obviously more important to differentiate between subjects at low to moderate risk, than to differentiate between subjects at high risk. Thus, we imposed this higher precision on lower Agatston scores. Still, the largest CVD miscategorizations were found in the lower risk categories. Miscategorization was predominantly caused by incorrect identification of CAC and aortic calcifications near the coronary artery ostia. Even manual classification of these calcifications can be very difficult when they spread from the aorta through an ostium into the coronary artery. It often involves inspecting multiple adjacent slices in 3-D. Thus, performance of the method might be improve by exploiting additional 3-D information in future work. Additionaly, performance might improve by increasing input image resolution. The current resolution was chosen based on the majority of chest CT images, being roughly half the resolution of cardiac CTs. Nevertheless, even though all cardiac CTs were downsampled a high performance was obtained in these CTs.
The proposed method shows near perfect agreement in CVD risk categorization compared to manual calcium scoring, even when trained with a relatively low number of scans from a single dataset. Interestingly, training on one type of data allowed the model to be applied to the other type of data without any modifications or transfer learning. However, we found that a model trained on only chest CT led to better results than a model trained only on cardiac CT. One potential reason for this may be the distribution of CAC in the datasets: the population of ex-heavy smokers typically have more CAC [@jacobs2010] than the population undergoing calcium scoring cardiac CT. However, Figure \[fig:histograms\] shows that the distribution of CAC in equally sized datasets of cardiac CT and chest CT is similar. An alternative reason could be the presence of motion artifacts, which are nearly absent in ECG-synchronized cardiac CT, but abundant in non-ECG-synchronized chest CT. Therefore, a model trained on chest CT may be more robust to such artifacts. While our experiments indicated that a ConvNet trained on the cardiac and chest CT datasets supplement each other, a calcium scoring ConvNet trained with only chest CTs almost matched performance of the best performing ConvNet. Additionally, we have shown that the method obtained near perfect CVD risk categorization results on cardiac CTs from the orCaScore challenge. The method did not require retraining on representative data from the different hospitals and vendors. Having a single system that can handle potentially any CT scan that visualizes the heart would be very practical in a routine radiology setting. In future work we will investigate whether the method could be readily applied on other types CTs, without requiring retraining or fine-tuning.
Additionally, we have shown that the method can provide per-artery calcium scores. While this is not required for CVD risk categorization, it might be interesting for clinical research. In terms of ICC [@koo2016], per-artery calcium scoring achieved *good reliability* ($>0.75$) in the LCX, and *excellent reliability* ($>0.90$) in the LAD and the RCA. In addition, determination of CVD risk using combined per-artery scores led to *almost perfect agreement* (${\kappa>0.90}$) [@mchugh2012]. Nevertheless, performance was slightly better when total calcium was directly determined. This difference in performance may be a consequence of increased complexity of the per-artery scoring task while using the same number of samples for training.
The proposed method can achieve a calcium score hundreds of times faster than previously proposed methods. This is mainly due to one-shot (i.e. non-iterative) registration, and direct quantification using regression. The direct calcium scoring method circumvents time-consuming intermediate segmentation. The method might also be suitable for e.g. determination of volume, (pseudo-)mass, or number of CAC; and for quantification of other lesions or diverse anatomical structures. However, the benefit of using a segmentation approach over direct scoring is that it provides immediate insight to the end-user. We mitigate this shortcoming of direct scoring, by providing decision feedback with a visual attention heatmap. In this way valuable feedback is still provided whenever an end-user requires it.
Conclusion
==========
We have presented an automatic method for direct calcium scoring in cardiac CT and chest CT. The method employs two ConvNets, one for atlas-registration to align the FOV of input images to an atlas image made from cardiac CTs and one for direct calcium scoring of input image slices using regression. The method achieves robust and accurate predictions of calcium scores in real-time. By providing visual feedback, insight is given in the decision process, making it readily implementable in a clinical and research settings.
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[^1]: Copyright © 2019 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. Bob D. de Vos, Jelmer M. Wolterink, Nikolas Lessmann, and Ivana Išgum are with the Image Sciences Institute of the University Medical Center Utrecht and Utrecht University, Utrecht, The Netherlands. Tim Leiner and Pim A. de Jong are with the Department of Radiology, University Medical Center Utrecht and Utrecht University, Utrecht, the Netherlands. This work is part of the research programme ImaGene with project number 12726, which is partly financed by the Netherlands Organisation for Scientific Research (NWO). The authors thank the National Cancer Institute for access to NCI’s data collected by the National Lung Screening Trial. The statements contained herein are solely those of the authors and do not represent or imply concurrence or endorsement by NCI.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this article, we classify all involutions on $S^6$ with 3-dimensional fixed point set. In particular, we discuss the relation between the classification of involutions with fixed point set a knotted 3-sphere and the classification of free involutions on homotopy ${\mathbb C}{{\mathrm{P}}}^3$’s.'
author:
- Martin Olbermann
title: 'Involutions on $S^6$ with 3-dimensional fixed point set'
---
Introduction
============
As a general assumption, we are interested in smooth involutions on connected, closed, smooth manifolds.
The study of group actions on very simple manifolds such as disks, spheres or Euclidean spaces has been a very active subject. In his MathSciNet review of [@Paw], Masuda notes: “Representations of groups provide examples of group actions on Euclidean spaces, disks or spheres, and an important natural question in transformation groups is to what extent arbitrary actions on those spaces resemble actions provided by representations." The first highlight theorems are due to P.A. Smith. For an involution on a sphere, Smith proved that the fixed point set is a ${\mathbb Z}_2$-homology sphere. There are various converses to this theorem which can be found in the literature, e.g. [@DW; @Paw] and references therein. However, the following theorem seems to be new:
[@confree] Every ${\mathbb Z}_2$-homology 3-sphere is the fixed point set of an involution on $S^6$.
Moreover, the method used in [@confree] generalizes to a classification of these involutions, which is the subject of the present paper. (A shorter proof of the above existence theorem can be given using Dovermann’s equivariant surgery approach along the lines of [@Sch] and [@DMS].)
Recently, a class of involutions called [*[conjugations]{}*]{} was defined in [@HHP] and various aspects of conjugations were studied in [@FP; @OlbThesis; @HHo; @HH]. Conjugations $\tau$ on topological spaces $X$ have the property that the fixed point set has ${\mathbb Z}_2$-cohomology ring isomorphic to the ${\mathbb Z}_2$-cohomology ring of $X$, with the slight difference that all degrees are divided by two. In the case of smooth involutions on (positive-dimensional) spheres, the conjugations are exactly the involutions on even-dimensional spheres with half-dimensional fixed point set. In dimension 2, the Schönflies theorem gives a classification: every conjugation is conjugate to the reflection of $S^2$ at the equator. In dimension 4, work of Gordon and Sumners shows that there are infinitely many non-equivalent conjugations on $S^4$. Hambleton and Hausmann recently reduced the study of such involutions to a non-equivariant four-dimensional knot theory question [@HH]. Knot theory of $k$-spheres in $S^{2k}$ is easier for $k>2$, so that the study of conjugations gets a different flavor for $k>2$.
The study of free involutions on simply-connected spin manifolds with the same homology groups as ${\mathbb C}{{\mathrm{P}}}^3$ was motivated by the question whether it is possible to define an “equivariant Montgomery-Yang correspondence". After Haefliger [@Hae] proved that the group $C^3_3$ of knotted $S^3$’s in $S^6$ (isotopy classes of smooth embeddings, or equivalently diffeomorphism classes relative to $S^3$) is isomorphic to ${\mathbb Z}$, Montgomery and Yang showed [@MY] that there is a natural bijection between $C^3_3$ and the set of diffeomorphism classes of homotopy ${\mathbb C}{{\mathrm{P}}}^3$’s. Wall’s classification of all simply-connected spin manifolds with the same homology groups as ${\mathbb C}{{\mathrm{P}}}^3$ [@Wal] also uses the bijection between diffeomorphism classes of such manifolds, together with a basis of $H^2$, and isotopy classes of framed $S^3$-knots in $S^6$. In the equivariant setting, Li and Lü [@LL] show that the existence of a free involution on a homotopy ${\mathbb C}{{\mathrm{P}}}^3$ implies the existence of an involution on $S^6$ with fixed point set the corresponding knotted $S^3$. Similarly, in our approach, the same surgery arguments apply in both cases. However, it is not possible to produce a nice bijection on the set of equivariant diffeomorphism classes of these, as claimed in [@Su]. Our classification results are:
\[Mthm\] Let $M$ be a smooth closed simply-connected spin manifold with $H_2(M)={\mathbb Z}$, and $H_3(M)=0$. Let $x\in H^2(M;{\mathbb Z})$ be a generator. We assume that $M\not \cong S^2\times S^4$.
- If $\left\langle \frac{p_1(M)x-4x^3}{24},[M]\right\rangle$ is odd, there exists no free involution on $M$.
- If $\langle x^3,[M]\rangle$ is odd, and $\left\langle \frac{p_1(M)x-4x^3}{24},[M]\right\rangle$ is even, there exist up to diffeomorphism exactly two free involutions on $M$.
- If $\langle x^3,[M]\rangle$ is even, and $\left\langle \frac{p_1(M)x-4x^3}{24},[M]\right\rangle$ is even, there exist up to diffeomorphism exactly five free involutions on $M$.
If $M\cong S^2\times S^4$, then the same classification holds for orientation-reversing involutions which act by $-1$ on $H^2(M)$.
In the case of homotopy ${\mathbb C}P^3$’s the classification of free involutions was given by Petrie [@Pet] (whose result contains a mistake, corrected by Dovermann, Masuda and Schultz), and Su [@Su]. Li and Su (unpublished) give the answer to the existence question in the case of odd $\langle x^3,[M]\rangle$. Our method reproves all these results in a different way, extends to a larger class of manifolds, and gives classification results in all cases.
\[thm13\] For every even element of $C^3_3$ there are (up to equivariant diffeomorphism relative $S^3$) exactly four involutions (conjugations) on $S^6$ with the knot as fixed point set. For every odd element of $C^3_3$, there is no involution on $S^6$ with the knot as fixed point set.
The new part of the theorem is the classification of these involutions. Li and Lü proved that the existence of an involution with fixed point set a given knot is equivalent to the existence of a free involution on the corresponding homotopy ${\mathbb C}{{\mathrm{P}}}^3$ under the Montgomery-Yang correspondence, so that together with Su’s work mentioned above, the existence question was answered.
The case of involutions on $S^6$ with fixed point set $S^3$ is especially interesting since, given another 6-manifold with an involution that has a 3-dimensional fixed point set, (equivariant) connected sum gives a possibly different involution on the same 6-manifold with same fixed point set. (This is in analogy with the fact that connected sum with a homotopy sphere gives a possibly new smooth structure on the same underlying topological manifold.) Similarly, connected sum with a conjugation on $S^6$ with fixed point set different from $S^3$ gives a new conjugation on the same 6-manifold, with different fixed point space.
Our main result is a classification of smooth involutions on $S^6$ with arbitrary three-dimensional fixed point set, using a recent classification of embeddings of closed oriented connected 3-manifolds into $S^6$ by A. Skopenkov [@Sko]. Skopenkov proves that for a 3-dimensional ${\mathbb Z}_2$-homology sphere $M$ the set of isotopy classes of embeddings $i:M\to S^6$ has a free action by $C_3^3$ and the orbits are in canonical bijection with $H_1(M)$.
\[embcl\] Let $M$ be a ${\mathbb Z}_2$-homology sphere of dimension 3. The set of isotopy classes of embeddings $i:M\to S^6$ which are the fixed point sets of involutions (conjugations) is contained in the orbit corresponding to $0\in H_1(M)$. Moreover, it is acted upon freely and transitively by $2{\mathbb Z}\subseteq {\mathbb Z}\cong C_3^3$. There are up to equivariant diffeomorphism relative to $i$ exactly four such conjugations for every such $i$.
In our case, we can also classify involutions without additional identification of the fixed point set with a given 3-manifold. However, we consider involutions together with an orientation of their fixed point sets, and the equivalence relation is equivariant diffeomorphism which respects the orientations of both $S^6$ and the fixed point set. Equipping the involution with an orientation of the fixed point set is necessary in order to perform connected sums. Thus it is more natural to determine this more structured set of equivariant diffeomorphism classes $Inv_{M}(S^6)$.
Since the action of the mapping class group of $M$ on the above set of equivariant diffeomorphism classes of involutions relative to the fixed point set $M$ is trivial, we get the same classification as in theorem \[embcl\]:
\[invcl\] Conjugations up to conjugation (preserving orientations) with fixed point set of a fixed oriented diffeomorphism type of ${\mathbb Z}_2$-homology 3-spheres are in bijection with ${\mathbb Z}\oplus {\mathbb Z}_4$. Under connected sum, $Inv_{S^3}(S^6)\cong {\mathbb Z}\oplus {\mathbb Z}_4$ is an isomorphism of groups, and $Inv_{S^3}(S^6)$ acts freely and transitively on $Inv_{M}(S^6)$ for every ${\mathbb Z}_2$-homology 3-sphere $M$.
In principle, using the machinery described in [@KrM] it is also possible to prove classification results for conjugations on $X^6$ with fixed point set $M^3$ in other cases as $X=S^6$. (However, the argument we use to show the surgery obstruction is trivial does not extend to the case of free involutions on other manifolds.)
One would compute the set of equivariant diffeomorphism classes of the involution together with an identification of the fixed point set with a prescribed 3-manifold $M$, i.e. the set $$Emb_{{\mathbb Z}_2}(M\to X)=\{ f:(M,id)\to (X,\tau) \} / \sim$$ where $f$ is an inclusion onto the fixed point set of $\tau$, and $f:(M,id)\to (X,\tau)$ is identified with $f':(M,id)\to (X',\tau')$ if there is an equivariant diffeomorphism $\phi:(X,\tau)\to
(X',\tau')$ making the obvious triangle commute. One of the difficulties to overcome is “due" to Wall’s classification: in order to determine which of the resulting 6-manifolds are diffeomorphic to $X$, one would need a good understanding of the isomorphism classes of the algebraic invariants (trilinear forms), and this problem seems to be very hard in general.
[**[Acknowledgements.]{}**]{} I would like to thank Diarmuid Crowley, Jean-Claude Hausmann, Matthias Kreck and Arturo Prat-Waldron for many helpful discussions and remarks. Special thanks to Yang Su, whose article [@Su] was the origin of this paper, and who shared with me the so far unpublished notes on a generalization of [@Su] due to himself and Banghi Li.
Preliminaries
=============
Modified surgery
----------------
We will use Kreck’s modified surgery theory [@Kre; @KrM] which is also suited to give classification results. By the equivariant tubular neighbourhood theorem, we can write $S^6=M\times D^3 \cup_\partial V$, where $V$ is a manifold with boundary $M\times S^2$ equipped with a free involution, which restricts to $(id,-id)$ on the boundary. (It is not hard to see that the normal bundle of $M$ in $S^6$ is trivial. See [@OlbThesis] for a proof.)
A classification of manifolds $V$ with free involutions $\tau$ up to equivariant diffeomorphism is the same as a classification of the quotient manifolds $W=V/\tau$ up to diffeomorphism. This is what modified surgery theory will give us.
We first determine the normal 2-type of the manifolds $W$ under consideration. The normal 2-type $B$ of a 6-manifold (see the precise definition \[normal2type\]) is a fibration $B\to BO$. It carries roughly the information of a 3-skeleton of the manifold together with the restriction of the normal bundle to this 3-skeleton. After computing the bordism group of manifolds with normal $B$-structure, we show that in every bordism class there exists a manifold (together with a map to $B$) which qualifies as the $W$ above. Moreover, we show that given two normally $B$-bordant manifolds $W$ as above, the obstruction, which a priori lies in the complicated monoid $l_7({\mathbb Z}_2,-1)$, for the existence of an s-cobordism (i.e. a diffeomorphism) is zero. The diffeomorphism classification of the manifolds $W$ under consideration is given by the set of orbits of the action of the group of fiber homotopy self-equivalences $B\to BO$.
Conjugations on manifolds
-------------------------
This section explains what conjugation spaces are and shows that the smooth involutions on $S^{2n}$ with $n$-dimensional fixed point set are exactly the smooth conjugations. The rest of the paper does not depend on the material in this section.
A conjugation on a topological space $X$ is an involution $\tau:X\to X$, which we consider as an action of the group ${\mathbb Z}_2\cong C= \{id, \tau\}$ on $X$, and which satisfies the following cohomological pattern: We denote the Borel equivariant cohomology of $X$ by $H^*_C(X;{\mathbb Z}_2)$. It is a module over $H^*_C(pt;{\mathbb Z}_2)={\mathbb Z}_2[u]$. The restriction maps in equivariant cohomology are denoted by $\rho: H^*_C(X;{\mathbb Z}_2)\to H^*(X;{\mathbb Z}_2)$ and $r: H^*_C(X;{\mathbb Z}_2)\to H^*_C(X^\tau;{\mathbb Z}_2)\cong H^*(X^\tau;{\mathbb Z}_2)[u]$.
[@HHP]\[conjsp\] $X$ is a conjugation space if
- $H^{odd}(X;{\mathbb Z}_2)=0$,
- there exists a (ring) isomorphism $\kappa: H^{2*}(X;{\mathbb Z}_2)\to H^*(X^\tau;{\mathbb Z}_2)$
- and a (multiplicative) section $\sigma: H^*(X;{\mathbb Z}_2)\to H^*_C(X;{\mathbb Z}_2)$ of $\rho$
- such that the so-called conjugation equation holds: $$r\sigma(x)=\kappa(x)u^k + \text{ terms of lower degree in }u.$$
One does not need to require that $\kappa$ and $\sigma$ be ring homomorphisms, it is a consequence of the definition. Moreover, the “structure maps" $\kappa$ and $\sigma$ are unique, and natural with respect to equivariant maps between conjugation spaces.
There are many examples of such conjugations: complex conjugation on the projective space ${\mathbb C}{{\mathrm{P}}}^n$ and on complex Grassmannians, natural involutions on smooth toric manifolds [@DJ] and on polygon spaces [@HK]. Every cell complex with the property that each cell is a unit disk in ${\mathbb C}^n$ with complex conjugation, and with equivariant attaching maps, is a conjugation space. Coadjoint orbits of semi-simple Lie groups with the Chevalley involution are conjugation spaces. Moreover, there are various constructions of new conjugation spaces out of other conjugation spaces.
A conjugation manifold is a conjugation space consisting of a smooth manifold $X$ with a smooth involution $\tau$. As a consequence, a closed conjugation manifold $X$ must be even-dimensional, say of dimension $2n$, and $M$ is of dimension $n$.
In [@OlbThesis] we proved that it is possible to give a definition of conjugation spaces without the non-geometric maps $\kappa$ and $\sigma$, which is moreover well-adapted to the case of conjugation manifolds, where the fixed point set has an equivariant tubular neighbourhood.
Every smooth involution on $S^{2n}$ with (non-empty) n-dimensional fixed point set is a conjugation.
Proof: Let $pt\in S^{2n}$ be a fixed point of the involution. Then $pt$ is a conjugation space and $(S^{2n},pt)$ is a conjugation pair, by the same proof as in Example 3.5 of [@HHP]. Then the extension property for triples, Prop. 4.1 in [@HHP], shows that $S^{2n}$ is a conjugation space. (A slightly different proof is given in [@HH].) [**q.e.d.**]{}
Free involutions on certain 6-manifolds
=======================================
We are considering smooth involutions on smooth closed simply-connected spin manifolds $M$ with $H_2(M)={\mathbb Z}$, and $H_3(M)=0$. The classification by Wall in [@Wal] also uses a generator $x\in H^2(M)$ and an orientation of $M$. Then pairs $(M,x)$ up to diffeomorphism are classified by the bordism class of $M\stackrel x\to {\mathbb C}{\mathrm{P}^{\infty}}\in \Omega_6^{Spin}({\mathbb C}{\mathrm{P}^{\infty}})\cong {\mathbb Z}^2$, and $(M,x)$ is mapped under the isomorphism to $$\left( \left\langle \frac{p_1(M)x-4x^3}{24},[M]\right\rangle , \langle x^3,[M]\rangle\right).$$ Both switching the sign of the generator of $H^2(M)$ and the orientation of $M$ induce multiplication with $-1$, so that diffeomorphism classes are in bijection with ${\mathbb Z}^2/-1$.
As observed in [@LL], the Lefschetz fixed point theorem implies:
If $\langle x^3,[M]\rangle$ is non-zero, then a free involution on $M$ must be orientation reversing, and act by $-1$ on $H^2(M)$.
If $\langle x^3,[M]\rangle=0$, we consider only orientation reversing free involutions which are $-1$ on $H^2(M)$. From Li and Su we learned that except for the case $M=S^2\times S^4$, these are all free involutions: If the involution acts by $-1$ on $H^4(M)$, the first Pontryagin class must be 0, and we use the classification. The remaining case is handled as above by the Lefschetz fixed point theorem.
Obviously, for $M=S^2\times S^4$, our classification of orientation reversing free involutions which are $-1$ on $H^2(M)$ does not include all free involutions.
The normal 2-type
-----------------
\[normal2type\] The normal 2-type of a compact manifold $N$ is a fibration $B_2(N)\to BO$ which is obtained as a Postnikov factorization of the stable normal bundle map $N\to BO$: There is a 3-connected map $N\to B_2(N)$, the fibration $B_2(N)\to BO$ is 3-coconnected (i.e. $\pi_i(BO,B_2(N))=0$ for $i>3$), and the composition is the stable normal bundle map. This determines $B_2(N)\to BO$ up to fiber homotopy equivalence.
Let $\tau$ be an involution on $M$ as above, and let $N=M/\tau$ be the the quotient space of the involution. The second space in a Postnikov tower for $N$ is either $P=({\mathbb C}{\mathrm{P}^{\infty}}\times S^\infty)/(c,-1)$, where $c$ is complex conjugation, or $Q=({\mathbb C}{\mathrm{P}^{\infty}}\times S^\infty)/(\tau,-1)$, where $\tau$ is fiberwise the antipodal involution on $S^2\to {\mathbb C}{\mathrm{P}^{\infty}}\to {\mathbb H}{\mathrm{P}^{\infty}}$.
Proof: The first space in the Postnikov tower is a $K({\mathbb Z}_2,1)$, and the second space is a $K({\mathbb Z},2)$ fibration over it, with $\pi_1$ acting nontrivially on $\pi_2$. Such fibrations are classified by their $k$-invariant in $H^3(K({\mathbb Z}_2,1);{\mathbb Z}_-)\cong {\mathbb Z}_2$. The spaces $P$ and $Q$ have the required properties, and they are not homotopy equivalent, as e.g. $H^2(P;{\mathbb Z}_2)\cong {\mathbb Z}_2^2$ and $H^2(Q;{\mathbb Z}_2)\cong {\mathbb Z}_2$. So they represent all isomorphism classes of fibrations. ($P$ has $k$-invariant 0, and $Q$ has nonzero $k$-invariant.) [**q.e.d.**]{}
\[cohQ\] The ${\mathbb Z}_2$ cohomology ring of $N$ is ${\mathbb Z}_2[q,t]/\langle t^3,q^2\rangle$, where $deg(q)=4, deg(t)=1$.
Proof: (This was proved in [@Su] in the case of homotopy ${\mathbb C}{{\mathrm{P}}}^3$’s, and we generalize his proof.) We consider the Serre spectral sequence of the fibration $M\to N \to {\mathbb R}{\mathrm{P}^{\infty}}$, with ${\mathbb Z}_2$ (and also sometimes with integral) coefficients. The first case is that $d_3:E_3^{0,2}\to E_3^{3,0}$ is non-trivial. Then by multiplicativity the $E_4$-term has exactly one ${\mathbb Z}_2$ in $E_4^{p,0}$ and $E_4^{p,4}$ for each $p=0,1,2$. There are no further differentials, and we get the above cohomology ring. The second case is that $d_3:E_3^{0,2}\to E_3^{3,0}$ is trivial. We will show that this leads to a contradiction. By multiplicativity, also $d_3:E_3^{0,6}\to E_3^{3,4}$ is trivial. If we remember that we need the limit to have no cohomology in degrees $>6$, then we see in the sequence with integral coefficients that there must be a nontrivial $d_3$-differential between the fourth and second line. Then the same must hold for the ${\mathbb Z}_2$ coefficient sequence. And we get a $d_7$-differential from the sixth to the zeroth line. The $E_\infty$-term has exactly one ${\mathbb Z}_2$ in $E_\infty^{p,0}$ for $p=0, \dots , 6$ and $E_\infty^{p,2}$ for $p=0,1,2$.
This gives a cohomology ring with a generator $t\in H^1(N;{\mathbb Z}_2)$, and another generator $x\in H^2(N;{\mathbb Z}_2)$. Since in this case $H^2(N;{\mathbb Z}_2)\cong {\mathbb Z}_2^2$, the second Postnikov space must be $P$, and we can choose $x\in H^2(N;{\mathbb Z}_2)$ coming from $P$; we choose the class in $H^2(P;{\mathbb Z}_2)$ which maps to 0 under a section of $P\to {\mathbb R}{\mathrm{P}^{\infty}}$. Note that $x$ maps nontrivially to $H^2(M;{\mathbb Z}_2)$.
We have the relations $t^7=t^3x=x^2+at^2x+t^4=0$, where $a\in{\mathbb Z}_2$. (We have $Sq^1x=tx$ as this is true in $P$, thus $t^3x=Sq^1(t^2x)=0$ since $H^5(N;{\mathbb Z})=0$. By Poincaré duality $t^2x^2$ can’t be zero. This implies that in $x^2+at^2x+bt^4=0$ we have $b=1$.)
We see that $Sq^1(t^5)=t^6, Sq^2(t^4)=0, Sq^2(t^2x)=t^2x^2\ne 0$. It follows that the first Wu class is $t$ and the second Wu class is $x$. But that implies that the second Stiefel-Whitney class of $N$ maps to a non-trivial class in $H^2(M;{\mathbb Z}_2)$. But since $M$ is spin, this image must be zero, as it is the second Stiefel-Whitney class of $M$. Contradiction. [**q.e.d.**]{}
The second space in a Postnikov tower for $N$ is $Q$.
Proof: This follows from the fact that $H^2(N;{\mathbb Z}_2)\cong {\mathbb Z}_2$. [**q.e.d.**]{}
If such a manifold $M$ with $\langle x^3,[M]\rangle$ odd has a free involution $\tau$, then the quotient space $N$ has normal 2-type $B = BSpin \times Q \to BO \times BO(1)\stackrel{\oplus}\to BO$.
Proof: If $\langle x^3,[M]\rangle$ is odd, then the map $H^4(Q;{\mathbb Z}_2)\to H^4(N;{\mathbb Z}_2)$ is a bijection, since then we have isomorphisms $H^4(Q;{\mathbb Z}_2)\to H^4({\mathbb C}{\mathrm{P}^{\infty}};{\mathbb Z}_2)\to H^4(M;{\mathbb Z}_2) \leftarrow H^4(N;{\mathbb Z}_2)$. Then $Sq^2:H^4(N;{\mathbb Z}_2)\to H^6(N;{\mathbb Z}_2)$ is zero, so is the second Wu class of $M/\tau$, and as a consequence the quotient space has a spin structure twisted by $L\to Q$. [**q.e.d.**]{}
\[normalbpro\] If such a manifold $M$ with trivial square $H^2(M;{\mathbb Z}_2)\to H^4(M;{\mathbb Z}_2)$ has a free involution $\tau$, then the quotient space $N$ has one of the following normal 2-types: $$\begin{aligned}
A & = & Q\times BSpin \to BO(1)\times BO(1)\times BO(1)\times BO\stackrel{\oplus}\to BO,\\
B & = & Q\times BSpin \to BO(1)\times BO\stackrel{\oplus}\to BO,\\\end{aligned}$$ where the maps to all $BO(1)$’s are the projections $p:Q\to {\mathbb R}{\mathrm{P}^{\infty}}=BO(1)$.
Proof: If $\langle x^3,[M]\rangle$ is even, then there is a second possibility for the normal 2-type. If we fix the second space in a Postnikov tower to be $Q$, the second Stiefel-Whitney class of $N$ can be $t^2$ or 0. Thus $N$ either admits spin structures twisted by $L$ or spin structures twisted by $L\oplus L \oplus L = 3L$. [**q.e.d.**]{}
\[normalBstr\] A normal $B$-structure on a manifold $N$ can be defined in three equivalent ways.
- It is a vertical homotopy class of lifts of the normal bundle map $N\to BO$ to $B$ (this is independent of the particular map $N\to BO$ coming from an embedding of $N$ into some Euclidean space).
- It is a map $f:N\to Q$ (up to homotopy) together with a spin structure on the bundle $\nu_N - f^*(L)$, where $L$ is the nontrivial real line bundle on $Q$.
- It is a map $f:N\to Q$ together with a homotopy $\eta$ (and this up to homotopy) in the following square: $$\xymatrix{
N
\ar[r]^\nu
\ar[d]^f
& BO
\ar[d]^{w_1\times w_2}
\\
Q
\ar@{=>}[ur]^{\eta}
\ar[r]_/-2em/{t \times 0}
& K({\mathbb Z}_2,1)\times K({\mathbb Z}_2,2)
}$$ Here we fix maps corresponding to the classes $w_1\in H^1(BO;{\mathbb Z}_2), w_2\in H^2(BO;{\mathbb Z}_2)$, the generator $t\in H^1(Q;{\mathbb Z}_2), 0\in H^2(Q;{\mathbb Z}_2)$.
(For fixed $f$, the homotopy classes of homotopies $\eta$ have a free and transitive action by $\pi_1((K({\mathbb Z}_2,1)\times K({\mathbb Z}_2,2))^N) \cong H^1(N;{\mathbb Z}_2)\times H^0(N;{\mathbb Z}_2)$.)
Similarly, normal $A$-structures are defined. (In the first definition, replace $B$ by $A$. In the second definition, replace $L$ by $3L$. In the third definition, replace $0$ by $t^2$.)
As a converse to proposition \[normalbpro\], a manifold with normal $B$-structure $N^6\to B$ which is a 3-connected map is the quotient of an involution on a closed simply-connected spin manifold $M$ with $H_2(M)={\mathbb Z}$, and $H_3(M)=0$ if and only if $H_3(N;\Lambda)=0$. Here $\Lambda={\mathbb Z}[{\mathbb Z}_2]$ is the group ring of the fundamental group. The same holds for normal $A$-structures.
Computation of the bordism groups
---------------------------------
We compute bordism groups of manifolds with normal $A$-structures (resp. $B$-structures). The (co)homology of $Q$ is described in [@OlbThesis]. To compute the bordism groups $\Omega_6^A$ and $\Omega_6^B$ we use the Atiyah-Hirzebruch spectral sequence (which computes the group up to an extension problem) and an Adams spectral sequence (which can help solve the extension problem). We get:
We have isomorphisms $\Omega_6^A \cong {\mathbb Z}^2 \oplus {\mathbb Z}_2$ and $\Omega_6^B \cong {\mathbb Z}^2 \oplus {\mathbb Z}_4.$
Proof: The case of $\Omega_6^B$ was proven in [@OlbThesis]. For $\Omega_6^A$, the Atiyah-Hirzebruch spectral sequence is $$H_p(Q;\underline{\Omega_q^{Spin}})\Rightarrow \Omega_{p+q}^A,$$ the $d^2$-differential $$E^2_{p,1}\cong H_p(Q;{\mathbb Z}_2)\to H_{p-2}(Q;{\mathbb Z}_2)\cong E^2_{p-2,2}$$ is the dual of $Sq^2+tSq^1+t^2Sq^0$, and the $d^2$-differential $$E^2_{p,0}\cong H_p(Q;{\mathbb Z}_-)\to H_{p-2}(Q;{\mathbb Z}_2)\cong E^2_{p-2,1}$$ is reduction modulo 2 composed with the dual of $Sq^2+tSq^1+t^2Sq^0$. (We obtain this using the Thom isomorphism $$\Omega_6^{Spin}(Q;3L)\cong \Omega_9^{Spin}(D(3L),S(3L))$$ and the Atiyah-Hirzebruch spectral sequence for the latter. See also [@Tei].) From the calculations in [@OlbThesis] it immediately follows that the differentials $d^2:E^2_{6,1}\to E^2_{4,2}$ and $d^2:E^2_{6,0}\to E^2_{4,1}$ are non-trivial. This implies that the nonzero terms on the sixth diagonal in the $E^\infty$-term are: $$E^{\infty}_{2,4}\cong {\mathbb Z},\quad E^{\infty}_{5,1}\cong {\mathbb Z}_2,\quad E^{\infty}_{6,0}\cong 2{\mathbb Z}.$$ Thus $\Omega_6^A \cong {\mathbb Z}^2 \oplus {\mathbb Z}_2$ or $\Omega_6^A \cong {\mathbb Z}^2$.
Now we consider the Adams spectral sequence $$Ext_\mathcal{A}^{s,t}(H^*(MSpin\wedge T(3L);{\mathbb Z}_2),{\mathbb Z}_2)\Rightarrow \Omega_{t-s-3}^A,$$ where $\mathcal{A}$ is the mod 2 Steenrod algebra. We compute the left hand side for $t-s-3\le 6$. We have $$H^*(MSpin\wedge T(3L);{\mathbb Z}_2)\cong H^*(MSpin;{\mathbb Z}_2)\otimes \tilde{H}^*(T(3L);{\mathbb Z}_2),$$ and $\tilde{H}^*(T(3L);{\mathbb Z}_2)$ is a free $H^*(Q;{\mathbb Z}_2)$-module on one generator $u_3$ of degree 3 (the Thom class). We have $$Sq(u_3)= w(3L)u_3 = u_3+tu_3+t^2u_3+t^3u_3.$$ All of this allows to write down the $\mathcal{A}$-module structure of $H^*(MSpin\wedge T(3L);{\mathbb Z}_2)$ in degrees $\le 10$. Then we compute a free $\mathcal{A}$-resolution (in low degrees). From this we get the $E^2$-term of the spectral sequence, which is displayed in the following diagram: $$\xy <1.8pc,0pc>:<0pc,1.8pc>::
(0.2,-0.05) ="A"; (-0.2,0.35)="B",(0.0,0.15) ="C",
(-1,0) *!{0},
(-1,1) *!{1},
(-1,2) *!{2},
(-1,3) *!{3},
(-1,4) *!{4},
(-1,5) *!{5},
(-1,6) *!{6},
(-1,7) *!{7},
(-1,8.2) *!{s},
(0,-1) *!{2},
(1,-1) *!{3},
(2,-1) *!{4},
(3,-1) *!{5},
(4,-1) *!{6},
(5,-1) *!{7},
(6,-1) *!{8},
(7,-1) *!{9},
(8,-1) *!{10},
(9.2,-1) *!{t-s},
(8,0) *!{\cdots},
(8,1) *!{\cdots},
(8,2) *!{\cdots},
(8,3) *!{\cdots},
(8,4) *!{\cdots},
(8,5) *!{\cdots},
(8,6) *!{\cdots},
(8,7) *!{\cdots},
(1,0)+"C" *{\bullet},
(3,8)+"C"; (3,7)+"C" *{\bullet} **@{.}; (3,6)+"C" *{\bullet} **@{-};(3,5)+"C"="id42" *{\bullet} **@{-};
(3,4)+"C"*{\bullet} **@{-}; (3,3)+"C" *{\bullet} **@{-}; (3,2)+"C" *{\bullet} **@{-}; (3,1)+"B" *{\bullet} **@{-},
(3,1)+"A" *{\bullet}; (4,2)+"C"="id61" *{\bullet} **@{-}; (5,3)+"C" *{\bullet} **@{-};
(5,2)+"C" *{\bullet} **@{-},
(5,0)+"C" *{\bullet},
(7,8)+"A"; (7,7)+"A" *{\bullet} **@{.}; (7,6)+"A" *{\bullet} **@{-}; (7,5)+"A" *{\bullet} **@{-};
(7,4)+"A" *{\bullet} **@{-};(7,3)+"A" *{\bullet} **@{-},
(7,8)+"B"; (7,7)+"B" *{\bullet} **@{.}; (7,6)+"B" *{\bullet} **@{-}; (7,5)+"B" *{\bullet} **@{-};
(7,4)+"B" *{\bullet} **@{-};(7,3)+"B" *{\bullet} **@{-};(7,2)+"B" *{\bullet} **@{-};
(7,1)+"B" *{\bullet} **@{-},
(7,1)+"A" *{\bullet},
(-1.5,-0.3) ; (9.5,-0.3) **@{.},
(-1.5,0.7) ; (9.5,0.7) **@{.},
(-1.5,1.7) ; (9.5,1.7) **@{.},
(-1.5,2.7) ; (9.5,2.7) **@{.},
(-1.5,3.7) ; (9.5,3.7) **@{.},
(-1.5,4.7) ; (9.5,4.7) **@{.},
(-1.5,5.7) ; (9.5,5.7) **@{.},
(-1.5,6.7) ; (9.5,6.7) **@{.},
(-0.5,-1.5) ; (-0.5,8.5) **@{.},
(0.5,-1.5) ; (0.5,8.5) **@{.},
(1.5,-1.5) ; (1.5,8.5) **@{.},
(2.5,-1.5) ; (2.5,8.5) **@{.},
(3.5,-1.5) ; (3.5,8.5) **@{.},
(4.5,-1.5) ; (4.5,8.5) **@{.},
(5.5,-1.5) ; (5.5,8.5) **@{.},
(6.5,-1.5) ; (6.5,8.5) **@{.},
(7.5,-1.5) ; (7.5,8.5) **@{.},
\endxy$$ Since there are no differentials starting or ending at $(t-s,s)=(9,1)$, we obtain $\Omega_6^A \cong {\mathbb Z}^2 \oplus {\mathbb Z}_2$. [**q.e.d.**]{}
Construction and classification up to normal $B$-bordism {#Con}
--------------------------------------------------------
By Wall’s classification, every bordism class in $\Omega_6^{Spin}(K({\mathbb Z},2))\cong {\mathbb Z}^2$ contains a unique normal 2-smoothing with $H_3=0$ (up to diffeomorphism relative $BSpin\times K({\mathbb Z},2)$).
\[freebord\] In every bordism class in $\Omega_6^A$ there is a unique manifold $W\to A$ (up to diffeomorphism relative to $A$) such that $W\to A$ is 3-connected and $H_3(W;\Lambda)=0$. The same is true if one replaces $A$ by $B$.
Proof: For the construction and classification of free involutions on these manifolds, we use surgery theory. The existence proof is a simplified version of the proof of the main theorem in [@confree].
We start with any 6-dimensional closed manifold with normal $A$-structure, and we do surgery to get manifolds $W$ such that the map $W\to A$ is 3-connected and $H_3(W;\Lambda)=0$.
Surgery below the middle dimension is always possible [@Kre]. It allows to modify any closed 6-manifold $W$ with normal $A$-structure into one with normal 2-type $A$. Let us denote this new manifold again by $W$. It remains to kill $H_3(W;\Lambda)$.
By the Hurewicz theorem (its extended version) we have a surjection $\pi_3(W)\to H_3(W;\Lambda)$, where $\Lambda={\mathbb Z}[{\mathbb Z}_2]$ is the group ring of the fundamental group, and $H_3(W;\Lambda)$ can be identified with the homology of the universal cover of $W$.
The map $H_3(W;\Lambda)\to H_3(W;{\mathbb Z}_2)$ factors through $H_3(W;\Lambda)\otimes_\Lambda {\mathbb Z}_2$, more precisely the relation is given by a universal coefficient spectral sequence $$\operatorname{Tor}_p^\Lambda(H_q(W;\Lambda),{\mathbb Z}_2)\Rightarrow H_{p+q}(W;{\mathbb Z}_2).$$ (This can also be interpreted as the Serre spectral sequence for the fibration $\tilde{W}\to W\to {\mathbb R}{\mathrm{P}^{\infty}}$.) Here the zeroth and the second row are related by non-trivial differentials: we compare with the corresponding situation for the space $Q$. As a result, $H_3(W;{\mathbb Z}_2)\cong H_3(W;\Lambda)\otimes_\Lambda {\mathbb Z}_2$.
By Poincaré duality, $H_3(W;\Lambda)\cong H^3(W;\Lambda)$, and this is free over ${\mathbb Z}$, as there is no ${\mathbb Z}$-torsion in $H_2(W;\Lambda)$. Since $H_3(W;\Lambda)$ is free over ${\mathbb Z}$, it is a sum of summands of the form $\Lambda$, ${\mathbb Z}_+$ and ${\mathbb Z}_-$ [@CR]. We also get that $H^3(W;\Lambda)\cong Hom_\Lambda(H_3(W;\Lambda),\Lambda)$, for example again from a universal coefficient spectral sequence. The map $H_3(W;\Lambda)\to H^3(W;\Lambda) \to Hom_\Lambda(H_3(W;\Lambda),\Lambda)$ describes the $\Lambda$-valued intersection form on $H_3(W;\Lambda)$. The ${\mathbb Z}_2$-valued intersection form on $H_3(W;{\mathbb Z}_2)$ is given by tensoring with ${\mathbb Z}_2$.
But this implies that for a class $x\in H_3(W;\Lambda)$ with $Tx=\pm x$, its image in $H_3(W;{\mathbb Z}_2)$ has intersection 0 with all other elements, hence it must be 0: if $Tx=\pm x$, then $T\lambda(x,y)=\lambda(Tx,y)=\lambda(\pm x,y)=\pm \lambda(x,y)$, so $\lambda(x,y)$ is a multiple of $(1 \pm T)$ and its reduction in ${\mathbb Z}_2$ is zero.
Hence $H_3(W;\Lambda)$ is a free $\Lambda$-module with non-degenerate intersection form. Since we are in dimension 3, there is a quadratic refinement in $\Lambda / \langle x+\bar{x} ,1 \rangle$ which is uniquely determined by the intersection form. Hence we obtain an element in $\tilde{L_6}(\Lambda,w=-)=0$, as $L_6\cong {\mathbb Z}_2$ is given by the Arf invariant. Thus it is possible (after stabilization) to do surgery which makes $H_3(W;\Lambda)=0$.
The argument shows that every class of $\Omega_6^A$ contains a manifold $W$ with normal 2-type $A$ and $H_3(W;\Lambda)=0$.
For the uniqueness result we take two such manifolds $W_0,W_1$, and assume there is a normal $A$-bordism between them. By Kreck’s theory [@Kre], p. 734, there is a surgery obstruction in $\tilde{l}_7(\Lambda,w=-1)$ for turning the normal $A$-bordism into an $s$-cobordism. By surgery below the middle dimension we may assume that the bordism $Y$ is equipped with a 3-connected map to $A$. Now Kreck defines the surgery obstruction using a construction of a certain disjoint union $U$ of submanifolds of $Y$ diffeomorphic to $S^3\times D^4$, and defines the surgery obstruction to be the kernel of $H_3(\partial U;\Lambda)\to H_3(Y\setminus int(U), W_0;\Lambda)$. He also notes that the orthogonal complement of this kernel is the kernel of $H_3(\partial U;\Lambda)\to H_3(Y\setminus int(U), W_1;\Lambda)$. But in our case $H_3(W_i;\Lambda)=0$ so that both these kernels are equal to the kernel of $H_3(\partial U;\Lambda)\to H_3(Y\setminus int(U); \Lambda)$.
This implies that the surgery obstruction just defined lies in the group $\tilde{L}_7(\Lambda,w=-1)$ which is trivial as computed by Wall. Hence we get as a result that $A$-bordant manifolds $W_i$ with normal 2-type $A$ and $H_3(W;\Lambda)=0$ are diffeomorphic (relative to $A$).
The proof for normal 2-type $B$ is obtained by replacing all occurrences of $A$ by $B$. [**q.e.d.**]{}
The transfer
------------
For the transfer (double cover) map $\Omega_6^A\to \Omega_6^{\tilde{A}}\cong \Omega_6^{Spin}({\mathbb C}{\mathrm{P}^{\infty}})$ we compare the Atiyah-Hirzebruch spectral sequences. We fix the isomorphism $$\begin{aligned}
\Omega_6^{Spin}({\mathbb C}{\mathrm{P}^{\infty}}) & \cong & {\mathbb Z}^2\\
\left[f:M\to {\mathbb C}{\mathrm{P}^{\infty}}\right] & \mapsto & \left( \left\langle \frac{p_1(M)f^*x-4f^*x^3}{24},[M]\right\rangle,
\langle f^*x^3,[M]\rangle \right).\end{aligned}$$ One computes the homology transfers using the long exact sequences coming from short exact coefficient sequences: $$\begin{aligned}
H_6(Q;{\mathbb Z}_-)&\stackrel{\cong}{\to}& H_6({\mathbb C}{\mathrm{P}^{\infty}}), \\
H_4(Q;{\mathbb Z}_2)&\stackrel{0}{\to}& H_4({\mathbb C}{\mathrm{P}^{\infty}};{\mathbb Z}_2), \\
H_2(Q;{\mathbb Z}_-)&\stackrel{\cong}{\to}& H_2({\mathbb C}{\mathrm{P}^{\infty}}).\end{aligned}$$ We see that the transfer gives a map of short exact sequences $$\xymatrix{
F_5 \cong {\mathbb Z}_2\oplus{\mathbb Z}\ar[d]_{(a,b)\mapsto 2b}
\ar[rr]^{(a,b)\mapsto (a,b,0)}
&&
\Omega_6^A \cong {\mathbb Z}_2\oplus {\mathbb Z}^2
\ar[rr]^{(a,b,c)\mapsto c}
\ar[d]^{tr}
&&
E^{\infty}_{6,0}\cong {\mathbb Z}\ar[d]^{c\mapsto 2c}
\\
\tilde{F}_5 \cong {\mathbb Z}\ar[rr]_{b\mapsto (b,0)}
&&
\Omega_6^{\tilde{A}} \cong {\mathbb Z}^2
\ar[rr]_{ (b,c)\mapsto c}
&&
\tilde{E}^{\infty}_{6,0}\cong {\mathbb Z}}$$ which shows (using the snake lemma) that $\Omega_6^A\to \Omega_6^{\tilde{A}}$ has a cokernel of order 4. But the composition of projection and transfer: $\Omega_6^{\tilde{A}} \to \Omega_6^A\to \Omega_6^{\tilde{A}}$ is multiplication by 2. So the image of the transfer consists exactly of all classes divisible by 2.
It also follows that one can find generators for the free summands in $\Omega_6^A$ as images of the generators of $\Omega_6^{\tilde{A}}$.
[@confree] The image of the transfer map $\Omega_6^B\to \Omega_6^{\tilde{B}}\cong \Omega_6^{Spin}({\mathbb C}{\mathrm{P}^{\infty}})$ is $2{\mathbb Z}\oplus {\mathbb Z}$.
Generators for the bordism group $\Omega_6^B$
---------------------------------------------
We use the Thom isomorphism for twisted spin bordism: $\Omega_6^B\cong \Omega_6^{Spin}(Q;L) \cong \Omega_7^{Spin}(DL,SL)$. Under this isomorphism, the boundary map $\Omega_7^{Spin}(DL,SL)\to \Omega_6^{Spin}$ corresponds to the transfer map $\Omega_6^B\to \Omega_6^{\tilde{B}}$. It follows that the torsion elements in $\Omega_6^B\cong \Omega_7^{Spin}(DL,SL)$ come from $\Omega_7^{Spin}(DL)\cong \Omega_7^{Spin}(Q)$. Moreover, $\Omega_7^{Spin}({\mathbb C}{\mathrm{P}^{\infty}})=0$, as one sees easily from the Atiyah-Hirzebruch spectral sequence. Thus $\Omega_7^{Spin}(Q)\cong {\mathbb Z}_4$ must be responsible for the torsion. Moreover, a study of the Atiyah-Hirzebruch spectral sequence shows that the map $\Omega_7^{Spin}({\mathbb C}{{\mathrm{P}}}^3/\tau)\to \Omega_7^{Spin}(Q)$ is an isomorphism. Now there is a bundle ${\mathbb R}{{\mathrm{P}}}^2 \to {\mathbb C}{{\mathrm{P}}}^3/\tau \to S^4$. So ${\mathbb C}{{\mathrm{P}}}^3/\tau - {\mathbb R}{{\mathrm{P}}}^2$ is an ${\mathbb R}{{\mathrm{P}}}^2$-bundle over $D^4$, so homotopy equivalent to ${\mathbb R}{{\mathrm{P}}}^2$. Again the Atiyah-Hirzebruch spectral sequence shows that $\Omega_7^{Spin}({\mathbb C}{{\mathrm{P}}}^3/\tau - {\mathbb R}{{\mathrm{P}}}^2)=\Omega_6^{Spin}({\mathbb C}{{\mathrm{P}}}^3/\tau -{\mathbb R}{{\mathrm{P}}}^2)=0$. Thus $\Omega_7^{Spin}({\mathbb C}{{\mathrm{P}}}^3/\tau)\cong \Omega_7^{Spin}({\mathbb C}{{\mathrm{P}}}^3/\tau, {\mathbb C}{{\mathrm{P}}}^3/\tau - {\mathbb R}{{\mathrm{P}}}^2)$, and the latter is isomorphic to $\Omega_3^{Spin}({\mathbb R}{{\mathrm{P}}}^2)\cong {\mathbb Z}_4$ by the Thom isomorphism. Again the Atiyah-Hirzebruch spectral sequence shows that the transfer $\Omega_3^{Spin}({\mathbb R}{{\mathrm{P}}}^2)\to \Omega_3^{Spin}(S^2)\cong {\mathbb Z}_2$ is surjective. Thus one can detect a generator of $\Omega_7^{Spin}({\mathbb C}{{\mathrm{P}}}^3/\tau)$ by the composition $$\Omega_7^{Spin}({\mathbb C}{{\mathrm{P}}}^3/\tau)\to \Omega_7^{Spin}({\mathbb C}{{\mathrm{P}}}^3/\tau, {\mathbb C}{{\mathrm{P}}}^3/\tau - {\mathbb R}{{\mathrm{P}}}^2)
\to \Omega_3^{Spin}({\mathbb R}{{\mathrm{P}}}^2)\to \Omega_3^{Spin}(S^2).$$
Now take the seven-dimensional manifold $({\mathbb C}{{\mathrm{P}}}^3 \times S^1)/(\tau,c)$, where $c$ is complex conjugation. The spin structure on ${\mathbb C}{{\mathrm{P}}}^3 \times S^1$ which restricts to the non-bounding one on $S^1$ is preserved by the involution $(\tau,c)$, so that we obtain a spin structure on the quotient. The map to ${\mathbb C}{{\mathrm{P}}}^3/\tau$ is projection on the first coordinate. This intersects ${\mathbb R}{{\mathrm{P}}}^2={\mathbb C}{{\mathrm{P}}}^1/\tau$ transversely, so that the Thom isomorphism maps it to the pullback $({\mathbb C}{{\mathrm{P}}}^1\times S^1)/(\tau,c)\to {\mathbb C}{{\mathrm{P}}}^1/\tau$ whose double cover is the projection ${\mathbb C}{{\mathrm{P}}}^1 \times S^1 \to {\mathbb C}{{\mathrm{P}}}^1$, which is a generator for $\Omega_3^{Spin}(S^2)$ since the Spin structure restricts to the non-boundant one on $S^1$.
We have to apply the Thom isomorphism $\Omega_7^{Spin}(DL|_{{\mathbb C}{{\mathrm{P}}}^3/\tau},SL|_{{\mathbb C}{{\mathrm{P}}}^3/\tau})\to \Omega_6^{Spin}({\mathbb C}{{\mathrm{P}}}^3/\tau;L)$ to the map $({\mathbb C}{{\mathrm{P}}}^3 \times S^1)/(\tau,c) \to {\mathbb C}{{\mathrm{P}}}^3/\tau \to ({\mathbb C}{{\mathrm{P}}}^3 \times D^1)/(\tau,-1)$. For this, we homotope the map to make it transversal to the zero section: take $$\begin{aligned}
({\mathbb C}{{\mathrm{P}}}^3 \times S^1)/(\tau,c) & \to & ({\mathbb C}{{\mathrm{P}}}^3 \times D^1)/(\tau,-1) \\
\ [ x , y ] & \mapsto & [ x,Im(y) ]\end{aligned}$$ and intersect it with the zero section: we obtain two copies of ${\mathbb C}{{\mathrm{P}}}^3/\tau$ with different $B$-structures.
It follows that the torsion ${\mathbb Z}_4$ in $\Omega_6^B$ is generated by the sum (or the difference) of two copies of ${\mathbb C}{{\mathrm{P}}}^3/\tau$ with different $B$-structures. Let us denote one of them by $\sigma$ and the other by $\sigma'$.
Thus we obtain as generators for $\Omega_6^B\cong \Omega_7^{Spin}(DL,SL)$:
- $X\times D^1 \stackrel{f\times id}\to {\mathbb C}{\mathrm{P}^{\infty}}\times D^1$, where $X$ is a simply-connected spin 6-manifold with $H^3(X)=0$, trivial cup product $H^2(X)\times H^2(X)\to H^4(X)$, and $f:X\to {\mathbb C}{\mathrm{P}^{\infty}}$ defines a generator of $H^2(X)$ such that the first Pontrjagin class of $X$ is equal to 24 times the generator of $H^4(X)$ which is dual to $f\in H^2(X)$.
- $(({\mathbb C}{{\mathrm{P}}}^3\times D^1)/(\tau,-1),\sigma)$,
- $(({\mathbb C}{{\mathrm{P}}}^3\times D^1)/(\tau,-1),\sigma) -(({\mathbb C}{{\mathrm{P}}}^3\times D^1)/(\tau,-1),\sigma')$.
The two former generators generate free summands, the latter generates the torsion summand ${\mathbb Z}_4$. In this basis, the map $\Omega_6^B \cong {\mathbb Z}^2 \oplus {\mathbb Z}_4 \to {\mathbb Z}^2\cong \Omega_6^{Spin}({\mathbb C}{\mathrm{P}^{\infty}})$ is given as $(a,b,c)\mapsto (2a,b)$.
The automorphism groups of $A$ and $B$ and their action on the bordism groups
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The automorphism group $Aut(B)_{BO}$ of fiber homotopy classes of fiber homotopy self-equivalences of $B$ acts on $\Omega_6^B$. We saw that $\Omega_6^B\cong \Omega_7^{Spin}(DL,SL)$, where $L$ is the nontrivial real line bundle $({\mathbb C}{\mathrm{P}^{\infty}}\times {\mathbb R})/(\tau,-1)$.
The set of equivariant oriented diffeomorphism classes of free involutions on 6-manifolds with $H_3=0$ and whose quotient spaces have normal 2-type $B$ are given as the orbits. Again, $B$ can be replaced by $A$ in the theorem.
Proof: Equivariant diffeomorphism classes of free involutions are the same as diffeomorphism classes of the quotients. Now the theorem follows from the uniqueness of the Postnikov decomposition, i.e. for a given manifold $W$ with normal 2-type $B$, the map $W\to B$ is uniquely determined up to fiber homotopy self equivalences of $B$ over $BO$. See also [@KrM]. [**q.e.d.**]{}
The restriction of the first component of a fiber homotopy self-equivalence of $Q\times BSpin$ to $Q$ is a self-homotopy equivalence of $Q$. There is a unique free homotopy class of maps $Q\times BSpin\to Q$ which is an isomorphism on $\pi_1$, and thus (using obstruction theory to extend homotopies from $Q$ to $Q\times BSpin$) a unique free homotopy class of maps $Q\times BSpin\to Q$ which are an isomorphism on $\pi_1$. The vertical homotopy classes of fiber homotopy equivalences thus correspond to the different choices (up to homotopy) of a homotopy $\eta$ from $Q\times BSpin \to Q \to K({\mathbb Z}_2,1)\times K({\mathbb Z}_2,2)$ to $Q\times BSpin\to BO \to K({\mathbb Z}_2,1)\times K({\mathbb Z}_2,2)$. So the group has four elements, and the action of the group on the set of normal $B$-structures on a manifold $f:M\to B$ (i.e. spin structures on $\nu_M-f^*L$) is given by just changing the spin structure $\sigma$ into $\sigma, -\sigma, \sigma +f^*t, -\sigma+f^*t$, where $t\in H^1(B;{\mathbb Z}_2)$ is the generator.
On $\Omega_7^{Spin}(DL,SL)$, the group $Aut(B)_{BO}$ acts in the following way: the negative spin generator acts by -1, and the spin flip generator acts by $$(({\mathbb C}{{\mathrm{P}}}^3\times D^1)/(\tau,-1),\sigma)\mapsto (({\mathbb C}{{\mathrm{P}}}^3\times D^1)/(\tau,-1),\sigma'),$$ and is the identity on $X$. Thus in the decomposition $\Omega_7^{Spin}(DL,SL)\cong {\mathbb Z}\oplus {\mathbb Z}\oplus {\mathbb Z}_4$ given by the above generators, the negative spin generator acts by $(a,b,c)\mapsto (-a,-b,-c)$, and the spin flip generator acts by $(a,b,c)\mapsto (a,b,b-c)$. We obtain orbits of the form $$\{ (a,b,c), (a,b,b-c),(-a,-b,-c), (-a,-b,-b+c)\}.$$
The group $Aut(A)_{BO}$ also has four elements which act on the set of normal $A$-structures on a manifold $f:M\to A$ (i.e. spin structures on $\nu_M-f^*(3L)$) by just changing the spin structure $\sigma$ into $\sigma, -\sigma, \sigma +f^*t, -\sigma+f^*t$, where $t\in H^1(A;{\mathbb Z}_2)$ is the generator.
This is either the identity or minus the identity on the generators for the free summands in $\Omega_6^A$ as they are images of the generators of $\Omega_6^{\tilde{A}}$ unde projection. All group elements must act by the identity on the torsion generator. We obtain orbits $$\{ (a,b,c), (-a,-b,-c)\}.$$
The group $Aut(K({\mathbb Z},2) \times BSpin)_{BO}$ also has four elements, which act on the bordism group by reversing the spin structure and/or the class in $H^2$. It follows that here the orbits are of the form $$\{ (a,b), (-a,-b) \}.$$
For the proof of theorem \[Mthm\], it is now sufficient to count preimages and orbits: An element of the form $(2a+1,b)\in \Omega_6^{Spin}({\mathbb C}{\mathrm{P}^{\infty}})$ has no preimages in $\Omega_6^A$ or $\Omega_6^B$. An element $(2a,2b+1)$ in $\Omega_6^{Spin}({\mathbb C}{\mathrm{P}^{\infty}})$ has four preimages $(a,2b+1,c)$ in $\Omega_6^B$ and no preimages in $\Omega^A_6$. These four preimages, together with the four preimages of $(-2a,-2b-1)$, decompose into two orbits. For $(2a,2b)$ in $\Omega_6^{Spin}({\mathbb C}{\mathrm{P}^{\infty}})$ we obtain four preimages $(a,2b,c)$ in $\Omega_6^B$. These four preimages, together with the four preimages of $(-2a,-2b)$, decompose into three orbits. The element $(2a,2b)$ in $\Omega_6^{Spin}({\mathbb C}{\mathrm{P}^{\infty}})$ has two preimages $(a,b,c)$ in $\Omega_6^B$. These two preimages, together with the two preimages of $(-2a,-2b)$, decompose into two orbits.
Involutions on $S^6$ with three-dimensional fixed point set
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Non-equivariant classification of embeddings in $S^6$
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Before we classify involutions on $S^6$, let us recall the non-equivariant results on embeddings of three-manifolds into $S^6$.
Naturally the most interesting case is the one of knotted 3-spheres in the six-sphere. Here the results are due to Haefliger. One could consider various equivalence relations on knotted $S^3$’s in $S^6$. In all cases there is an addition defined using connected sums:
First we can look at the group $C_3^3$ of isotopy classes of embeddings of $S^3$ into $S^6$. Second, the group $\Theta$ of diffeomorphism classes of embeddings of $S^3$ into oriented manifolds $X$ diffeomorphic to $S^6$, relative to $S^3$. (We require a diffeomorphism to be the identity on $S^3$ and to preserve orientations.) Third, the group $\Sigma$ of orientation-preserving diffeomorphism classes of pairs $(X,M)$, where the oriented manifold $X$ is diffeomorphic to $S^6$ and the oriented submanifold $M$ is diffeomorphic to $S^3$.
There are obvious surjective group homomorphisms $C_3^3\to \Theta\to\Sigma$. Haefliger showed that $C_3^3$ and $\Sigma$ are both isomorphic to ${\mathbb Z}$, so that both of the above maps are isomorphisms.
To prove that $C_3^3\to \Theta$ is an isomorphism, one needs to show that a diffeomorphism $S^6\to S^6$ relative the embedded $S^3$ can be replaced by an isotopy. This is true since $\pi_0(Diff(D^n,\partial))\to \pi_0(Diff(S^n))$ is surjective. This means it is always possible to modify the original diffeomorphism on a disk such that the resulting diffeomorphism is isotopic to the identity.
One explanation of the isomorphism $\Theta\to\Sigma$ is Cerf’s result that $Diff^+(S^3)$ is connected. Thus every orientation-preserving diffeomorphism of $S^3$ is isotopic to the identity, and this isotopy extends to an ambient equivariant isotopy of $S^6$.
The negative of an isotopy class is given by precomposing the embedding $S^3\to S^6$ with an orientation-reversing self-diffeomorphism of $S^3$.
Isotopy classes of framed embeddings $S^3\times D^3\to S^6$ are in bijection with ${\mathbb Z}^2$, the framing giving an additional integer invariant (the isomorphism to ${\mathbb Z}^2$ depends on a choice). Note also that in the PL category, all non-framed knots are trivial, but the isotopy classes of framed knots are in bijection with smooth isotopy classes of smooth framed knots.
For embeddings of general closed oriented connected 3-manifolds $M^3$ into $S^6$, the argument that diffeomorphism relative to the submanifold implies isotopy of the embeddings still holds. Isotopy classes of embeddings $Emb(M,S^6)$ have been classified by Skopenkov in [@Sko]. To an isotopy class of embeddings $i:M\to S^6$ he associates its Whitney invariant $W(i)\in H_1(N)$. (For the precise definition we refer to [@Sko], we give a description in special cases in remark \[WhitInv\].) The map $W:Emb(M,S^6)\to H_1(N)$ is surjective, and $C^3_3\cong {\mathbb Z}$ acts transitively on the fibers by connected sum. This action has non-trivial stabilizers in general: There is a bijective map (the Kreck invariant) from the fiber over $u\in H_1(N)$ to ${\mathbb Z}_{d(u)}$, where $d(u)$ is the divisibility of $\bar{u}\in H_1(M)/\{\text{torsion}\}$. In general, both the Whitney and the Kreck invariant depend on choices. For ${\mathbb Z}_2$-homology spheres $M$, the Whitney invariant does not depend on choices, and $C_3^3$ acts freely on the fibers, so that $Emb(M,S^6)$ is in non-canonical bijection with ${\mathbb Z}\times H_1(M)$. Instead of a map to ${\mathbb Z}$, the Kreck invariant describes an action of ${\mathbb Z}$ on $Emb(M,S^6)$ which leaves the Whitney invariant fixed.
What should we classify in the equivariant case?
------------------------------------------------
In the equivariant case, there are again various equivalence relations one can put on the set of embeddings respectively involutions. However, in order to get a well-defined connected sum operation, it is necessary to orient both the 6-manifold and the fixed point set.
The proof of the uniqueness of the non-equivariant connected sum construction can be generalized to show that the equivariant diffeomorphism type of the equivariant connected sum of two conjugation manifolds of dimension $2n$ only depends on the connected component of the set of equivariant isomorphisms of a tangent space at a fixed point with $({\mathbb R}^{2n},(1^n,-1^n))$. More generally, varying the chosen fixed point, we get a bundle of such isomorphisms over the fixed point set, and the equivariant diffeomorphism type depends only on the connected component in the total space of this bundle. (The total space has two components if the fixed point set is orientable, and one component if it is not.) See also Definition 1.1 and Lemma 1.2 of [@Loe]. In particular, the connected sum is unique up to equivariant diffeomorphism if we are provided with orientations of the conjugation manifolds and their fixed point sets, but in general depends also on an orientation of the fixed point sets. This answers a question in [@HHP], and it also explains why we are less interested in the classification of conjugations without an orientation of the fixed point set.
We fix $M^3$, allow various involutions $\tau$ on $S^6$, and consider equivariant embeddings $(M,id)\to (S^6,\tau)$ such that the image of $i$ is the fixed point set of $\tau$. Again there are several equivalence relations which we can put on this set.
- Two embeddings $i_0:(M,id)\to (S^6,\tau_0)$ and $i_1:(M,id)\to (S^6,\tau_1)$ are equivalent if there is an equivariant, orientation-preserving diffeomorphism $\phi:(S^6,\tau_0)\to (S^6,\tau_1)$ relative $M$, i.e. there is a commutative triangle $$\xymatrix{
(M,id)
\ar[r]^{i_0}
\ar[dr]^{i_1}
& (S^6,\tau_0)
\ar[d]^{\phi}
\\
& (S^6,\tau_1).
}$$ This classsifies involutions together with an identification of the fixed point set with the given 3-manifold $M$. We get a set $Emb_{{\mathbb Z}_2}(M,S^6)$.
- Two embeddings $i_0:(M,id)\to (S^6,\tau_0)$ and $i_1:(M,id)\to (S^6,\tau_1)$ are equivalent if there is an equivariant diffeomorphism $\phi:(S^6,\tau_0)\to (S^6,\tau_1)$ which restricts to some self-diffeomorphism $\phi_M$ of $M$. We require that both $\phi$ and $\phi_M$ are orientation-preserving. There is a commutative square $$\xymatrix{
(M,id)
\ar[r]^{i_0}
\ar[d]^{\phi_M}
& (S^6,\tau_0)
\ar[d]^{\phi}
\\
(M,id)
\ar[r]^{i_1}
& (S^6,\tau_1).
}$$ This classifies (up to orientation-preserving diffeomorphism) involutions plus an orientation of the fixed point set. One might call this the classification of conjugations up to conjugation. We get a set $Inv_M(S^6)$.
One could also define an equivariant version of isotopy classes of embeddings: we say that two equivariant embeddings $i_0:(M,id)\to (S^6,\tau_0)$ and $i_1:(M,id)\to (S^6,\tau_1)$ are equivalent if there is an equivariant embedding $i:(M\times I,id)\to (S^6\times I,\tau)$ such that $i(x,t)=(i_t(x),t)$ and $\tau(y,t)=(\tau_t(y),t)$, hence in particular it restricts on both ends to $i$ respectively $i'$. (We also require that the image of all the embeddings involved is the whole fixed point set.)
Analysis of involutions on $S^6$ with three-dimensional fixed point set
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We recall the classical result:
If an involution on $S^n$ has fixed points, the fixed point set is a ${\mathbb Z}_2$-homology sphere.
Comparing with Skopenkov’s classification of embeddings, our first result is the following.
If $i:M^3\to S^6$ is the embedding of a fixed point set of an involution, then the Whitney invariant of the embedding vanishes: $W(i)=0$.
Proof: Let $\tau$ be such an involution, and let $\sigma$ be the reflection of $S^6$ at the equator. In [@Sko] it is proved that $W(\sigma \circ i)=-W(i)$. But we have a commutative square $$\xymatrix{
M
\ar[d]^=
\ar[r]^{i}
& S^6
\ar[d]^{\sigma \circ \tau}
\\
M
\ar[r]^{\sigma \circ i}
& S^6
}$$ which shows that $i$ and $\sigma \circ i$ are isotopic. Thus $W(i)=-W(i)$, and since $H_1(M)$ consists of odd torsion only, we have $W(i)=0$. [**q.e.d.**]{}
\[WhitInv\] Up to a multiplicative factor, which is invertible in the case of ${\mathbb Z}_2$-homology spheres, the Whitney invariant can also be defined in the following way: Let $C_i=S^6\setminus i(M^3)$. By Alexander duality, $H^2(C_i)\cong {\mathbb Z}$ and $H^4(C_i)\cong H^2(M)\cong H_1(M)$. Then the invariant is given by the square of a generator of $H^2(C_i)$. In the case of a fixed point set of a conjugation, the involution acts by multiplication with $-1$ on both $H^2(C_i)$ and $H^4(C_i)$. It follows that the square of a generator of $H^2(C_i)$ must be 0.
Recall that by the equivariant tubular neigborhood theorem, for every involution on $S^6$ with fixed point set $M$ we can write $S^6=M\times D^3 \cup V$, such that the involution is $id\times -id$ on $M\times D^3$, and free on $V$. The quotient $W=V/\tau$ is a manifold with fundamental group ${\mathbb Z}_2$ and boundary $M\times {\mathbb R}{{\mathrm{P}}}^2$. The normal 2-type of $W$ is $B$. The inclusion of the boundary needs to induce an isomorphism $\pi_2({\mathbb R}{{\mathrm{P}}}^2)\to \pi_2(B)$.
In order to define the bordism set $\Omega_6^{(B,M\times {\mathbb R}{{\mathrm{P}}}^2)}$, we have to fix a normal $B$-structure on $M\times {\mathbb R}{{\mathrm{P}}}^2$. Since we are interested in a classification, we first consider all relevant normal $B$-structures.
For any such structure, the homotopy class of maps $M\times {\mathbb R}{{\mathrm{P}}}^2\to Q\to {\mathbb R}{\mathrm{P}^{\infty}}$ is the non-trivial class in $H^1(M\times {\mathbb R}{{\mathrm{P}}}^2;{\mathbb Z}_2)$. We have to lift this non-trivial map $M\times {\mathbb R}{{\mathrm{P}}}^2\to {\mathbb R}{\mathrm{P}^{\infty}}$ to $Q$. Any lift, together with a choice of a spin structure on $\nu_{M\times {\mathbb R}{{\mathrm{P}}}^2} - f^*(L)$, will then be a normal $B$-structure on $M\times {\mathbb R}{{\mathrm{P}}}^2$. It is easy to find a lift $f$ on ${\mathbb R}{{\mathrm{P}}}^2$, see [@OlbThesis], p.47, and composing any lift with the projection $M\times {\mathbb R}{{\mathrm{P}}}^2\to {\mathbb R}{{\mathrm{P}}}^2$ gives a lift $S^3\times {\mathbb R}{{\mathrm{P}}}^2\to Q$. Obstruction theory shows that pointed homotopy classes of lifts on ${\mathbb R}{{\mathrm{P}}}^2$ are classified by $H^2({\mathbb R}{{\mathrm{P}}}^2,{\mathbb Z}_-)\cong {\mathbb Z}$, and that every lift on ${\mathbb R}{{\mathrm{P}}}^2$ extends uniquely up to homotopy to a lift on $M\times {\mathbb R}{{\mathrm{P}}}^2$, see [@OlbThesis], pp.50-52.
But there is a further condition. Only two pointed homotopy classes of lifts induce an isomorphism $\pi_2({\mathbb R}{{\mathrm{P}}}^2)\to \pi_2(Q)$, and one obtains one from the other by precomposing with the nontrivial pointed homotopy class of maps ${\mathbb R}{{\mathrm{P}}}^2 \to {\mathbb R}{{\mathrm{P}}}^2$. However, this map is freely homotopic to the identity of ${\mathbb R}{{\mathrm{P}}}^2$. Thus we get up to (free) homotopy a unique map $f:{\mathbb R}{{\mathrm{P}}}^2\to Q$. There are four spin structures on $\nu_{M\times {\mathbb R}{{\mathrm{P}}}^2} - f^*(L)$, since spin structures on unoriented (but orientable and spin) bundles over $N$ are in bijection with $H^0(N;{\mathbb Z}_2)\times H^1(N;{\mathbb Z}_2)$. Thus there are four distinguished normal $B$-structures on $M \times {\mathbb R}{{\mathrm{P}}}^2$ which can be used in the construction.
For every distinguished normal $B$-structure on $M\times {\mathbb R}{{\mathrm{P}}}^2$, every element of the bordism set $\Omega_6^{(B,M\times {\mathbb R}{{\mathrm{P}}}^2)}$ contains (up to diffeomorphism relative to $B$ and the boundary) a unique manifold $W$ which produces a conjugation on $S^6$.
Actually we also see that $H_3(W;\Lambda)$ consisting just of odd torsion is a necessary and sufficient condition for this.
Proof: The proof of existence is a slight modification of the proof of theorem \[freebord\]. (For full details, see the proof of theorem 1.3 in [@confree].) Also the uniqueness extends from the proof of theorem \[freebord\]: If we take two manifolds $W_0,W_1$ with the same normal $B$-structure on the boundary, and which both produce conjugations on $S^6$, and assume that there is a normal $B$-bordism between them, we have to modify the argument from \[Con\] slightly to show that Kreck’s surgery obstruction is 0: For both $i=0,1$ the kernel of $H_3(\partial U;\Lambda)\to H_3(Y\setminus int(U),W_i;\Lambda)$ is equal to the kernel of $H_3(\partial U;\Lambda)\to H_3(Y\setminus int(U);\Lambda) /\{\text{torsion}\}$. So again the surgery obstruction lies in $\tilde{L}_7(\Lambda,w=-1)=0$. [**q.e.d.**]{}
Diffeomorphism classes of $W$ relative to $M\times {\mathbb R}{{\mathrm{P}}}^2$ and to $B$ (where $M\times {\mathbb R}{{\mathrm{P}}}^2 \to B$ is fixed) producing conjugations on $S^6$ are in bijection with $\Omega_6^{(B,M\times {\mathbb R}{{\mathrm{P}}}^2)}\cong {\mathbb Z}^2\oplus {\mathbb Z}_4$.
Diffeomorphism classes of $W$ relative to $M\times {\mathbb R}{{\mathrm{P}}}^2$ and to $B$ (where $M\times {\mathbb R}{{\mathrm{P}}}^2 \to B$ is not fixed) producing conjugations on $S^6$ are in bijection with the disjoint union of these four bordism sets.
Equivariant connected sum of two conjugations with fixed point sets $M_1$ respectively $M_2$ corresponds to a map of bordism sets defined by gluing along part of the boundary (or parametrized boundary connected sum): Choose disks $D^3$ in $M_1,M_2$ centered at the points where the connected sum is performed. Then there is a map $$\begin{aligned}
\Omega_6^{(B,M_1\times {\mathbb R}{{\mathrm{P}}}^2)} \times \Omega_6^{(B,M_2\times {\mathbb R}{{\mathrm{P}}}^2)} &\to& \Omega_6^{(B,(M_1\# M_2)\times {\mathbb R}{{\mathrm{P}}}^2)} \\
(W_1, W_2) & \mapsto & W_1\cup_{D^3\times {\mathbb R}{{\mathrm{P}}}^2} W_2\end{aligned}$$ This is equivariant with respect to the action of the bordism group $\Omega_6^B$. Hence it equips $\Omega_6^{(B,S^3\times {\mathbb R}{{\mathrm{P}}}^2)}$ with a group structure, and we obtain an action of $\Omega_6^{(B,S^3\times {\mathbb R}{{\mathrm{P}}}^2)}$ on $\Omega_6^{(B,M\times {\mathbb R}{{\mathrm{P}}}^2)}$ for any $M$.
To compare with the non-equivariant embedding results of Skopenkov, it suffices to consider the image under the transfer map: Comparing with [@Sko], we see that for the embeddings with Whitney invariant 0, the Kreck invariant describes an action of ${\mathbb Z}$ on $Emb(M,S^6)$ which corresponds precisely to the action of $$\Omega_6^{Spin}(K({\mathbb Z},2),\partial=S^3\times S^2)/(0\oplus{\mathbb Z}\subset \Omega_6^{Spin}(K({\mathbb Z},2)))$$ on the set $$\Omega_6^{Spin}(K({\mathbb Z},2),\partial=M\times S^2)/(0\oplus{\mathbb Z}\subset \Omega_6^{Spin}(K({\mathbb Z},2))).$$ We will see in the next section how the quotients arise also in the equivariant setting.
Equivariant diffeomorphism classes as a quotient by group actions
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Now we have to relate the set of relative diffeomorphism classes of the previous section to the set $Emb_{{\mathbb Z}_2}(M,S^6)$ of equivariant diffeomorphism classes of six-spheres with involution whose fixed point set is identified with $M$.
Basically, we forget the $B$-structure, we construct the equivariant inclusion $M\times D^3\to X$ from $M\times {\mathbb R}{{\mathrm{P}}}^2 \to W$, and we forget the tubular neighbourhood and the framing of the normal bundle.
More precisely, we have the following sets of equivalence classes:
1. \[item1\] The set $T_1$ of diffeomorphism classes of manifolds $W$ relative to $M\times {\mathbb R}{{\mathrm{P}}}^2$ and $BO$. An element is represented by $M\times {\mathbb R}{{\mathrm{P}}}^2 \to W \to B$, such that the first map is the inclusion of the boundary. Two representatives $W$ and $W'$ are equivalent if there is a diffeomorphism $W\to W'$ which commutes with the maps from $M\times {\mathbb R}{{\mathrm{P}}}^2$ and to $BO$.
2. \[item2\] The set $T_2$ of diffeomorphism classes of manifolds $W$ relative to $M\times {\mathbb R}{{\mathrm{P}}}^2$. An element is represented by $M\times {\mathbb R}{{\mathrm{P}}}^2 \to W$, which is the inclusion of the boundary. Two representatives $W$ and $W'$ are equivalent if there is a diffeomorphism $W\to W'$ which commutes with the maps from $M\times {\mathbb R}{{\mathrm{P}}}^2$.
3. The set $T_3$ of equivariant diffeomorphism classes of manifolds $V$ relative to $M\times S^2$. An element is represented by $(M\times S^2,(id,-id)) \to (V,\tau)$, which is the inclusion of the boundary, and where $\tau$ is a free involution. Two representatives $V$ and $V'$ are equivalent if there is a diffeomorphism $(V,\tau)\to (V',\tau')$ which commutes with the maps from $M\times S^2$.
4. The set $T_4$ of equivariant diffeomorphism classes of closed manifolds $X$ relative to $M\times D^3$. An element is represented by an equivariant embedding $(M\times D^3,(id,-id)) \to (X,\tau)$, where $\tau$ is free on the complement of the image. Two representatives $X$ and $X'$ are equivalent if there is a diffeomorphism $(X,\tau)\to (X',\tau')$ which commutes with the maps from $M\times D^3$.
5. The set $Emb_{{\mathbb Z}_2}(M,S^6)$ of equivariant diffeomorphism classes of manifolds $X$ relative to $M$. An element is represented by an equivariant embedding $(M,id) \to (X,\tau)$, where $\tau$ is free on the complement of the image. Two representatives $X$ and $X'$ are equivalent if there is a diffeomorphism $(X,\tau)\to (X',\tau')$ which commutes with the maps from $M$.
On the set $T_1$, the automorphism group $Aut(B)_{BO}$ of fiber homotopy classes of fiber homotopy self-equivalences of $B$ acts by post-composing $M\times {\mathbb R}{{\mathrm{P}}}^2 \to B$ with $B\to B$.
We saw that the group $Aut(B)_{BO}$ has four elements, and the action of the group on the set of normal $B$-structures on a manifold $f:M\to B$ (i.e. spin structures on $\nu_M-f^*L$) is given by just changing the spin structure $\sigma$ into $\sigma, -\sigma, \sigma +f^*t, -\sigma+f^*t$, where $t\in H^1(B;{\mathbb Z}_2)$ is the generator.
Thus the group $Aut(B)_{BO}$ acts freely and transitively on the set of distinguished normal $B$-structures on $M\times {\mathbb R}{{\mathrm{P}}}^2$. In particular each orbit of the action on the set in \[item1\] contains a unique element from each of the four bordism sets.
By uniqueness of the Postnikov decomposition, forgetting the map to $B$ is a bijection from the quotient of $T_1$ by $Aut(B)_{BO}$ to the set $T_2$. See also [@KrM]. Thus the set $T_2$ is in bijection with $\Omega_6^{(B,M\times {\mathbb R}{{\mathrm{P}}}^2)}\cong {\mathbb Z}^2\oplus {\mathbb Z}_4$.
The sets $T_2$, $T_3$ and $T_4$ are in bijective correspondence: One gets from $M\times D^3 \to X$ by restriction to $M\times S^2 \to V=X\setminus int(M\times D^3)$ and from $M\times S^2 \to V$ one takes the quotient by the involution to obtain $M\times {\mathbb R}{{\mathrm{P}}}^2 \to W=V/\tau$. Vice-versa, from $M\times {\mathbb R}{{\mathrm{P}}}^2 \to W$, take any non-trivial double covering $V$ of $W$ and any map $M\times S^2 \to V$ inducing the given $M\times {\mathbb R}{{\mathrm{P}}}^2 \to W$. The involution on $V$ is the non-trivial deck transformation. Up to equivalence, this does not depend on the choices made. And from $\phi:M\times S^2\to V$, obtain $M\times D^3\to X=M\times D^3\cup_\phi V$. The smooth structure on the latter depends on choices, but only up to equivalence.
On the sets $T_2$, $T_3$ and $T_4$, we have an action of the group of bundle automorphisms $Map(M,O(3))$ of $M\times D^3$: every bundle automorphism induces a self-diffeomorphism of the boundary $M\times S^2$ which induces a self-diffeomorphism of $M\times {\mathbb R}{{\mathrm{P}}}^2$, and we precompose with these diffeomorphisms. A bundle automorphism $f:M\to O(3)$ corresponds to the diffeomorphism $$\begin{gathered}
\phi_f:M\times {\mathbb R}{{\mathrm{P}}}^2 \to M\times {\mathbb R}{{\mathrm{P}}}^2\\
(x,\pm y)\mapsto (x,\pm f(x)\cdot y)\end{gathered}$$ for $y\in S^2$. So the bundle automorphism which is minus the identity on each fiber acts trivially on $M\times {\mathbb R}{{\mathrm{P}}}^2$, hence the action is trivial also on the other sets. (This corresponds in $T_4$ to the fact that the embeddings $i:M\times D^3 \to X$ and $M\times D^3\stackrel{id\times -id}\to M\times D^3 \stackrel i\to X$ are related by the equivariant diffeomorphism $\tau:(X,\tau)\to (X,\tau)$.) Thus we may restrict to orientation preserving bundle automorphisms.
The equivariant tubular neighbourhood of the fixed point set is unique up to isotopy and bundle automorphisms. But isotopies of embeddings can be enlarged to isotopies of the ambient space, so that they act trivially on the set of equivariant diffeomorphism classes. It follows that the action of the group of bundle automorphisms descends to an action of $[M,SO(3)]$, and that dividing out this action, we get the set $Emb_{{\mathbb Z}_2}(M,S^6)$.
The part which is a little more complicated is to see the action of ${\mathbb Z}\cong \pi_3(SO(3))\cong [M,SO(3)] \ni f$ on a bordism set $\Omega_6^{(B,M\times {\mathbb R}{{\mathrm{P}}}^2)}$. We can also precompose with the corresponding self-diffeomorphism $\phi_f$ of $M\times {\mathbb R}{{\mathrm{P}}}^2$, but this changes the precise map $M\times {\mathbb R}{{\mathrm{P}}}^2 \to B$. Still the normal $B$-structure is preserved: Up to homotopy, we may assume that $f:M\to SO(3)$ is trivial on a whole disk $D^3$, so that the normal $B$-structure on $D^3\times {\mathbb R}{{\mathrm{P}}}^2$ does not change. Spin structures on vector bundles over $M\times {\mathbb R}{{\mathrm{P}}}^2$ are in bijection with spin structures on their restrictions to ${\mathbb R}{{\mathrm{P}}}^2$ (using the natural framing of the normal bundle), and the latter are invariant. Hence these bundle automorphisms preserves the normal $B$-structure on $M\times {\mathbb R}{{\mathrm{P}}}^2$, and we may consider the action of $f$ on the bordism group as given by gluing a mapping cylinder of $f$.
Let $B\to BO$ be a fibration, let $N$ be an $(n-1)$-manifold with normal $B$-structure, $W_1, W_2, W_3$ be normal $B$-nullbordisms of $N$, let $\phi:N\to N$ be a diffeomorphism preserving the normal $B$-structure, let $C_\phi$ be the mapping cylinder of $\phi$, let $i_0,i_1:N\to C_{\phi}$ be the two natural inclusions, and let $T_\phi$ be the mapping torus of $\phi$. Then, in the bordism group $\Omega_n^B$, we have $$(W_1 \cup_{i_0} C_\phi \cup_{i_1} - W_2) - (W_1 \cup_{id_N} - W_2) = T_{\phi} = W_3 \cup_{i_0} C_\phi \cup_{i_1} -W_3.$$
Proof: $W\times I$ can be considered as a bordism between the manifolds with boundary $\partial W \times I$ and $-W \cup W$. The second equality in the statement follows from applying this to $W=W_3$. Similarly, the first equality is obtained by applying this to $W=W_1$ and $W=W_2$. [**q.e.d.**]{}
It follows that the action of $f$ on the bordism set $\Omega_6^{(B,S^3\times {\mathbb R}{{\mathrm{P}}}^2)}$ is the same as taking the disjoint sum with $D^4\times {\mathbb R}{{\mathrm{P}}}^2 \cup_{\phi_f} D^4\times {\mathbb R}{{\mathrm{P}}}^2$. The latter is an ${\mathbb R}{{\mathrm{P}}}^2$-bundle on $S^4$, and the corresponding double cover is a $S^2$-bundle over $S^4$ which can be identified with $S^2\to {\mathbb C}{{\mathrm{P}}}^3 \to {\mathbb H}{{\mathrm{P}}}^1$. Thus the induced action on $\Omega_6^{(\tilde{B},S^3\times S^2)}\cong{\mathbb Z}^2$ is by a generator for the first summand of $\Omega_6^{\tilde{B}}\cong {\mathbb Z}^2$.
It follows that the set $Emb_{{\mathbb Z}_2}(M,S^6)$ of orbits of the action on $T_4$ is in bijection with ${\mathbb Z}\oplus {\mathbb Z}_4$. In particular $Emb_{{\mathbb Z}_2}(M,S^6)$ is a group which acts freely and transitively on $Emb_{{\mathbb Z}_2}(M,S^6)$ for all $M$.
Comparing with the non-equivariant classification of embeddings in $S^6$, we see that forgetting the involution defines a map $Emb_{{\mathbb Z}_2}(M,S^6) \cong {\mathbb Z}\oplus {\mathbb Z}_4 \to Emb(M,S^6) \cong {\mathbb Z}\oplus H_1(M)$ which is equivariant with respect to the group homomorphism $Emb_{{\mathbb Z}_2}(S^3,S^6) \cong {\mathbb Z}\oplus {\mathbb Z}_4 \to Emb(S^3,S^6) \cong {\mathbb Z}$ given by $(a,b)\mapsto 2a$. In particular the image of $Emb_{{\mathbb Z}_2}(M,S^6)$ is acted upon freely by $2{\mathbb Z}\subseteq {\mathbb Z}$, and the map is 4-to-1.
For embeddings of ${\mathbb Z}_2$-homology spheres we saw that the elements $[i:M\to S^6] \in Emb(M,S^6)$ with vanishing Whitney invariant are acted freely and transitively upon by $C_3^3\cong {\mathbb Z}$. The subset of isotopy classes of embeddings which are the fixed point sets of conjugations are acted freely and transitively upon by $2{\mathbb Z}\subseteq {\mathbb Z}$. There are up to equivariant diffeomorphism relative to $i$ exactly four such conjugations for every $i$. This proves theorems \[thm13\] and \[embcl\].
From embeddings to submanifolds - proof of theorem \[invcl\]
------------------------------------------------------------
The more natural thing is to classify involutions without the additional identification of the fixed point set with a fixed 3-manifold $M$. The invariant of the involution should be its fixed point set, i.e. a submanifold of $S^6$. In order to get from embeddings up to diffeomorphism to submanifolds up to diffeomorphism, it suffices to divide out the action of the group of self-diffeomorphisms $Diff(M)$. Since isotopies extend to ambient isotopies (this also holds in this equivariant case, since it suffices to extend a vector field on the fixed point set to an equivariant vector field on the whole space), and these give equivariant diffeomorphisms, the action of $Diff(M)$ factors through the mapping class group of $M$.
Since in our case, the map $M\times {\mathbb R}{{\mathrm{P}}}^2 \to Q$ factors through ${\mathbb R}{{\mathrm{P}}}^2$, this map does not change, so that we get the same normal $B$-structure. Then the action of a self-diffeomorphism $f:M\to M$ on the bordism set $\Omega_6^(B,M\times {\mathbb R}{{\mathrm{P}}}^2)$ is by disjoint union with the mapping torus $T_{f\times id}$ of $f\times id: M\times {\mathbb R}{{\mathrm{P}}}^2 \to {\mathbb R}{{\mathrm{P}}}^2$. Now $T_{f\times id}=T_f \times {\mathbb R}{{\mathrm{P}}}^2$, and the map to $Q$ factors again through ${\mathbb R}{{\mathrm{P}}}^2$. Spin structures twisted by $L$ on $T_f\times {\mathbb R}{{\mathrm{P}}}^2$ are products of a spin structure on $M$ and a spin-structure twisted by $L$ on ${\mathbb R}{{\mathrm{P}}}^2$. Thus a spin-nullbordism of $T_f$ gives a normal $B$-nullbordism of $T_{f\times id}$. Recall that a four-dimensional spin manifold is zero bordant iff its signature is zero, and that the signature of a mapping torus is always zero. Hence the action of the mapping class group of $M$ on the bordism set is trivial. As a consequence, the mapping class group of $M$ acts trivially on $Emb_{{\mathbb Z}_2}(M,S^6)$. This proves theorem \[invcl\].
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[<span style="font-variant:small-caps;">Max-Planck-Institut für Mathematik, Bonn, Vivatsgasse 7, 53111 Bonn, Germany</span>]{}\
E-mail:
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper studies the statistical properties of the web of import-export relationships among world countries using a weighted-network approach. We analyze how the distributions of the most important network statistics measuring connectivity, assortativity, clustering and centrality have co-evolved over time. We show that all node-statistic distributions and their correlation structure have remained surprisingly stable in the last 20 years – and are likely to do so in the future. Conversely, the distribution of (positive) link weights is slowly moving from a log-normal density towards a power law. We also characterize the autoregressive properties of network-statistics dynamics. We find that network-statistics growth rates are well-proxied by fat-tailed densities like the Laplace or the asymmetric exponential-power. Finally, we find that all our results are reasonably robust to a few alternative, economically-meaningful, weighting schemes.'
author:
- Giorgio Fagiolo
- Javier Reyes
- Stefano Schiavo
date: November 2008
title: |
The World-Trade Web:\
Topological Properties, Dynamics, and Evolution
---
Introduction
============
In the last decade, a lot of effort has been devoted to the empirical exploration of the architecture of the World Trade Web (WTW) from a complex-network perspective [@LiC03; @SeBo03; @Garla2004; @Garla2005; @serrc07; @Bhatta2007a; @Bhatta2007b; @Garla2007; @FagioloClustArxiv; @FRS2007wp; @Communities2007; @fagic08]. The WTW, also known as International Trade Network (ITN), is defined as the network of import/export relationships between world countries in a given year. Understanding the topological properties of the WTW, and their evolution over time, acquires a fundamental importance in explaining international-trade issues such as economic globalization and internationalization [@StiglitzBook; @GlobBook]. Indeed, it is common wisdom that trade linkages are one of the most important channels of interaction between world countries [@Krugman1995]. For example, they can help to explain how economic policies affect foreign markets [@HelliwellPadmore1985]; how economic shocks are transmitted among countries [@Artis2003]; and how economic crises spread internationally [@Forbes2002]. However, direct bilateral-trade relationships can only explain a small fraction of the impact that an economic shock originating in a given country can have on another one, which is not among its direct-trade partners [@AbeFor2005]. Therefore, a complex-network analysis [@AlbertBarabasi2002; @DoroMendes2003; @Newman2003; @PastosVespignani2004] of the WTW, by characterizing in detail the topological structure of the network, can go far beyond the scope of standard international-trade indicators, which instead only account for bilateral-trade direct linkages [^1].
The first stream of contributions that have studied the properties of the WTW has employed a *binary-network analysis*, where a (possibly directed) link between any two countries is either present or not according to whether the trade flow that it carries is larger than a given lower threshold [@SeBo03; @Garla2004; @Garla2005]. According to these studies, the WTW turns out to be characterized by a high density and a right-skewed (but not exactly power-law) distribution for the number of partners of a given country (i.e., the node degree). Furthermore, there seems to be evidence of bimodality in the node-degree distribution. While the majority of countries entertain few trade partnerships, there exists a group of countries trading with almost everyone else [@FRS2007wp; @fagic08]. Also, the binary WTW is a very disassortative network (i.e., countries holding many trade partners are on average connected with countries holding few partners) and is characterized by some hierarchical arrangements (i.e., partners of well connected countries are less interconnected among them than those of poorly connected ones). Remarkably, these properties are quite stable over time [@Garla2005].
More recently, a few contributions have adopted a *weighted-network approach* [@Barr04; @Barr05; @Bart05] to the study of the WTW, where each link is weighted by some proxy of the trade intensity that it carries. The motivation is that a binary approach cannot fully extract the wealth of information about the trade intensity flowing through each link and therefore might dramatically underestimate the role of heterogeneity in trade linkages. Indeed, Refs. [@FagioloClustArxiv; @FRS2007wp; @fagic08] show that the statistical properties of the WTW viewed as a weighted network crucially differ from those exhibited by its unweighted counterpart. For example, the weighted version of the WTW appears to be weakly disassortative. Moreover, well-connected countries tend to trade with partners that are strongly-connected between them. Finally, the distribution of the total trade intensity carried by each country (i.e., node strength) is right-skewed, indicating that a few intense trade connections co-exist with a majority of low-intensity ones. This is confirmed, at the link level, by Refs. [@Bhatta2007b; @Bhatta2007a] who find that the distribution of link weights can be approximated by a log-normal density robustly across the years [@LiC03]. The main insight coming from these studies is that a weighted-network analysis is able to provide a more complete and truthful picture of the WTW than a binary one [@fagic08].
Additional contributions have instead focused on specific features of the structure and dynamics of the WTW. For example, Refs. [@Garla2004; @Garla2007] find evidence in favor of a hidden-variable model, according to which the topological properties of the WTW (in both the binary and weighted case) can be well explained by a single node-characteristic (i.e., country gross-domestic product) controlling for the potential ability of a node to be connected. Furthermore, Ref. [@serrc07] studies the weighted network of bilateral trade imbalances [^2]. The Authors note that also the international trade-imbalance network is characterized by a high level of heterogeneity: for each country, the profile of trade fluxes is unevenly distributed across partners. At the network level, this prompts to the presence of high-flux backbones, i.e. sparse subnetworks of connected trade fluxes carrying most of the overall trade in the network. The Authors then develop a method to extract (for any significance level) the flux backbone existing among countries and links. This turns out to be extremely effective in sorting out the most relevant part of the trade-imbalance network and can be conveniently used for visualization purposes. Finally, Ref. [@Communities2007] considers the formation of “trade islands”, that is connected components carrying a total trade flow larger than some given thresholds. The analysis of the evolution of the WTW community structure [@NewmanCommunities] finds mixed evidence for globalization.
In this paper we present a more thorough study of the topological properties of the WTW by focusing on distribution dynamics and evolution. More specifically, following the insights of Ref. [@fagic08], we employ a weighted network approach to characterize, for the period 1981-2000, the distribution of the most important network statistics measuring node connectivity, assortativity, clustering and centrality; as well as link weights. We ask three main types of questions: (i) Have (and, if so, how) the distributional properties of these statistics (and their correlation structure) been changing within the sample period considered? (ii) Can we make any prediction on the out-of-sample evolution of such distributions? (iii) Do the answers to the previous questions change if we play with a number of alternative, economically-meaningful weighting schemes (i.e., if we allow for different rules to weight existing links)?
The rest of the paper is organized as follows. Section \[Sec:Data\] presents the data sets and defines the statistics studied in the paper. Section \[Sec:Results\] introduces the main results. Finally, Section \[Sec:Conclusions\] concludes and discusses future extensions.
Data and Definitions {#Sec:Data}
====================
We employ international-trade data provided by [@GledData2002] to build a time-sequence of weighted directed networks. Our balanced panel refers to $T=20$ years (1981-2000) and $N=159$ countries. For each country and year, data report trade flows in current US dollars. To build adjacency and weight matrices, we followed the flow of goods. This means that rows represent exporting countries, whereas columns stand for importing countries. We define a “trade relationship” by setting the generic entry of the (binary) adjacency matrix $\tilde{a}_{ij}^t=1$ if and only if exports from country $i$ to country $j$ ($e_{ij}^t$) are strictly positive in year $t$.
Following Refs. [@LiC03; @Bhatta2007a; @Bhatta2007b; @Garla2007], the weight of a link from $i$ to $j$ in year $t$ is defined as $\tilde{w}_{ij}^t=e_{ij}^t$ [^3]. Thus, the sequence of $N\times N$ adjacency and weight matrices $\{\tilde{A}^t,\tilde{W}^t\}$, $t=1981,...,2000$ fully describes the within-sample dynamics of the WTW.
A preliminary statistical analysis of both binary and weighted matrices suggests that $(\tilde{A}^t,\tilde{W}^t)$ are sufficiently symmetric to justify an undirected analysis for all $t$. From a binary perspective, on average about 93% of WTW directed links are reciprocated in each given year. This means that, almost always, if country $i$’s exports to country $j$ are positive ($\tilde{a}_{ij}^t=1$), then $\tilde{a}_{ji}^t=1$, i.e. country $j$’s exports to country $i$ are also positive. To check more formally this evidence from a weighted perspective, we have computed the weighted symmetry index defined in Ref. [@FagioloSymmEcoBull]. The index ranges in the sample period between 0.0017 and 0.0043, signalling a relatively strong and stable symmetry of WTW weight matrices [^4]. We have therefore symmetrized the network by defining the entries of the new adjacency matrix $A^t$ so that $a_{ij}^t=1$ if and only if either $\tilde{a}_{ij}^t=1$ or $\tilde{a}_{ji}^t=1$, and zero otherwise. Accordingly, the generic entry of the new weight matrix $W^t$ is defined as $w_{ij}^t=\frac{1}{2}(\tilde{w}_{ij}^t+\tilde{w}_{ji}^t)$. This means that the symmetrized weight of link ${ij}$ is proportional to the total trade (imports plus exports) flowing through that link in a given year. Finally, in order to have $w_{ij}^t\in[0,1]$ for all $(i,j)$ and $t$, we have re-normalized all entries in $W^t$ by their maximum value $w^t_{\ast}=max_{i,j=1}^{N}w_{ij}^t$.
For each $(\tilde{A}^t,\tilde{W}^t)$, we study the distributions of the following node statistics:
- *Node degree* [@AlbertBarabasi2002; @Pastor2001], defined as $ND_i^t=A_{(i)}^t\textbf{1}$, where $A_{(i)}^t$ is the $i$-th row of $A^t$ and $\textbf{1}$ is a unary vector. ND is a measure of binary connectivity, as it counts the number of trade partners of any given node. Although we mainly focus here on a weighted-network approach, we study ND because of its natural interpretation in terms of number of trade partnerships and bilateral trade agreements.
- *Node strength* [@DeMontis2005], defined as $NS_i^t=W_{(i)}^t
\textbf{1}$, where again $W_{(i)}^t$ is the $i$-th row of $W^t$. While ND tells us how many partners a node holds, NS is a measure of weighted connectivity, as it gives us an idea of how intense existing trade relationships of country $i$ are.
- *Node average nearest-neighbor strength* [@DeMontis2005], that is $ANNS_i^t=(A_{(i)}^t W^t\textbf{1})/(A_{(i)}^t\textbf{1})$. ANNS measures how intense are trade relationships maintained by the partners of a given node. Therefore, the correlation between ANNS and NS is a measure of network assortativity (if positive) or disassortativity (if negative). It is easy to see that ANNS boils down to average nearest-neighbor degree (ANND) if $W^t$ is replaced by $A^t$.
- *Weighted clustering coefficient* [@Saramaki2006; @FagioloClustArxiv], defined as $WCC_i^t=({[W^t]}^{\left[\frac{1}{3}\right]})_{ii}^{3}/(ND_i^t(ND_i^t-1))$. Here $Z_{ii}^3$ is the $i$-th entry on the main diagonal of $Z\cdot Z\cdot Z$ and $Z^{\left[\frac{1}{3}\right]}$ stands for the matrix obtained from $Z$ after raising each entry to $1/3$. WCC measures how much clustered a node $i$ is from a weighted perspective, i.e. how much intense are the linkages of trade triangles having country $i$ as a vertex [^5]. Again, replacing $W^t$ with $A^t$, one obtains the standard binary clustering coefficient (BCC), which counts the fraction of triangles existing in the neighborhood of any give node [@WattsStrogatz1998].
- *Random-walk betweenness centrality* [@newm05; @five06], which is a measure of how much a given country is globally-central in the WTW. A node has a higher random-walk betweenness centrality (RWBC) the more it has a position of strategic significance in the overall structure of the network. In other words, RWBC is the extension of node betweenness centrality [@Scott2000] to weighted networks and measures the probability that a random signal can find its way through the network and reach the target node where the links to follow are chosen with a probability proportional to their weights.
The above statistics allow one to address the study of node characteristics in terms of four dimensions: connectivity (ND and NS), assortativity (ANND and ANNS, when correlated with ND and NS), clustering (BCC and WCC) and centrality (RWBC). In what follows, we will mainly concentrate the analysis on ND and the other weighted statistics (NS, ANNS, WCC, RWBC), but we occasionally discuss, when necessary, also the behavior of ANND and BCC.
We further explore the network-connectivity dimension by studying the time-evolution of the link-weight distribution $w^t=\{w_{ij}^t,
i\neq j=1,\dots,N\}$. In particular, we are interested in assessing the fraction of links that are zero in a given year $t$ and becomes positive in year $t+\tau$, $\tau=1,2,\dots$ and the percentage of links that are strictly positive at $t$ and disappear in year $t+\tau$. This allows one to keep track of trade relationships that emerge or become extinct during the sample period [^6].
Results {#Sec:Results}
=======
Shape, Moments, and Correlation Structure of Network Statistics {#SubSec:Shape}
---------------------------------------------------------------
We begin by studying the shape of the distributions of node and link statistics and their dynamics within the sample period under analysis. As already found in Refs. [@Bhatta2007a; @Bhatta2007b], link weight distributions display relatively stable moments (see Figure \[Fig:linkweights\_distr\_stats\]) and are well proxied by log-normal densities in each year (cf. Figure \[Fig:srp\_linkweights\] for an example). This means that the majority of trade linkages are relatively weak and coexist with few high-intensity trade partnerships. The fact that the first four moments of the distribution do not display remarkable structural changes in the sample period hints to a relatively strong stability of the underlying distributional shapes [^7]. We shall study this issue in more details below.
A similar stable pattern is detected also for the moments of the distributions of all node statistics under analysis, see Figure \[Fig:ns\_distr\_stats\] for the case of NS distributions. To see that this applies in general for node statistics, we have computed the time average (across 19 observations) of the absolute value of 1-year growth rates of the first four interesting moments of ND, NS, ANNS, WCC and RWBC statistics, namely mean, standard deviation, skewness and kurtosis [^8]. Table \[Tab:moments\_gr\] shows that these average absolute growth-rates range in our sample between 0.0043 and 0.0615, thus indicating that the shape of these distributions seem to be quite stable over time.
But how does the shape of node- and link-statistic distributions look like? To investigate this issue we have begun by running normality tests on the logs of (positive-valued) node and link statistics. As Table \[Tab:normality\] suggests [^9], binary-network statistic distributions are never log-normal (i.e., their logs are never normal), whereas all weighted-network statistics but RWBC seem to be well-proxied by log-normal densities. To see why this happens, Figure \[Fig:nd\_rsp\_plus\_kernel\_2000\] shows the rank-size plot of ND in 2000 (with a kernel density estimate in the inset) [^10]. It is easy to see that ND exhibits some bimodality, with the majority of countries featuring low degrees and a bunch of countries trading with almost everyone else. Figure \[Fig:srp\_ns\_2000\] shows instead, for year 2000, how NS is nicely proxied by a log-normal distribution. This is not so for RWBC, whose distribution seems instead power-law in all years, with slopes oscillating around -1.15, see Figures \[Fig:srp\_rwbc\_2000\] and \[Fig:power\_law\_rwbc\]. Therefore, being more central is more likely than having high NS, ANNS, or WCC (i.e., the latter distributions feature upper tails thinner than that of RWBC distributions; we shall return to complexity issues related to this point when discussing out-of-sample evolution of the distributions of node and link statistics). The foregoing qualitative statements can be made quantitative by running comparative goodness-of-fit (GoF) tests to check whether the distributions under study come from pre-defined density families. To do so, we have run Kolmogorov-Smirnov GoF tests [@Massey1951; @Owen1962] against three null hypotheses, namely that our data can be well described by log-normal, stretched exponential, or power-law distributions. The stretched-exponential distribution (SED) has been employed because of its ability to satisfactorily describe the tail behavior of many real-world variables and network-related measures [@SED1; @SED2]. Table \[Tab:kstest\] reports results for year 2000 in order to facilitate a comparison with Figures \[Fig:nd\_rsp\_plus\_kernel\_2000\]-\[Fig:srp\_rwbc\_2000\], but again the main insights are confirmed in the entire sample. It is easy to see that the SED does not successfully describe the distributions of our main indicators. On the contrary, it clearly emerges that NS, ANNS and RWBC seem to be well described by log-normal densities, whereas the null of power-law RWBC cannot be rejected. For ND, neither of the three null appears to be a satisfactory hypothesis for the KS test.
We now discuss in more detail the evolution over time of the moments of the distributions of node statistics. As already noted in Refs. [@Garla2004; @Bhatta2007a; @Bhatta2007b; @fagic08], the binary WTW is characterized by an extremely high network density $d^t=\frac{1}{N(N-1)}\sum_i \sum_j{a^t_{ij}}$, ranging from 0.5385 to 0.6441. Figure \[Fig:ave\_nd\_ns\] plots the normalized (by $N$) population-average of ND, which is equal to network density up to a $N^{-1}(N-1)$ factor, together with population-average of NS. While the average number of trade partnerships is very high and slightly increases over the years, their average intensity is rather low (at least as compared to NS conceivable range, i.e. $[0,N-1]$) and tends to decline [^11]. As far as ANND/ANNS, clustering and centrality are concerned, a more meaningful statistical assessment of the actual magnitude of empirical population-average statistics requires comparing them with expected values computed after reshuffling links and/or weights. In what follows, we consider two reshuffling schemes (RSs). For binary statistics, we compute expected values after reshuffling existing links by keeping fixed the observed density $d^t$ (hereafter, B-RS). For weighted ones, we keep fixed the observed adjacency matrix $A^t$ and re-distribute weights at random by reshuffling the empirical link-weight distribution $w^t=\{w_{ij}^t, i\neq j=1,\dots,N\}$ (hereafter, W-RS) [^12]. Figure \[Fig:expected\] shows empirical averages vs. expected values over time. Notice that empirical averages of ANND, BCC, ANNS, WCC, and RWBC are larger than expected, meaning that the WTW is on average more clustered; features a larger nearest-neighbor connectivity, and countries are on average more central than expected in comparable random graphs.
The relatively high clustering level detected in the WTW hints to a network architecture that, especially in the binary case, features a peculiar clique structure. To further explore the clique structure of the WTW we computed, in the binary case, the node k-clique degree (NkCD) statistic [@CliqueDeg1]. The NkCD for node $i$ is defined as the number of $k$-size fully connected subgraphs containing $i$. Since NkCD for $k=2$ equals node degree, and for $k=3$ is closely related to the binary clustering coefficient, exploring the properties of NkCD for $k>3$ can tell us something about higher-order clique structure of the WTW. Figure \[Fig:CliqueDeg\] reports —for year 2000— an example of the cumulative distribution function (CDF) of NkCDs for $k=4,5$ (very similar results hold also for the case $k=6$). In the insets, a kernel estimate of the corresponding probability distributions are also provided. We also compare empirical CDFs with their expected shape in random nets where, this time, links are reshuffled so as to preserve the initial degree sequence (i.e., we employ the edge-crossing algorithm, cf. Refs. [@CliqueDeg1; @CliqueDeg2] for details). The plots indicate that the majority of nodes in the WTW are involved in a very large number of higher-degree cliques, but that the observed pattern is not that far from what would have been expected in networks with the same degree sequence (if any, observed NkCDs place relatively more mass on smaller NkCDs and less on medium-large values of NkCDs). Notice that these findings are at odds with what commonly observed in many other real-world networks, where NkCD distributions are typically power law. However, this peculiar feature of the WTW does not come as surprise, given its very high average degree, and the large number of countries trading with almost everyone else in the sample. Note also that the high connectivity of the binary WTW makes it very expensive to compute NkCD distributions for $k>6$. In order to shed some light on the clique structure for larger values of $k$, we have employed standard search algorithms to find all Luce and Perry (LP) $k$-cliques [@CliqueDeg3], that is maximally connected k-size subgraphs [^13]. Notice that in general the WTW does not exhibit LP cliques with size smaller than 12 (this is true in any year). Furthermore, a large number of LP cliques are of a size between 50 and 60 (see Figure \[Fig:LP-Cliques\], left panel, for a kernel-density estimate of LP clique size distribution in 2000). Hence, the binary WTW, due to its extremely dense connectivity pattern, seems to display a very intricate clique structure. Additional support to this conclusion is provided by Figure \[Fig:LP-Cliques\] (right panel), where we plot a kernel-density estimate of the distribution of the number of agents belonging to LP cliques of any size. Although a large majority of countries belong to less than 2000 LP cliques, a second peak in the upper-tail of the distribution emerges, indicating that a non-negligible number of nodes are actually involved in at least 10000 different LP cliques of any size (greater than 12).
Correlation Structure and Node Characteristics
----------------------------------------------
To further explore the topological properties of the WTW, we turn now to examine the correlation structure existing between binary- and weighted-network statistics [^14]. As expected [@SeBo03; @Garla2004; @Garla2005], Figure \[Fig:corr\_stats\] shows that the binary version of the WTW is strongly disassortative in the entire sample period. Furthermore, countries holding many trade partners do not typically form trade triangles. Conversely, the weighted WTW turns out to be a weakly disassortative network. Moreover, countries that are intensively connected (high NS) are also more clustered (high WCC). This mismatch between binary and weighted representations can be partly rationalized by noticing that the correlation between NS and ND is positive but not very large (on average about 0.45), thus hinting to a topological structure where having more trade connections does not automatically imply to be more intensively connected to other countries in terms of total trade controlled. As to centrality, RWBC appears to be positively correlated with NS, signalling that in the WTW there is little distinction between global and local centrality [^15].
Another interesting issue to explore concerns the extent to which country specific characteristics relate to network properties. We focus here on the correlation patterns between network statistics and country per capita GDP (pcGDP) in order to see whether countries with a higher income are more/less connected, central and clustered. The outcomes are very clear and tend to mimic those obtained above for the correlation structure among network statistics. Figure \[Fig:corr\_pcgdp\] shows that high-income countries tend to hold more, and more intense, trade relationships and to occupy a more central position. However, they trade with few and weakly-connected partners, a pattern suggesting the presence of a sort of “rich-club phenomenon” [^16].
To further explore this evidence, we have firstly considered the binary version of the WTW and we have computed the rich-club coefficient $R^t(k)$, defined, for each time period $t$ and degree $k$, as the percentage of edges in place among the nodes having degree higher than $k$ (see, e.g., Ref. [@RichClub1]). Since a monotonic relation between $k$ and $R^t(k)$ is to be expected in many networks, due to the intrinsic tendency of hubs to exhibit a larger probability of being more interconnected than low-degree nodes, $R^t(k)$ must be corrected for its version in random uncorrelated networks (see Ref. [@RichClub2] for details). If the resulting (corrected) rich-club index $\tilde{R}^t(k)>1$, especially for large values of $k$, then the corresponding graph will exhibit statistically-significant evidence for rich-club behavior. In our case, the binary WTW does not seem to show any clear rich-club ordering, as Figure \[Fig:richclub\_binary\] shows for year 2000. This contrasts with, e.g., the case of scientific collaborations networks, but is well in line with the absence of any rich-club pattern in protein interaction and Internet networks [@RichClub2]. This may be intuitively due to the very high density of the underlying binary network, but also to the fact that, as suggested by Ref. [@RichClub2] and discussed above, any binary structure underestimates the importance of intensity of interactions carried by the edges. In fact, if studied from a weighted-network perspective, the WTW exhibits indeed much more rich-club ordering than its binary version. To see why, for any given year we have sorted in a descending order the nodes (countries) according to their strength, taken as another measure of richness. We have then computed the percentage of the total trade flows in the network that can be imputed to the trade exchanges occurring (only) among the first $k$ nodes of the NS year-$t$ ranking, i.e. a $k$-sized rich club. More precisely, let $\{i_1^t,i_2^t,...,i_N^t\}$ the labels of the $N$ nodes sorted in a descending order according to their year-$t$ NS. The coefficient for a given rich-club size $k>1$ is computed as the ratio between $\sum_{j=1}^{k}\sum_{h=1}^{k}{w_{i_j^t i_k^t}}$ and the sum of all entries of the matrix $W^t$. Figure \[Fig:richclub\_weighted\] shows how our crude “weighted rich-club coefficient” behaves in year 2000 for an increasing rich-club size. It is easy to see that the 10 richest countries in terms of NS are responsible of about 40% of the total trade flows (see dotted vertical and horizontal lines), a quite strong indication in favor of the existence of a rich club in the weighted WTW. This is further confirmed if one compares the empirical observations (very high) with their expected values under a W-RS reshuffling scheme (much lower, also after 95% confidence intervals have been considered) [^17].
In summary, the overall picture that our correlation analysis suggests is one where countries holding many trade partners and/or very intense trade relationships are also the richest and most (globally) central; typically trade with many countries, but very intensively with only a few very-connected ones; and form few, but intensive, trade clusters (triangles).
Furthermore, our correlation analysis provides further evidence to the distributional stability argument discussed above. Indeed, we have already noticed that the first four moments of the distributions of statistics under study (ND, NS, ANNS, WCC, RWBC) display a marked stability over time. Figure \[Fig:corr\_stats\] shows that also their correlation structure is only weakly changing during the sample period. This suggests that the whole architecture of the WTW has remained fairly stable between 1981 and 2000. To further explore the implications of this result, also in the light of the ongoing processes of internationalization and globalization, we turn now to a more in-depth analysis of the in-sample dynamics and out-of-sample evolution of WTW topological structure.
Within-Sample Distribution Dynamics {#SubSec:DistrDyn}
-----------------------------------
The foregoing evidence suggests that the shape of the distributions concerning the most important topological properties of WTW displays a rather strong stability in the 1981-2000 period. However, distributional stability does not automatically rule out the possibility that between any two consecutive time steps, say $t-\tau$ and $t$, a lot of shape-preserving turbulence was actually going on at the node and link level, with many countries and/or link weights moving back and forth across the quantiles of the distributions. In order to check whether this is the case or not, we have computed stochastic-kernel estimates [@ChungStocKern1960; @Futia1982] for the distribution dynamics concerning node and link statistics. More formally, consider a real-valued node or link statistic $X$. Let $\phi^{\tau}(\cdot,\cdot)$ be the joint distribution of $(X^t,
X^{t-\tau})$ and $\psi^{\tau}(\cdot)$ be the marginal distribution of $X^{t-\tau}$. We estimate the $\tau$-year stochastic kernel, defined as the conditional density $s^{\tau}(x|y)=\phi^{\tau}(x,y)/\psi^{\tau}(y)$ [^18].
Figures \[Fig:kernel\_ns\] and \[Fig:kernel\_linkweights\] present the contour plots of the estimates of the 1-year kernel density of logged NS and logged positive link-weights. Notice that the bulk of the probability mass is concentrated close to the main diagonal (displayed as a solid $45^\circ$ line). Similar results are found for all other real-valued node statistics (ANNS, WCC and RWBC) also at larger time lags. The kernel density of logged positive link weights, contrary to the logged NS one, is instead extremely polarized towards the extremes of the distribution range, whereas in the middle of the range it is somewhat flatter (Figure \[Fig:kernel\_linkweights\]). We will go back to the implications that this feature has on out-of-sample distributional evolution below.
This graphical evidence hints to a weak turbulence for the distributional dynamics of all node and link statistics under analysis. To better appreciate this point, we have estimated, for all five node statistics employed above (ND, NS, ANNS, WCC, RWBC), as well as link-weight distributions, the entries of $\tau$-step Markov transition matrices [@StuartOrd1994], where $\tau=1,2,\dots,T-1$ is the time lag. More formally, suppose that the distribution dynamics of the statistic $X$ can be well described using $K$ quantile classes (QCs) in every year $t$ (see footnote 72). Given the above stability results, we can assume that the process driving the distribution dynamics of $X$ is stationary and can be well represented by a discrete-state Markov process defined over such $K$ QCs. Let $n_{i,j}^{t-\tau,t}$ be the number of countries whose statistic $X$ was in QC $i$ in year $t-\tau$ and moved to QC $j$ in year $t$. Then the statistic: $$\hat{p}_{ij}^{\tau}=\frac{\sum_{t=\tau+1}^{T}{n_{i,j}^{t-\tau,t}}}{\sum_{t=2}^{T}\sum_{h=1}^{K}{n_{i,h}^{t-\tau,t}}}$$ can be shown to be the maximum likelihood estimators of the true, unobservable, $\tau$-step transition probability $p_{ij}^{\tau}$, i.e. the probability that $X$ belongs to QC $i$ at $t-\tau$ and to QC $j$ at $t$ [@AndersonGoodman1957].
In order to build a measure of persistence of distribution dynamics, we have computed for each node or link statistic $X$ the percentage mass of probability that lies within a window of $\omega$ quantiles from the main diagonal of the estimated $\tau$-step transition-probability matrix $\hat{P}^{\tau}=\{\hat{p}_{ij}^{\tau}\}$, defined as:
$$M_{\omega,K}^{\tau}(X)=\frac{1}{K}\sum_{h=1}^{K}\sum_{l:|l-h|\leq
w}^{K}{\hat{p}_{hl}^{\tau}},$$
where the window $\omega=0,1,\dots,K-1$. For example, when $\omega=0$, $M_{0,K}^{\tau}(X)$ boils down to the trace of $\hat{P}^{\tau}$ divided by $K$, whereas if $\omega=1$, $M_{1,K}^{\tau}(X)$ is the average of all the entries in the main diagonal and those lying one entry to the right and one entry to the left of the main diagonal itself. The statistic $M_{\omega,K}^{\tau}(X)\in [0,1]$ and increases the larger the probability that a country remains in the same (or nearby) QC between $t-\tau$ and $t$ (for any given choice of $\tau$, $w$ and $K$).
Table \[Tab:distr\_dynamics\_nodes\] shows for our main statistics (ND, NS, ANNS, WCC, RWBC) and $K=10$, the values of $M_{\omega,K}^{\tau}(X)$ as $\tau \in \{1,4,7,10\}$ and $\omega=0,1$ [^19]. The figures strongly supports the result obtained by looking at the estimated stochastic kernels. Indeed, the entries of $\hat{P}^{\tau}$ close to the main diagonal always represent a large mass of probability, thus hinting to a distribution dynamics that in the period 1981-2000 is characterized by a rather low turbulence. For example, more than 96% of countries are characterized by node statistics that either stick to the same QC between $t-\tau$ and $t$, or just move to a nearby QC of the distribution. This share is often close to 99%. To better statistically evaluate the figures in Table \[Tab:distr\_dynamics\_nodes\], we have also estimated the distribution of $M_{\omega,K}^{\tau}(X)$ under reshuffling scheme W-RS, i.e. in random graphs where we keep fixed the observed adjacency matrix $A^t$ and we re-distribute weights at random by reshuffling the observed link-weight distribution [^20]. This allows us to compute confidence intervals (at 95%) for $M_{\omega,K}^{\tau}(X)$. As reported in Table \[Tab:distr\_dynamics\_nodes\], the empirical values are always larger than the upper bound of these confidence intervals, thus confirming the relatively strong persistence found in WTW node-statistic dynamics.
The same analysis can be also applied to the link-weight distribution $w^t=\{w_{ij}^t, i\neq j=1,\dots,N\}$. In order not to treat the same way existing links (with strictly positive weight) and absent links (with a zero weight), we first define the two link sets $L_0^t=\{(i,j),i\neq j=1,\dots,N:w_{ij}^t=0\}$ and $L_{+}^t=\{(i,j),i\neq j=1,\dots,N:w_{ij}^t>0\}$ and we then separately study the within-sample dynamics of the associated link distributions $w_0^t=\{w_{ij}^t\in w^t: (i,j)\in L_0^t\}$ and $w_{+}^t=\{w_{ij}^t\in w^t: (i,j)\in L_{+}^t\}$. To begin with, notice that a strong persistence also characterizes the dynamics of transition from an absent link to an existing one (and back). Indeed, the estimated probability of remaining an absent link (zero weight) is 0.9191, while that of remaining a present link (positive weight) is 0.9496. Thus, the link birth-rate is on average about 8%, while the death-rate is around 5% [^21]. This means that in the period 1981-2000, the WTW has shown a slight tendency toward an increase in trade relationships. This is remarkable for two reasons. First, our panel of countries has been balanced in order to focus on a fixed number of nodes. Second, the density of the network was already very high at the beginning. Table \[Tab:distr\_dynamics\_linkweights\] shows instead the persistence measure $M_{\omega,K}^{\tau}(X)$ where $X$ are the distributions of positive link-weights $w_{+}^t$ and the number of QCs is set to $K=20$. Again, most of transitions occur within the same or nearby QCs, signaling that also the dynamics of weight distributions of existing links is rather persistent. Furthermore, as happens for node statistics, also in this case confidence intervals (at 95%) for randomly-reshuffled weights always lie to the left of the observed value of $M$. Very similar results are obtained computing the persistence measure $M$ to logged link-weight distributions.
Country-Ranking Dynamics
------------------------
The distributional-stability results obtained in the foregoing sections naturally hint to the emergence of a lot of stickiness in country rankings (in terms of node statistics) as well. To explore this issue, for each year $t=1981,\dots\,2000$ we have ranked our $N=159$ countries according to any of the five main statistics employed so far (ND, NS, ANNS, WCC, RWBC) in a descending order. The first question we are interested in is assessing to which extent also these rankings are sticky across time. We check stability of rankings by computing the time-average of Spearman rank-correlation coefficients (SRCC) [@Spearman; @HollanderWolfe1973] between consecutive years [^22]. More formally, let $r_{(i)}^t (X)$ be the rank of country $i=1,\dots,N$ in year $t$ according to statistic $X$, and $\rho_{t-1,t}(X)$ be the SRCC between rankings at two consecutive years $t-1$ and $t$, for $t=2,\dots,T$. Our ranking-stability index (RSI) for the statistic $X$ is defined as $$RSI(X)=\frac{1}{T-1}\sum_{t=2}^{T}{\rho_{t-1,t}(X)}.$$ Of course $RSI(X)\in [-1,1]$, where $RSI(X)=-1$ implies the highest ranking turbulence, whereas $RSI(X)=1$ indicates complete stability. The results for the WTW suggest that even rankings are very stable over time. Indeed, one has that $RSI(ND)=0.9833$, $RSI(NS)=0.9964$, $RSI(ANNS)=0.9781$, $RSI(WCC)=0.9851$ and $RSI(RWBC)=0.9920$. Notice that, since $\rho_{t-1,t}(X)\rightarrow N(0,N^{-1})$, our $RSI(X)$ should tend to a $N(0,[N(T-1)]^{-1})\cong N(0,3.3102E-4)$. Therefore, our empirical values are more than 50 standard deviations to the right of 0 (no average rank correlation).
The second issue that deserves a closer look concerns detecting which countries rank high according to different node statistics. Table \[Tab:rankings\] displays the top-20 countries in each given node-statistic ranking in 2000, which, given the stability results above, well represents the entire sample period. First note that, apart from ANNS, all “usual suspects” occupy the top-ten positions. Germany scores very high for all statistics but ANNS, while the U.S. and Japan are characterized by a very high rank for weighted statistics but not for ND. This implies that they have relatively less trade partners but the share of trade that they control, the capacity to cluster, and their centrality is very high. Conversely, countries like Switzerland, Italy and Australia have a more diversified portfolio of trade partners with which they maintain less intense trade relationships. Furthermore, it is worth noting that China was already very central in the WTW in 2000, despite its clustering level was relatively lower. India was instead not present among the top-20 countries as far as NS and WCC were concerned; it was only 14th according to centrality and 11th in the ND ranking. Notice how all top-20 countries in the ANNS are micro economies: they typically feature a very low NS and ND, but only tend connect to the hubs of the WTW. Table \[Tab:rankings\] also presents country rankings in 2000 according to GDP and per-capita GDP (expressed in US dollars per person). As expected, the group of countries topping the rankings based on weighted statistics (except ANNS) are also among those having highest GDP levels. This is due to the fact that link weights are expressed in terms of total trade, which is typically positively correlated with GDP, and node-statistics like NS, WCC and RWBC partly reflect this ex-ante correlation. This is not the case, however, if one compares GDP with ND rankings, meaning that top countries in terms of number of trade relations are not also those at the top of GDP rankings. A similar mismatch occurs between node-statistic and per-capita GDP rankings. Hence, after one washes away country-size effects (e.g., population size), top countries in terms of income —as measured by per-capita GDP— do not necessarily occupy the first positions of the rankings according to their intensity of connections, centrality, and clustering.
Notwithstanding the presence of a relatively high ranking stability, there are indeed examples of countries moving up or falling behind over the period 1981-2000. For example, as far as centrality is concerned, Russia has steadily fallen in the RWBC ranking from the 6th to the 22th position. A similar downward pattern has been followed by Indonesia (from 17th to 36th). South Africa has instead fallen from 23th (in 1981) to 32th (in 1990) and then has become gradually more central (16th in 2000). On the contrary, the majority of high-performing Asian economies (HPAE), have been gaining positions in the RWBC ranking. For example, South Korea went from the 24th to the 8th position; Malaysia from the 43th to the 21th; Thailand started from the 42th position in 1981 and managed to become the 18th best central country in 2000. This evidence strongly contrast with the recent experience of Latin American (LATAM) economies (e.g., Mexico and Venezuela) that have – at best – maintained their position in the ranking of centrality [@RFS2008wp].
Within-Sample Autocorrelation Structure and Growth Dynamics {#SubSec:Growth}
-----------------------------------------------------------
To further explore the properties of within-sample distribution dynamics, we now investigate autocorrelation structure and growth dynamics of node and link statistics. More precisely, let $X_i^t$ the observation of statistic $X$ for node or link $i$ at time $t$, where $i=1,\dots,I$, $t=1,\dots,T$, and $I$ stands either for $N$ (in case of a node statistic) or for $N(N-1)/2$ (in case of link weights). We first compute the (node or link) distribution of first-order autocorrelation coefficients (ACC) defined as:
$$\hat{r}_i(X)=\frac{\sum_{t=2}^{T}{(X_i^t-\bar{X}_i^0)(X_i^{t-1}-\bar{X}_i^1)}}{\sqrt{\sum_{t=2}^{T}{(X_i^t-\bar{X}_i^0)^2}\sum_{t=2}^{T}{(X_i^{t-1}-\bar{X}_i^1)^2}}}
\label{Eq:ACC}$$
where $\bar{X}_i^j=(T-1)^{-1}\sum_{t=2}^{T}{X_i^{t-j}}$, $j=0,1$.
Second, we compute the first-order ACC $\hat{r}(X)$ on the (node or link) distribution of $X$ pooled across years. To do so, we preliminary standardize the distributions $\{X_i^t,i=1,\dots,I\}$ for each $t$, so as to have zero-mean and unitary standard deviation in each year, and then we pool all $T$ year-distributions together [^23].
The left part of Table \[Tab:autocorr\] shows the values of $\hat{r}(X)$ together with the population mean and standard deviation of $\hat{r}_i(X)$ for our five node statistics and link weights. We also report the percentage of observations (nodes or links) for which the ACC $\hat{r}_i(X)$ turns out to be larger than zero. Both $\hat{r}(X)$ and the percentage of positive-ACF observations indicate a relatively strong persistence in the dynamics of both node and link statistics.
Given that the pooled ACC figures are very close to unity, we further check whether autoregressive dynamics governing the evolution of logged network statistics is close to a random-walk. In particular, we test whether a Gibrat dynamics (i.e., a multiplicative process on the levels $X_i^t$, where rates of growth of $X_i^t$ are independent on $X_i^t$) applies to our variables or not [@Gib31; @Sut97]. Notice that, under a Gibrat dynamics, $X_i^t$ should be in the limit log-normally distributed, which is what we actually observe in our sample for the majority of node statistics (see Section \[SubSec:Shape\]). More formally, we begin by fitting the simple model:
$$\Delta log(X_i^t)=\beta_i log(X_i^{t-1})+\epsilon_i^{t},
\label{Eq:Gibrat}$$
where $\Delta log(X_i^t)=log(X_i^t)-log(X_i^{t-1})$ is the rate of growth of $X_i^t$ and $\epsilon_i^{t}$ are white-noise errors orthogonal to $log(X_i^{t-1})$. If a Gibrat dynamics applies for a given node or link, then $\beta_i=0$. We also fit the model in to the time-pooled sample by setting $\beta_i=\beta$, where again we first standardize in each year our variables in order to wash away trends and spurious dynamics.
As the right part of Table \[Tab:autocorr\] shows, our data reject the hypothesis that network statistics follow a Gibrat dynamics. Indeed, both the population average of $\hat{\beta}_i$ and the pooled-sample estimate $\hat{\beta}$ are significantly smaller than zero, thus implying a process where small-valued entities (i.e., nodes and links characterized by small values of any given statistic) tend to grow relatively more than large-valued ones. This is further confirmed by the percentage of nodes or links for which $\hat{\beta}_i$ turns out to be significantly smaller than zero.
Rejection of a Gibrat dynamics also implies that the distributions of growth rates $\Delta log(X_i^t)$ should depart from Gaussian ones [@BnS06]. This is confirmed by all our pooled fits. Indeed, as Figure \[Fig:gr\_distr\_node\] shows for node statistics, pooled growth-rate distributions are well proxied by Laplace (fat-tailed, symmetric) densities. Furthermore, the pooled distribution of growth rates $g$ for positive link weights is nicely described by an *asymmetric exponential power* (AEP) density [@BnS06b]:
$$d(g;a_{l},a_{r},b_{l},b_{r},m)=\left\{
\begin{array}{cc}
\Upsilon^{-1}e^{-\frac{1}{b_{l}}|\frac{g-m}{a_{l}}|^{b_{l}}},& g<m \\
\Upsilon^{-1}e^{-\frac{1}{b_{r}}|\frac{g-m}{a_{r}}|^{b_{r}}},& g\geq m \\
\end{array}
\right. \label{Eq:subboasym},$$
where $\Upsilon=a_l b_l^{1/b_l}\Gamma(1+1/b_l)+a_r b_r^{1/b_r}
\Gamma(1+1/b_r)$, and $\Gamma$ is the Gamma function [^24]. Maximum-likelihood estimation of tail parameters indicate that link-weight growth rates display tails much fatter than Laplace ones. Moreover, the right tail is remarkably thicker than the left one (as $\hat{b}_l=0.5026>0.2636=\hat{b}_r$), see Figure \[Fig:gr\_distr\_link\]. Therefore, link weights are characterized by a relatively much higher likelihood of large positive growth events than of negative ones. This result brings further evidence in favor or the widespread emergence of fat-tailed growth-rate distributions in economics. In fact, recent studies have discovered that Laplace (and more generally AEP) densities seem to characterize the growth processes of many economic entities, from business companies [@Sea96; @Aea97; @Cea98; @Fuetall05] to world-country GDP and industrial production [@FagNapRov2007].
Out-of-Sample Evolution {#SubSec:OutofSample}
-----------------------
In the preceding sections, we have investigated the within-sample dynamics of the distributions of node and link statistics. Now we turn out attention to the out-of-sample (long-run) evolution of such distributions by estimating their limiting behavior. To do so, we employ kernel density estimates obtained above to compute ergodic densities, which represent the long-run tendency of the distributions under study [^25].
As already noted above, stochastic kernels of all node statistics are quite concentrated and evenly distributed along the $45^\circ$ line. Therefore, it is no surprising that also their limiting distributions look quite similar to the ones in year 1981. This can be seen in Figure \[Fig:ergodic\_ns\], where we exemplify this point by plotting initial vs. estimates of the ergodic distribution for the logs of NS. Both distributions present a similar shape. If any, the ergodic one exhibits a larger variability, a shift to the left of the lower tail and a shift to the right of the upper tail. This can be explained by noticing that the kernel density estimate (Figure \[Fig:kernel\_ns\]) shows a relatively larger probability mass under the main diagonal in the bottom-left part of the plot, whereas in the top-right part this mass was shifted above the main diagonal. Such shape-preserving shifts hold also for the other node statistics under analysis. In particular, the ergodic distribution for node RWBC roughly preserves its power-law shape, as well as its scale exponent.
On the contrary, the shape of the stochastic kernel for logged link weights hinted at a concentration of transition densities at the extremes of the range. Middle-range values presented instead a flatter and more dispersed landscape. This partly explains why we observe a radical difference between initial and ergodic distributions of logged link weights. Whereas the initial one is close to a Gaussian (i.e., link weights are well-proxied by a log-normal density), the ergodic distribution displays a power-law shape with very small exponent. This can be seen in Figure \[Fig:ergodic\_linkweights\], where the two plots have been superimposed.
These findings imply that the architecture of the WTW will probably evolve in such a way to undergo a re-organization of link weights (i.e., country total trade volumes) that is nevertheless able to keep unchanged the most important node topological properties. Such a re-organization seems to imply a polarization of link weights into a large majority of links carrying moderate trade flows and a small bulk of very intense trade linkages. The power-law shape of the ergodic distribution suggests that such a polarization is much more marked than at the beginning of the sample period, when the distribution of link weights was well proxied by a log-normal density. Furthermore, it must be noted that results on Gibrat dynamics in Section \[SubSec:Growth\] indicate that some catching-up between low- and high-intensity links is going on within our sample period. The findings on out-of-sample evolution discussed here, on the contrary, seem to imply that such a catching-up dynamics is not so strong to lead to some convergence between e.g. low-intensity and high-intensity link weights.
Robustness to Alternative Weighting Schemes {#SubSec:Robustness}
-------------------------------------------
All results obtained so far refer to a particular weighting procedure. To recall, the weight of a link from $i$ to $j$ is, after symmetrization, proportional to the total trade (imports plus exports) flowing through that link in a given year. This baseline weighting scheme is very common in the literature [@LiC03; @Bhatta2007a; @Bhatta2007b; @Garla2007], but treats the same way all countries irrespective of their economic importance. Are our findings robust to alternative weighting schemes? To address this issue, we have considered here two alternative economically-meaningful setups, where we wash away size effects by scaling directed link weights with the GDP of either the exporter or the importer country.
More formally, in the first alternative setup, each directed link from node $i$ to $j$ is now weighted by total exports of country $i$ to country $j$ and then divided by the country $i$’s GDP (i.e., the *exporter* country). Such a weighting setup allows one to measure how much economy $i$ depends on economy $j$ as a buyer. In the second setup, we still remove size effects from trade flows, but we now divide by the GDP of the *importer* country ($j$’s GDP). This allows us to appreciate how much economy $i$ depends on $j$ as a seller [^26].
All our main results turn out to be quite robust to these two alternatives. This is an important point, as a weighted network analysis might in principle be sensible to the particular choice of the weighting procedure. To illustrate this point, we first compare the symmetry index for the three weighting schemes across the years, cf. Figure \[Fig:robustness\_symmetry\]. If one scales exports by exporter’s or importer’s GDP the symmetry index still remains very low and close to the one found in the baseline weighting scheme. This indicates that under all three schemes an undirected-network analysis is appropriate. As a further illustration, Figure \[Fig:robustness\_qqplot\] reports the quantile-quantile plots of logged link-weight, GDP-scaled, distributions vs. baseline logged link weights in year 2000. It is easy to see that both alternative link-weight distributions are very similar to the baseline one. This results holds also for pooled distributions, as well as for node statistics ones. Finally, Figure \[Fig:robustness\] depicts some examples of the across-time correlation patterns between node statistics and pcGDP. Left panels refer to the first alternative weighting scheme (exports scaled by exporter GDP) whereas right panels shows what happens under the second alternative setup (exports scaled by importer GDP). All previous results (see Figures \[Fig:corr\_stats\] and \[Fig:corr\_pcgdp\]) are confirmed. Notice that GDP scaling results in weaker but still significantly different from zero correlation coefficients (especially for WCC-NS). Of course, we do not expect our results to hold irrespective of *any* weighting scheme to be adopted. In fact, the binary characterization of the WTW, where some of the weighted-network results are reversed, is itself a particular weighting scheme, one that assigns to each existing link the same weight [^27].
Conclusions {#Sec:Conclusions}
===========
In this paper we have explored, from a purely descriptive perspective, the within-sample dynamics and out-of-sample evolution of some key node and link statistic distributions characterizing the topological properties of the web of import-export relationships among world countries (WTW). By employing a weighted-network approach, we have shown that WTW countries holding many trade partners (and/or very intense trade relationships) are also the richest and most (globally) central; typically trade with many partners, but very intensively with only a few of them (which turn out to be themselves very connected); and form few but intensive-trade clusters. All the distributions and country rankings of network statistics display a rather strong within-sample stationarity. Our econometric tests show that node and link statistics are strongly persistent. However, Gibrat-like dynamics are rejected. This is confirmed also by the fact that the growth-rate distributions of our statistics can be well approximated by fat-tailed Laplace or asymmetric exponential-power densities. Furthermore, whereas the estimated ergodic distributions of all node-statistics are quite similar to the initial ones, the (positive) link-weight distribution is shifting from a log-normal to a power law. This suggests that a polarization between a large majority of weak-trade links and a minority of very intense-trade ones is gradually emerging in the WTW. Interestingly, such a process is likely to take place without dramatically changing the topological properties of the network.
Many extensions to the present work can be conceived. First, building on Refs. [@Garla2004; @Bhatta2007a], one may try to explore simple but economically-meaningful models of WTW dynamics that are able to reproduce the main stylized facts put forth by our purely empirical analysis. To do that, one may attempt to deduce the probability distributions of link and node statistics of interest by postulating some given growth model for link weights. For example, it is well-known that Gibrat-like multiplicative, statistically independent, processes at the level of links can easily generate —as their limiting distribution— log-normal densities such as the observed ones. Notice, however, that the discussion in Sections \[SubSec:DistrDyn\] , \[SubSec:Growth\] and \[SubSec:OutofSample\], points towards patterns of within-sample growth dynamics persistently deviating from the standard Gibrat model, and indicates that also out-of-sample evolution does not seem to be driven by such a simple mechanics. Furthermore, statistical independence among growth processes for different link weights (both belonging to a given country and among different countries) can be easily dismissed on simple economic arguments. This suggests that, in order to single out simple stochastic growth models accounting at the same time both for the observed link-weight dynamics, and for the ensuing statistical properties of node-statistics computed from such a dynamics, more complicated models might be conceived. A possibility, indeed the very next point in our agenda, would be to adapt existing models of weighted-network evolution [@Barr05] in such a way to allow for more plausible rules —gathered e.g. from international-trade literature— governing the emergence of trade relationships and the subsequent evolution of their intensity.
Second, one would like to explore in more details the topological properties of the WTW, both cross-sectionally and over time. Interesting questions here concern the role of geographical proximity in shaping the structure of international trade, the degree of fragility of the network, and so on. More specifically, trade flows could be disaggregated across product classes to explore how trade composition affects network properties.
Third, one could abstract from aggregate statistical properties and analyze at a finer level the role of single countries in the network structure. For instance, how does the dynamics of degree, strength, clustering, etc. behave for single relevant countries in different regions? Do country-specific network indicators display the same time-stationarity of their aggregate counterparts?
Finally, in line with Ref. [@KaRe07], one can ask whether node statistics characterizing connectivity, clustering, centrality and so on, can be employed as explanatory variables for the dynamics of country growth rates and development patterns.
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------ -------- --------- ---------- ----------
Mean Std Dev Skewness Kurtosis
ND 0.0143 0.0047 0.0375 0.0086
ANND 0.0079 0.0279 0.0116 0.0260
BCC 0.0043 0.0197 0.0615 0.0205
NS 0.0379 0.0412 0.0263 0.0317
ANNS 0.0479 0.0512 0.0223 0.0452
WCC 0.0544 0.0097 0.0274 0.0543
RWBC 0.0049 0.0107 0.0251 0.0556
------ -------- --------- ---------- ----------
: Average over time of absolute-valued 1-year growth rates of the first four moments of node statistics. Given the value of the node statistic X at time $t$ for country $i$ ($X_i^t$) and $M^k(\cdot)$ the moment operator that for $k=1,2,3,4$ returns respectively the mean, standard deviation, skewness and kurtosis, the time-average of absolute-valued 1-year growth rates of the $k$-th moment-statistic is defined as $\frac{1}{T-1}\sum_{t=2}^{T}{|(E^k(X_i^t)/E^k(X_i^{t-1})-1)|}$.[]{data-label="Tab:moments_gr"}
1981 1982 1983 1984 1985 1986 1987 1988 1989 1990
------ ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
ND $0.0000^{***}$ $0.0000^{***}$ $0.0010^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0010^{***}$ $0.0000^{***}$ $0.0010^{***}$ $0.0000^{***}$
ANND $0.0277^{**}$ $0.0261^{**}$ $0.0400^{**}$ $0.0403^{**}$ $0.0525^{*}$ $0.0309^{**}$ $0.0333^{**}$ $0.0197^{**}$ $0.1254^{}$ $0.1020^{}$
BCC $0.0060^{***}$ $0.0040^{***}$ $0.0040^{***}$ $0.0040^{***}$ $0.0050^{***}$ $0.0020^{***}$ $0.0050^{***}$ $0.0030^{***}$ $0.0080^{***}$ $0.0040^{***}$
NS $0.2925^{}$ $0.2046^{}$ $0.4021^{}$ $0.2870^{}$ $0.4344^{}$ $0.6804^{}$ $0.6238^{}$ $0.5300^{}$ $0.3496^{}$ $0.5343^{}$
ANNS $0.1118^{}$ $0.2500^{}$ $0.2724^{}$ $0.2463^{}$ $0.2816^{}$ $0.2532^{}$ $0.3243^{}$ $0.1633^{}$ $0.1666^{}$ $0.1065^{}$
WCC $0.5673^{}$ $0.2525^{}$ $0.2821^{}$ $0.2874^{}$ $0.2867^{}$ $0.2601^{}$ $0.3564^{}$ $0.2035^{}$ $0.2005^{}$ $0.1202^{}$
RWBC $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
ND $0.0010^{***}$ $0.0020^{***}$ $0.0010^{***}$ $0.0010^{***}$ $0.0020^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$
ANND $0.0810^{*}$ $0.0630^{*}$ $0.0900^{*}$ $0.0700^{*}$ $0.0580^{*}$ $0.0400^{**}$ $0.0480^{**}$ $0.0460^{**}$ $0.0400^{**}$ $0.0260^{**}$
BCC $0.0090^{***}$ $0.0040^{***}$ $0.0040^{***}$ $0.0020^{***}$ $0.0020^{***}$ $0.0060^{***}$ $0.0050^{***}$ $0.0050^{***}$ $0.0050^{***}$ $0.0050^{***}$
NS $0.5367^{}$ $0.2398^{}$ $0.2917^{}$ $0.2016^{}$ $0.3685^{}$ $0.4693^{}$ $0.6000^{}$ $0.6312^{}$ $0.5918^{}$ $0.5260^{}$
ANNS $0.2450^{}$ $0.0905^{*}$ $0.1402^{}$ $0.1133^{}$ $0.1269^{}$ $0.0574^{*}$ $0.0734^{*}$ $0.0899^{*}$ $0.0668^{*}$ $0.1385^{}$
WCC $0.2661^{}$ $0.1166^{}$ $0.1562^{}$ $0.1356^{}$ $0.1358^{}$ $0.0638^{*}$ $0.1206^{}$ $0.1095^{}$ $0.1072^{}$ $0.1583^{}$
RWBC $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$ $0.0000^{***}$
--------- ----------- ---------------- ----------- ---------------- ----------- ----------------
KS Test Statistic p-Value Statistic p-Value Statistic p-Value
$ND$ 0.1262 $0.0114^{**}$ 0.5326 $0.0000^{***}$ 0.4185 $0.0000^{***}$
$NS$ 0.0647 $0.5059^{}$ 0.3563 $0.0000^{***}$ 0.2862 $0.0000^{***}$
$ANNS$ 0.0911 $0.1351^{}$ 0.5141 $0.0000^{***}$ 0.1336 $0.0061^{***}$
$WCC$ 0.0528 $0.7658^{}$ 0.6493 $0.0000^{***}$ 0.2569 $0.0000^{***}$
$RWBC$ 0.2103 $0.0000^{***}$ 0.4830 $0.0000^{***}$ 0.1065 $0.0564^{}$
--------- ----------- ---------------- ----------- ---------------- ----------- ----------------
: Kolmogorov-Smirnov goodness-of-fit test results for year 2000 distributions. The three null hypotheses tested are that the observed distributions come from, respectively, log-normal, stretched exponential, or power-law pdfs. Asterisks: (\*) null hypothesis rejected at 10%; (\*\*) null hypothesis rejected at 5%; (\*\*\*) null hypothesis rejected at 1%.[]{data-label="Tab:kstest"}
$\omega=0$
----------------------------- ------------------- ------------------- ------------------- -------------------
Statistic 1 4 7 10
$M_{\omega,K}^{\tau}(ND)$ 0.8674 0.7794 0.7282 0.6943
$M_{\omega,K}^{\tau}(NS)$ 0.9346 0.8612 0.8234 0.7874
$M_{\omega,K}^{\tau}(ANNS)$ 0.8541 0.7577 0.7145 0.6656
$M_{\omega,K}^{\tau}(WCC)$ 0.8553 0.7490 0.6776 0.6377
$M_{\omega,K}^{\tau}(RWBC)$ 0.9004 0.8280 0.7899 0.7515
C.I. (Reshuffled) \[0.1854,0.2146\] \[0.1842,0.2159\] \[0.1823,0.2176\] \[0.1800,0.2200\]
$\omega=1$
Statistic 1 4 7 10
$M_{\omega,K}^{\tau}(ND)$ 0.9950 0.9881 0.9753 0.9640
$M_{\omega,K}^{\tau}(NS)$ 0.9997 0.9965 0.9923 0.9875
$M_{\omega,K}^{\tau}(ANNS)$ 0.9930 0.9835 0.9710 0.9579
$M_{\omega,K}^{\tau}(WCC)$ 0.9980 0.9918 0.9787 0.9686
$M_{\omega,K}^{\tau}(RWBC)$ 0.9990 0.9965 0.9952 0.9906
C.I. (Reshuffled) \[0.5020,0.5370\] \[0.5004,0.5385\] \[0.4981,0.5406\] \[0.4954,0.5436\]
: Distribution dynamics. Persistence measure $M_{\omega,K}^{\tau}(X)$ for the distributions of node statistics and for alternative choices of the window $\omega\in \{0,1\}$ and the time lag $\tau\in \{1,4,7,10\}$. All statistics refer to $K=10$ quantile classes. The lines labeled as “C.I. (Reshuffled)” contain confidence intervals (at 95%) for the mean of the distribution of the statistic $M$ in random graphs where the observed adjacency matrices $A^t$ are kept fixed and weights are re-distributed at random by reshuffling the observed link-weight distributions $w^t=\{w_{ij}^t, i\neq j=1,\dots,N\}$. Values of $M_{\omega,K}^{\tau}(X)$ close to one and to the right of confidence intervals indicate a strong persistence of the associated with-sample distribution dynamics.[]{data-label="Tab:distr_dynamics_nodes"}
------------------- ------------------- ------------------- ------------------- -------------------
1 4 7 10
$\omega=0$ 0.8116 0.7073 0.6464 0.6012
C.I. (Reshuffled) \[0.1974,0.2026\] \[0.1971,0.2029\] \[0.1969,0.2032\] \[0.1962,0.2037\]
$\omega=1$ 0.9910 0.9733 0.9562 0.9397
C.I. (Reshuffled) \[0.5175,0.5238\] \[0.5172,0.5241\] \[0.5170,0.5248\] \[0.5167,0.5253\]
------------------- ------------------- ------------------- ------------------- -------------------
: Distribution dynamics. Persistence measure $M_{\omega,K}^{\tau}(X)$ for the distributions of positive link weights and for alternative choices of the window $\omega\in
\{0,1\}$ and the time lag $\tau\in \{1,4,7,10\}$. All statistics refer to $K=20$ quantile classes. The lines labeled as “C.I. (Reshuffled)” contain confidence intervals (at 95%) for the mean of the distribution of the statistic $M$ in random graphs where the observed adjacency matrices $A^t$ are kept fixed and weights are re-distributed at random by reshuffling the observed link-weight distributions $w^t=\{w_{ij}^t, i\neq j=1,\dots,N\}$. Values of $M_{\omega,K}^{\tau}(X)$ close to one and to the right of confidence intervals indicate a strong persistence of the associated with-sample distribution dynamics.[]{data-label="Tab:distr_dynamics_linkweights"}
Rank ND NS ANNS WCC RWBC Real GDP pcGDP
------ ------------- ------------- ------------------- ---------------------- -------------- ------------- ----------------------
1 Germany USA Sao Tome-Principe USA USA USA Luxembourg
2 Italy Germany Kiribati Germany Germany Japan Switzerland
3 UK Japan Nauru Japan Japan Germany Japan
4 France France Tonga UK France UK Norway
5 Switzerland China Vanuatu China UK France USA
6 Australia UK Tuvalu France China China Denmark
7 Belgium Canada Burundi Italy Italy Italy Iceland
8 Netherlands Italy Botswana Netherlands S. Korea Canada Sweden
9 Denmark Netherlands Lesotho S. Korea Netherlands Brazil Austria
10 Sweden Belgium Maldives Singapore Belgium Mexico Germany
11 India S. Korea Solomon Islands Mexico Spain Spain Ireland
12 Spain Mexico Bhutan Belgium Australia India Netherlands
13 USA Taiwan Comoros Spain Singapore S. Korea Finland
14 China Singapore Seychelles Taiwan India Australia Qatar
15 Norway Spain Saint Lucia Canada Taiwan Netherlands Belgium
16 Japan Switzerland Guinea-Bissau United Arab Emirates South Africa Taiwan Singapore
17 Taiwan Malaysia Mongolia Saudi Arabia Brazil Argentina France
18 Malaysia Sweden Cape Verde Iraq Thailand Russia UK
19 Ireland Thailand Grenada Switzerland Saudi Arabia Switzerland China
20 Canada Australia Fiji Russia Canada Sweden United Arab Emirates
------------------ -------- -------- ---------- -------- --------- -------- ---------- ---------
Mean SD $\%(>0)$ Pooled Mean StdDev $\%(<0)$ Pooled
ND 0.6438 0.2222 0.8428 0.9859 -0.3636 0.2235 0.6667 -0.1330
NS 0.6795 0.1170 0.9623 0.9949 -0.3186 0.1176 0.7799 -0.1910
ANNS 0.6353 0.0686 0.9811 0.9539 -0.3609 0.0670 0.9811 -0.2235
WCC 0.6404 0.1414 0.9119 0.9855 -0.3596 0.1414 0.8302 -0.2866
RWBC 0.6203 0.2007 0.8050 0.9983 -0.3795 0.2007 0.7862 -0.3651
Pos Link Weights 0.4330 0.3069 0.4171 0.9940 -0.4368 0.2299 0.9196 -0.1422
------------------ -------- -------- ---------- -------- --------- -------- ---------- ---------
: First-order autocorrelation coefficient $\hat{r}_i(X)$ and $\hat{\beta}_i$ parameter in Gibrat regressions for node and link statistics. Mean and SD columns: Population average and standard deviation computed across node or links. Columns labeled by $\%(>0)$ or $\%(<0)$ report the percentage of nodes or links whose estimate is larger or smaller than zero. Columns labeled by “Pooled” report estimates for the time-pooled normalized sample (i.e., the sample obtained by first standardizing each observation by the mean and standard deviation of the year, and then stacking all years in a column vector).[]{data-label="Tab:autocorr"}
[^1]: For example, “openness to trade” of a given country is traditionally measured by the ratio of exports plus imports to country’s gross domestic product (GDP).
[^2]: That is, they weight each bilateral trade relation by the difference between exports and imports. Notice that, as happens also in Refs. [@Bhatta2007a; @Bhatta2007b], their across-year comparison may be biased by the fact that trade flows are expressed in current U.S. dollars and do not appear to be properly deflated.
[^3]: In Section \[SubSec:Robustness\] we explore what happens if we employ a few alternative definitions for link weights.
[^4]: The expected value of the statistic in a random graph where link weights are uniformly and independently distributed as a uniform in the unit interval is 0.5 [@FagioloSymmEcoBull]. Furthermore, the expected value computed by randomly reshuffling in each year the empirically-observed weights among existing links ranges in the same period from 0.0230 to 0.0410. Therefore, the empirical value is significantly smaller than expected.
[^5]: Cf. Ref. [@Saramaki2006] for alternative definitions of the clustering coefficient for weighted undirected networks. Here we employ the above formulation because it is the only one retaining two properties important to fully characterize clustering in trade networks, namely (i) $WCC_i^t$ takes into account the weight of all links in any given triangle; (ii) $WCC_i^t$ is invariant to weight permutations in one triangle.
[^6]: Notice that the fraction of strictly positive links (over all possible links) also defines network density. More on that below.
[^7]: If any, there seems to be some evidence towards declining higher-than-one moments.
[^8]: More formally, let $X_i^t$ be the value of the node statistic X at time $t$ for country $i$ and $M^k(\cdot)$ the moment operator that for $k=1,2,3,4$ returns respectively the mean, standard deviation, skewness and kurtosis. The time-average of absolute-valued 1-year growth rates of the $k$-th moment-statistic is defined as $\frac{1}{T-1}\sum_{t=2}^{T}{|(E^k(X_i^t)/E^k(X_i^{t-1})-1)|}$.
[^9]: Table \[Tab:normality\] reports p-values for the Jarque-Bera test [@jbtest1; @jbtest2], the null hypothesis being that the logs of positive-valued statistics are normally distributed with unknown parameters. Alternative normality tests (Lilliefors, Anderson-Darling, etc.) yield similar results.
[^10]: A rank-size plot is simply a transformation of a standard cumulative-distribution function (CDF) plot, where the behavior of the upper tail is accentuated. Indeed, suppose that $(x_1,\dots,x_N)$ are the available empirical observations of a random variable $X$, and sort the $N$ observations to obtain $(x_{(1)},\dots,x_{(N)})$, where $x_{(1)}\geq x_{(2)}\geq \dots \geq
x_{(N)}$. A rank-size plot graphs $\log(r)$ against $\log(x_{(r)})$, where $r$ is the rank. However, since $r/N=1-F(x_{(r)})$, then $\log(r)=\log[1-F(x_{(r)})]+\log(N)$. Since the existing literature on the WTW has discussed at length the upper-tail behavior of node and link distributions, we have preferred to visualize the shape of the distributions of interest using a rank-size plot instead of employing standard (two-tailed) CDF plots.
[^11]: At the extreme, if in every year $t$ the network were an Erdös-Renyi random-graph [@Bollo1985] with link probability equal to network density $d^t$ and link weights drawn from an i.i.d. uniform r.v. defined on the unit interval (U\[0,1\]) – uniformly weighted ER graph henceforth – the expected NS would have been $\frac{1}{2}(N-1)[d^t]^2$, that is a value ranging over time between 22.9079 and 32.7699.
[^12]: Notice that under if the network were an uniformly weighted ER graph (see above), one would have obtained $E(ANND_i^t)=1+(N-2)d^t$, $E(ANNS_i^t)=E(NS_i^t)=\frac{N-2}{2}(d^t)^2$, $E(BCC_i^t)=d^t$ and $E(WCC_i^t)=\frac{27}{64} d^t$; see [@FagioloClustArxiv].
[^13]: A subgraph is defined to be maximal with respect to some property whenever either it has the property or every pair of its points has the property and, upon the addition of any point, either it loses the property or there is some pair of its points which does not have the property. Thus Luce and Perry’s cliques are maximal complete subgraph such that every pair of points in the clique is adjacent, and the addition of any point to the clique makes it incomplete. Note that an agent belonging to a LP clique of order $k$ will automatically belong to all the sub-cliques of that one of order $h<k$ with that node as one vertex.
[^14]: More precisely, the correlation coefficient between two variables $X$ and $Y$ is defined here as the product-moment (Pearson) sample correlation, i.e. $\sum_{i}{(x_i-\overline{x})(y_i-\overline{y})}/[(N-1)s_X s_Y]$, where $\overline{x}$ and $\overline{y}$ are sample averages and $s_X$ and $s_Y$ are sample standard deviations.
[^15]: These results can be made more statistically sound by comparing empirically-observed correlation coefficients with their expected counterparts under random schemes B-RS and W-RS (see above for their definitions). Simulation results (not shown in the paper for the sake of brevity) indicate that almost all empirical correlation coefficients (in every year) are in absolute value larger than the absolute value of their expected counterpart under either B-RS or W-RS. This means that the magnitude of almost all observed correlations are bigger than expected. The only exception is the ANNS-NS correlation that, albeit positive, is not significantly larger than in W-RS. This indicates that whereas the binary WTW is strongly disassortative, the weak-disassortative nature of the weighted WTW is not statistically distinguishable from what we would have observed in comparable random graphs.
[^16]: Again, simulation results (not shown here) suggest that all empirical correlations are in absolute values larger than their expected level under reshuffling schemes B-RS and W-RS, except for the ANNS-pcGDP correlation.
[^17]: Additional research on the rich-club phenomenon in the WTW could entail a backbone-extraction analysis like the one performed in Ref. [@serrano-2008].
[^18]: Cf. Refs. [@df1999; @FiaschiLavezzi2007] for economic applications. Here and in what follows, Markovianity of the statistics under analysis has been assumed without performing more rigorous statistical tests [@Markovianity]. This is actually one of the next points in our agenda.
[^19]: Parameters outside these ranges and choices do not change the main implications of the analysis. Similar results also are obtained if one computes the statistic $M_{\omega,K}^{\tau}$ on logged distributions of $X$ (i.e., logs of node statistics and positive link weights) and/or one preliminary re-scales the data by removing the time-averages of the distributions in order to wash away possible trends.
[^20]: The distributions of $M_{\omega,K}^{\tau}(X)$ turn out to be well-proxied by Gaussian densities.
[^21]: Standard deviations of such estimates are quite small. Let $\hat{p}_{00}$ and $\hat{p}_{++}$, respectively, be the probability of remaining a zero and positive link weight. We find that $\sigma(\hat{p}_{00})=0.0034$ and $\sigma(\hat{p}_{++})=0.0023$. Similarly, let $\hat{p}_{0+}$ and $\hat{p}_{+0}$ be the probability of becoming a positive (respectively, zero) link weight. Since $\hat{p}_{00}=1-\hat{p}_{0+}$ and $\hat{p}_{++}=1-\hat{p}_{+0}$ by construction, then $\sigma(\hat{p}_{0+})=\sigma(\hat{p}_{00})$ and $\sigma(\hat{p}_{+0})=\sigma(\hat{p}_{++})$.
[^22]: The SRCC between two variables $(Z^1_i,Z^2_i)$, $i=1,\dots,N$ is simply the Pearson product-moment correlation coefficient defined above, now computed between $(z^1_i,z^2_i)$, where $z^j_i$ are the ranks of each observation $i$ according to the original variables $Z^j_i$, $j=1,2$. Therefore, the SRCC equals one if the $N$ observations are ranked the same under $Z^1$ and $Z^2$; it equals $-1$ if the ranks according to the two variables are completely reversed; and it is zero if there is no correlation whatsoever between the ranking of the observations according to $Z^1$ and $Z^2$. We focus here only on one-year lags between rankings. An interesting extension to the present analysis would be to check for stability of rankings across time lags of length $\tau>1$.
[^23]: We stop at first-order autocorrelation coefficients because of the few time observations available. Notwithstanding their low statistical significance, also second-order ACCs turn out to be positive albeit much smaller than first-order ones.
[^24]: The AEP features five parameters. The parameter $m$ controls for location. The two $a$’s parameters control for scale to the left ($a_{l}$) and to the right ($a_{r}$) of $m$. Larger values for $a$’s imply – *coeteris paribus* – a larger variability. Finally, the two $b$’s parameters govern the left ($b_{l}$) and right ($b_{l}$) tail behavior of the distribution. To illustrate this point, let us start with the case of a symmetric exponential power (EP), i.e. when $a_{l}=a_{r}=a$ and $b_{l}=b_{r}=b$. It is easy to check that if $b=2$, the EP boils down to the normal distribution. In that case, the correspondent HCE distribution would be log-normal. If $b<2$, the EP displays tails thicker than a normal one, but still not heavy. In fact, for $b<2$, the EP configures itself as a medium-tailed distribution, for which all moments exist. In the case $b=1$ we recover the Laplace distribution. Finally, for $b>2$ the EP features tails thinner than a normal one and still exponential.
[^25]: Given the real-valued statistic $X$, its ergodic distribution $\phi_{\infty}(\cdot)$ is implicitly defined for any given $\tau$ as $\phi_{\infty}(x)=\int{s^{\tau}(x|z)\phi_{\infty}(z)dz}$, where $s^{\tau}(x|z)$ is the stochastic kernel defined in Section \[SubSec:DistrDyn\]. See also Ref. [@df1999].
[^26]: We have also experimented with the weighting scheme where trade is scaled *by the sum* of importer’s and exporter’s GDPs without detecting any significant difference.
[^27]: In this respect, an interesting exercise would imply to find (if any) a proper re-scaling or manipulation of original trade flows that makes weighted and binary results looking the same.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Autoscaling system can reconfigure cloud-based services and applications, through various configurations of cloud software and provisions of hardware resources, to adapt to the changing environment at runtime. Such a behavior offers the foundation for achieving elasticity in modern cloud computing paradigm. Given the dynamic and uncertain nature of the shared cloud infrastructure, cloud autoscaling system has been engineered as one of the most complex, sophisticated and intelligent artifacts created by human, aiming to achieve self-aware, self-adaptive and dependable runtime scaling. Yet, existing Self-aware and Self-adaptive Cloud Autoscaling System (SSCAS) is not mature to a state that it can be reliably exploited in the cloud. In this article, we survey the state-of-the-art research studies on SSCAS and provide a comprehensive taxonomy for this field. We present detailed analysis of the results and provide insights on open challenges, as well as the promising directions that are worth investigated in the future work of this area of research. Our survey and taxonomy contribute to the fundamentals of engineering more intelligent autoscaling systems in the cloud.'
author:
- Tao Chen
- Rami Bahsoon
- Xin Yao
bibliography:
- 'references.bib'
title: 'A Survey and Taxonomy of Self-Aware and Self-Adaptive Cloud Autoscaling Systems'
---
<ccs2012> <concept> <concept\_id>10011007.10010940.10010971.10011120.10003100</concept\_id> <concept\_desc>Software and its engineering Cloud computing</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10011007.10010940.10011003.10011002</concept\_id> <concept\_desc>Software and its engineering Software performance</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
[^1]
Introduction
============
Modern IT companies, from small business to large enterprises, increasingly leverage cloud computing to improve their profits and reduce the costs. Throughout all the Software-as-a-Service (SaaS), Platform-as-a-Service (PaaS) and Infrastructure-as-a-Service (IaaS) levels, one of the pronounced benefits of the cloud is referred to as elasticity, which reflects the extent to which a system can adapt to the workload fluctuations by adjusting configurations and resource provisioning close to the demand. In certain predictable scenarios where the environmental condition has strong and stable seasonality, the configurations and resources can be approximately specified by human experts in advance. Nevertheless, for many other cases, for examples, unexpected workload changes, elasticity can be only enabled by runtime automatic scaling, or simply autoscaling: *a dynamic process, often operating on a Physical Machine (PM), that adapts software configurations (e.g., threads, connections and cache, etc) and hardware resources provisioning (e.g., CPU, memory, etc) on-demand, according to the time-varying environmental conditions.* The ultimate goal of autoscaling is to continually optimize the non-functional Quality of Service (QoS) (e.g., response time and throughput) and cost objectives for all cloud-based services[^2]; thus their Service Level Agreement (SLA) and budget requirements can be better complied with. In particular, autoscaling systems help to realize elasticity by providing timely and elastic adaptation in scales, which is one of the key benefit of cloud computing that attracts a wide range of practitioners [@lorido2014review][@qu2016auto]. Comparing with the other cloud resources management in general, autoscaling system has been specifically designed for: (i) scaling cloud-based applications in response to dynamic changes in load, uncertainties in operations, handling multitenancy while ensuring Service Level Agreement compliance etc. In contrast, other resource management tasks, e.g., resource scheduling, often work on planned and deterministic sequence of resource demand. (ii) Adapting both the software configurations and hardware resources (and their interplays) that span over all SaaS, PaaS and IaaS levels, whereas most of the other resource management considers hardware resources and IaaS only. (iii) Taking the QoS for cloud-based applications/services at the centre of the concerned objectives (explicitly or implicitly) while the other resource management tasks often focus on resource utilization.
From the literal meaning of the word autoscaling, it is obvious that the process is dynamic and requires the system to adapt itself subject to the dynamic and uncertain state of the services being managed and the environment. In such a way, the cloud-based services, running on a Virtual Machine (VM) or containers, can be expanded and shrink according to the environmental conditions at runtime. This characteristic has made autoscaling systems well-suited to the broad category of self-aware and self-adaptive systems [@Chen:2015:computer] [@roadmap] [@landscape]. However, given the unique characteristics of cloud, engineering Self-aware and Self-adaptive Cloud Autoscaling System (SSCAS) poses many challenges, including efficient autoscaling architectural styles, accurate model to predict the effects of autoscaling decision[^3] on the quality attributes, appropriate granularity of runtime control and effective trade-off decision making. In particular, a SSCAS should be able to handle various dimensions of QoS attributes, software configuration and hardware resources, in the presence of ***QoS interference*** [@fuzzy-vm-interference] [@qcloud] [@vm-fuzzy-MIMO] [@2014-decision-tree-software-CP-2014] where the quality of a single cloud-based service can be influenced by the dynamic behaviors of its neighbors on a PM, i.e., the other co-located services and co-hosted VMs or containers, under the sharing infrastructure of cloud.
In this article, we provide a survey and taxonomy for the landscape of SSCAS research to better understand the state-of-the-arts and to identify the open challenges in the field. In particular, we focus on cloud autoscaling researches with respect to the well-known principles of self-awareness [@epics] and self-adaptively [@landscape] in computing systems, as well as the fundamental approaches and techniques that realize them. Broadly speaking, we aim to answer the following research questions: (i) What are the levels of self-awareness and self-adaptivity that have been captured in SSCAS? (ii) What are the architectural patterns used for engineering SSCAS? (iii) What are the approaches used to model the quality related to SSCAS? (iv) What is the granularity of control in SSCAS? (v) What are the approaches used for decision making in SSCAS? In a nutshell, our key findings are:
- *Stimulus-*, *time-* and *goal-awareness* are the most commonly considered self-awareness levels in current SSCAS research while *self-configuring* and *self-optimizing* are more attractive than the others on self-adaptivity (Please refer to Section \[sec:s-aware\] and \[sec:s-adaptive\] for their definitions, respectively).
- It was found that the general and simple feedback loop architectural pattern (and its variations) has been prominent for engineering SSCAS.
- Analytical and machine learning based modeling have been the most widely used approaches for modeling the effects of autoscaling decision on quality attributes. But surprisingly, systematic selection of model’s input features and QoS interference are rarely considered for SSCAS.
- The level of service/application is the most popular granularity of control for SSCAS.
- Explicit optimization driven decision making is the most commonly used approach in SSCAS, but most of them have assumed single objective or using weighted sum aggregation of objectives. Further, we noted that QoS interference is again absent in many studies.
In addition, we obtain the following insights on the open challenges for future research of the field:
- There is a lack of considering required knowledge and its representation for SSCAS architecture with respect to the principles of self-awareness [@epics], thus urging further investigations. This can help to reason about and prevent improper design decisions, leading to better self-adaptivity.
- Despite that QoS interference has been found to be an important issue [@fuzzy-vm-interference] [@qcloud] [@vm-fuzzy-MIMO] [@2014-decision-tree-software-CP-2014], it is often overlooked in both the QoS modeling and decision making aspects of SSCAS. Therefore, we call for novel and effective approach to manage and mitigate QoS interference in cloud.
- Most studies attempt to scale hardware resources at the IaaS level only. However, a mature SSCAS should additionally consider the software configurations related to the cloud-based services and their interplay with the hardware resources, as found in recent studies [@2013-JRAO-most-closest-work-2013] [@software-RP-two-loops] [@2014-decision-tree-software-CP-2014].
- Instead of using fixed granularity of control (i.e., the boundary of decision making is on each service/application, VM/container, PM or cloud), future SSCAS should consider more flexible ones, e.g., dynamic and hybrid level, as discovered in recent studies [@MIMO-fuzzy-hill-climbing-2013] [@11icde_smartsla_full] [@Chen:2014:seams].
- The assumption of autoscaling bundles (e.g., the VM instances from Amazon EC2) has been made in a considerable amount of SSCAS studies. However, it is known that renting bundles cannot and does not reflect the interests of consumers and the actual demand of their cloud-based services [@rule-control-elasticity-cloud]. Thus, considering arbitrary and custom combinations of configurations and resources is an inevitable trend in the cloud computing paradigm.
- Considering multi-objectivity in SSCAS is a must in order to create better diversity and possibly better trade-off quality without the needs of weights specification. In addition, how to achieve balanced trade-off over the set of non-dominated solutions is worth investigating [@Chen:2015:tsc-pending].
- More real world cases and scenarios are needed as this can be the only way to fully verify the potentials, effectiveness and impacts of SSCAS.
This article is structured as follows: Section \[sec:back\] introduces the background and challenges. Section \[sec:related\] compares our work with the other reviews. A taxonomy and findings of the survey, with respect to different aspects of a SSCAS, are presented in Section \[sec:result\]. Section \[sec:diss\] discusses on the findings and presents the open challenges we learned. Section \[sec:con\] presents the conclusion.
Backgound {#sec:back}
=========
Autoscaling System in Cloud
---------------------------
![The conceptual design of an autoscaling system. (Note that this figure represents the conceptual design of an autoscaling system in cloud. Practical deployment of the autoscaling engine can be either centralized, or decentralized where there are different engines, each of which is running on a PM.[]{data-label="ch1:simple_arch"}](figures/simple-arch.pdf){width="7.3cm"}
Given that it is almost impossible to access the low level details of cloud-based services (e.g., their codes and algorithms) at runtime, an autoscaling system often consist of two physical parts: a managing part containing the autoscaling engine and a manageable part encapsulating services and VMs/containers running in the cloud. The two physical parts are seamlessly and transparently connected for realizing the entire autoscaling process, known as external adaptation [@roadmap] [@landscape].
The external adaptation of an autoscaling system is shown in Figure \[ch1:simple\_arch\]. As we can see, the core of an autoscaling system in the cloud is the autoscaling engine, which can consist of multiple logical aspects. A typical example of autoscaling system is a feedback loop that covers monitoring and scaling: the former gathers the service’s or application’s current state while the latter utilizes the information to decide an action after being analyzed and reasoned about by the autoscaling engine. Given the multi-tenant nature of cloud, cloud-based services often come with different QoS objectives, SLA requirements and budget constraints, etc. The ultimate goal of an autoscaling system is to adapt those cloud-based services, through scaling the related software configurations and hardware resources, in such a way that their objectives are continually optimized. To execute an autoscaling decision, the scaling actions could be vertical (scale up/down where changes occurs on a VM/container), horizontal (scale in/out that adds/removes other VMs/containers), or both.
Self-Adaptivity in Software Systems {#sec:s-adaptive}
-----------------------------------
The broad category of automatic and adaptive systems aim to deal with the dynamics that the system exhibited without human intervention; but this does not necessarily involve uncertainty, i.e., there are changes related to the system but it is easy to know when they would occur and the extent of these changes. Self-adaptivity, being a sub-category, is a particular capability of the system to handle both dynamics and uncertainty. Here, self-adaptive systems refer to the systems that are capable to adapt their behaviors according to the perception of the uncertain environment and its own state. To date, self-adaptivity in software systems remain an important and challenging research field [@femosaa][@2014-kalman-pair-wised-coupling-on-tiers-2014][@icpechen]. According to the adaptive behaviors, self-adaptivity can be regarded as the following four properties, each of which covers a specific set of goals, as discussed by [@landscape]:
- **Self-configuring:** “The capability of reconfiguring automatically and dynamically in response to change by installing, updating, integrating, and composing/decomposing software entities.” [@landscape]
- **Self-healing:** “This is the capability of discovering, diagnosing, and reacting to disruptions. It can also anticipate potential problems, and accordingly take proper actions to prevent a failure.” [@landscape]
- **Self-optimizing:** “This is also called self-tuning or self-adjusting, is the capability of managing performance and resource allocation in order to satisfy the requirements of different users, e.g., response time, throughput and utilization.” [@landscape]
- **Self-protecting:** “This is the capability of detecting security breaches and recovering from their effects. It has two aspects, namely defending the system against malicious attacks, and anticipating problems and taking actions to avoid them or to mitigate their effects.” [@landscape]
Self-Awareness in Software Systems {#sec:s-aware}
----------------------------------
In contrast, self-awareness is about the capability of a system to acquire knowledge about its current state and the environment. Such knowledge permits better reasoning about the system’s adaptive behaviors. Consequently, self-awareness is often seen as the lowest level of abstraction of self-adaptivity [@landscape], and thus it can improve the basic perceptions and self-adaptivity of a system [@2014_epics_handbook] [@7185305] [@epics_survey] [@Chen2016:book]. Inspired from the psychology domain, Becker et al. [@epics] have classified self-awareness of a computing system into the following general capabilities (they have used node to represent any conceptual part of a system being managed):
- **Stimulus-aware:** “A node is stimulus-aware if it has knowledge of stimuli. The node is not able to distinguish between the sources of stimuli. It is a prerequisite for all other levels of self-awareness.” [@epics]
- **Interaction-aware:** “A node is interaction-aware if it has knowledge that stimuli and its own actions form part of interactions with other nodes and the environment. It has knowledge via feedback loops that its actions can provoke, generate or cause specific reactions from the social or physical environment.” [@epics]
- **Time-aware:** “A node is time-aware if it has knowledge of historical and/or likely future phenomena. Implementing time-awareness may involve the node possessing an explicit memory, capabilities of time series modeling and/or anticipation.” [@epics]
- **Goal-aware:** “A node is goal-aware if it has knowledge of current goals, objectives, preferences and constraints. It is important to note that there is a difference between a goal existing implicitly in the design of a node, and the node having knowledge of that goal in such a way that it can reason about it. The former does not describe goal-awareness; the latter does.” [@epics]
- **Meta-self-aware:** “A node is meta-self-aware if it has knowledge of its own capability(ies) of awareness and the degree of complexity with which the capability(ies) are exercised. Such awareness permits a node to reason about the benefits and costs of maintaining a certain capability of awareness.” [@epics]
Comparison to Related Surveys {#sec:related}
=============================
Research on cloud autoscaling systems and the related topics have been reviewed in some other surveys. For example, Manvi and Shyam [@manvi2014resource] present a review on resource management in the cloud, particularly at the IaaS level. They have provided a board survey on different issues related to managing cloud resources, e.g., resource adaptation, resource mapping and resource brokering etc. While resource management has some similarities to autoscaling, they lie in different levels of abstraction: the latter is more specific than the former. In other words, cloud autoscaling is one, but probably the most important part of the board cloud resource management. Another review from Mana [@mann2015allocation] is explicitly concerned with VM to PM mapping problem which is also belong to the cloud resource management category, but is often regarded as a fundamentally different issue to cloud autoscaling. Ardagna et al. [@ardagna2014quality] present a survey on QoS modeling and its application in the cloud. Indeed, QoS is the major concern for a cloud autoscaling system, but its management can be governed by various different approaches other than autoscaling, e.g., load balancing and admission control, which are also covered in [@ardagna2014quality]. In contrast to the above, our survey has explicitly focused on automatically scaling software configuration and hardware provisioning in the cloud in order to change the capacity of cloud-based services to handle the dynamic workloads.
Al-Dhuraibi et al. [@al2017elasticity] present a review on elastic autoscaling approaches in the cloud, specifically focusing on the physical infrastructure level support, e.g., benchmarking and containerization techniques. Our survey, in contrast, is primarily concerned with the logical architecture and algorithmic level techniques for achieving different aspects of cloud autoscaling, e.g., modeling and decision making. Qu et al. [@qu2016auto] also survey autoscaling approaches for a special type of cloud application, i.e., web applications, with a coarse correlation to self-adaptivity, e.g., if an approach is self-adaptive or not; while our survey is application agnostic and we present finer correlation of an approach to different levels of self-adaptivity, e.g., self-optimization. The most related survey from the literature is probably the one by Lorido-Botran et al. [@lorido2014review], in which different category of algorithmic level techniques for QoS modeling and decision making in cloud autoscaling are reviewed. However, their survey differs from ours in the following three aspects: (i) they have not provided a comprehensive taxonomy on the cloud autoscaling problem; such a taxonomy (i.e., modeling, architecture, granularity and decision making), which we will present in the next section, is important as it clearly state the open problems and challenges related to different aspects of the cloud autoscaling domain, providing better clarifications and clearer directions for researchers on this research field. (ii) In addition, Lorido-Botran et al. [@lorido2014review] did not explicitly link the cloud autoscaling systems to different levels of self-awareness and self-adaptivity, which is one of the key contributions of our survey. (iii) Finally, we discuss the open problems and challenges of cloud autoscaling systems in a broader fashion.
In summary, our survey differs from the other similar reviews in the following:
- We present a focused survey on the logical architecture and algorithmic level techniques for cloud autoscaling which are application agnostic.
- We explicitly correlate the reviewed approaches with different levels of self-awareness, self-adaptivity and the required knowledge in a fine grained manner.
- We provide a clarified taxonomy that covers different logical aspects for engineering cloud autoscaling systems, and classify every study accordingly.
- We discuss the open problems and challenges of cloud autoscaling systems in a broader fashion.
Taxonomy and Survey Results for SSCAS {#sec:result}
=====================================
In this section, we present a taxonomy and survey results for the state-of-the-art SSCAS research obtained from our review process.
Review Process and Research Questions {#sec:rq}
-------------------------------------
The review is intended to create a broad scope to cover the landscape of SSCAS research. Particularly, the following research questions serve as the main drivers of this review:
- *RQ1:* What are the levels of self-awareness and self-adaptivity that have been captured in SSCAS?
- *RQ2:* What are the architectural patterns used for engineering SSCAS?
- *RQ3:* What are the approaches used to model the quality related to SSCAS?
- *RQ4:* What is the granularity of control in SSCAS?
- *RQ5:* What are the approaches used for decision making in SSCAS?
The following prominent indexing services were used during the review: IEEE Xplore, ACM Digital Library, Science Direct, ISI Web of Knowledge, and Google Scholar. The search term was “Cloud computing” AND “Autoscaling” AND (“QoS modeling” OR “Performance modeling” OR “decision making” OR “optimization” OR “Architecture” OR “Interference” OR “Resource allocation” ). After applying inclusion (e.g., considering only journal, conference, and workshop papers) and exclusion criteria (e.g., removing duplicate entries and considering only the extended version) to the initial search result, the review has ended with the total of 109 studies.
A Taxonomy of SSCAS Research
-----------------------------
The overall taxonomy, concluded from the extracted studies, is given in Figure \[fig:tax\]. As we can see, current research on SSCAS often require sophisticated designs in different highest leveled logical aspects of the autoscaling engine, which we have classified and discussed as the following:
![A taxonomy of SSCAS research.[]{data-label="fig:tax"}](figures/global.pdf){width="7.3cm"}
- ***Self-Awareness:*** This is concerned with the ability to acquire and maintain knowledge about the system’s own states and the environment, as specified in Section \[sec:s-adaptive\]. The key challenges here are which level(s) of knowledge is required for a SSCAS, what does it means for certain level in the problem context, (e.g., what does interaction refers to?) and what is the representation for different levels of knowledge, e.g., how do we represent goals in the SSCAS?
- ***Self-Adaptivity:*** This is about the ability to change the system’s own behavior with specific goals in mind, as specified in Section \[sec:s-aware\]. Often, the required levels of self-adaptivity depending on the requirements, but they could be also related to the specific levels of knowledge that the SSCAS is able to capture, e.g., the SSCAS has to be goal-aware to achieve self-optimization.
- ***Architectural Pattern:*** Autoscaling architecture is the most essential element of SSCAS. It describes the structure of the autoscaling process, the interaction between components and the modularization of the other important logical aspects in autoscaling. The challenge of architecting SSCAS is concerned with how to systematically capture different logical aspects (e.g., decision making) of SSCAS using a given architectural pattern. More importantly, how to encapsulate these aspects and the algorithms that realizes them into different components of the pattern.
- ***QoS Modeling:*** While modeling the cost incurred by cloud-based services is straightforward, modeling the QoS is often much more complex and challenging. Here, the QoS modeling is concerned with the sensitivity of QoS with respect to the environment conditions (e.g., workload) and the control knobs (e.g., software configurations and hardware resources). The resulting model is a powerful tool to assist the autoscaling decision making process. Without loss of generality, in this article, we use ***cloud primitives*** to refer to both control knobs and environmental conditions in the cloud. In particular, we further decompose the notion of primitives into two categories, termed ***control primitive*** and ***environmental primitive***. Control primitives refer to the internal control knobs that can be either software or hardware. They are the fundamental features that can be controlled by the cloud providers to support QoS. Specifically, software control primitives are the key cloud configurations at the software level, e.g., the number of threads in the thread pool, the buffer size and the cache size, etc. In contrast, hardware control primitives refer to the computational resources, such as CPU, memory and bandwidth, etc. Typically, software control primitives exist on the PaaS layer while the hardware control primitives present on the IaaS layer. It is worth noting that considering software control primitives when autoscaling in the cloud is a non-trivial task, as they have been proved to be important features that can significantly influence the QoS [@2013-JRAO-most-closest-work-2013] [@software-RP-two-loops] [@2014-decision-tree-software-CP-2014]. The environmental primitives, on the other hand, refer to those external stimuli that is uncontrollable but can cause dynamics and uncertainties in the cloud. These, for example, can be the workload, incoming data, node failure, etc. The above examples of primitives listed above are not exhaustive, Ghanbari et al. [@rule-control-elasticity-cloud] have provided a more completed and detailed list of the possible control primitives in cloud.
The challenges of QoS modeling include: (i) which primitives should be selected as model’s input features; (ii) how does the QoS change in conjunction with those primitives; (iii) how to incorporate the information of QoS interference into the model; (iv) whether the model is built offline or online; (v) and whether the model is dynamic, semi-dynamic or static.
- ***Granularity of control:*** Determining the granularity of control in the autoscaling engine is essential to ensure the benefits (e.g, QoS and cost objectives) for all cloud-based service. It is concerned with understanding whether certain objectives can be considered in isolation with some of the others, i.e., the boundary of decision making. This is because ***objective-dependency*** (i.e., conflicted or harmonic objectives) often exist in the decisions making process, which implies that the overall quality of autoscaling can be significantly affected by the inclusion of conflicted or harmonic objectives when making decision, hence rendering it as a complex task. This is especially important for the shared cloud infrastructure where objective-dependency exists for both intra- and inter-services. That is to say, objective-dependency is not only caused by the nature of objectives (intra-service), e.g, throughput and cost objective of a service; but also by the QoS interference (inter-services) due to the co-located services on a VM/container and co-hosted VMs/containers on a PM [@2013-JRAO-most-closest-work-2013] [@software-RP-two-loops] [@2014-decision-tree-software-CP-2014] [@qcloud].
Here, the challenges are which granularity of control to use, what is the basic entity to control (e.g., application or VM), and whether the control is in a centralized or decentralized manner.
- ***Decision making:*** The final logical aspect in autoscaling logic is the dynamic decision making process that produces the optimal (or near-optimal) decision, which consists of the newly configured values of the related control primitives, for all the related objectives. In the presence of objective dependency, autoscaling decision making requires to resolve complex trade-offs, subject to the SLA and budget requirements. The trade-off decision can be then executed using either vertical (scale up/down) and/or horizontal scaling actions (scale in/out), which adapt the cloud-based services and/or VMs/containers correspondingly.
The challenges of decision making in SSCAS include: (i) how to reason about and search for the effected adaptation decisions; (ii) what are the objectives, their representations and conflicting relations, if any; (iii) and which are the control primitives to tune.
In the following, we present our detailed findings in regards to the taxonomy of SSCAS.
The Levels of Self-Awareness and Self-Adaptivity in Cloud Autoscaling Systems {#sec:sscas}
-----------------------------------------------------------------------------
\[fig:sscas\]
[0.33]{} \[fig:self-aw\]
[0.33]{} \[fig:self-ad\]
[0.33]{} \[fig:knowledge\]
*RQ1: What are the levels of self-awareness and self-adaptivity that have been captured in SSCAS?*
It is worth noting that not all the studies have explicitly declared which levels of self-awareness/self-adaptivity that they have taken into account, therefore we identified this by looking at the studies in details with respect to the definitions of self-awareness/self-adaptivity. From Figure \[fig:self-aw\], we can see that *stimulus-awareness*, which is the fundamental level in self-awareness, has been considered in all the studies. The *time-* and *goal-awareness* have attracted relativity similar amount of attention. However, *interaction-* and *meta-awareness* has not been widely studied in recent SSCAS research. In Figure \[fig:self-ad\], we see that *self-configuring* and *-optimizing* have been predominately captured in the studies, whereas *self-healing* and *-protecting* receive little attentions. In particular, we found no study that explicit aims for *self-protecting* in SSCAS. Figure \[fig:knowledge\] illustrates whether the required knowledge representations at the SSCAS architecture level, e.g., knowledge of goal is required in the architecture, have been discussed, implicitly discussed or explicitly discussed in the studies. We can see that the majority of the studies surveyed do not attempt to declare what knowledge is required in SSCAS architecture, leaving only 33% of the studies have discussed the knowledge implicitly or explicitly.
Architectural Pattern {#sec:arch}
---------------------
*RQ2: What are the architectural patterns used for engineering SSCAS?*
We classified the predominantly applied architectural patterns for SSCAS into three categories based on their basic form; these are *Feedback Loop* [@Brun:2009], *Observe-Decide-Act* (ODA) [@self-aware-ML-adaptive-control] and *Monitor-Analysis-Plan-Execute* (MAPE) [@ibm].
From Figure \[fig:arch\], we see that the generic feedback loop has been the predominant architecture pattern in SSCAS, following by the MAPE pattern. Particularly, as we can observe form Table \[tb:arch\], single and close feedback loop are widely exploited in SSCAS. In the following, we specify some representative studies under each category in details.
### Feedback Loop
Feedback loop is the most general architectural pattern for controlling self-adaptive systems, including the autoscaling systems. It is usually a closed-form loop made up of the managing system itself and the path transmitting its origin (e.g., a sensor) to its destination (e.g., an actuator). Here, we further divide the pattern as *single* or *multiple* loops:
- *4.4.1.1 Single Loop:* Single loop is the simplest, yet the most commonly used pattern for SSCAS due to its flexibility. The most common practice with single loop is to build a closed feedback control where the core is the decision making component and an optional QoS modeling component, e.g., Ferretti et al. [@05557978] , CloudOpt [@fine-grained-servce-LQM-MIP-power], SmartSLA [@11icde_smartsla_full] and CLOUDFARM [@2014-linear-centralized-decision-making-2014], etc. Some other studies have included an additional component for workload or demand prediction based on either offline profiling, e.g., Jiang et al. [@typical-navie-scaling-VM-number] and Fernandez et al. [@2014-profiling-decision-tree-scaling-2014], or online learning, e.g., Kingfisher [@2011-cost-aware-v-h-scaling-2011] and PRESS [@signal-resource-trend-prediction].
Open feedback exists for single loop, as presented in Cloudine [@2013-elasticity-primitives-2013], where the scaling actions are partially triggered by user requests. In particular, they use a centralized *Resource and Execution Manager* to handle all the scaling actions. Apart from the general autoscaling architecture, other efforts are particularly designed upon specific cloud providers. For example, Zhang et al. [@MPC-price] as well as Kabir and Chiu [@cache-static-ANN-bi-obj-2012] propose to use a simple feedback loop for architecting autoscaling system, which is heavily tied to the properties of Amazon EC2.
[|P[2.4cm]{}|P[0.8cm]{}|P[1.1cm]{}|X|]{} ***Architectural Pattern***&***Style***&***Open/Close***&***Representative Examples***\
&&close&[@05557978], [@fine-grained-servce-LQM-MIP-power],[@11icde_smartsla_full], [@HPL-2008-123R1-mimo], [@2014-eplison-GA-weigh-h-scaling-2014], [@06119056], [@2014-linear-centralized-decision-making-2014], [@2014-kalman+rule-based-2014], [@typical-navie-scaling-VM-number], [@2014-profiling-decision-tree-scaling-2014], [@2011-cost-aware-v-h-scaling-2011], [@kriging-controller], [@2013-self-organising-map-2013], [@signal-resource-trend-prediction], [@2014-fuzzy-compare-to-JRao-2014], [@2013-software-CP-only-2013],[@acdc09], [@compare-to-Rao-fuzzy-2013], [@full-simulation-model], [@GA-full-simulation] [@CloudSim], [@CloudAnalysis], [@DCSim], [@sla-provision], [@parallel-RL-vertical-QoS-2013], [@MASCOTS11], [@E3-R-extended], [@wosp10sla], [@queue-VM-group], [@3-stages-game-theory], [@2013-rule-based-multi-elasiticity-2013], [@linear-mapping-CP-bundles-2012], [@qcloud], [@vm-fuzzy-MIMO], [@2014-decision-tree-software-CP-2014], [@dynamic-model-comparison], [@change-point], [@2013-single-learner-filter-wrapper-LR-2013], [@kalman-AR], [@kalman-clustering], [@ILP-cost-only-scaling-2013], [@2014-ARMA-single-agg-objective-2014], [@scale-rule-based], [@IWQOS11], [@2013-scaling-select-one-predictor-and-error-correlation-2013], [@2010-demand-pattern-match-2010], [@ensemble-prediction-VM-2014-full], [@seelam2015polyglot], [@souza2015using], [@farokhi2016hybrid], [@shariffdeen2016workload], [@gandhi2017model], [@qu2016reliable], [@sun2016roar], [@da2016autoelastic], [@rameshan2016augmenting], [@baresi2016discrete], [@zhang2016container], [@li2015rest], [@sun2017automated], [@awada2017improving]\
&&open& [@2013-elasticity-primitives-2013], [@MPC-price], [@cache-static-ANN-bi-obj-2012]\
&multiple&close&[@2014-kalman-pair-wised-coupling-on-tiers-2014], [@2012-app-VM-mapping-2012], [@EMA-CPU-memory-PD], [@MIMO-fuzzy-hill-climbing-2013], [@Arc-cover-all-controller], [@ICDCS2011], [@PCA-model-scaling-2013], [@2014-2-SVM-workload-type-2014], [@06032254], [@fuzzy-2-loop-control], [@MASCOTS11-bu-software-CP-full] [@2013-JRAO-most-closest-work-2013], [@TR-10-full-version], [@Chen:2013:iccs], [@Chen:2015:tsc-pending], [@Chen:2015:computer], [@multitier-resalloc-Cloud11], [@profit-cent-local-search], [@2014-navie-ANN-GA-2014], [@JSS_Kousiouris], [@Chen:2013:seams], [@MOCS2014-full], [@Chen:2014:ucc], [@Chen:2015:tse-pending], [@ma2016auto]\
OAD&single&close&[@van2015mnemos],[@herbein2016resource], [@HuBrKo2011-SEAMS-ResAlloc]\
&single&close&[@li-opt-clouds-4], [@cloud_computing_2010_5_20_50060-ext], [@Compsac_2010_I_Brandic], [@SEASS_2010_Michael_Maurer], [@tse1], [@fuzzy-vm-interference], [@queue-prediction], [@Emeakaroha_CloudComp2010], [@HPCS_IWCMC_Vincent], [@self-sla-and-resource], [@megahed2017stochastic], [@lakew2017kpi], [@aslanpour2017auto], [@gill2017chopper]\
&multiple&close&[@software-RP-two-loops], [@BRGA-resource],[@Chen:2014:seams]\
- *4.4.1.2 Multiple Loops:* It is possible to use multiple loops and controllers for autoscaling in the cloud. Here, multiple feedback loops operate in different levels of the architecture, e.g., one operates at the cloud level while the others operate on each VM. The benefit is that multiple loops provide low coupling in the design of the loops for SSCAS. Notably, multiple loop control can be used to separate global and local controls. Among others, Kalyvianaki et al. [@2014-kalman-pair-wised-coupling-on-tiers-2014] apply multiple decentralized feedback loops for autoscaling CPU in the cloud. Although it aims to exploit one loop per individual application, the controllers actually operate on each tier of an application. Chen and Bahsoon [@Chen:2015:tsc-pending] also leverage multiple feedback loop to auto-scale cloud services, where each PM maintains a loop. Unlike classic feedback loop where the adaptations occur only on the manageable part of SSCAS, their adaptations also happen on the manging part. Multiple loop control is also effective for isolating the logical aspects of autoscaling and management in the cloud. For instance, Wang, Xu and Zhao [@fuzzy-2-loop-control] propose a two layered feedback control for autoscaling in the cloud. The first layer, termed guest-to-host optimization, controls the hardware resources, e.g., CPU and memory. Subsequently, the host-to-guest optimization adapts the software configuration accordingly.
### Observe-Decide-Act
Observe-Decide-Act (ODA) loop [@self-aware-ML-adaptive-control] is considered as an extended pattern of the generic feedback loop. As specified in the SEEC framework [@self-aware-ML-adaptive-control], ODA is unique in the sense that it decouples multiple loops to different roles (i.e., application developer, system developer, and the SEEC runtime decision infrastructure) in the development life-cycle, each role focuses on one or more steps in ODA. In such a way, ODA links the effects of human activities on the adaptive behaviors.
Among others application of OAD in SSCAS, MNEMOS [@van2015mnemos] has relied on OAD to realize an integrated, datacenter-wide architecture for autoscaling resources in the cloud, in which the *System Monitor* acts as the observer, the *Portfolio Scheduler* acts as the decider, and the *VM Manager* acts as the executioner. Huber et al. [@HuBrKo2011-SEAMS-ResAlloc] also use ODA for self-aware autoscaling resources in the cloud. However, unlike traditional ODA loop, it has an additional *Analysis* step which is used to detect the type of problems that trigger adaptation.
### Monitor-Analyze-Plan-Execute
Another pattern extended from the generic feedback loop, namely Monitor-Analyze-Plan-Execute (MAPE), is firstly proposed by IBM for architecting self-adaptive systems. In such pattern, the *Decide* step in OAD is further divided into two substeps, these are *Analyze* and *Plan*, where the former is particularly designed to determine the causes for adaptations, e.g., SLA violation; the latter, on the other hand, is responsible for reasoning about the possible actions for adaptation. MAPE sometime can be extended by a Knowledge component (a.k.a. MAPE-K) which maintains historical data and knowledge used by the system for better adaptation. MAPE can be also realized as either *single* or *multiple* loops:
- *4.4.3.1 Single Loop:* MAPE (or MAPE-K) is also widely applied for SSCAS. For example, the architecture of the FoSII project [@Compsac_2010_I_Brandic] [@SEASS_2010_Michael_Maurer] leverages single MAPE-K to realize the self-management interface, aiming to prevent SLA violations in cloud by devising the related actions. They also use the additional *Knowledge* (K) component to record cases and the related solutions, which can assist the autoscaling decision making. Chen and Bahsoon [@Chen:2014:seams] have realized MAPE as a single loop where the adaptations occur on both the managing and manageable parts of the SSCAS.
- *4.4.3.2 Multiple Loops:* Realizing multiple MAPE loops for SSCAS is also possible. Zhang et al. [@software-RP-two-loops] introduce an architecture for autoscaling using two nested MAPE loops. The first loop is responsible for adapting the software primitives while the other loop is used to change the hardware primitives. These two loops run sequentially upon autoscaling, that is, adapting the software control primitives before changing the hardware control primitives. Similarly, BRGA [@BRGA-resource] utilizes MAPE to realize a framework for autoscaling in the cloud. Such solution consists of both the local and global view of the cloud-based application.
QoS Modeling {#sec:qos}
------------
*RQ3: What are the approaches used to model the quality related to SSCAS?*
QoS modeling, or performance modeling, is a fundamental research theme in cloud computing and it can serve as useful foundations for addressing many research problems in the cloud [@lorido2014review], including autoscaling. The QoS models correlate the QoS attributes to various control primitives and environmental primitives. Clearly, these models are particularly important for SSCAS, since they are powerful tools that can assist the reasoning about the effects of adaptation on objectives in the autoscaling decision making process. Note that although QoS model can provide great helps to the decision making in SSCAS, not all of the studies have considered QoS modeling as part of their solutions. In fact, some of them rely on model-free solution, e.g., control theoretic approach, which we will review in Section \[sec:dm\].
Typically, QoS modeling consists of two phases: the primitives selection phase and the QoS model construction phase. More precisely, the primitives selection phase determines *which* and *when* the cloud primitives correlate with the QoS; while the QoS model construction phase identifies *how* these primitives correlate with the QoS, i.e., their magnitudes in the correlation. The QoS models might come as three forms: (i) static models where the models’ expression and their structure (e.g., the number of inputs and the coefficients) do not change over time; (ii) dynamic models which permits those changes; or (iii) semi-dynamic models in the sense that the expression (e.g., coefficients) could be dynamically updated but the input features do not. Further, those models can be built online at system runtime, or offline at the design phase of the system. In this section, we classified the studies mainly based on the modeling methods applied to the QoS model construction phase, since we found that the primitives selection is often conduced using manual and static approaches in the studies. As we can see in Figure \[fig:qos\], the majority of the studies has exploited analytical model (43%) and machine learning model (38%) to predict QoS. In contrast, simulation and hybrid model receives much less attention. From Table \[tb:qos\], we can obtain the following observations:
[|P[1.3cm]{}|P[1.7cm]{}|P[0.7cm]{}|P[1.3cm]{}|P[1.2cm]{}|X|]{} ***Modeling Approach***&***Type***&***Built***&***QoS Interference***&***State***&***Concrete Models***\
& Queuing model&offline&no&static& **S-QUEUE (5):** [@MPC-price], [@06119056], [@E3-R-extended], [@queue-VM-group], [@typical-navie-scaling-VM-number] **M-QUEUES (3):** [@multitier-resalloc-Cloud11], [@queue-prediction], [@wosp10sla] **LQN (4):** [@3-stages-game-theory], [@fine-grained-servce-LQM-MIP-power], [@li-opt-clouds-4], [@sla-provision]\
&\[-0.05cm\]&online&no&semi& **MDP (1):** [@tse1] **MODEL@RUNTIME (1):** [@2014-eplison-GA-weigh-h-scaling-2014]\
&&offline&no&static& **PCM (1):** [@HuBrKo2011-SEAMS-ResAlloc] **GRAPH (1):** [@2013-rule-based-multi-elasiticity-2013]\
&&offline&no&dynamic&**PCA (1):** [@PCA-model-scaling-2013]\
&&offline&no&static& **EMPIRICAL-MODEL (9):** [@06032254], [@Compsac_2010_I_Brandic], [@2014-linear-centralized-decision-making-2014], [@Emeakaroha_CloudComp2010], [@HPCS_IWCMC_Vincent], [@profit-cent-local-search], [@cache-static-ANN-bi-obj-2012], [@BRGA-resource], [@shariffdeen2016workload], [@megahed2017stochastic], [@gill2017chopper], [@awada2017improving]\
&&offline&yes&static& **EMPIRICAL-MODEL (1):** [@ma2016auto]\
Simulation& &offline&no&semi& **PROFILING (2):** [@2014-profiling-decision-tree-scaling-2014], [@sun2016roar] **SIMULATOR (4):** [@full-simulation-model], [@CloudSim], [@CloudAnalysis], [@DCSim], [@GA-full-simulation]\
&&online&no&semi& **LR (3):** [@acdc09], [@software-RP-two-loops], [@linear-mapping-CP-bundles-2012] **ARMA (1):** [@HPL-2008-123R1-mimo], [@MIMO-fuzzy-hill-climbing-2013] **KF (1):** [@2014-kalman-pair-wised-coupling-on-tiers-2014]\
&&online&no&dynamic& **LR (1):** [@ISPASS07]\
&&online&yes&semi& **MIMO (4):** [@fuzzy-vm-interference], [@qcloud], [@vm-fuzzy-MIMO], [@2014-decision-tree-software-CP-2014]\
&&online&no&semi& **KM (1):** [@kriging-controller] **RT (1):** [@11icde_smartsla_full] **ANN (4):** [@dynamic-model-comparison] [@2014-navie-ANN-GA-2014] [@JSS_Kousiouris] [@Chen:2013:seams] **SVM (2):** [@2014-2-SVM-workload-type-2014] [@dynamic-model-comparison] **CHANGE-POINT (1):** [@change-point]\
&&online&yes&semi& **SVM (1):** [@rameshan2016augmenting]\
&&online&no&semi& **ARMA+SVM (1):** [@TR-10-full-version] **ANN+ANN (1):** [@MOCS2014-full]\
&&online&no&dynamic& **KNN+LR:+RT (1):** [@2013-single-learner-filter-wrapper-LR-2013]\
&&online&yes&dynamic& **ARMA+ANN+RT (2):** [@Chen:2014:ucc], [@Chen:2015:tse-pending]\
&&semi&no&semi& **LQN+KF (2):** [@2014-kalman+rule-based-2014], [@kalman-AR] **LQN+KF+K-MEAN (1):** [@kalman-clustering] **S-QUEUE+ARMA (1):** [@ICDCS2011] **M-QUEUE+KF (1):** [@gandhi2017model]\
&&offline&yes&static& **EMPIRICAL-MODEL+PROFILING (1):** [@sun2017automated]\
1. Despite the high importance of QoS interference, it has not received much attention when modeling the QoS (only 13%).
2. Truly dynamic QoS modeling, i.e., changing both the input features and their coefficients, is still minority (7%). Other studies have merely considered changing coefficients of the model while ignoring the input features’ dynamics (54% semi-dynamic), or none at all (39% static).
3. The concrete modeling methods applied for machine learning model is more diverse than the methods in other categories.
Additionally, from Table \[tb:qos-inout\], we can observe that:
1. Although most studies (65%) have only considered certain inputs/output during their experiments, they have claimed that their model is compatible with any given inputs (i.e., any cloud control primitives) and/or output (i.e., any QoS attributes).
2. The most widely considered input dimension is CPU while the most common output is response time (except the general one, i.e., QoS attribute).
3. The most explicitly modeled number of outputs is three, while the most explicitly considered number of inputs is four.
In the following, we specify some representative studies on QoS modeling for SSCAS in details.
[|P[3cm]{}|X|]{} ***Outputs***&***Cloud Primitives***\
QoS attributes& **CPU (1):** [@HuBrKo2011-SEAMS-ResAlloc] **number of VM (1):** [@2014-eplison-GA-weigh-h-scaling-2014], [@2014-profiling-decision-tree-scaling-2014] **configurations (1):** [@2014-2-SVM-workload-type-2014], [@2014-decision-tree-software-CP-2014] **resources (2):** [@2014-navie-ANN-GA-2014], [@qcloud] **CPU and bandwidth (1):** [@vm-fuzzy-MIMO] **CPU and memory (4):** [@kriging-controller], [@MIMO-fuzzy-hill-climbing-2013], [@fuzzy-vm-interference], [@GA-full-simulation] **configurations and resources (6):** [@PCA-model-scaling-2013], [@software-RP-two-loops], [@tse1], [@Chen:2014:ucc], [@Chen:2015:tse-pending], [@Chen:2013:seams] **configurations, CPU and memory (1):** [@TR-10-full-version] **resources and workload (8):** [@ISPASS07], [@JSS_Kousiouris], [@kalman-AR], [@full-simulation-model], [@CloudSim], [@CloudAnalysis], [@DCSim], [@gill2017chopper] **CPU, memory and disk (2):** [@2013-rule-based-multi-elasiticity-2013], [@HPL-2008-123R1-mimo] **CPU, memory and bandwidth (3):** [@2014-linear-centralized-decision-making-2014], [@dynamic-model-comparison], [@self-sla-and-resource] **CPU, storage and bandwidth (2):** [@06032254], [@Emeakaroha_CloudComp2010] **configurations, resources and workload (1):** [@2013-single-learner-filter-wrapper-LR-2013] **CPU, bandwidth, storage and number of VM (1):** [@Compsac_2010_I_Brandic] **CPU, memory, workload and number of VM (1):** [@11icde_smartsla_full] **CPU, memory, workload and bandwidth: (1)** [@linear-mapping-CP-bundles-2012] **CPU, memory, storage and bandwidth (1):** [@HPCS_IWCMC_Vincent] **CPU, number of VMs and workload (1):** [@ma2016auto] **workload and interference index (1):** [@rameshan2016augmenting]\
Response time& **CPU (1):** [@2014-kalman-pair-wised-coupling-on-tiers-2014] **number of VM (1):** [@multitier-resalloc-Cloud11] **workload and number of VMs (4):** [@3-stages-game-theory], [@queue-prediction], [@queue-VM-group], [@sla-provision] **CPU and memory (2):** [@E3-R-extended], [@MPC-price] **workload and CPU (2):** [@li-opt-clouds-4], [@ICDCS2011] **thread and CPU (1):** [@wosp10sla] **CPU, memory, workload and number of VMs (2):** [@fine-grained-servce-LQM-MIP-power], [@2014-kalman+rule-based-2014] **CPU, workload and number of VMs (1):** [@gandhi2017model]\
Response time and workload& **number of VMs (1):** [@cache-static-ANN-bi-obj-2012] **CPU and memory (1):** [@profit-cent-local-search] **workload and number of VMs (1):** [@typical-navie-scaling-VM-number] **workload, number of VMs and VM type (1):** [@sun2017automated]\
Response time and utilization & **number of VMs and VM type (1):** [@awada2017improving]\
Response time and throughput & **CPU and memory (1):** [@06119056] **CPU, memory, number of VMs and VM type (1):** [@sun2016roar]\
CPU utilization & **workload and number of VM (1):** [@acdc09]\
QoS attributes and hardware demand& **configurations and resources (1):** [@MOCS2014-full] **CPU and workload (1):** [@kalman-clustering]\
QoS attributes and workload& **workload and number of PM (1):** [@change-point]\
QoS attributes and overhead& **resources (1):** [@BRGA-resource]\
Cost& **workload and number of VM (2):** [@shariffdeen2016workload], [@megahed2017stochastic]\
### Analytical Modeling
Analytical modeling approaches rely on a closed-form structure to model the cloud-based service. These models are often built offline based on theoretical principles and assumptions. Next, we further divide the analytical modeling approach into *queuing theory*, *dependability models* and *black box models*.
- *4.5.1.1 Queuing theory:* Queuing model and queuing network are widely applied for QoS modeling in the cloud. They model the cloud-based services as a single queue or a collection of queues that are interacting through a mixture of request arrivals and completes. Specifically, a single queue has been used to model the correlation of response time (or throughput) to CPU, number of VM and workload. For example, depending on the assumption of the distribution on arrival and service rate, the model can be built as M/G/m queue[^4] by Zhang et al. [@MPC-price], M/G/m queue by Jiang et al. [@06119056], M/M/1 queue by E$^3$-R [@E3-R-extended] and JustSAT [@queue-VM-group], and M/M/m queue by Jiang et al. [@typical-navie-scaling-VM-number]. To create more detailed modeling with respect to the internal structure of cloud-based services, multiple queues can be used to create QoS models, for example, Goudarzi and Pedram [@multitier-resalloc-Cloud11] apply multiple queues to model the response time for cloud-based multi-tiered applications with respect to number of VM and workload. Their work calculates average response time for the queue in the forward direction throughout the tiers.
Unlike classical queuing model and queuing network, the Layer Queuing Network (LQN) additionally model the dependencies presented in a complex workflow of requests to cloud-based services and applications. For instance, Zhu et al. [@sla-provision] have also used LQN where the authors employ a global M/M/m queue for the entire on-demand dispatcher and then a M/G/1 queue on each tier of an application. The former queue correlates the response time to number of VMs, while the latter queue models the relationship between response time and CPU of the VM that contains the corresponding tier.
- *4.5.1.2 Dependability models:* Dependability models focus on the modeling of various states for QoS attributes. For example, in QoSMOS [@tse1], the authors analytically solve the Markov Models (Discrete-Time Markov Chain and Markov Decision Process) to model the QoS for services in an application. The model correlates QoS attributes with hardware resources and workload. Huber et al. [@HuBrKo2011-SEAMS-ResAlloc] uses Palladio Component Model (PCM) as architecture-level QoS model since it permits to explicitly model different usage profiles and resource allocations.
- *4.5.1.3 Black box models:* Black-box models handle the QoS using empirical and historical domain knowledge. Among others, the CLOUDFARM framework [@2014-linear-centralized-decision-making-2014] uses a empirical QoS model where the correlation between certain QoS values and the required resource is captured (i.e., CPU). In particular, the authors assumed that the magnitudes of resources to the QoS values is known, as specified by the cloud service or application provider. Another study from Emeakaroha et al. [@Emeakaroha_CloudComp2010] [@HPCS_IWCMC_Vincent] propose an empirical model that maps the expected QoS values with CPU, memory, bandwidth and storage based on the assumptions of the system that being managed.
### Simulation Based Modeling
QoS models can be also generated by various simulators; here, conducting simulations is usually a complex and expensive process and thus they are used in an offline manner. In practice, simulation is required to be setup by the domain experts, who will often need to analyze, interpret and profile the data collected after simulation runs. Specifically, Fernandez et al. [@2014-profiling-decision-tree-scaling-2014] have relied on a profiling approach that builds the QoS model for each bundle of VM offline. The process is similar to a simulation modeling approach. CDOSim [@full-simulation-model] is a framework that simulates the actual application in the cloud to restrict the search-space for autoscaling and to steer the exploration towards promising decisions. CloudSim [@CloudSim] is a simulation toolkit that models QoS attributes (of VM) with respect to resource allocation. It supports both single cloud and multiple clouds scenarios. As an extension of CloudSim, CloudAnalyst [@CloudAnalysis] allows the simulation of QoS attributes for the application deployed on geographically-distributed datacenters. Similarly, DCSim [@DCSim] simulates the overall quality of resource autoscaling for the entire cloud.
### Machine Learning Based Modeling
The increasing complexity of cloud-based services has rendered the modeling process an extremely difficult task for human experts. To this end, recent studies have exploited the advances of machine learning algorithms and theory to create more reliable QoS models. In the following, we survey the key studies that apply machine learning approaches for QoS modeling in the cloud. In particular, we have further classified them into two categories, these are: *linear* and *nonlinear* modeling.
- *4.5.3.1 Linear modeling:* Learning algorithms based on linear models for QoS modeling in the cloud can handle linear correlation between a selected set of cloud primitives (e.g, CPU, memory, number of VM, workload etc) and output (i.e., QoS attributes), and they are sometime very efficient. Simple linear models most commonly rely on linear regression, where each primitive input is associated with a time-varying weight, e.g., Lim et al. [@acdc09], Zhang et al. [@software-RP-two-loops] and Collazo-Mojica et al. [@linear-mapping-CP-bundles-2012]. More advanced forms exist, e.g., Padala et al. [@HPL-2008-123R1-mimo] have used Auto-Regressive-Moving-Average (ARMA) model that is trained continually by Recursive Least Squares (RLS). The authors claim that the second-order linear ARMA model is easy to be estimated online and can simplify the corresponding controller design problem with adequate accuracy.
We found that there are limited studies, which attempt to capture the information of QoS interference in the linear QoS model and they only focus on the VM-level [@fuzzy-vm-interference], [@qcloud], [@vm-fuzzy-MIMO], [@2014-decision-tree-software-CP-2014]. As an example, Q-Cloud [@qcloud] has explicitly considered QoS interference by using the hardware control primitives of all co-hosted VMs as inputs, rendering it in a Multi-Inputs-Multi-Output (MIMO) model, which is trained by Least Mean Square (LMS) method.
- *4.5.3.2 Nonlinear modeling:* Learning algorithms based on nonlinear models for QoS modeling in the cloud is able to capture complex and nonlinear correlation, in addition to the linear one. However, it can also produce relatively large overhead than the linear modeling. Here, existing studies often aim to model the correlation between hardware control primitives (e.g., CPU, memory and bandwidth) and QoS. The nonlinear modeling can be relied on kriging model [@kriging-controller], Regression Tree (RT)[@11icde_smartsla_full], Artificial Neural Network (ANN) [@dynamic-model-comparison] [@2014-navie-ANN-GA-2014] [@JSS_Kousiouris] [@Chen:2013:seams], Support Vector Machine (SVM) [@2014-2-SVM-workload-type-2014] [@dynamic-model-comparison], change-point detection [@change-point]. For example, SmartSLA [@11icde_smartsla_full] employs Regression Tree (RT) and boosting to model the QoS. RT partitions the parameter space in a top-down fashion, and organizes the regions into a tree style. The tree is then trained by M5P where the leaves are regression models. The study from Kunda et al. [@dynamic-model-comparison] presents sub-modeling based on ANN and SVM for correlating QoS with hardware control primitives in the cloud. Instead of building a single model for a QoS attribute, they train *n* sub-models, whereby *n* is determined by performing k-mean clustering based on the similarity between data values of QoS, creating more accurate and finer grained models.
- *4.5.3.3 Ensemble modeling:* Examples exist for cases where multiple linear and/or nonlinear machine learning algorithms are explored together. Among others, Chen, Bahsoon and Yao [@Chen:2014:ucc] [@Chen:2015:tse-pending] exploit a bucket of learning algorithms (both linear and non-linear models). The model accuracies are tracked continually at runtime, considering QoS interference. The best model for a given input values, according to both local and global errors, will be used to make prediction.
- *4.5.3.4 Comparison of different learning algorithms:* Given the various types of machine learning algorithms, it can be difficult to determine which one(s) are the appropriate algorithms for QoS modeling in the cloud, with respect to both accuracy and overhead. There are researches that have conducted empirical comparisons of different possible learning algorithms for QoS modeling in the cloud [@2013-offline-profiling-2013] [@determine-CP-compare-models] [@tp38].
### Hybrid Modeling
We discovered that linear machine learning algorithms are also commonly used with analytical approaches to form QoS models. Specifically, Grandhi et al. [@2014-kalman+rule-based-2014] and Zheng et al. [@kalman-AR] have proposed hybrid model: to model the multi-tiered application, they have relied on a modified LQN containing some time-varying coefficients. The authors then employ the Kalman filter as an online parameter estimator to continually estimate those coefficients. The approach proposed by Xiong et al. [@ICDCS2011] has relied on a combined model, where a M/G/1 queue is used to model the correlation of workload to response time; while ARMA is used to model the relationship between response time and CPU.
### Dynamic Primitives Selection
We noticed that the majority of the aforementioned studies regard the primitives selection as a manual and offline process, most commonly, they have relied on empirical knowledge and heavy human analysis to select the important primitives as the input features of QoS models. Although not many, there are some studies that explicitly consider dynamic process in primitives selection, which tends to be more accurate and can be easily applied [@PCA-model-scaling-2013] [@ISPASS07] [@2013-single-learner-filter-wrapper-LR-2013] [@Chen:2015:tse-pending]. As an example, vPerfGuard [@2013-single-learner-filter-wrapper-LR-2013] is a framework that correlates QoS attributes with respect to software control primitives, hardware control primitives and environmental primitives. The authors achieve primitive selection based on both filter (relevance based correlation coefficient) and wrapper (i.e., hill-climbing comparison for different learning algorithms). Chen and Bahsoon [@Chen:2015:tse-pending] dynamically select primitives that maximize both information relevance (between a primitive and quality) and minimize redundancy (between already selected primitives). While explicitly modeling the effects of QoS interference, the authors propose a fully self-adaptive approach that selects primitives that improve prediction accuracy given a learning algorithm.
### Workload and Demand Modeling
We found that some existing studies (e.g., [@2011-cost-aware-v-h-scaling-2011], [@ILP-cost-only-scaling-2013], [@JSS_Kousiouris]) attempts to model the workload and demand for assisting autoscaling decision making. In those cases, the modeling is reduced to a single dimension, where the core is to model the trend of the workload or demand using its historical data. However, unlike QoS modeling which is often multi-dimensional, the single dimension in workload or demand models do not offer the ability to reason about the effects of autoscaling decisions and the possible trade-offs.
Granularity of Control {#sec:gra}
----------------------
*RQ4: What is the granularity of control in SSCAS?*
The ultimate goal of autoscaling is to optimize the QoS and cost objectives, which are referred to as benefit, for all cloud-based services. To this end, the granularity of control in autoscaling plays an integral role, since it determines the boundary of decision making: which and how many objectives should be considered in a decision making process of autoscaling. In the following, we classify existing SSCAS studies depending on what level of granularity they operate at.
As we can see from Figure \[fig:gra\], the predominant granularity of control is at the service/application level where the boundary of decisions making is grouped by each service/application. Notably, controlling at the cloud level tends to be the second most popular, leaving the other levels being minority. Generally, the finer granularity of control implies that it is harder to achieve globally-optimal benefit but likely to generate smaller overhead. On the other hand, globally-optimal benefit can be easier reached with large overhead if the granularity of control is coarser. According to Table \[tb:gra\], we can obtained the following observations:
[|P[1.6cm]{}|P[1.3cm]{}|P[1.6cm]{}|X|]{} ***Granularity***&***Entity***&***Style***&***Representative Examples***\
&Application&Decentralized& [@2011-cost-aware-v-h-scaling-2011], [@queue-VM-group], [@2014-profiling-decision-tree-scaling-2014], [@linear-mapping-CP-bundles-2012], [@TR-10-full-version], [@wosp10sla], [@06119056], [@cloud_computing_2010_5_20_50060-ext], [@signal-resource-trend-prediction], [@PCA-model-scaling-2013], [@acdc09], [@ILP-cost-only-scaling-2013], [@typical-navie-scaling-VM-number], [@multitier-resalloc-Cloud11], [@2014-decision-tree-software-CP-2014], [@2014-aco-app-to-VM-consolidation-2014], [@scale-rule-based], [@2014-kalman+rule-based-2014], [@ICDCS2011], [@IWQOS11], [@compare-to-Rao-fuzzy-2013], [@HPL-2008-123R1-mimo], [@souza2015using], [@farokhi2016hybrid], [@shariffdeen2016workload], [@megahed2017stochastic], [@lakew2017kpi], [@aslanpour2017auto]\
&Service&Decentralized& [@2013-rule-based-multi-elasiticity-2013], [@2014-ARMA-single-agg-objective-2014], [@Compsac_2010_I_Brandic], [@SEASS_2010_Michael_Maurer], [@kriging-controller], [@tse1], [@2014-eplison-GA-weigh-h-scaling-2014], [@E3-R-extended], [@GA-full-simulation], [@2013-scaling-select-one-predictor-and-error-correlation-2013], [@2010-demand-pattern-match-2010], [@cache-static-ANN-bi-obj-2012], [@HuBrKo2011-SEAMS-ResAlloc]\
& Application&Decentralized& [@fuzzy-2-loop-control], [@parallel-RL-vertical-QoS-2013], [@2013-software-CP-only-2013], [@software-RP-two-loops], [@2014-2-SVM-workload-type-2014], [@vm-fuzzy-MIMO], [@qcloud], [@gandhi2017model],\
&Application&Centralized& [@gill2017chopper], [@sun2017automated], [@awada2017improving]\
&VM&Decentralized& [@2014-fuzzy-compare-to-JRao-2014], [@2014-kalman-pair-wised-coupling-on-tiers-2014], [@ensemble-prediction-VM-2014-full]\
&VM&Centralized& [@qu2016reliable], [@da2016autoelastic], [@rameshan2016augmenting]\
&Container&Decentralized& [@baresi2016discrete], [@li2015rest]\
&Container&Centralized& [@zhang2016container]\
PM level&Application&Decentralized&[@MASCOTS11], [@MASCOTS11-bu-software-CP-full], [@2013-JRAO-most-closest-work-2013], [@2014-navie-ANN-GA-2014], [@fuzzy-vm-interference]\
&Application&Centralized&[@3-stages-game-theory], [@2012-app-VM-mapping-2012], [@2013-elasticity-primitives-2013], [@profit-cent-local-search], [@2014-linear-centralized-decision-making-2014], [@li-opt-clouds-4], [@sla-provision], [@06032254], [@fine-grained-servce-LQM-MIP-power], [@MPC-price], [@EMA-CPU-memory-PD], [@BRGA-resource], [@self-sla-and-resource], [@van2015mnemos], [@herbein2016resource], [@2013-self-organising-map-2013]\
&Application&Decentralized&[@05557978],[@Arc-cover-all-controller]\
&Application&Decentralized&[@MIMO-fuzzy-hill-climbing-2013], [@11icde_smartsla_full], [@ma2016auto]\
&Application&Centralized&[@seelam2015polyglot]\
&Service&Decentralized&[@Chen:2013:iccs], [@Chen:2014:seams], [@Chen:2015:tsc-pending], [@Chen:2015:computer]\
1. Most of the studies (72%) see each application as the basic entity regardless to the granularity of control in SSCAS.
2. Decentralization (74%) is the most popular approach for all granularity of control, except the cloud level where centralized (or partially centralized) control is predominately exploited.
In the following, we specify each granularity of control for SSCAS in details.
### Controlling at Service and Application Level
Service/Application level is the finest level of control in a SSCAS. It is worth noting that by service, we refer to any conceptual part of the system being managed. As a result, control granularity at the service/application level may refer to independently controlling/scaling an application, a tier of an application or a cloud-based service.
We found that most of the studies have focused on controlling each cloud-based application. These approaches have relied on controlling the QoS and/or cost for each individual application in isolation, and therefore, they sometimes regard an application as a service. Examples of such include: Lim et al. [@acdc09] control the application and its required VM, in which case an application is regarded as a service. Sedaghat et al. [@ILP-cost-only-scaling-2013] regard application as a service, and considered the required number of VMs and the fixed VM bundles for such service.
There are studies that explicitly controls cloud-based service in general. Among others, Copli et al. [@2013-rule-based-multi-elasiticity-2013] control the QoS, cost and their elasticity for each service deployed in the cloud. Yang et al. [@2014-ARMA-single-agg-objective-2014] control the cost of individual cloud-based services. The FoSII project [@Compsac_2010_I_Brandic] controls individual cloud-based service, their QoS and cost. Gambi et al. [@kriging-controller] control at the service level, where the controller decides on the optimal autoscaling decision for cloud-based service in isolation.
### Controlling at Virtual Machine and Container Level
VM hypervisor and container are two fundamental infrastructure that underpin cloud computing. In particular, VM and container differ in the sense that the former requires a full Operating System to be installed on a VM while the latter does not. This fact allows the container to set naively with the host PM, providing much faster creation and removal time of VM image. However, such benefit comes in the expenses of weaker security guarantees and potentially greater chances of interference, given that the container instances have less isolation. Despite such a difference, the two infrastructures are conceptually similar as they both aim to provide certain level of isolation on top of the hosting PM, and thus they can be regarded as the same granularity of control.
VM level means that the control and decision making operate at each VM in the SSCAS. In particular, certain studies assumes a one-to-one mapping between application (or a tier) and VM and thus they can be categorized as either service level or VM level granularity. To better separate them from the pure service/application level granularity of control, these studies are regarded as VM level granularity. Specifically, FC2Q [@2014-fuzzy-compare-to-JRao-2014] regards application tier and VM interchangeably, therefore controlling each tier of an application is equivalent to control each individual VM. Similarly, Kalyvianaki et al. [@2014-kalman-pair-wised-coupling-on-tiers-2014] control a tier of an application that resides on a VM, and the authors only focus on CPU allocation of a VM.
### Controlling at Physical Machine Level
Autoscaling decision making on each PM independently is referred to as PM level control in the SSCAS. The primary intention of PM level control is to manage the QoS interference caused by co-hosted VMs. Among the others, Xu et al. [@MASCOTS11] [@MASCOTS11-bu-software-CP-full] control the VMs collectively at the PM level, thus the autoscaling promotes better management of QoS interference at the VM level.
### Controlling at Cloud Level
The most coarse level of control granularity is at the cloud level for SSCAS. The majority of the studies achieves autoscaling at the cloud level by using a centralized and global controller, with an aim to manage utility ([@3-stages-game-theory], [@2012-app-VM-mapping-2012], [@6008793], [@2013-elasticity-primitives-2013], [@profit-cent-local-search], [@2014-linear-centralized-decision-making-2014]), profits ([@li-opt-clouds-4], [@sla-provision]) and availability ([@06032254]). Among others, Ferretti et al. [@05557978] control the QoS for all cloud-based services in a global manner. However, the actual deployment can be either centralized or decentralized. Similarly, CRAMP [@EMA-CPU-memory-PD] uses a centralized and global controller, it controls the entire cloud for cost and QoS. CloudOpt [@fine-grained-servce-LQM-MIP-power] also controls the entire cloud using centralized control, as the considered optimization involves all the PM in the cloud.
Some of the studies have relied on a decentralized manner where a consensus protocol is employed for controlling at the cloud granularity. For example, Wuhib et al. [@Arc-cover-all-controller] aim to control the entire cloud, and thus the QoS and the overall power consumption of cloud can be collectively managed. Further, they have relied on decentralized deployment, which can reduce the overhead of cloud-level control.
### Controlling at Hybrid Level
We found that it is also possible for SSCAS to operate at multiple and hybrid levels, with an aim to better manage the overhead and global benefit. For example, Minarolli and Freisleben [@MIMO-fuzzy-hill-climbing-2013] combine both PM level and cloud level control, where the PM level is decentralized and the objective is to optimize the utility locally. Similarly, SmartSLA [@11icde_smartsla_full] aims to control the resource allocation for all the cloud-based services, therefore it utilizes a global, cloud-level control in addition to the decentralized local control on each VM. Different to the others, Chen and Bahsoon [@Chen:2014:seams] exploit a dynamic schema where the multiple simultaneously presented granularity are changed at runtime, according to the objective-dependency.
Trade-off Decision Making {#sec:dm}
-------------------------
*RQ5: What are the approaches used for decision making in SSCAS?*
The final important logical aspect in cloud autoscaling is the challenging decision making process, with the goal to optimize QoS and cost objectives. It is even harder to handle the trade-off between possibly conflicting objectives. Such decision making process is essentially a combinatorial optimization problem where the output is the optimal (or near-optimal) decision containing the newly configured values for all related control primitives. In the following, we survey the key studies on the decision making for SSCAS. In particular, we classify them into three categories, these are *rule based control*, *control theoretic approach* and *search-based optimization*.
As we can see from Figure \[fig:dm\], while rule-based control and control theoretic approach share similar popularity in the studies, search-based optimization receives much more attentions than the other two. From Table \[tb:dm\], we can observe that:
1. Most of the studies (63%) in SSCAS do not attempt to consider the trade-off between objectives during the decision making, or they handle such trade-off in the way that different objectives are aggregated using weighted sum (29%), which essentially converting the multiple objectives into a single one.
2. Explicit consideration of QoS interference (16%) is still rare during the decision making.
From Table \[tb:dm-inout1\] and \[tb:dm-inout2\] we can see that:
1. Most of the studies (78%) claim that they can work on any given objectives, thus the QoS attributes and cost being the most popular objectives to be improved during the decision making process of SSCAS.
2. Hardware resources, particularly CPU and memory, are the most commonly considered dimension of control primitives to be tuned in SSCAS. However, there are little studies (9%) that consider the interplay between software and hardware control primitives.
3. Considerable amount of studies (34%) has assumed bundles on the autoscaling decision, which will reduce the search space but might negatively constrain the quality of decision making.
Observations from Table \[tb:dm-vh\] shows us that:
1. The majority of the studies (66%) has considered both vertical and horizontal scaling.
2. Horizontal scaling receives much more attentions than the vertical one.
In the following, we specify some of the decision making approaches for SSCAS in details.
[|P[2cm]{}|P[0.8cm]{}|P[1.5cm]{}|P[1.2cm]{}|Xp[2.5cm]{}|]{} ***Decision Making Approach***&***Form***&***Trade-off***&***QoS Interference***&***Concrete Methods***&\
&&None&No&\
&&None&Yes&\
&&None&No&**PDC (3):** [@2012-app-VM-mapping-2012], [@acdc09], [@EMA-CPU-memory-PD] **KC (2):** [@2014-kalman-pair-wised-coupling-on-tiers-2014], [@2014-kalman+rule-based-2014] **FC (3):** [@2014-fuzzy-compare-to-JRao-2014], [@compare-to-Rao-fuzzy-2013], [@fuzzy-2-loop-control]&\
&&Weighted sum&No&**FC (1):** [@IWQOS11]&\
&&None&No&**PIC+LA (1):** [@ICDCS2011] **MPC+MA (1):** [@farokhi2016hybrid] **PIC+ILP (1):** [@baresi2016discrete] & **ANN+FC (1):** [@2013-software-CP-only-2013] **MIMO(1):** [@lakew2017kpi]\
&&Weighted sum&No&**FC+QPS (1):** [@MIMO-fuzzy-hill-climbing-2013] & **PIDC+RL+ES (1):** [@TR-10-full-version]\
&&Weighted sum&Yes& **MPC+QP (1):** [@fuzzy-vm-interference]&\
&&None&Yes& **FUZZY-MIMO (1):** [@vm-fuzzy-MIMO] & **MIMO (1):** [@qcloud]\
&&None&Yes&**RL (3):**[@MASCOTS11-bu-software-CP-full], [@MASCOTS11], [@2013-JRAO-most-closest-work-2013]&\
&&None&No&\
&&Single&No&**HEURISTIC (2):** [@2014-ARMA-single-agg-objective-2014], [@ma2016auto] **DP (1):** [@MPC-price] **ES (7):** [@PCA-model-scaling-2013], [@queue-VM-group], [@linear-mapping-CP-bundles-2012], [@HuBrKo2011-SEAMS-ResAlloc], [@shariffdeen2016workload], [@megahed2017stochastic], [@zhang2016container] **ILP (2):** [@2011-cost-aware-v-h-scaling-2011], [@ILP-cost-only-scaling-2013] & **MIP (1):** [@fine-grained-servce-LQM-MIP-power] **ACO (1):** [@2014-aco-app-to-VM-consolidation-2014] **LA (1):** [@2014-2-SVM-workload-type-2014]\
&&Single&Yes& **HEURISTIC+DT (1):** [@2014-decision-tree-software-CP-2014]&\
&&Weighted sum&Yes&**ES (2):** [@3-stages-game-theory], [@gandhi2017model] **RS (1):** [@Chen:2014:seams] & **HEURISTIC (1):** [@sun2017automated]\
&&Weighted sum&No&**ES (6):** [@tse1], [@kriging-controller], [@typical-navie-scaling-VM-number], [@Compsac_2010_I_Brandic], [@SEASS_2010_Michael_Maurer], [@awada2017improving] **NFM (1):** [@li-opt-clouds-4] **FDS (1):** [@multitier-resalloc-Cloud11] **BS (1):** [@cache-static-ANN-bi-obj-2012] **DP (1):** [@2014-linear-centralized-decision-making-2014] & **LS (1):** [@profit-cent-local-search] **GS (1):** [@11icde_smartsla_full] **DT (1):** [@2014-profiling-decision-tree-scaling-2014] **QP (1):** [@HPL-2008-123R1-mimo]\
&&Weighted sum&No&**TS (1):** [@sla-provision] **GA (3):** [@software-RP-two-loops], [@2014-navie-ANN-GA-2014], [@BRGA-resource] **PSO (1):** [@software-RP-two-loops]&\
&&Pareto&No&**SMS-MOEA (1):** [@wosp10sla] **NSGA-II (3):** [@2014-eplison-GA-weigh-h-scaling-2014], [@E3-R-extended], [@GA-full-simulation]&\
&&Pareto&Yes&**MOACO (3):** [@Chen:2013:iccs], [@Chen:2015:tsc-pending], [@Chen:2015:computer] &\
### Rule Based Control
Rule-based control is the most classic approaches for making decision in SSCAS. Commonly, one or more conditions are manually specified and mapped to a decision, e.g., increase CPU and memory by *x* if the throughput is lower than *y*. Therefore, the possible trade-off is often implicitly handled by the conditions and actions mapping. Specifically, Cloudline [@2013-elasticity-primitives-2013] allows programmable elasticity rules to drive autoscaling decisions. It is also possible to modify these rules at runtime as required by the users. Copil et al. [@2013-rule-based-multi-elasiticity-2013] handle the decision making process by specifying different condition-and-actions mapping for autoscaling in the cloud. In addition, the rules can be defined at different levels, e.g., PaaS and IaaS. Similarly, Ferretti et al. [@05557978] allow to setup mapping between QoS expectation and actions using XML like notations.
[|P[3cm]{}|P[1.2cm]{}|X|]{} ***Objective***&***Bundles***&***Control Primitives***\
&Yes&**number of VMs (3):** [@2014-aco-app-to-VM-consolidation-2014], [@shariffdeen2016workload], [@qu2016reliable] **CPU and memory (2):** [@2011-cost-aware-v-h-scaling-2011], [@MPC-price] **CPU, memory and number of VM (1):** [@ILP-cost-only-scaling-2013], [@fine-grained-servce-LQM-MIP-power]\
&No&**number of VMs (1):** [@megahed2017stochastic] **configurations (1):** [@2014-2-SVM-workload-type-2014] **CPU (1):** [@ICDCS2011]\
&Yes&**number of VMs (3):** [@3-stages-game-theory], [@typical-navie-scaling-VM-number], [@cache-static-ANN-bi-obj-2012] **resources (1):** [@2014-ARMA-single-agg-objective-2014]\
&No&**number of VM (1):** [@sla-provision] **CPU, memory and bandwidth (1):** [@scale-rule-based] **CPU, memory and number of VM (1):** [@2014-kalman+rule-based-2014]\
&Yes&**number of VMs (1):** [@2012-app-VM-mapping-2012] **configurations and resources (1):** [@PCA-model-scaling-2013] **CPU and memory (1):** [@2014-profiling-decision-tree-scaling-2014] **CPU, memory and number of VMs (2):** [@2013-elasticity-primitives-2013], [@EMA-CPU-memory-PD] **CPU, memory and disk (1):** [@2013-elasticity-primitives-2013] **CPU, memory and bandwidth (2):** [@2014-linear-centralized-decision-making-2014], [@linear-mapping-CP-bundles-2012]\
&No&**CPU (4):** [@2014-fuzzy-compare-to-JRao-2014], [@li-opt-clouds-4], [@IWQOS11], [@compare-to-Rao-fuzzy-2013] **number of VM (2):** [@multitier-resalloc-Cloud11], [@2014-eplison-GA-weigh-h-scaling-2014] **resources (2):** [@tse1], [@2014-navie-ANN-GA-2014] **CPU and memory (6):** [@kriging-controller], [@MIMO-fuzzy-hill-climbing-2013], [@parallel-RL-vertical-QoS-2013], [@MASCOTS11], [@fuzzy-vm-interference], [@GA-full-simulation] **configurations and resources (4):** [@Chen:2013:iccs], [@Chen:2015:tsc-pending], [@Chen:2015:computer], [@Chen:2014:seams] **CPU, memory and disk (1):** [@HPL-2008-123R1-mimo] **CPU, memory and configurations (1):** [@TR-10-full-version] **CPU, storage and bandwidth (1):** [@06032254] **CPU, memory and thread (2):** [@fuzzy-2-loop-control], [@MASCOTS11-bu-software-CP-full] **CPU, bandwidth, storage and number of VM (1):** [@Compsac_2010_I_Brandic] **CPU, thread, session, buffer and memory (1):** [@2013-JRAO-most-closest-work-2013].\
QoS attributes&No& **configurations (1):** [@2013-software-CP-only-2013] **configurations and resources (1):** [@software-RP-two-loops] **CPU, memory and bandwidth (1):** [@05557978] **thread, CPU and memory (1):** [@cloud_computing_2010_5_20_50060-ext] **CPU and memory (1):** [@lakew2017kpi] **resources (1):** [@gill2017chopper] **number of VMs (1):** [@rameshan2016augmenting]\
&No&**CPU (1):** [@2014-kalman-pair-wised-coupling-on-tiers-2014] **memory (1):** [@farokhi2016hybrid] **CPU and bandwidth (2):** [@vm-fuzzy-MIMO], [@qcloud] **CPU and thread (1):** [@wosp10sla] **CPU and number of VMs (1):** [@gandhi2017model] **CPU, memory and number of VMs (1):** [@li2015rest]\
&Yes& **CPU, memory and number of VMs (1):** [@baresi2016discrete]\
[|P[5.8cm]{}|P[1cm]{}|X|]{} ***Objective***&***Bundles***&***Control Primitives***\
CPU utilization&No&**CPU and number of VMs (1):** [@acdc09]\
CPU utilization&Yes&**number of VMs (1):**[@da2016autoelastic] **CPU and number of VMs (1):** [@ma2016auto]\
Throughput&Yes&**number of VMs (1):** [@seelam2015polyglot]\
General utilization&No& **configurations (1):** [@2014-decision-tree-software-CP-2014] **number of VMs (1):** [@2013-scaling-select-one-predictor-and-error-correlation-2013] **resources (1):** [@signal-resource-trend-prediction] **CPU (2):** [@2010-demand-pattern-match-2010], [@herbein2016resource] **CPU and memory (1):** [@van2015mnemos]\
General utilization&Yes&**number of VMs (1):** [@ensemble-prediction-VM-2014-full] **CPU, memory, number of VMs and bandwidth (1):** [@zhang2016container]\
QoS attributes and power&Yes&**CPU, memory and number of VM (1):** [@Arc-cover-all-controller]\
Response time and utilization&Yes&**number of VM (1):** [@aslanpour2017auto] **number of VM and VM type (1):** [@awada2017improving]\
Response time, cost and availability&No&**CPU and memory (1):** [@profit-cent-local-search]\
VM consumption&Yes&**number of VM (2):** [@queue-prediction], [@queue-VM-group]\
SLA penalty&No&**CPU, memory and number of VM (1):** [@11icde_smartsla_full]\
Response time, throughput, CPU utilization and cost&No&**CPU and memory (1):** [@06119056]\
Response time, throughput, CPU utilization&Yes&**number of VMs and VM type (1):** [@sun2017automated]\
SLA and power&No&**resources (1):** [@2013-self-organising-map-2013]\
SLA and power&No&**CPU (1):** [@souza2015using]\
Response time, throughput and cost&No&**CPU and memory (1):** [@E3-R-extended]\
Benefits and overhead&No&**CPU and memory (1):** [@BRGA-resource]\
QoS attributes, utilization and cost&No&**CPU (1):** [@HuBrKo2011-SEAMS-ResAlloc]\
QoS attributes, utilization, number of actions&No&**CPU, memory and bandwidth (1):** [@self-sla-and-resource]\
[|P[1.3cm]{}|X|]{} ***Scaling***&***Decision Making Approaches***\
Vertical& **RULES (1):** [@souza2015using] **CONTROL THEORY (4):** [@2014-kalman-pair-wised-coupling-on-tiers-2014], [@2013-software-CP-only-2013], [@farokhi2016hybrid], [@lakew2017kpi] **SEARCH-BASED OPTIMIZATION (6):** [@parallel-RL-vertical-QoS-2013], [@tse1], [@wosp10sla], [@06119056], [@2010-demand-pattern-match-2010], [@2014-decision-tree-software-CP-2014]\
Horizontal& **RULES (6):** [@herbein2016resource], [@Arc-cover-all-controller], [@seelam2015polyglot], [@aslanpour2017auto], [@da2016autoelastic], [@rameshan2016augmenting] **CONTROL THEORY (1):** [@2012-app-VM-mapping-2012] **SEARCH-BASED OPTIMIZATION (13):** [@3-stages-game-theory], [@MPC-price], [@multitier-resalloc-Cloud11], [@queue-VM-group], [@sla-provision], [@typical-navie-scaling-VM-number], [@cache-static-ANN-bi-obj-2012], [@2014-aco-app-to-VM-consolidation-2014], [@2013-scaling-select-one-predictor-and-error-correlation-2013], [@ensemble-prediction-VM-2014-full], [@shariffdeen2016workload], [@megahed2017stochastic], [@sun2017automated]\
Both& **RULES (12):** [@2013-elasticity-primitives-2013], [@2013-rule-based-multi-elasiticity-2013], [@05557978], [@06032254], [@Compsac_2010_I_Brandic], [@scale-rule-based], [@self-sla-and-resource], [@cloud_computing_2010_5_20_50060-ext], [@van2015mnemos], [@qu2016reliable], [@gill2017chopper], [@li2015rest] **CONTROL THEORY (14):**[@2014-fuzzy-compare-to-JRao-2014], [@acdc09], [@EMA-CPU-memory-PD], [@fuzzy-2-loop-control], [@MIMO-fuzzy-hill-climbing-2013], [@2014-kalman+rule-based-2014], [@ICDCS2011], [@IWQOS11], [@TR-10-full-version], [@fuzzy-vm-interference], [@compare-to-Rao-fuzzy-2013], [@vm-fuzzy-MIMO], [@qcloud], [@baresi2016discrete] **SEARCH-BASED OPTIMIZATION (35):** [@MASCOTS11-bu-software-CP-full], [@MASCOTS11], [@2011-cost-aware-v-h-scaling-2011], [@2014-ARMA-single-agg-objective-2014], [@2014-linear-centralized-decision-making-2014], [@fine-grained-servce-LQM-MIP-power], [@kriging-controller], [@li-opt-clouds-4], [@PCA-model-scaling-2013], [@profit-cent-local-search], [@software-RP-two-loops], [@2014-2-SVM-workload-type-2014], [@2014-navie-ANN-GA-2014], [@2014-profiling-decision-tree-scaling-2014], [@HPL-2008-123R1-mimo], [@linear-mapping-CP-bundles-2012], [@2014-eplison-GA-weigh-h-scaling-2014], [@BRGA-resource], [@E3-R-extended], [@GA-full-simulation], [@HuBrKo2011-SEAMS-ResAlloc], [@2013-JRAO-most-closest-work-2013], [@signal-resource-trend-prediction], [@2013-self-organising-map-2013], [@ILP-cost-only-scaling-2013], [@Chen:2014:seams],[@SEASS_2010_Michael_Maurer], [@11icde_smartsla_full], [@Chen:2013:iccs], [@Chen:2015:tsc-pending], [@Chen:2015:computer], [@gandhi2017model], [@ma2016auto], [@zhang2016container], [@awada2017improving]\
### Control Theoretic Approach
Advanced control theory is another widely investigated approach for autoscaling decision making in SSCAS because of its low latency and dynamic nature. Studies in this category could be either *standard*, i.e., they rely solely on the classic control theory; or *extended* where additional methods are considered. Among the others, standard controllers (e.g., Proportional-Derivative control [@2012-app-VM-mapping-2012] [@acdc09] [@EMA-CPU-memory-PD], Kalman control [@2014-kalman-pair-wised-coupling-on-tiers-2014] [@2014-kalman+rule-based-2014] and Fuzzy control [@2014-fuzzy-compare-to-JRao-2014] [@compare-to-Rao-fuzzy-2013] [@fuzzy-2-loop-control] ) are commonly designed as a sole approach to make autoscaling decisions in the cloud. Specifically, ARUVE [@2012-app-VM-mapping-2012] and CRAMP [@EMA-CPU-memory-PD] utilizes a Proportional-Derivative (PD) controller, where the proportional and derivative factors are not sensitive to a concrete QoS model while supporting proactive autoscaling of cloud services and applications in a shared hosting environment. Anglano et al. [@2014-fuzzy-compare-to-JRao-2014] and Albano et al. [@compare-to-Rao-fuzzy-2013] apply fuzzy control that is updated by fuzzy rules at runtime. The aim is to optimize both QoS, cost and energy by autoscaling hardware resources. Although the authors claim they can cope with any hardware resources, only CPU tuning is explored. They have also ignored the QoS interference.
We have also found that control theoretic approaches can be sometime used with other algorithms to better facilitate the autoscaling decision making, forming extended controller: [@ICDCS2011], [@MIMO-fuzzy-hill-climbing-2013], [@2013-software-CP-only-2013], [@IWQOS11], [@TR-10-full-version], [@fuzzy-vm-interference]. Particularly, the gains in the controllers can be further tuned by optimization and/or machine learning algorithms, and this is especially useful for Model Predictive Control (MPC). Among others, Zhu and Agrawal [@TR-10-full-version] utilize a Proportional-Integral-Derivative (PID) and reinforcement learning controller for decision making with respect to adapting software control primitives. Such result is then tuned in conjunction with the hardware control primitives using exhaustive search. The QoS attributes and cost are formulated as weighted-sum relation. The autoscaling decision making process in APPLEware [@fuzzy-vm-interference] have relied on Model Predictive Control (MPC), with the aim to optimize a cost function that represents the local objectives and resource constraints at a point in time. The state of an application, together with the other autoscaling decisions from the neighboring VMs, are collectively considered in a quadratic programming solver.
### Search-Based Optimization
A large amount of existing studies of SSCAS relies on search-based optimization, in which the decisions and trade-offs are extensively reasoned in a finite, but possibly large search space. Depending on the algorithms, search-based optimization for autoscaling decision making in the cloud can be either *explicit* or *implicit*the former performs optimization as guided by explicit system models; while this process is not required for the later.
- *4.7.3.1 Implicit search:* The implicit and search-based optimization approaches for autoscaling decision making do not use QoS models. Similar to the control theoretic approaches, the implicit search is also limited in reasoning about the possible trade-offs. For example, the study from Xu et al. [@MASCOTS11] , [@MASCOTS11-bu-software-CP-full] applies a model-free Reinforcement Learning (RL) approach for adapting thread, CPU and memory for QoS and cost. The approach is however implicit, providing that there is neither explicit system models nor explicit optimization. The authors have considered QoS interference during autoscaling. Similarly, VScale [@parallel-RL-vertical-QoS-2013] utilizes RL for making autoscaling decisions, which are then achieved by vertical scaling. The RL is realized by using parallel learning, that is to say, the authors intend to speed up agent’s learning process of approximated model by learning in parallel, without visiting every state-action pair in a given environment.
The approaches that rely on demand prediction (e.g., the Autoflex [@2013-scaling-select-one-predictor-and-error-correlation-2013], PRESS [@signal-resource-trend-prediction], [@ILP-cost-only-scaling-2013], [@2010-demand-pattern-match-2010], [@2013-self-organising-map-2013] and [@ensemble-prediction-VM-2014-full]) are also regarded as implicit search. This is because the autoscaling decision is directly predicted by the demand models, without the needs of reasoning and optimization.
- *4.7.3.2 Explicit search:* In search-based optimization category, the explicit approaches for autoscaling decision making rely on the explicit QoS models to evaluate and guide the search process. Depending on the different formulations of the decision making problem for autoscaling in the cloud, explicit search can reason about the effects of decisions and the possible trade-offs in details. According to our survey, we found three most commonly used formulations, these are single objective optimization, weighted-sum optimization, and pareto-based optimization.
We discovered that it is not uncommon to optimize only a single objective (e.g., cost or profit) for SSCAS, providing that the requirements of the other objectives are satisfied (i.e., they are often regarded as constraints). For example, Kingfisher [@2011-cost-aware-v-h-scaling-2011] and Sedaghat et al. [@ILP-cost-only-scaling-2013] use Integer Linear Programming (ILP) to optimize the cost for scaling the CPU and memory for VMs of an application while regarding the demand for satisfying QoS as constraint.
To apply explicit search-based optimization for SSCAS, the most widely solution for handling the multi-objectivity is to aggregate all related objectives into a weighted (usually weighted-sum) formulation, which converts the decision making process into a single objective optimization problem. The search-based algorithm include: exhaustive search [@3-stages-game-theory] [@tse1] [@kriging-controller] [@typical-navie-scaling-VM-number], auxiliary network flow model [@li-opt-clouds-4], force-directed search [@multitier-resalloc-Cloud11], binary search [@cache-static-ANN-bi-obj-2012]. For example, the FoSII project [@Compsac_2010_I_Brandic] [@SEASS_2010_Michael_Maurer] regards the autoscaling decision making as case based reasoning process, where the decision is made by looking for the similar historical cases using exhaustive search. The solution of the most similar case is reused o solve the current one. We noted that some studies have relied on more advanced and nonlinear search algorithms, ranging from relatively simple ones: dynamic programming [@2014-linear-centralized-decision-making-2014] and local-search strategy [@profit-cent-local-search], to more complex forms: grid search [@11icde_smartsla_full], decision tree search [@2014-profiling-decision-tree-scaling-2014] [@2014-decision-tree-software-CP-2014] and quadratic programming [@HPL-2008-123R1-mimo]. For example, CLOUDFARM [@2014-linear-centralized-decision-making-2014] addresses the decision making based on a weighted-sum utility function of all cloud-based application and services. The decision making process is formulated as a knapsack problem, which can be resolved by dynamic programming. We have also found that metaheuristic algorithms are popular for autoscaling decision making in SSCAS, because they can often efficiently address NP-hard problems with approximated results under no assumptions of the problem. The most common algorithms include: Tabu Search [@sla-provision], Genetic Algorithm (GA) [@software-RP-two-loops] [@2014-navie-ANN-GA-2014] [@BRGA-resource] , Particle Swarm optimization (PSO) [@software-RP-two-loops]. As an example, Zhu et al. [@sla-provision] formulate the autoscaling decision making as optimize a weighted-sum formulation of response time and cost. To optimize the objectives, the authors apply a hybrid Tabu Search, which relied on iterative gradient descent.
Finally, pareto relation can explicitly handle multi-objectivity for autoscaling in the cloud without the need to specify weights on the objectives [@2014-eplison-GA-weigh-h-scaling-2014], [@wosp10sla], [@E3-R-extended], [@GA-full-simulation], [@Chen:2015:tsc-pending]. For example, in E$^3$-R [@E3-R-extended], the decision making problem is formulated using Pareto relation, where it is resolved by using Non-dominated Sorting Genetic Algorithm-II (NSGA-II). Further, the approach applies objective reduction technique with an aim to remove the objectives, which are not significantly conflicted with the others, from the decision making process. Chen and Bahsoon [@Chen:2015:tsc-pending] exploit Multi-Objective Ant Colony Optimization (MOACO) for trade-off decision making when autoscaling cloud-based services. The authors consider the trade-off between naturally conflicting objectives and between competing services (i.e., QoS interference). Further, a compromise-dominance mechanism is proposed to find well-compromised trade-off decision.
Experimental Evaluation on SSCAS
--------------------------------
Another important step in SSCAS research is to quantitatively evaluate the proposed solution, which is often achieved via experimental analysis. To this end, setting the experiments are of high importance to researchers in this field. Table \[tb:exp\] summarizes the common infrastructure, benchmarks and workload trace from the considered studies. As we can see that there are studies chose to use simulator for controlled experiments and ease of complexity. However, simulators may not fully capture the realistic environment. As a result, custom private cloud and public cloud are also exploited for both controlled and open experiments. Notably, custom private cloud can be much more flexible on choosing the underlying software, e.g., one may utilize the hypervisor or container directly or choose to use higher level software such as OpenStack [@open]. It is worth noting that custom private and public cloud can share similar underlying software and tools, but they may require expertise on different levels of abstraction. For example, one may choose to deploy SSCAS on a custom private cloud that makes use of Xen [@xen]the same hypervisor that underpins Amazon EC2 [@iaas] which contains additional high-level interfaces and restrictions. Benchmarks are not required for simulator, but it is crucial for both private and public cloud infrastructure. A wide range of benchmarks have been exploited to evaluate SSCAS, from simple web hosting to complex multi-tier software. Finally, the workload traces can be either synthetic in which a fixed pattern is generated by the workload generator (e.g., JMeter [@jmeter]); or real where recorded traces form different real domains are used to stress the benchmark and SSCAS.
Implementation of Scaling
-------------------------
The actual implementation of scaling depends on the underlying scenarios, e.g., the type of hypervisor/container, the cloud-based applications and the actual cloud control primitives among the others. Existing virtulization techniques have provided readily available commands and tools to support autoscaling at runtime. For example, if the underlying hypervisor was Xen [@xen], then resource such as CPU and memory of a VM, as well as create/destroy VMs can be scaled dynamically using the $xm$ command. Regarding the actual scaling methods, vertical scaling will have trivial effects on the states of the cloud-based applications, thus they can be directly applied using the command support by hypervisor/container. For horizontal scaling, making new replicas or removing old one needs consistency guarantee on stateful applications/services, which can be ensured by the underlying hypervisor through various readily available protocols. For example in Xen, horizontal scaling can be achieved via *primary-backup replication* or *asynchronous checkpointing*, *etc*.
[|p[2cm]{}|p[2cm]{}|X|]{} &***Software and Tools***\
&Simulator&CloudSim [@CloudSim], CDOSim [@full-simulation-model], CloudAnalyst [@CloudAnalysis], DCSim [@DCSim]\
&Private Cloud&Xen [@xen], VMWare ESXi [@vmware], KVM [@kvm], Docker [@docker], OpenStack [@open] , Eucalyptus [@euc]\
&Public Cloud&Amazon EC2 [@iaas], RackSpace [@rack], Azure [@azure], Google Compute Engine [@google]\
&RUBiS [@rubis], RUBBoS [@rubbos], TPC-W [@tpcw], WikiBench [@wikibench]\
&Synthetic trace, FIFA98 [@f98], Wikipedia [@wikipedia], ClarkNet [@clark]\
Reflections and Open Challenges for SSCAS Research {#sec:diss}
==================================================
In this section, we reflect on the finding of our survey and taxonomy; state the open challenges as well as discuss industrial situation and pricing strategy for SSCAS.
Discussion and Comparison on Existing SSCAS Research
----------------------------------------------------
We now discuss the most noticeable observations by reviewing the existing SSCAS research. We carefully position our discussions in light of the different logical aspects of SSCAS.
### The Levels of Self-Awareness and Self-Adaptivity in Cloud Autoscaling Systems {#the-levels-of-self-awareness-and-self-adaptivity-in-cloud-autoscaling-systems}
*Stimulus-awareness* has been considered in all the 109 studies as it is the most fundamental levels in self-awareness principles, because it is the basic requirement for a software system to be able to adapt. *Time-* (52%) and *goal-awareness* (52%) receives same attention due to the fact that the objective models often contain historical information and they can be used to reason about goals. In contrast, handling *interaction-* (13%) and *meta-self-awareness* (7%) are less popular as the former requires to handle QoS interference while the latter often come with extra complexity.
While *self-optimizing* and *self-configuring* have been the major themes for SSCAS, we found very little studies that considered *self-healing* (5%) and only one work targets for *self-protecting*. This is obvious as the fundamental idea of autoscaling is not for security related purposes but for the performance related quality, which is often much more appealing for cloud consumers.
Also, 67% of the studies has ignored the importance of specifying the required knowledge at the architecture level, entailing the risk of limited awareness [@epics].
### Architectural Pattern {#architectural-pattern}
The generic feedback (82%), particularly the single and close loop, has been the predominant architectural pattern for SSCAS. MAPE (15%) is ranked the second and OAD (3%) being the much less popular one, as OAD often assume the involvement of human decisions maker which is difficult in the case of SSCAS.
The reason could be due to the fact that the feedback loops are flexible and simple to be realized, providing the basic components to achieve self-adaptivity. However, such design can limit the consideration of required knowledge for the autoscaling system to perform adaptations, or the consideration is rather simple and coarse-grained. In contrast, such an issues has been relaxed by OAD and MAPE as they are more stricted by predefining components to capture different aspects of a SSCAS. We see that MAPE is clearly more popular than OAD because the former can be good for separation of concepts (e.g., *Analyze*, *Plan* and *Knowledge*) and for expressing the sequential interactions between those concepts while the latter fails to capture runtime aspects of the SSCAS, as it is mainly designed for decoupling loops of different human activities. Nevertheless, these architectural patterns lack of fine-grained representation of the required knowledge. Thus, it is not immediately intuitive what level(s) of the knowledge is required by each logical aspect of a SSCAS.
### QoS Modeling {#qos-modeling}
Both analytical model and machine learning model, included in 43% and 38% of the studies respectively, are widely exploited in QoS modeling for SSCAS while the simulation (10%) and hybrid (9%) approach are clearly less popular. This could be because analytical model is good for runtime efficiency, simplicity, interoperability, and they could be very effective if all of their assumptions are satisfied. However, analytical approaches generally require in-depth knowledge about the likely behaviors of the system being modeled, i.e., some knowledge about the system’s internal structure or environmental conditions. Such an issues is resolved by using machine learning model which are often assumptions free, and more importantly, they are able to continually evolve themselves at runtime in order to cope with dynamics and uncertainty. Nevertheless, depending on the learning algorithm, the resulting overhead can be high (e.g., the nonlinear ones) and the accuracy is sensitive to the situation (e.g., fluctuation of the data trend).
In contrast, simulation exhibits static nature and it is restricted by a wide set of assumptions, including e.g., the distribution of workload and the effects of QoS interference, etc. needs complex human intervention and assumptions. However, it is believe that simulations model could be the most accurate way to model QoS when the all assumptions are satisfied [@full-simulation-model]. Hybrid model, as in 6 of the studies surveyed, could potentially combine the strengths from different models.
We noted that only 13% of the studies intend to address QoS interferences when modeling the QoS in SSCAS. This might be because considering QoS interference will significantly increase the dimensionality in the model, which in turn, rendering the problem much more complex. Such a complexity makes human analysis very difficult, if not impossible. As a result, for those studies that do consider QoS interference, machine learning algorithms are often exploited.
There are plenty of studies (61%) that consider dynamic (or semi-dynamic) structure of a QoS model (i.e., those that denoted as both *dynamic* and *semi* in Table \[tb:qos\]), however, the dynamic related to the input features have been rarely researched simultaneously, i.e, only 7% of the studies (for those that denoted as *dynamic* in Table \[tb:qos\]). Indeed, changing the inputs of a model could be useful only when the dimensionality of a model is high; that is to say, changing the input features might not cause significant difference if the considered total number of inputs is around, e.g., less than five. The majority of the *dynamic* (or *semi-dynamic*) modeling are machine learning based, leaving the *static* ones are largely analytical or simulation based. This is obvious, as the nature of those modeling approaches determine the extents to which they can be changed when they are built.
A considerable amount of studies (65%) have claimed that their QoS models can work on any given inputs and/or output, as shown in Table \[tb:qos-inout\]. This happens mostly for machine learning and simulation approaches. However, during experimental analysis, the highest number of QoS that being modeled and the number of inputs were both four [@Chen:2014:ucc] [@Chen:2015:tse-pending]. The most commonly considered output is response time and the inputs are hardware resources, particularly CPU, memory and number of VM. This complies with the current trend in the cloud computing market.
### Granularity of Control {#granularity-of-control}
In SSCAS research, the service/application level of granularity is the most popular one, which yields 45% of the studies. This is because focusing on the finest granularity of control can achieve the maximum level of scalability, which particularly fits the cloud. However, fine granularity of control is achieved in the expenses of the globally optimal quality of the cloud, since no interactions between service/application are considered. In contrast, focusing on cloud-level is another extreme, which trades scalability for global optimality. Considering multiple levels could be a solution to reach a better trade-off as discussed in a small amount of studies (8%), but how and when to select the levels to consider imposes additional challenges.
Notably, most of the studies (72%) has relied on the control of each application in cloud regardless to the actual granularity of control. The reason being could be due to the fact that cloud-based application (or a collection of services) is the most crucial unit for consumers to experience the benefit of cloud, which is a common interest for both cloud consumers and providers.
### Decision Making
Generally, rule based control is a highly intuitive approach for autoscaling decision making, and it also has negligible overhead. However, the static nature of the rules requires to assume all the possible conditions and the effects of those decisions that are mapped to the conditions, which is highly depending on the assumptions. To resolve such an issue, control theoretic approach appear to be an effective solutions as it is also efficient while require very little assumptions. However, the major drawback of control theoretic approaches is that they require to make many actuations on the physical system, in order to collect the ’errors’ for stabilizing itself. This means that amateur decisions are very likely to be made. In addition both approaches lack of handling multi-objectivity and the trade-off; they often fail to cope with the problem where there is a large number of autoscaling decisions, which is common for cloud. In contrast, search-based optimization, especially the explicit search, makes loose assumption about the number of autoscaling decisions and is able to find optimality (or near-optimality) under highly dynamic and uncertain environment. Therefore as we have shown, search-based optimization, either implicit or explicit, is the most popular approach for making decisions in SSCAS.
We noted that there is a considerable amount studies (63%) that do not attempt to explicitly consider trade-off during decision making of SSCAS, as they assumed single objective or rely completely on human preferences. The reason might be attributed to the fact that such formalization is simple and straightforward, which can work well when there is a strong preference on an objective. The rest studies handle multi-objectivity via either weighted sum or pareto relation.
QoS interference is again absent in many studies (84%) due to the fact that considering it will unavoidably increase the dimensionality (i.e., objectives) during the decisions making, leading to a more complex problem. This would make the problem unsolvable by many existing approaches.
We found that most of the studies (78%) have claimed that their decision making approach could handle any given objectives, thus they have considered arbitrary QoS attributes and cost as the objectives in SSCAS since these are the most critical indicator for cloud-based services and applications. The highest number of objectives that were considered during the experiments are five [@Chen:2015:tsc-pending] though. CPU, memory and number of VM are the most popular control primitives in decision making because they are the most straightforward dimension to be scaled. However, only as little as 9% of the studies consider the interplay between software and hardware control primitives. To apply solution that can not handle a large search space of the decision making, one often reduce the search space by introducing fixed bundles, which is an assumption made by a considerable amount of studies, i.e., 34%. However, such reduction has the risk that some good solution can be ruled out during the process. Finally, while both vertical and horizontal scaling have been considered in the majority of the studies (66%), focusing solely on horizontal scaling is more popular than the vertical one as the former is more widely supported by major cloud vendors.
Open Problems and Challenges for SSCAS research
-----------------------------------------------
Drawing on the survey and taxonomy, in the following, we specify the open problems and challenges for future SSCAS research and make suggestions for potential research directions where appropriate.
- *Explicit Knowledge Representations are Required in SSCAS Architecture:* As we can see from Section \[sec:sscas\], only 33% of the studies intend to discuss the required knowledge at the architecture level. This means that, in the remaining 67% work, it is more difficult to capture more complex and advanced levels of knowledge, as evident by the fact that most work does not go beyond the basic *stimulus-awareness*. Indeed, studies [@epics] [@2014_epics_handbook] [@7185305] [@epics_survey] [@Chen2016:book] have found that, for self-aware and self-adaptive software systems in general, the absence of explicit consideration for the fine-grained representation of the knowledge in the architecture can results in, e.g., improper inclusion of unnecessary knowledge and/or missing important knowledge that can improve adaptation quality when developing autoscaling systems. 67% studies which do not discuss knowledge at the architecture level implies that such an issue is often overlooked and it is remain unresolved in the SSCAS context, urging the need of further investigations.
The challenge here lies in the fact of how can one systematically distinguish different levels of knowledge and how they can be architected into SSCAS in a principal way. We argue that the required levels of knowledge and their representations can be declared in light with the formal principle of self-awareness. In particular, a potential way is to follow the handbook [@2014_epics_handbook] for mapping different levels of knowledge into a concrete SSCAS architecture.
- *Multiple Loops can Create More Benefit for SSCAS Architecture:* From Section \[sec:arch\] we noted that the majority of the studies has considered single loop, which could cause the problem of high coupling in the design of SSCAS. The multiple loops, on the other hand, helps to achieve better separation between different aspects of SSCAS, leading to fine-grained and localized adaptation. The challenge here is how many and at what levels of abstraction one should place the loops within the SSCAS Architecture. We suggest that designing multiple looped SSCAS architecture with respect to what levels of knowledge the system required could be a neat solution [@Chen:2015:computer].
- *QoS Interference Should be Explicitly Handled in QoS Modeling and Decision Making Process:* Our survey results (see Section \[sec:qos\] and \[sec:dm\]) indicate that only less than 16% of the studies took QoS interference into account. Missing QoS interference in the model and decision making could lead to incorrect or misleaded autoscaling decisions, as the cloud-based services would be unavoidably affected by the dynamic behaviors of its neighbors. However, incorporating QoS interference rises the challenge of dimensionality which causes the model and decision making much more complex. Therefore, this challenge calls for novel approach to reduce the dimensionality, or mitigate its negative effects, during the model and decision making in SSCAS [@Chen:2015:tse-pending] [@Chen:2015:tsc-pending].
- *The Interplay Between Software Configuration and Hardware Resources is Important:* Most existing studies of SSCAS focus on hardware resources as IaaS level only. However, as shown in [@2013-JRAO-most-closest-work-2013] [@software-RP-two-loops] [@2014-decision-tree-software-CP-2014] [@Chen:2015:tse-pending], various software configurations at PaaS level could interplay with each other and the hardware resources, which in turn, affect the QoS of cloud-based services. The challenge is how to create a holistic approach that combines both PaaS and IaaS level in SSCAS.
- *Dynamic Feature Selection is Required for QoS Modeling in SSCAS:* Section \[sec:qos\] indicates that the majority of the existing studies have ignored primitive selection in the QoS modeling or have been relying on manual approach, because the assumed dimensions of inputs is rather limited. However, when both QoS interference and software configurations are considered, selecting the most significant features in the model becomes a crucial task since there could be an explosion of the primitives space [@Chen:2015:tse-pending] [@2013-single-learner-filter-wrapper-LR-2013]. Challenge here lies in how to evaluate the effectiveness of feature combination on the model accuracy while generating reasonable overhead. Given the arbitrary types of feature, which calls for generic and efficient feature selection design for SSCAS.
- *More Flexible Granularity of Control in SSCAS is Needed:* Single, static and fixed granularity of control is predominately exploited in existing SSCAS research. To better handle dynamic and uncertainty, it could be more beneficial to introduce multiple granularity and/or dynamically adjust the granularity of control at runtime [@11icde_smartsla_full] [@Chen:2014:seams], as the granularity of control implies a trade-off between the global optimality of SSCAS and the imposed overhead. The challenge is how to explicitly capture the objective-dependency when designing granularity of control.
- *The Assumptions on the Bundles of Resources Needs to be Relaxed:* From Section \[sec:dm\] we noted that while most of the studies have not constrained the possible autoscaling decisions with respect to the fixed bundles, there are still certain amount of studies that heavily rely on the fixed types of bundles, e.g., a search space of 57 VM instance types on Amazon EC2. However, renting bundles cannot and does not reflect the interests of consumers and the actual demand of their cloud-based services. We argue that future cloud autoscaling would inevitably needs to take arbitrary combination of software configurations and resources into account, as what has already been supported in Google Compute Engine [@google]. As a result, autoscaling decision making imposes a challenging problem that faces with an explosion of decision space (e.g, millions of alternatives), calling for novel and efficient approach to achieve optimal or near-optimal quality.
- *The Trade-off Between Conflicting Objectives Should be Explicitly Handled:* As shown in Section \[sec:dm\], the approaches have mostly ignored trade-off. There is also certain amount of explicit search-based optimization studies has assumed only single objective. However, this can restrict the applicability of SSCAS as the decision making would fail in identifying good trade-off points or strongly bias to the single objective. Alternatively, there is also considerably large amount of studies exploit weighted sum objective aggregation, which embed the trade-off in a single representation. However, it is well-known that the relative weights are difficult to be tailored and a single aggregation could restrict the search, causing limitation when searching for good decisions spread over the search space. Further, achieving balanced trade-off have only being explored in very limited studies, e.g., [@Chen:2015:tsc-pending]. The challenge here is how to search for decisions that contain good convergence and diversity, and eventually selecting the one that has the most balanced trade-off for scaling. We advocate that stochastic optimization approach, particularly nature inspired algorithms, can be promising in addressing such a challenge.
- *More Real World Case Studies of SSCAS are Needed:* We found that real world cases and scenarios of SSCAS, especially those with large scale and practical application, are absent in many studies. Indeed, those studies impose many challenges beyond the perspective of research, but they can be the only way to fully verify the potentials, effectiveness and impacts of SSCAS.
Current Industrial Situation of SSCAS
-------------------------------------
Industrial cloud providers (e.g., Amazon [@iaas] and RightScale [@rightscale]) have been relying on model-free, simple rule and policy based autoscaling approaches for decades. These approaches leave the difficult problem of how to specify rules to cloud consumers, which may work well in the beginning when the demand and complexity of cloud-based applications are simple and straightforward. However, recently the level of complexity (e.g., in terms of the number of cloud control primitives) of cloud is changing to a state that makes human analysis very difficult, especially under conflicting objectives and a large number of alternative autoscaling decisions [@2014-eplison-GA-weigh-h-scaling-2014], [@wosp10sla], [@E3-R-extended], [@Chen:2015:tsc-pending]. Specifically, as discussed in Section 5.1.5, those approaches suffer two significant pitfalls. (i) They requires understanding of the application and domain knowledge to determine the mapping between conditions and actions, which can significantly affect the quality of scaling [@al2013impact], [@software-RP-two-loops]. (ii) They cannot adapt to dynamically changing workload or state of the applications [@2014-navie-ANN-GA-2014], [@Chen:2015:tsc-pending].
As a result, engineering advanced SSCAS is an inevitable trend in this area; the reason why current big cloud providers have not yet widely implemented them could be due to the fact that SSCAS itself is not mature to the state which it can be reliably adopted. However, as our survey reports, researchers and practitioners have been working on overcoming these challenges for almost a decade. There has been some attempts to apply SSCAS commercially, for example, Microsoft Azure [@azure] has recently benefited from Aneka [@vecchiola2009aneka], a research effort supporting high level framework, which contains a more advanced and complicated SSCAS that relies on search-based optimization as part of its subsystems. Aneka’s work is an evidence of how pending industrial challenges had informed research; the results are now incorporated in Azure. Nevertheless, we envision more progress on enhanced, scalable and cost-effective effective solutions for both the cloud providers and consumers.
Discussion on the Pricing for Cloud Autoscaling
-----------------------------------------------
Indeed, more advanced autoscaling approaches in SSCAS (e.g., machine learning and search-based optimization) may impose additional computational resources. Furthermore, advances in autoscaling cannot be done in isolation of pricing (dynamic metering and pricing in particular), as both the cost and revenue are acknowledged among the drivers for the industrial need of more advanced solutions. However, upfront investment in additional recourse can be arguably paid off, in situations where scale and dynamic demand is effectively enabled. This can be observed through greater pay off through better utilization and SLA guarantee, which in turn, improve the overall reputation and thus attracting more consumers. As analyzed in numerous existing work [@lorido2014review][@qu2016auto][@GA-full-simulation][@Chen:2015:tse-pending][@Chen:2015:tsc-pending], the additional resources spent are actually marginal compared with the savings obtained through more accurate, effective autoscaling. In case the cloud provider wishes to charge the consumers for the services provided by SSCAS, there could be two ways to achieve this: (i) charging the computation utilized by the SSCAS through existing pricing schema, e.g., the *reserved* or *spot instance* from Amazon EC2. Here, the SSCAS is an optional service which would be priced as normal instance for the consumers’ application/services. (ii) The charge of SSCAS is combined with the normal price per time unit in existing pricing schema, e.g., instead of charging \$1/hour of an instance, it can be priced as \$1.3/hour where the extra \$0.3/hour is for the SSCAS. Here, the SSCAS is a default and mandatory service to the cloud consumers.
Conclusion {#sec:con}
==========
In this article, we survey the state-of-the-art research on SSCAS and provide a taxonomy based on our findings. Specifically, we review the literature with respect to the research questions presented in Section \[sec:rq\]. According to our survey, the key findings are:
- *Stimulus-*, *time-* and *goal-awareness* are the most widely considered levels of knowledge in SSCAS. *Self-configuring* and *Self-optimizing* are the most popular self-adaptivity notions in SSCAS.
- Feedback loop is the most commonly exploited architectural pattern for engineering SSCAS.
- Analytical model and machine learning based model are prominent for QoS modeling in SSCAS.
- Controlling at the level of service/application is the mostly applied granularity.
- Search-based optimization is the most common approach for making autoscaling decisions.
Apart from those observations, we also gain many insights on the open problems and challenges for future SSCAS research. The most noticeable ones are:
- Explicit knowledge representations are required in SSCAS architecture.
- Multiple loops can create non-trivial Benefits for SSCAS architecture.
- QoS Interference should be explicitly handled in QoS modeling and decision making process.
- The interplay between software configurations and hardware resources is non-trivial.
- Dynamic Feature selection is required for QoS modeling in SSCAS.
- More flexible granularity of control in SSCAS is needed:
- The assumptions on the bundles of resources needs to be relaxed.
- The trade-off between conflicting objectives should be explicitly handled.
- More real world cases and scenarios of SSCAS are needed.
We hope that our survey and taxonomy will motivate further research for more intelligent cloud autoscaling system and its interactions with the other problems in the cloud computing paradigm.
[^1]: This work was supported by the Ministry of Science and Technology of China (Grant No. 2017YFC0804003), Science and Technology Innovation Committee Foundation of Shenzhen (Grant No. ZDSYS201703031748284), and EPSRC (Grant No. EP/J017515/01 and EP/K001523).
[^2]: Service could refer to an entire application, or any conceptual part within an application.
[^3]: Each autoscaling decision is a specific combination of configurations and/or resource provisions that achieves certain outcomes on the targeted objectives.
[^4]: In queuing theory, $M$ denote Poisson distribution; $G$ denotes arbitrary distribution. A term $M/G/m$ refers to Poisson distribution of arrival rate, arbitrary distribution of service rate and there exists $m$ servers.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension of the maps associated with these algorithms; prove that these natural extensions are in fact conjugate to the first return map of the geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost every real number has an infinite number of common approximants for both algorithms.'
address:
- 'Institut de Mathématiques de Luminy (UMR6206 CNRS), 163 Avenue de Luminy, case 907, 13288 Marseille cedex 09, France'
- |
Oregon State University\
Corvallis, OR 97331
author:
- Pierre Arnoux
- 'Thomas A. Schmidt'
date: 03 September 2013
title: Commensurable continued fractions
---
[^1]
Introduction
============
[*How does one compare two continued fraction algorithms related to commensurable Fuchsian groups?*]{} This question naturally arose as we considered the continued fractions that Arnoux and Hubert [@AH] introduced and called [*Veech continued fractions*]{}. For each $q=2n\ge 8$ this is a continued fraction algorithm expressing the (inverse of) slopes of lines through the origin in $\mathbb R^2$ in terms of a geometric expansion related to the regular $q$-gon. The algorithm was inspired by work of Veech on flat surfaces, and related to this is the fact that these continued fractions are naturally expressed in terms of elements of a Fuchsian triangle group of signature $(n, \infty, \infty)$. But, each Fuchsian group of this signature is conjugate to a subgroup of index two of the Hecke group $H_q$ for which Rosen some 50 years earlier gave related continued fractions. The natural question here is: Did they rediscover an old algorithm? And, if not, can one measure in some reasonable sense how closely related the two algorithms are for each fixed index $q$?
Here we make minor adjustments to the two algorithms. First, we use a symmetric form of the Rosen algorithm, multiplying by $-1$ for $x<0$. This slight change, which leaves unchanged the approximations produced for any real $x$, gives an algorithm that is expressed directly in terms of the fractional linear action of the corresponding Fuchsian group. Second, we modify the Veech continued fraction so as to expand slopes of lines.
Note that H. Nakada [@N] introduced the above variant of the Rosen fractions, and indeed Mayer and Stromberg [@MS] refer to these as Hurwitz-Nakada continued fractions. We refer to them as the [*symmetrized Rosen continued fractions*]{}.
The authors first heard A. Haas suggest using variants of the Rosen fractions in the study of natural extensions, in particular using so-called backwards fractions, see his later joint work with Gröchenig [@GH]. See [@DKS] for an infinite family of variants for each index $q\,$. See also [@SS] for an application of variants of the Rosen fractions to the study of length spectra of the hyperbolic surfaces uniformized by the Hecke groups.
Each of the algorithms is given by a piecewise Möbius (equivalently, increasing fractional linear) interval map and thus locally leaves invariant the equivalence class of measures that are absolutely continuous with respect to Lebesgue measure. Our approach is to find for each of these interval maps a dynamical system defined on a closed domain of $\mathbb R^2$ that fibers over the interval of definition and for which there is an obvious invariant measure that is absolutely continuous with respect to Lebesgue measure there. The marginal measure, given by integrating along fibers, is then invariant for the interval map. One can then reasonably hope that the two dimensional system is (a model of) the natural extension of the one dimensional system. In our setting, as in [@AS], the various Möbius transformations associated to each interval map generate a Fuchsian group, and the two dimensional system is in fact isomorphic to a map given by returns to a cross-section for the geodesic flow on the unit tangent bundle of the appropriately corresponding hyperbolic surface. It then follows that this cross-section is a natural extension of the interval map, and also that the interval map is ergodic. Composing the Rosen map with itself and appropriately conjugating, allows a direct comparison for each index of the natural extensions of the two interval maps, in the form of cross-sections for the geodesic flow on the unit tangent space of the same surface.
To be precise, we give abbreviated forms of the definitions of the maps. Full definitions and further discussion are given below.
Let $q = 2 n$ be an even integer greater than 4. Let $U$ denote the counter-clockwise rotation by $\pi/q$, and let $R=U^2$. Let $\mu = \mu_q= 2 \cot \pi/q$ and $I = [\, - \mu/2, \mu/2\,]$. Let $\text{sign}(x) = \pm 1$ be the usual signum function.
For a real matrix $M$, we denote by $M\cdot x$ the Möbius action of the matrix $M$ on an (extended) real $x$; that is, $M\cdot x = \frac{ax+b}{cx+d}$, where $M= \begin{pmatrix}
a&b\\c&d\end{pmatrix}$.
\[Conjugate Rosen map\] Let $k : I\to I$ be the map that sends $x \in I$ to its image under the rotation $U^{-\text{sign}(x)}$ followed by the unique translation by an integer multiple of $\mu$ such that this image lies in $I$. The conjugate Rosen map $r : I\to I$ is the square of $k$.
\[Veech map\] Let $a : I\to I$ be the map that sends $x\in I$ to its image under the unique power of $R^{-1}$ that maps $x$ outside of $I$ followed by the the unique translation by an integer multiple of $\mu$ bringing this image back into $I$. The Veech map $v:I\to I$ be the parabolic acceleration of $a$ at the end points of $I$; that is, for $x$ such that $a(x) = -\mu + R\cdot x$, let $v(x) = a^k(x)$ with $k$ minimal such that $a^{k+1}(x) \neq -\mu + R\cdot a^k(x)$, and similarly for $x$ such that $a(x) = \mu - R\cdot x$.
\[r:theMaps\] We show in Section \[s:conjugRosen\] that the map $r(x)$ is conjugate by way of an explicit element of $\slr$ to the square of the symmetrized Rosen map. The map $v(x)$ is the multiplicative Veech map of [@AH] except that it expresses an action on slopes as opposed to inverse slopes.
We find that the continued fraction map of [@AH] is certainly not simply a variant of the Rosen map, but still that the two are truly comparable. The main result of the paper is the following theorem.
\[t:comp\] Suppose that $q = 2 n$ is an even integer greater than four. Each of the maps $r$ and $v$ is a factor of the first return map to a respective cross-section of the geodesic flow on the unit tangent bundle of the hyperbolic surface uniformized by a Fuchsian triangle group of signature $(q, \infty, \infty)$. Furthermore, these two cross-sections intersect in a region of finite measure. In particular, for almost every $x\in I$, the $r$- and $v$-orbits agree infinitely often, and thus the respective sequences of approximants agree infinitely often.
In fact, the area of intersection of the two cross-sections limits when $q$ tends to infinity to be $1/3$ of the area of the cross-section factoring over $v(x)$, but is asymptotically of zero relative size in that over $r(x)$. See Section \[s:comparison\].
Natural extensions and geodesic flows {#ss:NatExtGeoFloIntro}
-------------------------------------
We consider a measure-preserving dynamical system $(T, I, \mathscr B, \nu)$, where $T: I \to I$ is a measurable map on the measurable space $(I,\mathscr B)$ which preserves the measure $\nu$ ($\nu$ is usually a probability measure).
A natural extension of the dynamical system $(T, I, \mathscr B, \nu)$ is an invertible system $(\mathcal T, \Omega,\mathscr B', \mu)$ with a surjective projection $\pi: \Omega \to I$ making $(T, I, \mathscr B, \nu)$ a factor of $(\mathcal T, \Omega, \mathscr B', \mu)$, and such that any other invertible system with this property has its projection factoring through $(\mathcal T, \Omega, \mathscr B', \mu)$. The natural extension of a dynamical system exists always, and is unique up to measurable isomorphism, see [@Roh]. Informally, the natural extension is given by appropriately giving to (forward) $T$-orbits an infinite (in general) past; an abstract model of the natural extension is easily built using inverse limits.
There is an efficient heuristic method for explicitly determining a geometric model of the natural extension of an interval map when this map is given (piecewise) by Möbius transformations. If the map is given by generators of a Fuchsian group of finite covolume, then one can hope to realize the natural extension as a factor of a section of the geodesic flow on the unit tangent bundle of the hyperbolic surface uniformized by the group. To do this explicitly, we expand the method explained and applied in [@AS], derived from [@Ar], as we briefly summarize here.
Using the Möbius action of $\text{SL}_2(\mathbb R)$ on the Poincaré upper-half plane $\mathbb H$, by identifying a matrix with the image of $z=i$ under it, we can identify $\text{SL}_2(\mathbb R)/ \text{SO}_2(\mathbb R)$ with $\mathbb H$. Similarly, $\text{PSL}_2(\mathbb R) = \text{SL}_2(\mathbb R)/\pm I$ can be identified with the unit tangent bundle of $\mathbb H$. The geodesic flow acts on a surface’s unit tangent bundle: Given a time $t$ and a unit tangent vector $v$, follow the unique geodesic to which $v$ is tangent for arclength $t$ in the direction of $v$. The image, $g_t(v)$, under the geodesic flow is the unit vector tangent to the geodesic at the end point of the geodesic arc. The hyperbolic metric on $\mathbb H$ corresponds to an element of arclength satisfying $ds^2 = (dx^2 + dy^2)/y^2$ with coordinates $z = x + i y$. In particular, for $t>0$, the points $z = i$ and $w = e^t i$ are at distance $t$ apart. Since $\text{SL}_2(\mathbb R)$ acts by isometries on $\mathbb H$, the geodesic flow on its unit tangent bundle is given by sending $A\in \text{PSL}_2(\mathbb R)$ to $A g_t$, where $g_t = \begin{pmatrix} e^{t/2}&0\\0&e^{-t/2}\end{pmatrix}$.
(In all that follows, we commit the standard abuse of notation of representing a class in $\pslr$ by a matrix of $\slr$ in this class.) The following subset provides a transversal to the geodesic flow on $\mathbb H$, and is central to all that follows.
$$\label{eq:thatsA}
{\mathcal A} = \left\{ \begin{pmatrix} x&xy-1\\1&y\end{pmatrix} \,\right\}
\subset \text{PSL}(2,{\mathbb R}).$$
Now fix a Fuchsian group $\Gamma$ of finite covolume. Given $A \in {\mathcal A}$ and $M = \begin{pmatrix}
a&b\\c&d\end{pmatrix} \in \Gamma$, suppose that $cx+d >0$ and let $t_0 = -2 \log (c x + d)$. Then $$MA g_{t_0} = \begin{pmatrix} M\cdot x&*\\1&(cx+d)^2y- c(cx+d)\end{pmatrix}$$ Here as defined above, $M\cdot x = \frac{ax+b}{cx+d}$.
Focusing on the diagonal entries of the above matrix, we find the following transformation. $$\label{eq:2Daction}
{\mathcal T}_M: (x,y) \to (\,M \cdot x,(cx+d)^2y- c(cx+d)\,)\,.$$ An elementary calculation shows that the Jacobian matrix of ${\mathcal T}_M$ has determinant one. Thus, ${\mathcal T}_M$ is Lebesque measure preserving on ${\mathbb R}^2$.
\[d:pwMoebiusMap\] An interval map $f: I \to I$ is called a [*piecewise Möbius map*]{} if there is a partition of $I$ into intervals $I=\bigcup \, I_\alpha$ and a set ${\mathcal M}:=\{M_\alpha\}$ of elements $\pslr$ such that the restriction of $f$ to $I_{\alpha}$ is exactly given by $ I_\alpha \ni x \mapsto
M_\alpha \cdot x\,$. We call the subgroup of $\pslr$ generated by ${\mathcal M}$ the [*group generated by $f$*]{}, and denote if by $\Gamma_f\,$.
Each of the interval maps that we consider is piecewise Möbius and locally expanding almost everywhere. Given such an interval map $f: I\to I$ for some interval $I$, we consider the transformation $\mathcal T$, defined as ${\mathcal T}_M$ on fibers above the subinterval where $f(x) = M\cdot x$. Upon identifying a set of positive measure $\Omega \subset \mathbb R^2$ fibering over the interval such that $\mathcal T$ is bijective (up to measure zero) on $\Omega$, using that the ${\mathcal T}_{M}^{-1}$ are expansive on $y$-values, one could confirm that one has found a model of the natural extension of $(f, I, \mathscr B, \nu)$, where $\nu$ is the marginal measure given by integrating along fibers, and $\mathscr B$ denotes the standard Borel $\sigma$-algebra. (See, say, the proof of Theorem 1 of [@KSS], on p. 2219 there.)
An alternative proof uses the Anosov property of the geodesic flow. Since the map $\mathcal T_M$ is defined by way of the transversal $\mathcal A$, the existence of an invariant compact set of positive measure $\Omega$ as above implies that $(f, I, \mathscr B, \nu)$ is a factor of a map $\Phi$ given by returns to a subset $\Sigma$ (included in the projection of $\mathcal A$) of the unit tangent bundle of $\Gamma_f\backslash \mathbb H$. The set $\mathcal A$ is fibered over the real $x$-line by elements corresponding to horocycles. Since the geodesic flow is exponentially contracting on these horocycles, the inverse of $\Phi$ is expansive on the “$y$”-coordinate. Suppose that we have two bi-infinite orbits $(x_i,y_i)$ and $(x_i,y'_i)$ which project to the same orbit of $T$ (i.e., for all $i\in {\mathbb{Z}}$, $x_{i+1}=T(x_i)$). Then $(x_i,y_i)$ and $(x_i,y'_i)$ belong to the same horocycle, and must be equal by the expansiveness of the inverse of $\Phi$. Hence there is exactly one two sided $\Phi$-orbit which projects to a given bi-infinite orbit of $T$, and thus we have indeed found the natural extension. Finally, if this $\Phi$ is actually given by [*first*]{} returns to $\Sigma$, we have that $f$ is a factor of the cross-section given by $\Sigma$.
Ergodicity to discover the domain of the natural extension
----------------------------------------------------------
One of the goals of the present work is to give non-trivial examples illustrating the method sketched just above. In practice, in order to discover an explicit domain $\Omega$ on which $\mathcal T$ is bijective, one uses the ergodicity of this map on the purported domain of definition. Informally stated, taking (a sufficiently large initial portion of) the ${\mathcal T}$-orbit of a sufficiently general point in $\Omega$ will trace out (a sufficiently large portion of) $\Omega$. (And, in fact, $\Omega$ is appropriately attractive for ${\mathcal T}$ (see our [@AS3]), so that one can begin with virtually any point of the plane.) Moreover, since Lebesgue measure is invariant, almost any orbit will cover the set $\Omega$ uniformly.
The interval maps that we consider in this work are defined in an algorithmic fashion and are of an arithmetic nature. It is easily verified that for any of these, any real transcendental $x$ in its interval of definition must have an infinite, non-periodic orbit. Thus, our Figures \[extNatRosSymFig\], \[natExtVeechAddFig\], \[natExtVeechMultFig\], \[natExtOverlayFig\] each show an initial portion of the orbit of $(\pi/10, 0)$ for the respective transformations. Of importance is now the fact that for non-zero real $\delta$, $$\label{eq:TMonDelta}
\mathcal T_M: (x,1/(x - \delta)\,) \mapsto (M\cdot x, 1/(M\cdot x - M\cdot \delta)\,)\,,$$ as one verifies by considering Equation while writing the relevant matrices in the form $\begin{pmatrix} x&y \delta\\1&y\end{pmatrix}$. The regions so sketched are sufficiently clear to guess equations of the form $y = 1/(x - \delta)$ for the boundary pieces in these examples. Generalization of these formulas to the families of interval maps considered is an easy matter. One can verify that $\mathcal T$ is bijective on a candidate $\Omega$ by careful use of the definition of the interval map at hand. The (Lebesgue measure) area of this region gives the normalizing constant so as to give the probability measure on $\Omega$ with respect to which $\mathcal T$ is ergodic; integrating along vertical fibers gives the invariant measure for the interval map.
Comparing algorithms
--------------------
With the natural extensions in hand, we can move towards comparing “commensurable” interval maps. There are two problems here; while having distinct conjugate group is not a major problem (conjugate Fuchsian groups uniformize the same hyperbolic surface, and one just need to use a suitable change of coordinates for one of the maps), interval maps associated with strict subgroups can create difficulties. But in this case, the second problem is easy to solve: the Veech map is associated to a group conjugate to a subgroup of index two of the Hecke group $H_q$, and each matrix associated with the Rosen map is contained in the complement of this subgroup of index two; hence the square of the Rosen map is defined, up to a conjugacy, in terms of the same group as the Veech map. Hence we find that we can actually intersect the natural extensions (which are given as cross-sections of the geodesic flow on the unit tangent bundle of the shared hyperbolic surface). The relative size of this intersection to the original sections gives a reasonable measure of how related the interval maps are.
In particular, we find that the Veech continued fractions are new, as (for each fixed $n$) the intersection of the natural extension with that of the corresponding Rosen-type continued fractions is a proper subset of each (of course, we describe much more detail than this). Still, because of the ergodicity of the maps and the fact that this intersection has positive measure, for almost every $x$, the sequences of approximants defined by the two continued fraction algorithms agree infinitely often.
Outline
-------
In the next section we discuss the symmetrized Rosen fractions, and in particular describe the process of solving for the planar model of the natural extension. In Section \[s:VeechFracs\], we treat the Veech continued fractions, beginning with an additive version, with infinite invariant measure, and proceeding to a multiplicative version, by explicitly accelerating (that is, repeatedly composing the map with itself) in the vicinity of the two parabolic fixed points. We give the domain of the planar model. Section \[s:conjugRosen\] is devoted to doubling and conjugating the Rosen maps so that they are defined over the same Fuchsian group as the Veech maps. Here also, explicit domains of planar models are determined. In \[s:firstReturn\], we turn to the idea of first return to cross-sections of the geodesic flow on the unit tangent bundle of the hyperbolic surface uniformized by the Fuchsian group at hand. In particular we give a more direct proof than in our [@AS2] of the fact that the product of the entropy times the area of the planar natural extension is greater than or equal to the volume of this unit tangent bundle and that equality holds if and only if the natural extension map is induced by the first return under the geodesic flow to the cross-section. In Subsection \[ss:oursFirst\], we show that for each index, both of our maps are of first return type. Section \[s:comparison\] then gives the comparison; the two natural extension domains meet in a region of positive measure, with relative area limiting to $1/3$ of that for the Veech maps, and to zero in the Rosen maps. Furthermore, for $q$ divisible by four, we give explicit regions showing that the unboundedness of the number of returns to its natural extension of the Veech map before returning to the intersection.
Thanks
------
The present work remained in nascent form for several years. Indeed, results of [@AS], long a subsection and appendix of early versions, are referred to in [@HS]. Similarly, portions of [@AS2] formed a part of early versions. We thank P. Hubert for prompting and support.
The symmetrized Rosen algorithm
===============================
The Rosen algorithm and its symmetrization
------------------------------------------
The Rosen algorithm [@R] aims to give a representation in terms of continued fractions with partial quotients rational integral multiples of the fixed number $\lambda$ in the form
$$x = \cfrac{\varepsilon_1}{b_1 \lambda +\cfrac{\varepsilon_2}{b_2 \lambda + \cfrac{\varepsilon_3}
{\ddots}}}\;,$$ where each $\varepsilon_i$ is $1$ or $-1$ and the $b_i$ are natural numbers. This makes sense when $0<\lambda<2$, and is particularly interesting when $\lambda = 2 \cos \pi/n\,$ with $n\ge 3$, because in that case it satisfies a Markov property, and it is related to the Hecke group $H_n$ (for $n=3$ we recover the nearest integer continued fraction). The case of interest for us is where $\lambda = \lambda_q = 2 \cos \pi/q\,$, with $q = 2n$ an even natural number, of value at least $4$.
Setting $ x = \varepsilon_1/( \, b_1 \lambda + f(x)\,)\,$, we see that $\varepsilon_1= \text{sign}(x)$, $b_1=b(x)= \bigg\lfloor \, \dfrac{1}{\vert\, x\, \vert\, \lambda} + \dfrac{1}{2}\,\bigg\rfloor$, and we find an interval map $f$ on $J = J_q = [-\lambda/2, \lambda/2]\,$:
$$\begin{aligned}
f: J &\to J\\
x &\mapsto \dfrac{1}{\vert\,x\,\vert} - \bigg\lfloor \, \dfrac{1}{\vert\, x\, \vert\, \lambda} + \dfrac{1}{2}\,\bigg\rfloor\, \lambda\;.
\end{aligned}$$
One computes the Rosen expansion of $x$ as $b_n=b(f^{n-1}(x))$ and $\varepsilon_n=\text{sign}(f^{n-1}(x))$.
The absolute value in Rosen’s algorithm introduces an asymmetry that unnecessarily complicates the study of $f\,$, since for positive $x$ it invokes fractional linear maps of the form $1/x - b \lambda\,$, given by matrices of determinant $-1\,$. We thus discuss an algorithm that is almost that of Rosen, in that it determines the same set of partial quotients, but which enjoys more symmetry. This [*symmetric Rosen map*]{} is the following: $$\label{e:rosenSymDef}
\begin{aligned}
h: J &\to J\\
x &\mapsto \dfrac{-1}{x} - \bigg\lfloor \, \dfrac{-1}{x\, \lambda} + \dfrac{1}{2}\,\bigg\rfloor\, \lambda\;.
\end{aligned}$$ The corresponding continued fraction gives a representation of $x$ in the form:
$$x = \cfrac{-1}{a_1 \lambda -\cfrac{1}{a_2 \lambda - \cfrac{1} {\ddots}}}\;,$$ where the $a_i$ are nonzero (positive or negative) integers. The geometry of this map are straightforward: one sends $x$ to $-1/x$, and then translates by an integral multiple of $\lambda$ to bring the result into the interval $J\,$. This translation is uniquely defined except for when the image lies at an endpoint of $J\,$, this set of measure zero we systematically ignore except as otherwise stated.
Setting $a(x)=\bigg\lfloor \, \dfrac{-1}{ x\, \lambda} + \dfrac{1}{2}\,\bigg\rfloor$, one checks that $a_n=a(h^{n-1}(x))$, and also that $a_n=(-1)^n \varepsilon_1\ldots \varepsilon_n b_n$, so the two maps give the same expansion.
Dynamics of the Rosen map
-------------------------
The two functions $f$ and $h$ agree for $x <0\,$; note that other than when $\frac{-1}{x \lambda} + \frac{1}{2}$ is integral, $h(-x) = - h(x)$. The symmetrized function has the advantage of being given, on each subinterval of continuity, by the standard action of elements of a Fuchsian subgroup of $\text{PSL}(2, \mathbb R)$, the Hecke triangle group of index $q$, denoted here $H_q\,$:
$$H_q = \langle \, S, \, T\, \rangle\,, \text{where}\; S = S_q = \begin{pmatrix}1 &\lambda\\0&1\end{pmatrix},\, T = \begin{pmatrix}0 &-1\\1&0\end{pmatrix}\;.$$ These two generators satisfy the relations $T^2 = \text{Id}$ and $(ST)^q = \text{Id}$ in $\text{PSL}(, 2, \mathbb R)\,$. Note that $S \cdot x = x + \lambda$ and $T \cdot x = -1/x\,$. Thus if $j =a(x)$, then $h(x) = S^{-j} T \cdot x\,$.
The [*cylinders*]{} of the map $h$ are the sets $\Delta_j = \{x \in J \,|\, a(x)=j\}$. On $\Delta_j$ one has $h(x) = S^{-j} T\cdot x$. If $j >1$ (resp. $j<-1$), then $\Delta_j=[\frac{-2}{(2j-1)\lambda}, \frac{-2}{(2j+1)\lambda})$ (resp. $[\frac{-2}{(2j-1)\lambda}, \frac{-2}{(2j+1)\lambda})$ ); each of these cylinders is [*full*]{}, in the sense that $h$ maps it onto the full interval $J$, as one can see on Fig. \[rosenSymFig\].
The dynamics of the cylinders $\Delta_{\pm 1}$ are more complicated, and we recall some observations of [@BKS], where the original Rosen algorithm is addressed; of course, this and the symmetrized algorithm treat negative $x$ in the same way. As in [@BKS], the orbit of $-\lambda/2$ is the key to the dynamics of the map; setting $\phi_j = f^j(-\lambda/2)$, an easy computation shows that $\phi_j=\frac{-\cos(j+1)\pi/q}{\cos j \pi/q}$ for $1\le j<n$, and in particular $\phi_{n-1} = 0\,$ (recall that $q = 2n\,$), after which the orbit is no longer defined. This finite sequence satisfies $\phi_0 = -\lambda/2< \phi1< \dots< \phi_{n-2} = -1/\lambda<-2/3\lambda<\phi_{n-1}=0$. Hence the cylinder $\Delta_1$ is partitioned by the intervals $[\phi_j,\phi_{j+1})$, for $0\le j<n-3$, and the interval $[\phi_{n-2}, 2/3\lambda)$, with $h$ sending each interval to the next, and the last interval to $[0,\lambda/2)$.
These values are visible on Figure \[extNatRosSymFig\] — they are the $x$-coordinates of the discontinuities along the upper boundary of $\mathcal E\,$.
Solving for the planar extension of the symmetrized Rosen fractions
-------------------------------------------------------------------
Associate to the piecewise linear fractional map $h$ the two-dimensional function $\mathcal T_h(x,y) = \mathcal T_M(x,y)$ whenever $h(x) = M\cdot x$. In this subsection, we indicate how to pass from a plot of the orbit of a point under $\mathcal T_h$ to the exact determination of $\mathcal E$, the domain fibering over $J$ on which this function is bijective.
From the simple shape of the matrices involved, equation becomes $$\mathcal T_{S^{-j} T}\,:(x, y) \mapsto (\, S^{-j} T \cdot x,\, x^2 y - x\,)\,.$$ Since $h(-x) = - h(x)$ almost everywhere, we find that the function $\mathcal T_h(-x,-y) =- \mathcal T_h(x, y) $ almost everywhere, and hence that $\mathcal E$ must be symmetric with respect to the origin.
Figure \[extNatRosSymFig\] allows one to guess that $\mathcal E$ is bounded by $y = 1/(x+1)$ above $(0, \lambda/2)$ and by $y=1/(x-1)$ below $(-\lambda/2, 0\,)$. From these, we solve for the boundaries of $\mathcal E\,$.
We denote by $\mathcal E_j$ the subset of $\mathcal E$ fibering over the cylinder $\Delta_j$. Numerical experiments show that the region where $x<0$ (and thus the various $j$ are positive) is mapped by $\mathcal T_h$ to lie above the $x$-axis. Similarly, the right hand side has image below the $x$-axis.
Therefore, with our assumption that $\mathcal E$ lies within the bounds $y = 1/(x \pm 1)$, keeping Equation in mind, we find that that the $\mathcal T_h$-image of $\mathcal E_j$ lies above that of $\mathcal E_{j+1}\,$. Assuming that $\mathcal E$ is connected, we can thus solve for the upper boundary $y = 1/(x - \delta)$ of $\mathcal E_j$ when $j \ge 2$ — since $\mathcal T_{S^{-j}T}$ sends $y = 1/(x+1)\,$ to $y = 1/(x + j \lambda + 1)\,$ solving for equality with the image of $y = 1/(x - \delta)$ by $\mathcal T_{S^{-j-1}T}$ gives $\delta = -1/(\lambda - 1)\,$.
To solve for the various pieces of the upper boundary of $\mathcal E_1\,$, as in [@BKS], we use the remarks above on the orbit of $-\lambda/2=\phi_0$. Suppose that the upper boundary above $[\phi_j, \phi_{j+1}\,)$ is of the form $y = 1/(x - \delta_j\,)\,$. Since $\mathcal T_{S^{-1}T}$ sends these boundaries one to the other, we find that $\delta_{j+1} = S^{-1}T\cdot \delta_j = - \lambda -1/\delta_j\,$. Since we already have found that $\delta_{n-2} = -1/(\lambda-1)\,$, all of these values are determined. Indeed, we have that $S^{-1}T \cdot \delta_{n-2} = -1\,$, and thus $\delta_0 = (TS)^{n-1}\cdot -1\,$. Now, from the proof of Lemma 3.1 of [@BKS] (beware: there one uses $q = 2p\,$), one has that $(ST)^n \cdot 1 = -1\,$. Since $\delta_0 = (TS)^{n-1}\cdot -1 = T (ST)^{n-1}T \cdot -1 = T(TS^{-1})(ST)^n\cdot 1\,$, we conclude that $\delta_0 = - \lambda - 1\,$.
In summary, we find that the dynamics of $\mathcal T_h$ on the left hand side of $\mathcal E$ can be described as follows. For $j\ge 2\,$, the strip $\mathcal E_j\,$ — of vertical sides $x = -2/(\, ( 2 j -1) \lambda)$ and $x = -2/(\, ( 2 j + 1) \lambda)$ and curvilinear upper boundary $y = 1/(x + 1/(\lambda-1)\,)$ and lower boundary $y = 1/(x-1)$ — is sent to the curvilinear rectangle whose boundaries are of equation $x = \pm \lambda/2\,$, $y = 1/(x + j \lambda +1)\,$, and $y = 1/(x + (j -1) \lambda +1)\,$.
The subset $\mathcal E_1$ is more complicated. The vertical boundaries are $x = -\lambda/2$ and $x = -2/3\lambda\,$. The lower boundary is $y = 1/(x-1)$ and the upper boundary is given as above by the various $y = 1/(x-\delta_i)\,$. The image of $\mathcal E_1$ consists of a curvilinear rectangle with boundaries $x=0$, $x= \lambda/2$ and the two hyperbolas $y = 1/(x+1)$ and $y = 1/( x + \lambda + 1)\,$, as well as an image on the left side of $\mathcal E$, that lies between $x =\phi_1= 2/\lambda - \lambda$ and $x=0$ and above $y = 1/(x + \lambda + 1)\,$, the upper boundary is given by $n-2$ hyperbolic segments.
Symmetry explains the remainder of the dynamics. One can now easily verify that $\mathcal T_h$ does act so as to send $\mathcal E$ bijectively to itself (modulo, as usual, sets of measure zero corresponding to boundaries of cylinders). The upper part of Fig. \[cylImageRosenFig\] shows the cylinder $\mathcal E_1$ and its image, in the case $q=8$, where the cylinder is divided in 3 parts with different upper boundaries; the lower part shows the cylinders $\mathcal E_j\,$ of $j>2$ and their images; the cylinders of negative index and their images are deduced by symmetry.
Note that we find the Markov condition of the continued fraction development — there can be no more than $n-1$ successive partial quotients of value 1 (or -1), and if this limit is attained, then the next partial quotient is of the opposite sign. These constraints are directly related to the relations which hold amongst our generators of the group $H_q\,$.
Veech algorithm {#s:VeechFracs}
===============
Background
----------
W. Veech showed [@V; @V2] that the translation surface obtained by identifying, via translation, opposite sides of the regular Euclidean $q=2n$-gon has a non-trivial group of affine diffeomorphisms. In local coordinates, these functions are given by affine maps of the plane $v \mapsto Av + b$ with $A \in \text{SL}(2, \mathbb R)$, and furthermore the matrix part $A$ is constant throughout the collection of local coordinates giving the atlas of the translation surface. The Veech group, $V_q\,$, is the group generated by these matrix parts, which we will considered projectively. Thus, $V_q \subset \text{PSL}(2, \mathbb R)\,$.
Letting $$\mu := \mu_q = 2 \cot \pi/q\,,$$ this group $V_q$ is generated by the parabolic $P = \begin{pmatrix} 1 & \mu\\0&1\end{pmatrix}$ and the rotation $R$ of angle $2 \pi/q\,$. It is clear that $R$ induces an automorphism of the surface; perhaps less clear is the fact that $P$ also arises from an affine diffeomorphism. Figure \[parabOctFig\] suggests how in fact the image under $P$ of the regular polygon can be cut into a finite number of pieces which can then translated back, respecting identifications, to reform the polygon. (In fact, $P$ arises from taking Dehn twists in appropriate parallel cylinders decomposing the surface.)
![The action of a parabolic element, $q=8$[]{data-label="parabOctFig"}](Parabol)
Arnoux and Hubert [@AH] used the action of the Veech group on the set of foliations by parallel lines to define a family of additive and multiplicative continued fraction algorithms for developments of real numbers. These algorithms are related to the Teichmüller geodesic flow, as explained in [@AH].
Their algorithms are given in terms of the inverse of the slope of the foliation. Here, it is more convenient to study the same action of the Veech group on the foliations, but rather in terms of the slopes themselves. We next give a brief sketch of the action and describe a natural algorithm.
Warning — note that the standard action of a matrix $\begin{pmatrix}a & b\\c&d\end{pmatrix}$ sends a line of direction vector $\begin{pmatrix} 1 \\y\end{pmatrix}$ to a line of direction vector $\begin{pmatrix} 1 \\\frac{c + d y}{a + by}\end{pmatrix}\,$. Thus, the matrix $\begin{pmatrix}a & b\\c&d\end{pmatrix}$ acts on real slopes $z$ by the standard fractional linear transformation of $\begin{pmatrix}d & c\\b&a\end{pmatrix}\,$! Fortunately, this twisting is simplified in the setting of $V_q\,$, in particular, the action of $R$ on slopes is given by the Möbius action of $R^{-1}$ on real values; similarly, the Möbius action of the transpose of $P$ (which is conjugate in $V_q$ to $P$, see [@AH]) gives the geometric action of $P$ on slopes.
Additive algorithm
------------------
We first give a purely geometric description of the algorithm. Consider a regular polygon in the complex plane with $q=2n$ sides, centered at zero, and with a vertex at the point $i \in \mathbb C\,$.
As an element of the projective group, the rotation $R$ is of order $n\,$; since the perpendicular bisectors of the sides adjacent to the vertex at $i$ have slope $\mu/2$ and $-\mu/2$, for each real slope $x$ of absolute value at most $\mu/2\,$, there is exactly one power (modulo $n$) of $R$ such that $R^j\cdot x$ is not contained in the interval $(\, -\mu/2, \mu/2\,)\,$ (indeed, this is the power of $R$ such that the image of the line of slope $x$ passes closer to the vertex $i$ than the image under all other powers of $R\,$). Of course, for any real $x$, there exists a unique $k$ such that $P^k \cdot x \in [\, -\mu/2, \mu/2\,)\,$.
The [*additive algorithm*]{} is defined on $I := I_q = [-\mu/2, \mu/2\,]\,$; for a given $x$, we apply the power of $R$ that sends $x$ outside of $I$, and then apply the unique power of $P$ that brings this value back into the interval.
We now more precisely give the corresponding function. Let $d_j := R^{-j}\cdot \frac{\mu}{2}$ for $j = 1, \dots, n\,$, and $c_j := R^{-j}\cdot \infty\,$. One has $$-\infty < d_1 = -\dfrac{\mu}{2}< c_1 < d_2 < \, \cdots\, < c_{n-1} < d_n = \dfrac{\mu}{2} < \infty\,.$$ The application is defined on $I = (d_1, d_n)$ except for the countable number of points of discontinuity (including in particular the $d_j$ and $c_j$). On each subinterval $(d_j, c_j)\,$, the function is given by $x \mapsto P^k R^j\cdot x$ where $k$ is a strictly negative integer (depending on $x$); on $(c_j, d_{j+1})\,$, the function is $x \mapsto P^k R^j\cdot x$ where now $k$ is a strictly positive.
We thus define the [*partial quotients*]{} $(k,j)$ and use $\Delta(k,j)$ to denote the corresponding cylinder, thus the subinterval on which the function is given by applying $P^k R^j\,$. Here, each $k$ is non-zero, and $j$ is in $\{1, \dots, n-1\}\,$. Note that here all of the cylinders are full and the map is clearly Markov. This reflects the fact that the elements $R$ and $P$ generate cyclic groups whose free product gives all of $V_q\,$.
The standard ordering of the reals $x \in \bigcup\, \Delta(k,j)$ induces an ordering on the partial quotients $(k,j)$ as follows: $$\begin{aligned}
&(-1, 1) < (-2,1) < \cdots < (-\infty, 1)< (\infty, 1) < \cdots < (2,1) < (1,1) \\&
< (-1, 2) < (-2,2) < \cdots< (-\infty, 2) < (\infty, 2) < \cdots < (1,2)\\
&< (-1,3) < \cdots < (-1, n-1) < (-2, n-1) < \cdots< (-\infty, n-1) \\
&< (\infty, n-1) < \cdots < (2, n-1) < (1, n-1) = (1, -1)\;.
\end{aligned}$$
For explicit calculations, it is helpful to have $R$ in terms of $\mu\,$. Applying the classical formulas relating the sine and cosine to the tangent of the half-angle, one has
$$R = \dfrac{1}{\mu^2 +4} \, \begin{pmatrix} \mu^2 -4&-4\mu\\
4\mu &\mu^2 -4\end{pmatrix}\;.$$
The action of $R$ on the endpoints of the interval is of fundamental importance for the following calculations. By direct geometric argument (beware of the change in actions requiring a passage from $R$ to $R^{-1}$ !), or simply by direct calculation one finds $$R \,\cdot(- \mu/2) = \mu/2\,;\;\; \;\;R^{-1}\, \cdot \mu/2 = -\mu/2\;.$$
The additive Veech map has an infinite invariant measure, we thus “accelerate” it to give a “multiplicative” algorithm.
Multiplicative algorithm
------------------------
The reason that the additive map has an [*infinite*]{} invariant measure is that it has two (indifferent) fixed points: each of the parabolic elements $P^{-1}R$ and $P R^{-1}$ is applied in a subinterval whose boundary contains its fixed point. We accelerate the map by appropriately grouping together powers of each of these parabolic elements and thus define our multiplicative Veech map, $v : I \to I\,$.
### Acceleration near $\mu/2$
The fixed point of $PR^{-1}$ is $\mu/2\,$. Let $$\alpha := R P^{-1} \cdot (- \mu/2) = \dfrac{\mu}{2}\; \dfrac{3 \mu^2 - 4}{5 \mu^2 + 4}\,.$$ Thus $\alpha$ is the left endpoint of $\Delta(\,(1, -1)\,)$ the domain of $P R^{-1}$ for the additive algorithm. We now define the image of $x \in (\alpha, \mu/2]$ under the multiplicative map as $(P R^{-1})^{\ell}\cdot x\,$, with $\ell$ the smallest power such that $(P R^{-1})^{\ell}\cdot x$ lies outside of $(\alpha, \mu/2]\,$.
To determine this power $\ell$, it is simpler to conjugate by a matrix that sends $0$ to $\mu/2$. Let $C = \begin{pmatrix} 1 & \mu/2 \\ 0 & 1\end{pmatrix}$, and consider the matrix $C^{-1} PR^{-1} C$; it is a parabolic element which preserves 0. Computation shows that, in the projective group, it is equal to $D = \begin{pmatrix} 1 & 0 \\
4 \mu/(\mu^{2}+4) & 1\end{pmatrix}$. We look for the smallest $\ell$ such that $(P.R^{-1})^{\ell}.x$ is smaller than $\alpha$, which is equivalent to $D^{\ell} .(x-\mu/2 ) < \alpha-\mu/2$. The smallest such $\ell$ is
$$\ell = \bigg\lceil \, \frac{\mu^{2}+4}{\mu} \,\frac{1}{\mu-2 \alpha}
\,\frac{x-\alpha }{\mu-2 x }\bigg\rceil \; .$$
Thus, we define $v(x) = (P R^{-1})^{\ell}\cdot x$ with $\ell$ as above, for any $x \in (\alpha, \mu/2]\,$.
### Acceleration near $-\mu/2$
The additive Veech function being odd, it is clear how we proceed here. For $x \in [\, - \mu/2, - \alpha\,)\,$ we let $v(x) = (P^{-1}R)^k\,$ where $k(x)=\ell(-x)$, that is $$k = \bigg\lceil \, \frac{\mu^{2}+4}{\mu} \,\frac{1}{\mu-2 \alpha }
\,\frac{-\alpha - x}{2 x + \mu}\bigg\rceil \; .$$
### No acceleration in central section
For $x \in [-\alpha, \alpha)\,$, the multiplicative and additive maps are the same. Thus, $v(x) = P^k R^j \cdot x$ in this subinterval, with these exponents as in the previous subsection.\
The graph of the multiplicative function is given in Figure \[multVeechFig\]. The function has an infinite number of intervals of continuity, accumulating at the endpoints $\pm \mu/2\,$ and at $n-1$ further points in the $V_q$-orbit of infinity. As the figure indicates, these intervals form a countable Markov partition: indeed, $-\alpha$, $\alpha$ are points of discontinuity, and each interval of continuity has image one of the three intervals: $I = [\,-\mu/2, \mu/2)$, $[\,-\mu/2, \alpha)$, and $[\, -\alpha, \mu/2)\,$.
Planar transformations for the Veech maps
-----------------------------------------
Because of its straightforward combinatorics, the planar extension of the additive algorithm is particularly easy to find. Computational experiment (see Figure \[natExtVeechAddFig\]) shows that the domain is bounded by the graphs of Möbius transformations, each with a pole at an endpoint of $I\,$.
One is thus lead to conjecture that the domain $\Omega_a$ of planar extension of the additive map is $$\Omega_a = \{ \,(x,y)\;\vert\, x \in I\,,\, \dfrac{1}{x - \mu/2} < y < \dfrac{1}{x + \mu/2} \,\}\,.$$ It is easy to prove that the planar extension map is bijective on this domain. Let $\mathcal D(k,j)\subset \Omega_a $ be the region projecting to the cylinder $\Delta(k,j)\,$. Thus, restricted to $\mathcal D(k,j)\,$, the planar extension map is given by $\mathcal T_M$ with $M = P^k R^j\,$. Using Equation , one verifies that
$$\begin{matrix} P^{k}R^{j} \;\;\text{sends} \;\;\; y = \dfrac{1}{x + \mu/2} \; \;\text{to}\; \;\; y = \dfrac{1}{x - k \mu -
R^{j} .\frac{-\mu}{2}} \\
\\
P^{k}R^{j} \;\;\text{sends} \;\;\; y = \dfrac{1}{x - \mu/2} \; \;\text{to}\; \;\; y = \dfrac{1}{x - k \mu -
R^{j} .\frac{\mu}{2}} \end{matrix} \;.$$ Now, the orbit of $\mu/2$ under powers of $R$ remains in $I$ and as the powers $k$ above are nonzero, $k \mu + R^j \cdot \dfrac{\mu}{2}$ and $k \mu + R^j \cdot \dfrac{-\mu}{2}$ are always outside of $I$, and of sign that of $k\,$. That is, the two dimensional cylinder $\mathcal D(k, j)$ is sent to the upper half-plane when $k$ is negative, and to the lower half-plane when $k$ is positive.
In the case that $k <0\,$, the image of $\mathcal D(k, j)$ is bounded above by $ y = \dfrac{1}{x - k \mu -
R^{j} .\frac{-\mu}{2}}$ and bounded below by $y = \dfrac{1}{x - k \mu -
R^{j} .\frac{\mu}{2}} \,$. Since $R \cdot \frac{-\mu}{2} = \frac{\mu}{2}\,$, whenever $j< n-1$ the lower boundary of the image of $\mathcal D(k, j)$ is the upper boundary of the image of $\mathcal D(k, j+1)\,$. If $j = n-1\,$, we have $$\begin{aligned}
k \mu + R^{n-1}\cdot \frac{\mu}{2} &= k \mu + R^{-1}\cdot \frac{\mu}{2} \; = k \mu + \frac{\mu}{2}\\
&= (k+1) \mu - \mu/2 = \; (k+1) \mu + R\cdot \frac{-\mu}{2}\,,
\end{aligned}$$ thus the lower boundary of the image of $\mathcal D(k, n-1)$ is also the upper boundary of the image of $\mathcal D(k+1, 1)\,$. In summary, when $k <0\,$, the images of these two dimensional cylinders are stacked with respect to the vertical in increasing order in accordance with the following ordering of indices. $$\begin{aligned}
(-1,1) < (-1,2) < \cdots (-1, n-1) < (-2, 1), (-2, 2) < \cdots < (2, n-1)\\
< (-3, 1) < \cdots < (k, 1) < (k,2) < \cdots < (k, n-1) < (k-1,1) < \cdots \;.
\end{aligned}$$ Note that in particular the upper boundary of the image of $\mathcal D(-1,1)$ is $y = 1/(\, x + \mu/2\,)$ and its lower boundary is given by $y = 1/(x + \gamma)\,$ with $\gamma = \mu - R \cdot \frac{\mu}{2}\,$. This cylinder is of course of infinite (Lebesgue) measure.
By symmetry, one completes the study of the dynamics of the planar extension; in particular, the cylinder $\mathcal D(1,n-1)$ — corresponding to the second parabolic fixed point — is also of infinite measure.
The planar extension for the multiplicative algorithm is now easy to deduce. Indeed, this is the same as for the additive algorithm, except that at each point of the two cylinders $\mathcal D(-1,1)$ and $\mathcal D(1,n-1)$ we must iterate the corresponding $\mathcal T_M$ so as to exit the cylinder. We find that the domain of $\mathcal T_v$ has each of its upper and lower boundaries of a single point of discontinuity, of $x$-coordinate $-\alpha$ and $\alpha\,$, respectively.
Recall that $\alpha = \dfrac{\mu}{2}\; \dfrac{3 \mu^2 - 4}{5 \mu^2 + 4}$ and let $\gamma = PR^{-1} \cdot \frac{\mu}{2} = \dfrac{\mu}{2}\; \dfrac{5 \mu^2 + 4}{3 \mu^2 - 4}\,$. We have thus have the following.
\[p:VeechMultNat\] Let $q = 2 n$ be an even integer greater than 4, and let $\mu = 2 \cot \pi/q\,$. Let $v(x)$ be the function on $I = [\, - \mu/2, \mu/2\,]$ defined by the multiplicative Veech algorithm of index $q$ on slopes.
Let $b_{+}, b_{-}$ be the two functions defined by $$\begin{cases}
b_{+}(x) = \frac{1}{x + \gamma}&\text{if}\;\; x \in [-\mu/2,-\alpha)\,;\\
b_{+}(x) = \frac{1}{x + \mu/2}&\text{if}\;\; x \in [-\alpha, \mu/2\,]\,;\\
b_{-}(x) = \frac{1}{x - \mu/2}&\text{if}\;\; x \in [-\mu/2,\alpha)\,;\\
b_{-}(x) = \frac{1}{x - \gamma}&\text{if}\; \; x \in [\alpha, \mu/2\,]\;.
\end{cases}$$ The planar extension $\mathcal T_v$ of $v(x)$ is defined on the domain $$\Omega_v = \{\,(x,y)\,\vert\, x \in I, \, b_{-}(x) < y < b_{+}(x)\,\}\,.$$ The area of $\Omega_v$ is $c_v := 2 \, \log 8 \cos^2\, \frac{\pi}{q}\,$.
All that remains to justify is the value of $c_v\,$. We have $$\begin{aligned}
c_v &= 2 \;\left( \int_{0}^{\alpha} \dfrac{1}{x+\mu/2}-
\dfrac{1}{x-\mu/2}\,dx +
\int_{\alpha}^{\mu/2} \dfrac{1}{x+\mu/2}- \dfrac{1}{x-\gamma}\,dx
\right)\\
\\
&= 2 \;\log \left( \dfrac{\alpha + \mu/2}{\mu/2}\;
\dfrac{\mu/2}{\mu/2-\alpha}\;\dfrac{\mu}{\alpha+\mu/2}\;
\dfrac{\gamma-\alpha}{\gamma-\mu/2}\right)\\
\\
&= 2 \; \log \dfrac{\mu\,
(\gamma-\alpha)}{(\mu/2-\alpha)(\gamma-\mu/2)}\\
\\
&= 2\; \log \dfrac{\mu\, (\mu/2)(\, \frac{5\mu^2 +4}{3 \mu^2
-4}-\frac{3 \mu^2 -4}{5 \mu^2 + 4}\,)}
{ (\mu/2)^2(\, 1-\frac{3 \mu^2 -4}{5
\mu^2 + 4}\,) (\, \frac{5\mu^2 +4}{3 \mu^2 -4}-1\,)}\\
\\
&= 2\; \log 2\,\dfrac{(5\mu^2 +4)^2 - (3 \mu^2 -4)^2}
{ [\, 5 \mu^2 + 4-(3 \mu^2 -4)\,]^2}\\
\\
&= 2\; \log \,\dfrac{8\mu^2}
{ \mu^2 +4} \;= \; 2\; \log \,2\mu^2\, \dfrac{4}
{ \mu^2 +4}\\
\\
&= 2\; \log \,8\cos^2\frac{\pi}{q}\,.
\end{aligned}$$
Conjugation of the Rosen algorithm {#s:conjugRosen}
==================================
In order to directly compare the symmetric Rosen algorithm with our Veech algorithm, the groups generated by the fractional linear transformations giving the respective maps should be the same. That is not the case, so we first make a conjugation to come closer to this equality. However, the resulting interval map is defined on $[\, 0, \mu\,]\,$, thus we cut and translate appropriately so as to find an interval map defined on $I\,$.
The conjugated Rosen map
------------------------
It is easy to prove that the Hecke group, acting on the upper half-plane, admits a fundamental domain bounded by the vertical lines $x=\pm \lambda/2$ and the unit circle centered at the origin; it contains a subgroup of index two with a fundamental domain bounded by the same vertical lines, and two circles of radius $1/\lambda$ and center $(\pm 1/\lambda,0)$ (see Fig. \[domfondFig\]). On the other hand, the Veech group admits a fundamental domain bounded by the vertical lines $x=0$ and $x=\mu$, and two circles of radius $1/\sin(2\pi/q)$ and center $(\mu/2\pm 1/\sin(2\pi/q),0)$; from this, one obtains easily a conjugacy between the Veech group and the subgroup of the Hecke group.
More explicitly, the Fuchsian group $V_q$ generated by $P = \begin{pmatrix} 1 & \mu \\
0 & 1\end{pmatrix}$ and $R$ is an index two subgroup of the group generated by $P$ and $U$, the rotation of angle $\pi/q$, which is $\text{PSL}(2, {\mathbb R})$- conjugate to the Hecke group of index $q\,$. As shown in [@AH], this conjugation is given by the matrix $$\label{eq:conjMatM}
Q = \frac{1}{\sqrt{\sin\, \pi/q}} \begin{pmatrix} 1 & \cos\, \pi/q \\
0 & \sin\, \pi/q
\end{pmatrix}\,.$$ Note that $Q$ sends $[-\lambda/2, \lambda/2)$ to $ [\,0, \mu\,)$. Thus, in order to transfer the symmetric Rosen map to the interval $ I = [\,-\mu/2, \mu/2\,]\,$, for any $x \in [0, \mu/2)$ we simply use $x \mapsto Q\cdot h(Q^{-1}\cdot x)$; for those $x \in [\,-\mu/2, 0)$ we first apply $x \mapsto P\cdot x = x + \mu$, and then act in the analogous way.
To discuss this explicitly, we use the following easily checked identities.
\[l:conjugates\] The following equalities hold.
1. $Q SQ^{-1} = P$;
2. $QTSQ^{-1} = U$;
3. $QTSTQ^{-1} = RP^{-1}$.
Recall that $T \cdot x= -1/x\,$. We have the following “conjugated” version of the symmetric Rosen map.
\[lemRosenSymOnI\] Let $\widetilde T := Q T Q^{-1}\,$. Then the map, as described above, induced on $I$ by the symmetric Rosen algorithm is given by $$\begin{aligned}
&h_{Q}: \;\; [-\mu/2, \mu/2\,) \;\;\; \to \;\;\; [-\mu/2,
\mu/2\,)\\
\\
&x \mapsto
\begin{cases} \widetilde T \cdot x - \lfloor \, (\widetilde T \cdot x)/\mu +
1/2\, \rfloor \,\mu &
\text{if}\; \; x \in (\, 0, \mu/2\,) ;\\
\\
\widetilde T P \cdot x - \lfloor\, (\widetilde T P \cdot x)/\mu +
1/2\, \rfloor \,\mu &
\text{if}\;\; x \in [\, -\mu/2, 0\,) \; .
\end{cases}
\end{aligned}$$
Although the map $h_Q$ may seem to arise artificially, it is in fact easily described geometrically.
Let $k$ be the interval map on $I$ defined by the following geometric algorithm:
— a positive $x\in I$ is sent to its image under the rotation $U^{-1}$, if this lies outside of $I$ then the final image is given by applying the exact power of $P$ that brings it back into $I\,$;
— a negative $x\in I$ is sent to its image under the rotation $U$, if this lies outside of $I$ then the final image is given by applying the exact power of $P$ that brings it back into $I\,$.
\[propRosenSymIsGeom\] The interval maps $h_Q$ and $k$ are equal.
We have $$\widetilde T P = QTQ^{-1} P = QTQ^{-1} QSQ^{-1} = U\,.$$ The equality of the two functions on $[-\mu/2, 0)$ thus holds.
Since $U^{-1} = P^{-1} \widetilde T$ and for each $x\ge 0$ there is an integral $j$ such that $h_q(x) = P^{j} \widetilde T \cdot x$, the equality also holds on $(0, \mu/2)$.
Despite the obvious similarities of this algorithm with the additive Veech algorithm, whereas the additive Veech map is of infinite invariant measure, this version of the Rosen map is of finite invariant measure.
Doubling the conjugated Rosen map
---------------------------------
We now consider the function $r(x) := k^2(x)$. This is a map that we can directly compare with the multiplicative Veech map, $v(x)\,$. One easily finds that $r(x) = R^{-1} \cdot x$ when $x \in [\,0, R \cdot \mu/2\,)$ and also that $r(x) = R \cdot x$ for $x \in [\,R^{-1} . (-\mu/2)\,, 0\,)$.
We next explicitly give $r(x)$ for negative values, symmetry allows the reader to extend this to all of the interval $I\,$.
For $x \ge 0\,$ the map $r(x)$ is given as follows.
$$r(x) = \begin{cases} R^{-1} \cdot x & \text{if}\;\; x\in [0, R \cdot \mu/2)\;;\\
\\
P^l (PR^{-1})^{k} \cdot x & \text{if}\; \;x\in
(RP^{-1})^{k}P^{-l}(\;[\;-\mu/2, \mu/2)\;)\;;\\
\\
(PR^{-1})^{k} \cdot x& \text{if}\;\; x \in (RP^{-1})^{k}(\;[-\mu/2, R \cdot
0)\;)\;;\\
\\
R^{-1}(PR^{-1})^{k} \cdot x & \text{if}\;\; x \in
(RP^{-1})^{k}R(\;[R^{-1} \cdot 0, \mu/2)\;)\;;\\
\\
P^{-j}R^{-1}(PR^{-1})^{k} \cdot x\;\;\;\; & \text{if}\;\; x \in
(RP^{-1})^{k}RP^{j}([\;-\mu/2, \mu/2)\;)\;,
\end{cases}$$ with $j,k,l \in {\mathbb N}\,$.
We treat several cases, determined by the value of $k(x)$.
[**case 1 : $\;\;k(x) \in [0, U \cdot \mu/2)$**]{}
In this case, there are three possibilities, according to the value of $x$:
$$r(x) = \begin{cases} R^{-1} \cdot x \;;\\
\\
U^{-1}P^{-k} U^{-1} \cdot x \;;\\
\\
U^{-1}P^{l} U \cdot x \;.
\end{cases}$$
Now, using that fact that $T^2 = \text{Id}$ and the identities of Lemma \[l:conjugates\], we find that the final two of these possibilities are also as claimed. $$\begin{aligned}
U^{-1}P^{-k} U^{-1} &= Q S^{-1}TQ^{-1} \cdot QS^{-k}Q^{-1} \cdot Q
S^{-1}T M^{-1}\\
\\
&= Q S^{-1}TS^{-1}T
TS^{-k} T Q^{-1}\\
\\
&= Q S^{-1}TS^{-1}T
Q^{-1} \cdot TS^{-k} T Q^{-1}\\
\\
&= R^{-1} (PR^{-1})^k\\
\\
&= P^{-1}
(PR^{-1})^{k+1}\;.
\end{aligned}$$
[**case 2 : $\;\;k(x) \in [U^{-1} \cdot (-\mu/2)\,,0)$**]{}
Here again there are three possibilities, and similar reasoning leads to $$r(x) = \begin{cases} R \cdot x \;;\\
\\
UP^{-k} U^{-1} \cdot x \;;\\
\\
UP^{l} U \cdot x \;.
\end{cases}$$
Now one verifies that $UP^{-k} U^{-1} =
(PR^{-1})^{k}$, and that $UP^{l} U = (RP^{-1})^{l}R\,$.
[**case 3 : $\;\;k(x) \notin [U^{-1} . (-\mu/2)\,,U \cdot
\mu/2)$**]{}
If $k(x)$ is of sign differing from $x$, then $r(x)$ is either $P^{-k} U^{-1}P^{l} u\cdot x$ or $P^{l} UP^{-k}
U^{-1}\cdot x\,$. On finds that
$$P^{-k} U^{-1}P^{l} U = P^{-k} (P^{-1}R)^l \;;\;\;\; P^{l}
UP^{-k} U^{-1}= P^l (PR^{-1})^k\;.$$
If $k(x)$ has the same sign as $x$, then $r(x)$ is either $P^{-j} U^{-1}P^{-k} U^{-1} \cdot x$ or $P^{m} UP^{l} U \cdot x\,$. On finds that $$\begin{aligned}
P^{-j} U^{-1}P^{-k} U^{-1} &= P^{-(j+1)}
(PR^{-1})^{k+1}\,;\\
P^{m} UP^{l} U &=
P^{m-1} (P^{-1}R)^{l+1}\,.
\end{aligned}$$
The proof is completed by verifying that the domains of the pieces of the function are indeed the subintervals claimed.
The planar domain
-----------------
From our definitions, we have that $\mathcal T_r(x,y) = \mathcal
T_{k}^{2}(x,y)$ and as $\mathcal T_k(x,y)$ is Lebesgue almost everywhere surjective, the domains of definition of these two transformations are the same. Since we already have the domain of the planar map for the symmetric Rosen map, the calculation of the equations for the boundary of the planar domain for $k(x)$ is rather straightforward.
We first collect some identities. Recall that $\phi_j = f^j(-\lambda/2) = (S^{-1}T)^{j}\cdot(-\lambda/2)$; $\delta_{j+1} = S^{-1}T\cdot \delta_j = - \lambda -1/\delta_j\,$, with $\delta_0 = - \lambda - 1\,$ and thus $\delta_{n-2} = -1/(\lambda-1)\,$.
\[lemUsefulIds\] Let $q=2n\,$, $\mu=\mu_q\,$ and $Q$ the matrix of conjugation given in Equation . Then the following equalities hold.
1. $Q\cdot\phi_j = \tan j \pi/q \;\; \text{for}\; \; j\ge 0\;$;
2. $Q\cdot \delta_j = -U^{j} \cdot(Q\cdot1)\;\; \text{for}\; \; j\ge 0\;$;
3. $P^{-1}Q\cdot(-\delta_j) = U^{j} \cdot(Q\cdot1) \;\; \text{for}\; \; j\ge
0\;$.
Since $\phi_0 = -\lambda/2$, direct calculation gives $Q\cdot\phi_0 = 0\,$. Thus, $$Q \cdot \phi_j = Q(S^{-1}T)^j\cdot (-\lambda/2) = Q(S^{-1}T)^jQ^{-1} \cdot 0 = U^{-j}\cdot 0\,.$$ Since $U$ is the rotation of angle $\pi/q$, the first equality holds.
Since $\delta_0 = -1 -2 \cos \pi/q$, direct calculation gives $Q\cdot(\delta_0) =
-Q\cdot 1\,$. Since $\delta_j = (S^{-1}T)^j$, as for the previous identity, we find $Q\cdot(\delta_j) = U^{-j} \cdot (- Q\cdot 1)$. But, $-(\Lambda\cdot( -x)\,) = \Lambda \cdot x$ for any rotation $\Lambda$ and any real $x$, and the second equality holds.
For the third identity, the case of $j=0$ can be directly verified. For $j>0$, we rely on the fact that for any $y$, $-(S^{-1}T \cdot y) = -(-\lambda -1/y) = ST\cdot(-y)$. By induction, we find $- (\, (S^{-1}T)^j \cdot y) = (ST)^j\cdot(-y)$. Therefore, $$\begin{aligned}
P^{-1}Q\cdot(-\delta_j) &= P^{-1}Q (ST)^j\cdot (-\delta_0) = P^{-1}QS Q^{-1} \, Q (TS)^j S^{-1}\cdot (-\delta_0))\\
&= U^j QS^{-1}\cdot (-\delta_0)) = U^j P^{-1}Q\cdot (-\delta_0))\,
\end{aligned}$$ and the result holds.
The reader may wish to compare the following description of the boundary of $\Omega_r$ with Figure \[natExtOverlayFig\].
\[lemNatExtDouble\] Let $\Omega_{r}$ let be the domain of the planar system, determined as usual, of $r(x)\,$. Then the boundary of $\Omega_r$ is given by $$\begin{cases}
y = 1/(x - Q \cdot 1\,) & x \in [\,-2/\mu,\mu/2\,)\;;\\
\\
y = 1/(x + Q \cdot 1\,)& x \in [\,-\mu/2, 2/\mu\,)\;;\\
\\
y = 1/(x - U^{j} Q \cdot 1)\,)& x \in [\,-\tan(j+1)\pi/q,-\tan j \pi/q \,)\,\;;\\
\\
y = 1/(x + U^{j} Q \cdot 1)\,)& x \in [\,\tan j
\pi/q,\tan (j+1) \pi/q)\;,\\
\end{cases}$$ where $j\in\{1, \dots, n-2 \}$.
(Sketch) The first step is the conjugation by the matrix $Q$. The image of points $(x,y= 1/(x-\
\delta)\,)$ being $(Q \cdot x, y=1/(Q \cdot x - Q \cdot
\delta)\,)$, we find that the boundary above $(\mu/2, \mu)$ is given by $y = 1/(x + Q \cdot 1)$, and that under $(0,\mu/2)$ is $y = 1/(x - Q \cdot 1\,)$; the remaining boundaries follow as easily.
Secondly, we cut and translate by letting $P^{-1}$ act on the right half. Since the $(2,1)$-element of $P^{-1}$ is $c=0\,$, one finds that only the $x$-coordinate values change. Of course, the equations giving the new boundaries change accordingly.
For the study of the intersections of the planar extensions of $r(x)$ and $v(x)$ it is helpful to note that after $y = 1/(x+ Q \cdot 1)$ the next lowest piece of the upper boundary of $\Omega_{k}\,$ is of equation $y =
1/(x - Q \cdot (-\delta_1)\,\;)$. Equivalently, this is $y=1/(x+U Q \cdot 1)\,)$. Furthermore, $Q \cdot (-\delta_1) = Q \cdot
(-\lambda +1/(\lambda+1)\;) = \mu/2 +(-\mu +\rho)$, with $\rho = 1/(\;
(\sin \pi/q)(\lambda + 1)\;)$. Thus, $-Q \cdot (-\delta_1) = \mu/2
- \rho$ and $\rho$ is positive. (From this, we will conclude that $y=1/(x+\mu/2)$ gives a piece of the upper boundary of the intersection of the domains of the planar extensions.)
\[lemAreaNatExt\] The area of $\Omega_{r}$ is $$c_r = 2 \log\cot
\frac{\pi}{2q}\;.$$
From the discussion above, it suffices to find the Lebesgue area of $\Omega_{k}\,$, the planar extension of $k(x)\,$. By symmetry, we easily find that this area is $$\aligned
c_r &= 2 \, \log \; \dfrac{2/\mu +
Q\cdot1}{Q\cdot1-\mu/2}\,\prod_{j=1}^{n-2}\,
\dfrac{Q\cdot\phi_{j+1}-Q\cdot(-\delta_j)}{Q\cdot\phi_j-Q\cdot(-\delta_j)}\,
\dfrac{\vert \,Q\cdot\phi_j - Q\cdot1\,\vert }{\vert \,Q\cdot\phi_{j+1}-
Q\cdot1\,\vert}
\\
&= 2 \, \log \; \dfrac{2/\mu + Q\cdot1}{Q\cdot 1-\mu/2}\,
\dfrac{\vert \,Q\cdot\phi_1 - Q\cdot 1\,\vert }{\vert \,Q\cdot\phi_{n-1}-
Q\cdot1\,\vert}\; \prod_{j=1}^{n-2}\,
\dfrac{Q\cdot\phi_{j+1}-Q\cdot(-\delta_j)}{Q\cdot\phi_j-Q\cdot(-\delta_j)}\\
\\
&= 2 \, \log \; \dfrac{2/\mu + Q\cdot1}{1/(\sin \pi/q)}\;
\prod_{j=1}^{n-2}\, \dfrac{\tan\frac{(j+1)\pi}{q}+U^{j}Q\cdot 1)}
{\tan\frac{j \pi}{q}+U^{j}Q\cdot 1}\;.
\endaligned$$
But, $Q\cdot 1 = \mu/2 - 1/(\sin\pi/q)$, that is $Q \cdot 1 = (\cos \pi/q +
1)/(\sin\pi/q)$. Its image under $U^{j} = \begin{pmatrix} \cos j\pi/q & -
\sin j\pi/q\\
\sin j\pi/q & \cos
j\pi/q\end{pmatrix}$ is thus $\frac{\cos \frac{j \pi}{q} + \cos \frac{(j+1)\pi}{q}}{\sin \frac{j
\pi}{q} + \sin \frac{(j+1)\pi}{q}}\, $. Let $W = \prod_{j=1}^{n-2}\,
\dfrac{\tan\frac{(j+1)\pi}{q}+U^{j}Q\cdot 1} {\tan\frac{j
\pi}{q}+R^{j/2}Q\cdot 1}$, then
$$\aligned
W &=\dfrac{\cos \pi/q}{\cos \frac{(n-1)\pi}{q}}\,\prod_{j=1}^{n-2}\,
\dfrac{\sin\frac{(j+1)\pi}{q}+\cos\frac{(j+1)\pi}{q}\, \frac{\cos
\frac{j \pi}{q} + \cos \frac{(j+1)\pi}{q}}{\sin \frac{j \pi}{q} +
\sin \frac{(j+1)\pi}{q}}}
{\sin\frac{j\pi}{q}+\cos\frac{j\pi}{q}\, \frac{\cos \frac{j \pi}{q} +
\cos \frac{(j+1)\pi}{q}}{\sin \frac{j \pi}{q} + \sin
\frac{(j+1)\pi}{q}}}\\
\\
&= \dfrac{\cos \pi/q}{\sin \pi/q}\,
\prod_{j=1}^{n-2}\, \dfrac{ \sin\frac{(j+1)\pi}{q}\sin\frac{j\pi}{q}
+1 +
\cos\frac{(j+1)\pi}{q}\cos \frac{j \pi}{q} }
{1 +\sin\frac{j\pi}{q}\sin\frac{(j+1)\pi}{q} + \cos \frac{j \pi}{q}\cos\frac{(j+1)\pi}{q}}\\
\\
&= \dfrac{\cos \pi/q}{\sin \pi/q}\;.
\endaligned$$
Since $c_r = 2 \, \log \; \dfrac{2/\mu + Q\cdot 1}{1/(\sin
\pi/q)}\cdot W$, we have that $c_r = 2 \, \log \; (2/\mu + Q\cdot 1)(\cos
\pi/q)\,$. This evaluates to $2\, \log [\sin \pi/q + (\cos^2
\pi/q+\cos \pi/q)/(\sin \pi/q)]\,$ that is, to $2 \log\cot
\frac{\pi}{2q}$. One could express this as $c_r
= 2 \log (Q\cdot1)$.
First return type {#s:firstReturn}
=================
Natural extensions provide a basic tool for comparing continued fraction type interval maps. We show that each of the interval maps of interest to us has a model of its natural extension given by the first return map to a subset of the unit tangent bundle of a corresponding hyperbolic surface.
First return type defined
-------------------------
We give definitions allowing us to formalize the notion of first return type.
For $M \in \slr$ and $x \in \mathbb R\,$ such that $M \cdot x \neq \infty\,$, let $$\tau(M,x) := -2 \,\log |c x + d\,|\,\,$$ where $(c,d)$ is the bottom row of $M\,$ as usual.
As usual, we consider the projective group $\text{PSL}(2, \mathbb R)$ in lieu of $\text{SL}(2, \mathbb R)$. Elementary calculation shows that $\tau$ induces on $\text{PSL}(2, \mathbb R)$ a cocycle, in the following sense: $$\tau(MN, x) = \tau(M, Nx) + \tau(N, x)$$ whenever all terms are defined and we choose each projective representative such that the corresponding $cx + d$ is positive. (In all that follows, the set where such a $c x +d$ is zero can be avoided.)
\[d:returnTimeAndGamma\] Suppose that $f$ is a piecewise Möbius interval map, say defined on an interval $I$, with $I=\bigcup \, I_\alpha$ and for each $\alpha$, $f$ on $I_{\alpha}$ given by $x \mapsto
M_\alpha \cdot x\,$. For each $x \in I$, the [*return time*]{} of $x$ is $\tau_f(x) := \tau(M_\alpha\,,\, x)\,$. Finally, let $\Gamma_f$ be the group generated by the set of the $M_{\alpha}$.
In the following, we rely on terms and notation introduced in Section \[ss:NatExtGeoFloIntro\], see especially Equation . Recall that a Fuchsian group is a discrete subgroup of $\text{SL}(2, \mathbb R)$ (or of $\text{PSL}(2, \mathbb R)$, depending upon context).
\[defReturnType\] For $f$ as above, on $I \times \mathbb R$ we have the piecewise defined map $\mathcal T_f$ given by taking each transformation $\mathcal T_{M_{\alpha}}$ above $I_{\alpha}$. We say that $f$ [*has a positive planar model*]{} if there is a compact set $\Omega_f \subset I \times \mathbb R$ also fibering over $I$, and of positive Lebesgue measure, say $c_f$, such that $\mathcal T_f( \Omega_f )= \Omega_f$. One can prove (cf. [@AS3]) that such a set is unique, and $\mathcal T_f$ is then a measurable automorphism of $\Omega_f$. We then refer to the marginal measure of $(1/c_f) \,dx \, dy$ (that is, the measure on $I$ obtained by integrating along the fibres) simply as [*the marginal measure*]{}.
Further, we say that $f$ is of [*first return type*]{} if:
1. $f$ has a positive planar model;
2. $\Gamma_f$ is a Fuchsian group;
3. for almost every $x \in I$ we have $\tau_f(x)> 0$; and,
4. for almost every $(x,y) \in \Omega_f$ and every non-trivial $M \in \Gamma_f$ with $\mathcal T_M(x,y) \in \Omega_f$ and $\tau(M, x)\ge 0$, we have $\tau_f(x) \le \tau(M, x)$.
Recall that, with our usual notation, the derivative of $M\cdot x$ at $x$ is $(cx +d)^{-2}\,$. The positivity of the values $\log 1/(c x + d)$ shows that any (non-trivial) piecewise fractional linear map of first return type is expanding almost everywhere.
Cross-section as natural extension
----------------------------------
In tersest terms, a measurable [*cross-section*]{} for the geodesic flow is a subset of the unit tangent bundle through which almost every geodesic passes tranversely and infinitely often, equipped with the transformation defined by first return under this geodesic flow. Any flow invariant measure then induces an invariant measure on the cross-section and thus one can speak of the corresponding dynamical system. Similarly, a [*factor*]{} of a dynamical system is a second dynamical system with a measurable map from the first to the second underlying spaces such that the corresponding diagrams of spaces and maps commute in the measurable category.
Recall that a [*natural extension*]{} of a system $(f, I, \mathscr B, \nu)$ is a dynamical system $(\mathcal T, \Sigma, \mathscr B', \mu)$ such that with the projection map $\pi: \Sigma\to I$ to the first coordinate, four criteria are satisfied: (1) $\pi$ is a surjective and measurable map that pulls-back $\nu$ to $\mu$; (2) $\pi \circ \mathcal{T} = f \circ \pi$; (3) $\mathcal{T}$ is an invertible transformation; and, (4) any invertible system that admits $(f, I, \mathscr B, \nu)$ as a factor must itself be a factor of $(\mathcal T, \Sigma, \mathscr B', \mu)$. A standard method to verify this minimality criterion is to verify that $\mathscr{B}' = \bigvee_{n\ge 0} \mathcal{T}^{n} \pi^{-1} \mathscr{B}$. (In our setting, the $\sigma$-algebras are always the appropriate Borel algebras.) In the setting that $f$ is expanding, it suffices to show that $\mathcal T^{-1}$ is expanding for $y$-values (that is, that a.e. $(x,y) \in \Sigma$ has a neighborhood in which $\mathcal T^{-1}$ has this property); see, say, the proof of Theorem 1 of [@KSS] (on p. 2219 there).
It is well known that the geodesic flow on the unit tangent bundle of $\mathbb H$ is an Anosov flow (indeed, sometimes called a “hyperbolic flow”) — there is a splitting into the direct sum of three invariant subbundles, with one tangent to the flow, one expanded exponentially, one contracted exponentially; see [@M].
\[t:firstReturnTypeIsFirstReturn\] If $f$ is of first return type, then there is a cross-section for the geodesic flow on the unit tangent bundle of $\Gamma_f\backslash \mathbb H$ such that the first return map to this cross-section is a model of the natural extension of the system defined by $f$ and the marginal measure.
We first prove that the domain $\Omega_f$ projects to a cross-section for the geodesic flow on $\Gamma_f\backslash\text{PSL}(2, \mathbb R)$, then we prove that the first-return map to this section is conjugate to $\mathcal T_f$.
There is an injective map $\iota: \Omega_f \to \mathcal A \subset \text{PSL}(2, \mathbb R)$ given by sending $(x,y) \mapsto A = \begin{pmatrix} x&xy-1\\1&y\end{pmatrix}$. (We use the standard abuse of using a matrix to represent its class in the projective group.)
Just as the unit tangent bundle of $\mathbb H$ is given by $\text{PSL}(2, \mathbb R)$, that of $\Gamma_f\backslash \mathbb H$ is given by $\Gamma_f\backslash\text{PSL}(2, \mathbb R)$. Furthermore, the geodesic flow on this quotient is simply of the (local) form $[A] \mapsto [A g_t]$, where each $[B]$ here denotes the coset $\Gamma_f B$ and $g_t$ denotes the usual diagonal matrix.
Let $\Sigma_f = \{ \, [A]\,|\, A = \begin{pmatrix} x&xy-1\\1&y\end{pmatrix} \; \mbox{for}\;\; (x,y) \in \Omega_f\;\}\subset \Gamma_f\backslash\text{PSL}(2, \mathbb R)$. We claim that the map $\Omega_f \to \Sigma_f$ given by sending $(x,y)$ to $[\iota(x,y)]$ is almost everywhere injective. For this, let $A, A'$ be in $\iota(\Omega_f)$, suppose that there is $M \in \Gamma_f$ such that $M = A' A^{-1}$. Now, $$A' A^{-1} = \begin{pmatrix} 1 + x'(y-y')&(x'-x) - xx'(y-y')\\y-y'&1 - x(y-y')\end{pmatrix}\,.$$ If $y=y'$ then $M$ is a translation matrix $\begin{pmatrix} 1&x'-x\\0&1\end{pmatrix}$, such that $\mathcal T_M(x,y)=(x',y)$; but obviously, $\tau(M,x)=0$, and items three and four of the definition of the first-return type preclude the existence of such a matrix, except for a set of measure zero.
On the other hand, if $y-y' \neq 0$, then we first find that $y-y'$, being the $(2,1)$-element of the matrix $M\in \Gamma_f$, belongs to a countable set; hence $x'= (a-1)/(y-y')$, where $a$ is the $(1,1)$-element of $M$, belongs to a countable set, as does $x$, and almost everywhere injectivity holds.
Recall that Liouville measure on $T^1 \mathbb H$ is given as the product of the hyperbolic area measure on $\mathbb H$ with the length measure on the circle of unit vectors at any point. In [@AS2] (see especially Section 3 there), we show that $\mathcal A$ gives a (measurable) transversal to the geodesic flow on $T^1 \mathbb H$ in the sense that almost every geodesic meets $\mathcal A$ exactly once and that Liouville measure factors as $dx\,dy\,dt$ where $t$ is the variable for geodesic flow.
It follows from the third and fourth items of the definition of first return type that the map $\Phi_f: \Sigma_f \to \Sigma_f$, given by $[A] \mapsto [M A g_{t_0}]$ with the various $M$ satisfying $M \cdot x = f(x)$ and the $t_0 = -2 \log (c x + d)$ correspondingly defined, is in fact a [*first*]{} return map to the cross-section $ \Sigma_f $.
Since $A\cdot i = (x i + xy-1)(i + y)$, one finds that $\mathcal A_x = \{ A\in \mathcal A\,|\, x \;\mbox{is fixed}\}$ corresponds to the horocycle of $\mathbb H$ of Euclidean radius $1/2$ based at $x$. Therefore, the fiber of $\Sigma_f$ above any $x \in I$ lies on a horocycle, and in particular lies in the strong unstable manifold of the geodesic flow, viewed as an Anosov flow. In other words, the geodesic flow is contracting in this “$y$”-direction. Since $\Phi_f$ is a return map of this flow, it follows that $\Phi_{f}^{-1}$ acts as an expanding map on the $y$-values. For the $x$-values, $\Phi_f$ is expanding, as it agrees with $f$ on these.
\[c:firstReturnIsErgodic\] If $f$ is of first return type and $\Gamma_f$ is of finite covolume, then $f$ is ergodic with respect to the marginal measure.
By the well-known results of Hopf, when $\Gamma_f$ is of finite covolume the geodesic flow on unit tangent bundle of $\Gamma_f\backslash \mathbb H$ is ergodic. But, this ergodicity then implies that of the induced map, $\phi: \Sigma_f \to \Sigma_f$. Finally, the ergodicity is a property shared by a transformation and its natural extension.
Not every $f$ with a planar model of a natural extension is of first return type, as we showed in [@AS2]. Rather, the entropy of $f$ must accord with the measure of $\Omega_f$, as the following shows.
\[p:entropyFirstReturn\] Suppose that $f$ is a piecewise Möbius interval map with a positive planar model of its natural extension, that the group $\Gamma_f$ is Fuchsian of finite covolume, and that $f$ is ergodic with respect to the marginal measure. Then $f$ is of first return type if and only if the volume of the unit tangent bundle of the surface uniformized by $\Gamma_f$ equals the product of the entropy of $f$ with the Lebesgue measure of $\Sigma_f$.
For ease of typography, let $\Sigma= \Sigma_f$.
By Rohlin’s formula for the entropy of an ergodic interval map, see say [@DK], and the fact that locally $f(x) = (a x + b)/(c x + d)$, we have
$$\begin{aligned}
h(T) &= \int_I \log |f'(x)|\, d\nu \\
&= \int_I -2 \log |c x + d| \,d \nu\\
&= \int_I \tau_f(x) \,d \nu\\
& = \dfrac{1}{c_f} \, \int_{\Sigma} \tau_f(x) \,dx\,dy\\
& = \dfrac{ \int_{\Sigma} \tau_f(x)\, dx \, dy}{ \lambda(\Sigma)}\\
&\ge \dfrac{\text{vol}(T^1(\Gamma\backslash \mathbb H))}{ \lambda(\Sigma)}\,.
\end{aligned}$$
The final inequality holds since the [*first*]{} return map to $\Sigma$ is given by following the unit tangent vectors along the geodesic arcs with unit tangent vectors in $\Sigma$ until a first return to $\Sigma$. The fact that $\mathcal A$ is a transversal implies that these geodesic arcs sweep out the unit tangent bundle (up to measure zero). Hence, equality holds if and only if $f$ is of first return type.
In the case of our previous paper [@AS2], some of the interval maps treated were piecewise given by matrices of determinant $-1$, causing the cross-section to consist of two copies of the natural domain of the underlying interval map, and hence there is then a factor of 1/2 in the above. We have simplified the argument given there for the analogous result — there we relied on a result of Sullivan stating that the entropy of the geodesic flow on the unit tangent bundle of the uniformized surface (with respect to normalized Liouville measure) equals one. Here, we see that the existence of a first return time interval map $f$, combined with Abramov’s formula, shows that the entropy of this flow on the unit tangent bundle of $\Gamma_f\backslash \mathbb H$ must equal one.
Our continued fraction maps are of first return type {#ss:oursFirst}
----------------------------------------------------
\[p: RosenSymmIsFirst\] The symmetric Rosen fraction map $h: J \to J$, as defined in Equation , is of first return type.
Nakada [@N] has shown that this map has the correct entropy value so that we can apply Proposition \[p:entropyFirstReturn\].
Our use of the geodesic flow on $\mathcal A \subset \slr$ rather quickly identifies the region $\Omega_f\,$, as opposed to the often quite tedious approach of directly building a cross-section for which the first return map induces the function, see [@Se], [@AF], [@MS]. On the other hand, in our approach one is left to prove that it is the [*first*]{} return map that induces $f$. One can do this by evaluating integrals as in Proposition \[p:entropyFirstReturn\]. However, this can be quite challenging, as these integrals naturally involve polylogarithmic functions. Indeed, in general the existence of an interval map of first return type implies an identity of polylogarithmic functions, see [@F]. Here we use specific aspects of maps to show that they must indeed be of first return type.\
\[propVeechAddIsFirst\] Suppose that $q = 2n$ and let $\mu = \mu_q = 2 \cot \pi/q\,$. Let $\mathbb I = (-\mu/2, \mu/2)\,$, and $P, R$ be the elements of $\text{PSL}(2, \mathbb R)$ representing translation by $\mu$ and the rotation standard rotation by $2\pi/q\,$. Then $V_q\,$, the group generated by $P, R\,$, is the (internal) free product of the cyclic groups generated by these two elements; in particular, this is isomorphic to $\mathbb Z \star \mathbb Z/n \mathbb Z\,$.
For $j \in \{1, n-1\}$, let $d_j = R^{-j}\cdot \mu/2\,$. Each interval $ (d_j, d_{j+1}\,)$ is contained in $\mathbb I\,$. The [*additive Veech interval map*]{} is $$\begin{aligned}
f : \mathbb I &\to \mathbb I\\
x &\mapsto P^k R^j\cdot x\,,
\end{aligned}$$ whenever $x \in (d_j, d_{j+1}\,)$, where $k \in \mathbb Z$ is the unique power of $P$ such that the image lies in $\mathbb I\,$.
The map $f$ is of first return type.
Recall that $\Delta(k, j)$ denotes the [*cylinder*]{} of $P^k R^j$. It is easily checked that each $x \mapsto P^k R^j \cdot x$ is expanding almost everywhere on $\Delta(k, j)\,$. Thus, $\tau(f,x) \ge 0$ for all $x \in \mathbb I\,$.
Recall that the action of $M$ sends the curve of equation $y = 1/(x - \delta)$ to that of the equation $y = 1/(x - M\cdot \delta)$, and $(x,y) \in \Omega_{a}$ implies that $\delta \in \mathbb R \setminus \mathbb I$ (or $\delta = \pm \mu/2$).
By the structure of $V_q\,$ we can uniquely express $M$ in the following form $$M = P^{t_m} R^{s_m} \cdots P^{t_1}R^{s_1}\,$$ with $s_i \in \{1, \dots, n-1\}$ if $i>1\,$; $s_1 \in \{0, \dots, n-1\}\,$; $t_i \in \mathbb Z \setminus \{0\}$ if $i<m$; and, $t_m \in \mathbb Z\,$.
We thus fix $(k,j)$, a subinterval $J \subset \Delta(k,j)$, and $M$ in the above form that sends $J$ into $\mathbb I\,$. We let $\mathcal J \subset \Omega_{a}$ denote the subset fibering over $J$.
[**Case: $s_1 \notin \{0, j\}$.**]{} Since $s_1 \neq j$, we have that for any $x \in J$, $R^{s_1}\cdot x \in \mathbb I$. On the other hand, $R^{s_1}$ maps all $\delta$ in the complement of $\mathbb I$ into $\mathbb I$. That is, each $(x,y) \in \mathcal J$ has image under $R^{s_1}$ that lies outside of $\Omega_{a}$ due to the value of its $y$-coordinate. Clearly, $M \neq R^{s_1}$ and hence $t_1 \neq 0$; but each $(x,y) \in \mathcal J$ has image under $P^{t_1} R^{s_1}$ that lies outside of $\Omega_{a}$ now due to the value of its $x$-coordinate. Since the expansion of $M$ as a word in $P$ and positive powers of $R$ has finite length, these alternating types of obstruction persist, and we conclude that there is no $M$ with $s_1 \notin \{0, j\}$ that can send all of $\mathcal J$ into $\Omega_{a}$.
[**Case: $s_1 = 0$.**]{} Of course here $t_1 \neq 0$, but then $P^{t_1}$ sends $J$ outside of $\mathbb I$ and maps a subset of the $\delta \in \mathbb R \setminus \mathbb I$ back into $ \mathbb R \setminus \mathbb I$. For these values, we then proceed by invoking the alternating obstruction argument of the previous case. For those $\delta \in \mathbb R \setminus \mathbb I$ that $P^{t_1}$ maps into $\mathbb I$, there could be some $R^{s_2}P^{t_1}$ mapping the corresponding set of $(x,y)$ into $\Omega_a$; however, this map is the inverse of $P^{-t_1}R^{-s_2}$, and under the assumption that the return time under $M$ is positive, the return time of $MP^{-t_1}R^{-s_2}$ is less than that of $P^{-t_1}R^{-s_2}$. That is, this second possibility reduces to replacing $x$ by $R^{s_2}P^{t_1}\cdot x$.
[**Case: $s_1 = j$.**]{} In this case, we again cannot have $t_1 = 0$. If nonzero $t_1 \neq k$, then an alternating failure of new $x$- or $y$-coordinate again appears. Thus, we can only have that $t_1 = k$. It is then trivial (by the cocycle property of return times) that $M$ has at least the return time of $P^{k}R^j$.
Let $f$ be a piecewise fractional linear map on some interval $I\,$. With notation as above, suppose that for some index value $\alpha$ one has $M_{\alpha}$ parabolic and that the fixed point of $M_{\alpha}$ lies in $I_{\alpha}\,$. Then the [*parabolic acceleration*]{} of $f$ with respect to $M_{\alpha}$ is the piecewise fractional linear map on $I$ that agrees with $f$ except on $I_{\alpha}$, and such that for $x \in I_{\alpha}$ this new map is given by applying the appropriate power of $M_{\alpha}$ to escape $I_{\alpha}\,$.
The following is easily proved.
Suppose that $f$ is a piecewise fractional linear map on some interval $I\,$. Suppose that $g$ is a parabolic acceleration of $f$ that generates the same Fuchsian group. Then $g$ is of first return type if and only $f$ is.
The multiplicative Veech map is of first return type.
By construction, the multiplicative Veech map is the (two-fold) parabolic acceleration of the Veech additive map in the indifferent fixed points. Therefore, the result follows from the previous lemma and Proposition \[propVeechAddIsFirst\].
The comparison {#s:comparison}
==============
The intersection, $\overline{\Omega} := \Omega_v\,\cap
\,\Omega_r$
-------------------------------------------------------
Let $q\ge 8$ be an even natural number, and let $\Omega_r$ and $\Omega_v$ be the domains of the natural extensions of $r(x)$ and $v(x)$, respectively. Let $\overline{\Omega} = \Omega_r \cap \Omega_v$. The upper boundary of $\overline{\Omega}$ is given by :
$$\begin{cases}
y = \frac{1}{x+M \cdot 1} & \text{if} \;\; x\in[-\mu/2,\;2/\mu)\;;\\
\\
y = \frac{1}{x+\mu/2} & \text{if} \;\; x\in[\;2/\mu,\;\mu/2)\;;\\
\end{cases}$$ and the lower boundary of $\overline{\Omega}$ by : $$\begin{cases}
y = \frac{1}{x-\mu/2}& \text{if} \;\; x\in[-\mu/2,-2/\mu)\;;\\
\\
y = \frac{1}{x-M \cdot 1} & \text{if} \;\; x\in[-2/\mu,\;\mu/2)\;.
\end{cases}$$
The Lebesgue area of $\overline{\Omega}$ is $\lambda(\overline{\Omega}) = 2 \, \log
\left(\, \cos \frac{\pi}{q}(1+\cos \frac{\pi}{q})\right)$.
The boundaries of $\Omega_v$ are given in Proposition \[p:VeechMultNat\], those of $\Omega_r$ in Lemma \[lemNatExtDouble\]. In particular, the upper boundary of $\Omega_v$ above the left end of $I$ is of equation $y = 1/(x+\gamma)$, whereas the leftmost upper boundary of $\Omega_r$ is given by $y = 1/(x+ M . 1)\,$. Recall that $M . 1 = \mu/2 + 1/(\sin \pi/q)$ whereas $\gamma = \mu/2 \cdot
(5\mu^2 +4)/(3 \mu^2 -4)$. As $1/(\sin \pi/q) = \sqrt{4+\mu^2}/2$ and $\gamma = \mu/2 +
\mu(\mu^2+4)/(3 \mu^2 - 4)$, elementary calculations show that as soon as $q \ge 8\,$, one has $M . 1 >
\gamma\,$. Now, it is easily checked that (for all of our $q$), $y = 1/(x+\gamma)$ does give the lowest piece of the upper boundary of $\Omega_v\,$. Thus, we conclude that $y = 1/(x+ M . 1)\,$ gives the upper boundary of $\overline{\Omega}$ above all of $[-\mu/2,\;2/\mu)\,$.
For $x>2/\mu$, the lowest piece of the upper boundary of $\Omega_r$ is of equation $y=1/(x- M \cdot( -\delta_1)\;)\,$. Using Lemma \[lemUsefulIds\], one finds that $-M \cdot ( -\delta_1)$ is smaller than $\mu/2\,$. Thus, one finds that the second piece of the boundary of $\overline{\Omega}$ is as claimed. Symmetry completes the description of this boundary.
The area is calculated as in Proposition \[p:VeechMultNat\] and Lemma \[lemNatExtDouble\].
![Overlay of the domains $\Omega_v$ and $\Omega_r\,$, $q=8\,$.[]{data-label="natExtOverlayFig"}](overlays2.jpg)
Were the two algorithms the same, we would of course find that the quotient of the area of their respective domains of planar natural extensions by the area of the intersection would equal one. Rather, in the limit we find that the intersection accounts for $1/3$ of the area of the natural extension domain for the multiplicative Veech map, but for a negligible amount of that of the doubled symmetric Rosen map.
As $q$ goes to infinity the limit of the ratio of the area of the intersection to the original domain of natural extensions is: $$\lim_{q \to \infty} \dfrac{\lambda(\, \overline{\Omega}\,)}{ \lambda(\, \Omega_v\,)} = 1/3
\;\;\; \mbox{and}\;\;\;\lim_{q \to \infty} \dfrac{\lambda(\, \overline{\Omega}\,)}{ \lambda(\, \Omega_v\,)} = 0\,.$$
Unbounded return times {#ss:unbounded}
----------------------
Since both the Rosen and the Veech maps are of first return type, the return maps to the intersection $\overline{\Omega}$ by iterations of ${\mathcal T}_r$ and ${\mathcal T}_v$, respectively, actually define the same map on this intersection. We now briefly consider the number of iterations required for these return maps.
Let $F: \Omega\to\Omega$ and $\Lambda \subset
\Omega$. For $p \in \Omega$, the [*induction index* ]{} with respect to $\Lambda$ of $F$ at $p$ is the smallest positive exponent $n$ such that $F^n(p) \in \Lambda$.
We now determine the region where where both maps have induction index equal to one.
For $(x,y) \in \overline{\Omega}$, the equality
$${\mathcal T}_r(x,y)={\mathcal T}_v(x,y)$$
if and only if either
1. $$\pm x \in [R \cdot \mu/2, \beta)\,;$$ or,
2. there is an $n\in {\mathbb N}$ such that $$\pm x \in (RP^{-1})^n(\; [-\mu/2, R \cdot 0)\;) \;.$$
In particular, the induction index of either of the functions equals one at any of these points.
It is clear that the subregions where the maps agree are defined simply in terms of $x$. Furthermore, the only possibility for such an equality is where both $v(x)$ and $r(x)$ are given by the same element of $\text{PSL}(2, {\mathbb R})$. By symmetry, it suffices to check for such equality when $x >0$.
For $x \in [0, R \cdot \mu/2)$, $r(x) = R^{-1} \cdot x$ whereas $v(x) =
P^k R^{-n} \cdot x$ with $n\ge 2$. That is, there is no equality here.
For $x \in [R \cdot \mu/2, \beta[$, both $r(x)$ and $v(x)$ are equal to $P^kR^{-1} \cdot x$ and furthermore, one verifies that the exponents $k$ are the same for these two maps.
If $x\in [\beta, \mu/2)$, then $v(x) = (PR^{-1})^k \cdot x$ for an appropriate value of $k$. However, $r(x)$ is only of this form on (a subinterval of each of) its non-full cylinders — thus, where $x \in
(RP^{-1})^n(\; [-\mu/2, R \cdot 0)\;)$.
Since the two functions take the same values at these point, we do indeed find that both have induction index there.
In Lemma \[l:slowReturn\] we will show that the induction index of $v(x)$ is unbounded. The following is helpful in the proof of that result.
\[l:anId\] The following equality holds for $Q$ the conjugation matrix of Equation . $$\mu+\frac{1}{Q \cdot 1} = Q\cdot 1\;.$$
By expressing all values in terms of elementary trigonometric functions, the result follows immediately.
\[l:slowReturn\] Suppose that $q$ is divisible by $4$. Then for any $(x,y) \in \overline{\Omega}$ such that $x \in
(R^{q/4}P^{-1})^k(\;[-\mu/2, \mu/2)\;)$, the induction index of ${\mathcal T}_v$ at $(x,y)$ is at least $k$.
The multiplicative Veech interval map is such that the subinterval upon which $x \mapsto P R^{q/4} \cdot x$ defines a full cylinder. Thus, for each $k \in \mathbb N$, there is a non-empty subinterval for which the $v$-orbit of $x$ begins with $x$ and then $( P R^{q/4} \cdot x, \dots, (P R^{q/4})^k \cdot x)$. (Of course, these $x$ are exactly those of our hypothesis: $x \in (R^{q/4}P^{-1})^k(\;[-\mu/2, \mu/2)\;)$.) We show that the strip lying above this interval has image under ${\mathcal T}_v, {\mathcal T}^{2}_v, \dots, {\mathcal T}^{k-1}_v$ exterior to the intersection domain $\overline{\Omega}$.
When $q$ is divisible by $4$, as a projective element, $R^{q/4}$ has order $2$. In fact, $R^{q/4}\cdot x = -1/x$. By Lemma \[l:anId\], $PR^{q/4} \cdot (-Q \cdot 1) = \mu +1/(Q \cdot 1) = Q \cdot 1$. Recall that $R^{q/4}P^{-1}(\;[-\mu/2,
\mu/2)\;)$ is a subinterval of $[0,R \cdot \mu/2)$. Therefore the intersection of the vertical fibres with $\overline{\Omega}$ lie between the curves of equation $y = 1/(x \pm Q \cdot 1)$. By Equation , $PR^{q/4}$ sends the upper boundary curve of equation $y = 1/(x + Q \cdot 1)$ to the lower boundary curve, of equation $y = 1/(x - Q \cdot 1\;)$.
We now consider further the orbit of $\delta = - (Q\cdot 1)$ under powers of $PR^{q/4}$. First note that $PR^{q/4}$ sends $0$ to $\infty$ and fixes $ \frac{\mu\pm \sqrt{\mu^2-4}}{2}$. Since $PR^{q/4}$ sends $\delta = -Q \cdot 1$ to the value $Q \cdot 1 = \frac{\mu+\sqrt{\mu^2+4}}{2}$ which is greater than the larger of the fixed points of this hyperbolic matrix, the values of $(PR^{q/4})^{1+j}\cdot (-Q\cdot 1)$ decrease with positive $j$, but remain greater than the fixed point value $ \frac{\mu + \sqrt{\mu^2-4}}{2}$.
Again applying Equation , we find that the region of $\overline{\Omega}$ fibering over the interval $(R^{q/4}P^{-1})^k(\;[-\mu/2, \mu/2)\;)$ has image under the $\mathcal T_{P(R^{q/4})^j}$ with $1 \le j < k$ lying exterior to $\overline{\Omega}$. It then follows that the induction index of ${\mathcal T}_v$ at any $(x,y)$ with $x \in
(R^{q/4}P^{-1})^k(\;[-\mu/2, \mu/2)\;)$ is at least $k$.
[DKS]{}
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[^1]: The second-named author thanks in particular the Université d’Aix-Marseille for friendly hospitality during the completion of this work.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Networks exhibiting “accelerating" growth have total link numbers growing faster than linearly with network size and can exhibit transitions from stationary to nonstationary statistics and from random to scale-free to regular statistics at particular critical network sizes. However, if for any reason the network cannot tolerate such gross structural changes then accelerating networks are constrained to have sizes below some critical value. This is of interest as the regulatory gene networks of single celled prokaryotes are characterized by an accelerating quadratic growth and are size constrained to be less than about 10,000 genes encoded in DNA sequence of less than about 10 megabases. This paper presents a probabilistic accelerating network model for prokaryotic gene regulation which closely matches observed statistics by employing two classes of network nodes (regulatory and non-regulatory) and directed links whose inbound heads are exponentially distributed over all nodes and whose outbound tails are preferentially attached to regulatory nodes and described by a scale free distribution. This model explains the observed quadratic growth in regulator number with gene number and predicts an upper prokaryote size limit closely approximating the observed value.'
author:
- 'M. J. Gagen and J. S. Mattick'
title: 'Inherent size constraints on prokaryote gene networks due to “accelerating" growth'
---
Introduction
============
The rapidly expanding field of network analysis, reviewed in [@Dorogovtsev_02_10; @Albert_02_47], has provided examples of networks exhibiting “accelerating" network growth, where link number grows faster than linearly with network size. For instance, the Internet [@Faloutsos_99_25] appears to grow by adding links more quickly than sites though the relative change over time is small and the Internet appears to remain scale free and well characterized by stationary statistics [@Vasquez_02_066130]. Similarly, the number of links per substrate in the metabolic networks of organisms appears to increase linearly with substrate number [@Jeong_00_65], the average number of links per scientist in collaboration networks increases linearly over time [@Dorogovtsev_00_33; @Vasquez_00_0006132; @Barabasi_01_0104162; @Barabasi_02_590; @Vasquez_03_056104], and languages appear to evolve via accelerated growth [@Dorogovtsev_01_2603].
In the main, the chief focus of these studies has been on locating parameter regimes allowing accelerating networks to maintain scale free statistics and thereby to allow continued unconstrained growth. For example, an early study considered a growing network receiving $N^\alpha$ new links for $\alpha>0$ when the network size is at $N$ nodes, but restricted analysis to the case $\alpha\leq 1$ as “Obviously, $\alpha$ cannot exceed 1 (the total number of links has to be smaller than $N^2/2$ since one may forbid multiple links)." and “The density of connections in real networks remains rather low all the time, so one may reasonably assume that $\alpha$ is small." [@Dorogovtsev_01_025101]. Equivalent limits were considered in Ref. [@Sen_0310513]. In such restricted parameter regimes networks could maintain scale free statistics, though this result carries the implicit but unexamined finding that alternate parameter regimes permit transitions from stationary to nonstationary statistics. This paper builds on these implicit findings.
Accelerating networks are more prevalent and important in society and in biology than is commonly realized—see the survey in Ref. [@Gagen_03_accel_survey]. In fact, any network that requires functional integration and organization (where the activity of any given node is dependent on the state of the network or different subnetworks) is by definition an accelerating network, that is, as the network expands, the proportion of the network devoted to control and regulation expands disproportionately. This in turn means that all such networks, sooner or later, must be limited in their size and complexity, which limitations can only be breached by changing either the physical nature of the control architecture (a state transition) or by reducing the functional integration. In the latter case, where networks are hitting a complexity limit, further growth in network size will likely display structural transitions from randomly connected, to scale free statistics, to densely connected and perhaps finally to fully connected statistics. Should such networks be unable to successfully complete these transitions for any reason, then it is likely that network growth must cease entirely or that either a transition to a nonaccelerating structure is required to permit further growth or novel technologies must appear allowing the continuation of accelerated growth. Exemplar accelerating networks displaying such size limits or structural transitions include (a) all forms of economic markets where the latest price offered by any participant instantly affects all other participants, (b) industrial companies and sectors implementing a Just-In-Time business model where any worker can halt the entire production system, (c) error propagation networks linking an error source with all affected nodes as studied in software analysis and in models of the propagation of diseases, bushfires, cracks, and electricity grid failures, (d) in any dynamical system dependent on relative quantities so changes in one node instantly affects every other node such as relative transcription factor binding probabilities or relative evolutionary fitness, (e) in computer hardware and in cluster and grid supercomputer networks, and (f) in organizational networks [@Gagen_03_accel_survey]. In fact, it is well understood that social networks only take on small world statistics when the network is large enough—in small towns everyone one knows everyone else so social networks are accelerating, and social networks make a transition to small world statistics only as individual nodes saturate their connectivity limits [@Watts_99_493]. Similar observations can be made about the scale free Internet and World Wide Web—when sufficiently small, these networks were likely accelerating until connectivity capacities saturated forcing a transition to scale free structures to permit further growth [@Gagen_03_accel_survey].
This paper develops an accelerating network model of prokaryotic (single celled) gene regulatory networks to investigate size and complexity limits inherent in the adoption of an accelerating architecture. Because our focus is on structural transitions, we explicitly do not need to restrict the degree of acceleration to low values of $\alpha\approx 0$. Rather, we permit this parameter to take on any value including $\alpha>1$ and ensure that the network is not saturated by making link formation probabilistic. The resulting novel “probabilistic" accelerating networks grow by adding on average $pN^{\alpha}$ new links with $\alpha>0$ and otherwise arbitrary provided the probability of adding a link is suitably constrained $p\ll 1$ so that total link number remains less than of order $N^2$.
The gene regulatory model presented here is motivated by comparative genomics findings that the total number of regulatory proteins controlling gene expression (links) scales quadratically with the number of genes or operons (nodes) in prokaryotes [@vanNimwegen_03_479; @Croft_03_unpub]. This quadratic growth results as the number of links made by a regulator exploiting homology dependent (sequence specific) interactions scales proportionally to the number of randomly drifting promotor sequences or effectively, with gene number [@Croft_03_unpub]. Hence, gene regulatory networks are inherently accelerating—the probable number of links per regulator $pN^{\alpha}$ increases linearly with node number with $\alpha=1$, so consequently, the total number of links scales quadratically as $pN^{\alpha+1}$. In small and sparsely connected networks, most links come from different regulators suggesting that regulator number also scales quadratically with gene number, $pN^{\alpha+1}$. Such an accelerating network would be characterized initially by sparse connectivity at low gene numbers and subsequently by denser connectivity at high gene numbers as networks attempt a transition to a densely connected regime. If the evolving networks can successfully make this transition, the evolutionary record will display a transition in network statistics for some critical network size $N_c$. Conversely, if these networks, optimized by evolution in the sparse regime, are unable to make the transition to the densely connected regime, the evolutionary record would show a strict size limit $N\leq N_c$ at some critical network size. But this is exactly what is observed. All prokaryotic gene numbers and genomes are indeed of restricted size (less than about 10,000 genes with genomes of between 0.5 and 10 megabases [@Casjens_98_33]), in contrast to the genomes of multicellular eukaryotes (with for humans, about 30,000 genes and a genome of about 3 gigabases [@Int_Human_genome_01_86; @Venter_01_13]). Ref. [@Croft_03_unpub] predicted the size limit $N_c\leq
20,000$ genes as continued genome growth requires the number of new regulators to exceed the number of nonregulatory nodes.
A satisfactory model of prokaryotic gene regulatory networks requires some novel features. As mentioned above, we introduce probabilistic link formation to allow rapid accelerated growth and correspondingly stricter size limits. (A different but related mechanism was introduced in Refs. [@Liu_02_036112; @Goh_02_108701] which considered the effects of stochastic fluctuations in the number of added links with each additional node.) In addition, we employ directed links and partition nodes into two classes where “regulators" can source outbound regulatory links to regulate other nodes (both regulators and non-regulators), while “non-regulators" cannot source outbound links. (Ref. [@Cheng_02_066115] has previously considered networks of distinguishable nodes.) Further, experimental evidence presented below indicates that the distribution of inbound links is compact and exponential while the distribution of outbound links is long-tailed and likely scale-free. As a result, the heads and tails of our directed links are placed according to two distinct distributions. Altogether, these features allow the reproduction of the observed features of prokaryote gene regulatory networks and satisfactorily predicts the maximum prokaryotic gene count.
Our approach reproducing accelerating network statistics for growing prokaryote genomes complements and informs alternate networking approaches seeking to deduce or simulate the regulatory networks of particular organisms from gene perturbation and microarray experiments [@Wagner_01_1183; @Bhan_02_1486; @Lukashin_03_1909].
In Section \[sect\_overview\_prok\_networks\] we canvass the available literature to characterize the statistics of prokaryote gene regulatory networks. This then allows the construction of accelerating growth network models in Section \[sect\_accelerating\_models\] where we use the continuous approximation and simulations to analyze network statistics. The size constraints inherent in accelerating prokaryote regulatory networks are modelled in Section \[sect\_size\_constraints\].
Overview of prokaryote gene networks {#sect_overview_prok_networks}
====================================
Ongoing genome projects are now providing sufficient data to usefully constrain analysis of the gene regulatory networks of the simpler organisms. Ref. [@vanNimwegen_03_479] first noted the essentially quadratic growth in the class of transcriptional regulators ($R$) with the number of genes ($N_g$) in bacteria with the observed results $$R\propto \left\{
\begin{array}{ll}
N_g^{1.87\pm0.13}, & \mbox{transcriptional regulation} \\
& \\
N_g^{2.07\pm0.21}, & \mbox{two component systems} \\
& \\
N_g^{2.03\pm0.13}, & \mbox{transcriptional regulation} \\
& \\
N_g^{2.16\pm0.26}, & \mbox{transcriptional regulation}. \\
\end{array}
\right.$$ Here, the top two lines refer to different classes of regulators while the bottom two lines are the results of a crosschecking analysis of two alternate databases. Quoted intervals reflect 99% confidence limits [@vanNimwegen_03_479]. The explanation for this quadratic growth was that each additional transcription factor doubles the number of available dynamical states which, it was posited, allows for a doubling in the fixation probabilities for this class of genes.
As noted above, Ref. [@Croft_03_unpub] provides an alternate theoretical analysis predicting quadratic growth in any regulatory network exploiting homology dependent interactions and analyzed 89 bacterial and archeael genomes to determine the relations $$\label{eq_Croft_results}
R= \left\{
\begin{array}{ll}
aN_g^b=(1.6\pm0.8)10^{-5}N_g^{1.96\pm0.15} & (r^2=0.88) \\
& \\
pN_g^2=(1.10\pm0.06)10^{-5}N_g^{2} & (r^2=0.87) \\
& \\
cN_g=(0.055\pm0.004)N_g & (r^2=0.75). \\
\end{array}
\right.$$ In this paper, accelerating networks will be based on the quadratic second line (while nonaccelerating models presented in later work will work with the linear third line [@Gagen_0312022]). In all cases, the limits reflect 95% confidence levels. For completeness, the data is shown in Fig. \[f\_prok\_reg\_data\]. The observed quadratic growth implies an ever growing regulatory overhead so there will eventually come a point where continued genome growth requires the number of new regulators to exceed the number of nonregulatory nodes, and based on this, Ref. [@Croft_03_unpub] predicted an upper size limit of about 20,000 genes, within a factor of two of the observed ceiling.
![*Double-logarithmic plot of regulatory protein number ($R$) against total gene number ($N_g$) for bacteria (circles) and archaea (triangles), adapted from Ref. [@Croft_03_unpub]. The log-log distribution is well described by a straight line with slope $1.96\pm 0.15$ ($r^2=
0.88$, 95% confidence interval indicated), corresponding to a quadratic relationship between regulator number and genome size. The inset shows the same data before log-transformation [@Croft_03_unpub]. Dashed lines show the best linear fit to the data $R=(0.055\pm0.004)N_g$ ($r^2=0.75$).*[]{data-label="f_prok_reg_data"}](gagen_fig1a.eps "fig:"){width="0.9\columnwidth"}![*Double-logarithmic plot of regulatory protein number ($R$) against total gene number ($N_g$) for bacteria (circles) and archaea (triangles), adapted from Ref. [@Croft_03_unpub]. The log-log distribution is well described by a straight line with slope $1.96\pm 0.15$ ($r^2=
0.88$, 95% confidence interval indicated), corresponding to a quadratic relationship between regulator number and genome size. The inset shows the same data before log-transformation [@Croft_03_unpub]. Dashed lines show the best linear fit to the data $R=(0.055\pm0.004)N_g$ ($r^2=0.75$).*[]{data-label="f_prok_reg_data"}](gagen_fig1b.eps "fig:"){width="0.5\columnwidth"}
Earlier surveys of bacterial genomes noted that larger genomes harboured more transcription factors per gene than smaller ones [@Cases_03_248], with this trend attributed to the need in larger genomes for a more complex network of regulatory proteins to achieve coordinated expression of a larger set of cellular functions, and to selection in complex environments leading to enrichment in transcription factors allowing regulation of gene expression and signal integration. A similar upward trend in the proportion of regulators as a fraction of genome size with increasing genome size was observed in Ref. [@Stover_00_959] attributed to a need for an increasing responsiveness in diverse environments, with confirming observations in Ref. [@Bentley_02_141].
Prokaryotes typically group their DNA encoded genes in operons, co-regulated functional modules of average size 1.70 genes each in [*E. coli*]{} which value we treat as typical though in reality, operon size decreases slightly with genome size [@Cherry_03_40]. Each operon can be either unregulated and so constitutively or stochastically expressed or subject to combinatoric regulation by multiple regulatory protein transcription factors binding to each operon’s promotor sequence.
Again assuming that [*E. coli*]{} is typical, any given regulatory protein affects an average of about 5 operons with this distribution being long tailed [@Shen_Orr_02_64] so the majority of regulators affect only one operon though some regulators (CRP) can affect up to 71 operons or 133 genes [@Thieffry_98_43]. (This latter reference estimated that each regulator controls on average 3 genes.) More recent estimates have the transcription factor CRP, a global sensor of food levels in the environment, regulating up to 197 genes directly and a further 113 genes indirectly via 18 other transcription factors [@MandanBabu_03_1234]. (To observe the long tailed distribution, see Fig. 2 of Ref. [@Thieffry_98_43] and Fig. 4 of Ref. [@MandanBabu_03_1234].)
However, the number of inputs taken by an operon is characterized by a compact exponential distribution with a rapidly decaying tail so the majority of regulated operons are controlled by a single regulator while very few regulated operons are controlled by four, five, six or seven regulators [@Thieffry_98_43; @Shen_Orr_02_64; @MandanBabu_03_1234]. In particular, Ref. [@Thieffry_98_43] examined 500 regulatory links from about 100 regulators to almost 300 operons to estimate that each regulated operon takes on average 2 inputs though Fig. 2 of this reference suggests an average input number of about 1.5. Similarly, Ref. [@Shen_Orr_02_64] suggests that 424 regulated operons receive 577 links giving an average input number of 1.4, while Ref. [@MandanBabu_03_1234] estimates that 327 regulated operons receive 524 links giving an average input number of 1.6.
Accelerating prokaryote network models {#sect_accelerating_models}
======================================
We extend the gene network model of Ref. [@Thieffry_98_43] to construct an accelerating network model of prokaryote regulatory gene networks. Prokaryotes typically pack their $N_g$ genes into a lesser number of $N=N_g/g_o$ co-regulated operons where we assume that operons contain exactly $g_o=1.70$ genes. Of the existing operons, $O_r$ are regulated operons and $O_u=N-O_r$ are unregulated operons. Of the total number of operons, there are $R$ regulatory operons whose regulatory interactions are directed links from regulatory operons to regulated operons. Under the assumption that there is only one regulatory gene per regulatory operon, the observed quadratic relation of Eq. \[eq\_Croft\_results\] becomes $$\label{eq_reg_No}
R=pg_o^2N^2.$$ When regulators and regulatory links are very rare, i.e. when genomes are small, it is likely that every new link is associated with a new regulator so the number of links varies roughly quadratically with operon number. We write $$\label{eq_link_No}
L=lN^2,$$ where $l$ denotes the probability of forming a particular beneficial link per operon. The value for $l$ will be approximately $pg_o^2$, but the exact relation must be derived from the details of the implemented model.
Each regulatory link between nodes is directed, and characterized by two distinct distributions describing respectively the placement of the heads and tails of each link. Only a relatively few nodes are regulatory, and of these, the number of outbound link tails per regulatory node are described by a size dependent long-tailed distribution with average about $\langle
t\rangle\approx 5$. Such a long-tailed distribution requires that link tails be preferentially attached to an existing regulatory operon or equivalently, the associated regulated operon must possess one promotor binding site (among others) that binds that particular regulator. Consequently, the preferential selection of regulators means that the promotor sequences of newly regulated nodes cannot be randomly chosen—randomly drifting promotor sequences would be as likely to match any one regulator as another. A plausible physical explanation for the preferential attachment of link tails to existing regulators is that newly fixated operons come largely from gene duplication events [@Yanai_00_2641] where some of the duplicated promotor binding sites are under strong selective constraint while other binding sites and the operon genes can drift freely. Gene duplication then implies that in a genome of size $N$ operons, if some regulator $n_j$ has $t_{jN}$ outbound regulatory links to approximately $t_{jN}$ regulated operons, then the probability that a newly fixated operon is also regulated by $n_j$ is simply the proportion of such regulated operons in the genome, or $t_{jN}/N$. This implements the required preferential attachment as the resulting rate of growth in the number of links attached to node $n_j$ is also then proportional to $t_{jN}$. If there is also some probability of the appearance of novel promotor sequences, these combined processes suffice to produce the observed scale free distributions. This model is roughly consistent with recent estimates of the relative contributions to prokaryote genome growth which suggest that horizontal gene transfer rates $\gamma_h$ are roughly one third of gene loss rates $\gamma_h=\gamma_l/3$ and roughly one half of vertical inheritance or gene genesis rates $\gamma_h=\gamma_v/2$ leading to roughly constant sized genomes over long times (as $\dot{N}\approx
\gamma_h+\gamma_v-\gamma_l\approx0$), while “it is remarkable that phylogenetic distributions of at least 60% \[and up to 75%\] of protein families can be explained merely by vertical inheritance." [@Kunin_03_1589]. Similarly, three quarters of examined transcription factors in Ref. [@MandanBabu_03_1234] were two-domain proteins with shared domain architectures leading to the estimate that about 75% of transcription factors have arisen as a consequence of gene duplication (though the joint duplication of regulatory regions and of regulated genes or of transcription factors together with regulated genes is more rare). A further implication of these gene duplication processes is that, in the main, regulators can only appear on entry to the genome—a potential regulator lacking any target matches in a given genome will never form any links when most operons arise from promotor preserving duplication events. This allows us to considerably simplify our model, and hereafter, we only allow regulators to appear on their entry to the genome. Of course, more realistic but considerably more complicated models are possible.
In contrast to the relatively small number of regulatory nodes, all nodes can themselves be regulated by inbound links and in fact, can be multiply regulated as promotor regions can contain more than one binding site. Further, the many used and unused promotor region binding sites broadly sample the space of possible binding sites so only a small fraction of nodes will be regulated by any one regulator. As a result, the number of inbound link heads per node is described by a size dependent exponential distribution with a low average of $\langle h\rangle\approx 1.5$ as typically results from the random or non-preferential attachment of inbound links to operon promotor sequences.
![*An example statistically generated E. coli genome using the later results of this paper where (for convenience only) operon nodes numbered $n_1, \dots, n_N$ are placed sequentially counterclockwise on a circle in their historical order of entry into the genome. The filled points on the outer circle locate regulators and have radius indicating the number of outbound regulatory links. The open points on the middle circle locate regulated operons and have radius indicating the number of inbound regulatory inputs. The arrows in the inner circle show all directed regulatory links.* []{data-label="f_e_coli_network"}](gagen_fig2.eps){width="\columnwidth"}
We suppose that the operon network grows by the sequential addition of numbered nodes $n_k$ for $1\leq k\leq N$, and that at network size $k$, node $n_i$ ($1\leq i\leq k$) has $t_{ik}$ outbound tails and $h_{ik}$ inbound heads. We do not model the many trials of potential genes over many generations and merely include fixated genes in our count—that is, drifting sequence is not counted as part of the fixated genome. This further implies the sequence of established nodes is under severe selective constraint and unable to drift so consequently new links cannot be added between existing nodes. (If a proportion $fN$ of existing nodes can explore novel sequence space in time $dt$, then the number of new regulators increases as $dR\propto fN^2dt$, and as $N$ is itself a function of time, this integrates to generate a non-quadratic relation between regulator and operon number which is not observed.)
For clarity, Fig. \[f\_e\_coli\_network\] preempts later calculations and depicts a statistically generated version of an [*E. coli*]{} genome where nodes are placed sequentially counterclockwise in a circle (for convenience only). Alternative genome models may be distinguished by the age distribution of regulators, regulated operons and their link numbers, and these are indicated in this figure. In particular, Fig. \[f\_e\_coli\_network\] shows a highly nonuniform distribution of regulators and outbound link numbers with gene age in contrast to a uniform distribution of regulated operons and of inbound link numbers. (It will turn out that these latter age-independent distribution are only present when regulator number grows quadratically with genome size.)
These distributions result from the physical processes underlying the formation of regulatory links in prokaryotes. As discussed above, a substantial proportion of the gene regulation network of prokaryotes is enacted via homology dependent interactions as when sequence specified protein transcription factors bind to specific promoter sequences. The undirected nature of evolutionary searches means that gene regulatory networks fundamentally exploit the same sequence matching algorithms used in comparative genetics where the probability of obtaining matches between a single given trial sequence of some small fixed length and an entire genome scales proportionately to genome length—doubling genome length doubles the probability of a match. Hence, the expected number of links formed per regulator scales linearly with present genome size. As the number of source trial sequences also scales with genome length, the expected number of matches between all regulators and all regulated operons scales quadratically with genome length, or effectively, with operon number assuming constant sized operons over the evolutionary record.
As a consequence, on entry into the genome, each new gene has some probability of being a regulator dependent firstly on its suitability to bind DNA and secondly on the linearly increasing expected number of acceptable binding targets present in the genome on entry (or at later times). As discussed above, the predominance of vertical gene genesis events allows a simplified model wherein the probability of a new node being regulatory is determined solely by the number of available links present at the time of entry. We assume then that on entry into the genome each new node $n_k$ can form $2k-1$ links with nodes $n_1, \dots, n_k$ consisting of a single self-regulatory link from node $n_k$ to itself with probability $l$, $(k-1)$ regulatory outbound links to the existing nodes each with equal probability $l$, and, provided that sufficient regulators already exist, $l(k-1)$ inbound regulatory links from some subset of the existing regulators chosen according to preferential attachment. (For consistency, we can only add $\approx lk$ distinct regulatory links to node $n_k$ provided there are at least this many regulators in existence. From Eq. \[eq\_reg\_No\], the average number of regulators $pg_o^2k^2$ must be greater than the number of regulatory links $lk$, and this will be satisfied for $k>l/(pg_o^2)\approx 1$.) As a result, the total number of heads or tails attached to node $n_k$ on entry to the genome ranges between $0$ and $k$, with each link formed with probability $l$. Hence, the respective probabilities that the initial number of heads $h_{kk}=j$ or the initial number of tails $t_{kk}=j$ for node $n_k$ is $$\label{eq_P_j_k_dist}
P(j,k) = {k \choose j} l^j (1-l)^{k-j},$$ with the proviso that all the inbound links can only be added to node $n_k$ if there is a sufficient number of regulators among the nodes $n_1, \dots, n_k$. The average number of inbound and outbound links is identical, $\langle t_{kk}\rangle=\langle
h_{kk}\rangle=lk$ showing linear growth in link number with increasing network size. The addition of node $n_k$ and its links will increase the probable number of heads attached to earlier nodes $n_j$ for $1\leq j\leq (k-1)$ so $h_{jk}\geq h_{jj}$, while the probable number of tails outbound from node $n_j$ increases $t_{jk}\geq t_{jj}$ if and only if that node is regulatory with $t_{jj}>0$.
As regulators can only be created on entry to the genome, the distribution of regulators at any time is specified by the distribution $P(j,k)$ for $t_{kk}$. Using Eq. \[eq\_P\_j\_k\_dist\], the probability that node $n_k$ is a regulator is $1-P(0,k)$, so for a network of $N$ nodes, the predicted total number of regulators is $$\begin{aligned}
\label{eq_reg_density}
R &=& \sum_{k=1}^{N} \left[ 1- (1-l)^k \right] \nonumber \\
&=& N- \frac{1-l-(1-l)^{N+1}}{l} \nonumber \\
&\approx & \frac{l}{2}N(N+1).\end{aligned}$$ The exact top line shows the expected behaviour for the number of regulators in the respective limits $l\rightarrow 0$ giving $R\rightarrow 0$, and $l\rightarrow 1$ giving $R\rightarrow N$. The approximate relation in the third line can be compared to the observed Eq. \[eq\_reg\_No\] and immediately suggests $l\approx
2pg_o^2$, while a fit to the more accurate top line gives the connection probability as $$\label{eq_l_assignment}
l = 1.15 \times 2 p g_o^2 \;=\; 7.31 \times 10^{-5}.$$ This probability value suggests an average promotor binding site length of $-\log_4 l=6.9$ bases. The average number of links per regulator using the second line of Eq. \[eq\_reg\_density\] is then approximately $L/R\approx 2$, while the more accurate top line with $N=2528$ operons for [*E. coli*]{} [@Cherry_03_40] gives $L/R=2.12$, about a factor of two from the observed value of $5$ for [*E. coli*]{} [@Shen_Orr_02_64].
Random distribution of regulated operons
----------------------------------------
The distribution of link heads for all nodes (with possession of a link head designating a regulated node), can be straightforwardly calculated under the assumption that the $t_{kk}\approx lk$ new tails added with node $n_k$ are randomly distributed across the $k$ existing nodes so on average, each existing node receives $l$ links. To build insight, it is useful to consider the general case where $t_{kk}\approx h_{kk}\approx lk^\alpha$ for $\alpha\geq
0$. Setting $\alpha=0$ adds with some probability a constant number of links with each new node, $\alpha=1$ adds a linearly growing number of probable links with each new node, $\alpha=2$ adds a quadratically growing number of probable links with each new node, and so on. The total number of links present in the network is then $$\label{eq_total_links}
\int_0^N 2lk^\alpha = \frac{2lN^{\alpha+1}}{\alpha+1}$$ The continuous approximation [@Barabasi_99_17; @Barabasi_99_50; @Dorogovtsev_01_056125] for links randomly distributed over $k$ existing nodes determines the number of inbound head links for node $n_j$ according to $$\begin{aligned}
\label{eq_continuum_heads}
\frac{\partial h_{jk}}{\partial k}&=& \frac{t_{kk}}{k} \nonumber \\
& = & l k^{\alpha-1}.\end{aligned}$$ This can be integrated with initial conditions $h_{jj}\approx
lj^\alpha$ at time $j$ and final conditions $t_{jN}\approx
lN^\alpha$ at time $N$ to give $$\label{eq_hjn}
h_{jN} = \left\{
\begin{array}{cc}
l + l \ln\frac{N}{j} & \mbox{if }\alpha=0 \\
& \\
\frac{l}{\alpha}N^\alpha + \frac{l(\alpha-1)}{\alpha}j^\alpha & \mbox{if }\alpha>0. \\
\end{array}
\right.$$ Integration of these link numbers over all node numbers $j$ gives the required total number of links as in Eq. \[eq\_total\_links\]. For $0\leq\alpha<1$, the number of links per node is monotonically decreasing with node number. However, for $\alpha=1$ and only in this case, the final distribution is independent of node number $j$ because earlier nodes receive exactly enough links from latter nodes to balance the initially biased distribution of heads $h_{jj}\approx lj$, so in the end, all nodes receive on average the same number of inbound regulatory links $\langle
h_{jN}\rangle=lN$ for $1\leq j\leq N$. For faster acceleration rates, $\alpha>1$, the number of links per node is monotonically increasing as later nodes receive a greater number of links on entry to the genome and this imbalance is not corrected.
The possibility of monotonically increasing numbers of links with node number in accelerating networks has not previously been considered. This possibility requires modifying the usual continuum approach [@Barabasi_99_17; @Barabasi_99_50; @Dorogovtsev_01_056125] so the final inbound link distribution is obtained via $$\begin{aligned}
\label{eq_final_link_dist}
H(k,N) &=& \frac{1}{N} \int_0^N dj \; \delta(k-h_{jN})\nonumber \\
& = & \pm \frac{1}{N} \left( \frac{\partial h_{jN}}{\partial j}
\right)^{-1} \mbox{at }[j=j(k,N)],\end{aligned}$$ where $j(k,N)$ is the solution of the equation $k=h_{jN}$. The top line is used when all nodes possess the same average link number while the second line is applicable with the plus (negative) sign when the average numbers of links per node is monotonically increasing (decreasing) with node number. Non-monotonic cases require alternate approaches.
Under quadratic growth in total link number when $\alpha=1$, and only in this case, the final distribution of link heads is independent of node number and evaluated using Eq. \[eq\_final\_link\_dist\] to give $$\begin{aligned}
H(k,N) &=& \frac{1}{N} \int_0^N dj \; \delta(k-lN) \nonumber \\
&=& \delta(k-lN).\end{aligned}$$ As expected, a compact final link distribution results when all nodes have an average of $t_{jN}=lN$ inbound regulatory links at time $N$. This distribution calculated under the continuous approximation equates to one where in reality, each node receives a controlling head with probability $l$ from every other node (though in practise, the total number of received links is of order unity). Hence, for any node in a network of size $N$, the actual probability of having $k$ heads is $$\label{eq_h_kNo_final}
H(k,N) = {N \choose k} l^k (1-l)^{N-k}.$$ A network simulation with linear growth of link numbers per node model (Eq. \[eq\_P\_j\_k\_dist\]) serves to validate this predicted final distribution. Fig. \[f\_simulation\_heads\] compares the predicted distribution $H(k,N)$ against observed distributions for typical simulated networks of various sizes with negligible discrepancies.
For a network of size $N$, the probability that any given operon is unregulated is $H(0,N)$ so the expected number of unregulated operons summed over all $N$ nodes is $$\label{eq_unreg_operons}
O_u= N(1-l)^N.$$ This determines the number of regulated operons as $$\label{eq_reg_operons}
O_r=N-O_u=N\left[1- (1-l)^N\right],$$ showing the expected behaviour as $l\rightarrow 1$ giving $O_r\rightarrow N$ and $l\rightarrow 0$ giving $O_r\approx lN^2=L$ as each of the sparsely distributed links hits a distinct regulated operon. We note that random gene duplication and deletion events will not change the $H(k,N)$ distribution (other than changing $N$) as all nodes are identically connected on average. The $H(k,N)$ distribution appears in Fig. \[f\_e\_coli\_network\] which shows a uniform (age-independent) distribution of regulated nodes over the genome, and this uniformity is only expected for $\alpha=1$ corresponding to linear growth in link numbers per node and quadratic growth in regulator numbers.
![*A comparison of the predicted distribution of inbound link numbers per node $H(k,N)$ (solid lines) against that observed in simulated networks of various sizes (indicated points) with quadratic growth in the total probable number of randomly attached links. \[f\_simulation\_heads\]*](gagen_fig3.eps){width="\columnwidth"}
![*The predicted proportions $P_h(k)$ of the regulated operons of E. coli taking multiple regulatory inputs for a genome of $N=2528$ operons. This distribution closely approximates that observed for E. coli in Fig. 2(d) of Ref. [@Cherry_03_40] and of Fig. 5 of Ref. [@MandanBabu_03_1234]. \[f\_heads\_per\_reg\_operon\]*](gagen_fig4.eps){width="\columnwidth"}
These predictions can be compared to observation. For the [*E. coli*]{} network of size $N=2528$ operons or 4289 genes [@Cherry_03_40], the predicted proportion of regulated operons receiving $k>0$ inputs is $$P_h(k) = \frac{H(k,N)}{1-H(0,N)},$$ and is shown in Fig. \[f\_heads\_per\_reg\_operon\]. Here, the calculated distribution closely approximates the compact exponential distribution observed for [*E. coli*]{} shown in Fig. 2(d) of Ref. [@Cherry_03_40] and of Fig. 5 of Ref. [@MandanBabu_03_1234], though it underestimates the numbers of regulated operons with 4, 5, 6 and 7 inputs—essentially no regulators are predicted to have 5 or more inputs for genomes of size $N=2528$ operons. In addition, the average number of inbound regulatory links per operon (for all operons) is $\langle
k\rangle=L/N=lN=0.19$, while the average number of inbound regulatory links for regulated operons is $\langle
k_r\rangle=L/O_r\approx 1$. A more accurate calculation using the specific values for [*E. coli*]{} gives $\langle
k_r\rangle=L/O_r=1.10$, very close to the [*E. coli*]{} value of 1.5 or 1.6 noted in Refs. [@Thieffry_98_43; @Shen_Orr_02_64; @MandanBabu_03_1234].
Scale-free distribution of regulator operons
--------------------------------------------
On entry into the genome, node $n_k$ sources on average $lk$ outbound regulatory links and this linear growth in link number means that more recent nodes are more likely to be immediately regulatory and more likely to be highly connected on genome entry. However, node $n_k$ will also receive on average $lk$ inbound regulatory links whose tails will be preferential attached to existing regulators. The final distribution of link number with age will depend on the rate at which earlier nodes under preferential attachment can attract links relative to the linearly increasing link numbers of later regulatory nodes.
On entry at time $k$, node $n_k$ receives $h_{kk}\approx lk$ inbound links from existing regulatory nodes in the set $n_1,\dots,n_k$. As previously, we gain insight by considering the general case where $t_{kk}\approx h_{kk}\approx lk^\alpha$ for $\alpha\geq 0$ (though we continue to use the distribution $P(j,k)$ of Eq. \[eq\_P\_j\_k\_dist\] to determine both the number of links $j$ prior to exponentiation and regulatory probability so consequently the number of regulators continues to increase quadratically according to Eq. \[eq\_reg\_density\]). As a result, the need to ensure that all regulatory links to node $n_k$ are distinct requires that new link number $lk^\alpha$ be less than the number of existing regulators $lk^2$ requires $\alpha\leq 2$. The $h_{kk}$ new tails added with node $n_k$ are preferentially attached to the existing regulatory nodes $n_j$ with probability proportional to the number of existing regulatory links for that node at time $k$, i.e. $t_{jk}$. Using the continuous approximation [@Barabasi_99_17; @Barabasi_99_50; @Dorogovtsev_01_056125], the rate of growth in outbound link number for node $n_j$ is then approximately $$\label{eq_continuum_tails}
\frac{\partial t_{jk}}{\partial k} =
h_{kk}\frac{t_{jk}}{\int_{0}^{k} t_{jk} \; dj}.$$ The denominator here is a probability weighting to ensure normalization and is the total number of outbound links for all nodes. Following [@Dorogovtsev_02_10], we can evaluate the denominator using the identity $$\frac{\partial}{\partial k} \int_0^k t_{jk} \;dj =
\int_0^k \frac{\partial}{\partial k} t_{jk} \; dj + t_{kk}.$$ This can be evaluated using Eq. \[eq\_continuum\_tails\] noting $t_{kk}\approx h_{kk}\approx lk^\alpha$ giving $$\frac{\partial}{\partial k} \int_0^k t_{jk} \;dj = 2lk^\alpha,$$ which can be integrated determining the denominator of Eq. \[eq\_continuum\_tails\] to be $$\int_0^k t_{jk} \;dj = \frac{2l}{\alpha+1}k^{\alpha+1}.$$ This is in agreement with Eq. \[eq\_total\_links\]. Substituting this value into Eq. \[eq\_continuum\_tails\] gives $$\frac{\partial t_{jk}}{\partial k} =
\frac{\alpha+1}{2} \frac{t_{jk}}{k}.$$ Finally, this can be integrated with initial conditions $t_{jj}\approx lj^\alpha$ at time $j$ and final conditions $t_{jN}$ at time $N$ to give $$\label{eq_tails_t_jn}
t_{jN} = l N^{\frac{\alpha+1}{2}} j^{\frac{\alpha-1}{2}}.$$ Again we find that the respective choices $\alpha<1$ and $\alpha>1$ lead to monotonically decreasing and increasing numbers of links per node as a function of node number, while setting $\alpha=1$ ensures the number of links per node is independent of node number. In this case, the preferential attachment of links to earlier nodes does indeed act to cancel the initial bias in link number towards later nodes. It is also apparent that when $\alpha=1$, the limit $l\rightarrow 1$ implies all nodes possess exactly $N$ links as expected for a fully connected regular network. (Preferential attachment cannot distort connectivity numbers in this case as all nodes have an equal number of links.) Additionally, in the limit $l\rightarrow 0$ we have $t_{jN}=0$ as required for an entirely disconnected network. The case $\alpha=0$ duplicates results found for growing networks which add a constant number of links with each new node subject to preferential attachment [@Albert_02_47].
As previously, it is straightforward to calculate the final outbound link distribution in the case $\alpha=1$ using Eq. \[eq\_final\_link\_dist\]. This gives $$\begin{aligned}
T(k,N) &=& \frac{1}{N} \int_0^N dj \; \delta(k-lN) \nonumber \\
&=& \delta(k-lN).\end{aligned}$$ Again, we find the expected compact distribution resulting when all nodes possess the same average number of links. This raises the question however, of how it is that a probabilistic accelerating network subject to preferential attachment can end up with all nodes possessing the same average number of links? The answer lies in our use of two classes of distinguishable nodes, regulators and non-regulators, which requires that we take into account the known distribution of regulators with node number over the genome. The average link number per node at node $n_j$ (Eq. \[eq\_tails\_t\_jn\]) equates to the product of the average number of link tails per regulator at node $n_j$, denoted $t_r(j,N)$, and the average number of regulators per node at node $n_j$, denoted $\rho(j)$. This latter density is $\rho(j)=dR(j)/dj\approx lj$ by Eq. \[eq\_reg\_density\], so by definition, we have $$t_{jN}= t_r(j,N) \rho(j),$$ giving $$\label{eq_t_ii_dist}
t_r(j,N)= N^{\frac{\alpha+1}{2}} j^{\frac{\alpha-3}{2}}.$$ Hence, for $\alpha<3$, the average number of links per regulator is a decreasing function of node number $j$ as the growing number of links added to recent nodes is insufficient to outweigh the effects of preferential attachment which more rapidly increases the number of links attached to early nodes. In particular, for $\alpha=1$ with the addition of a linearly increasing number of links per node, the average number of regulatory links per regulator scales inversely with node number $j$. In other words, the density of regulators is very low at small node numbers $j$ while the very few regulatory nodes in this stretch of the genome are heavily connected due to preferential attachment so as to maintain the constant average of Eq. \[eq\_tails\_t\_jn\]. (See Fig. \[f\_e\_coli\_network\].)
The $t_r(j,N)$ distribution contains information about both node connectivity and node age and so approximates genome statistics (simulated or observed) when all of this information is available. However, it is usually the case that node age information is unavailable necessitating calculation of connectivity distributions that are not conditioned on node age. This effectively requires binning together all nodes irrespective of their age to obtain a final link distribution. In the case of linearly growing number of links per node, $\alpha=1$, the delta function of Eq. \[eq\_final\_link\_dist\] is resolved by the equality $j=N/k$ giving the final distribution as $$\begin{aligned}
\label{eq_k_minus_two}
T(k,N) &=& -\frac{1}{N} \left( \frac{\partial j}{\partial k}\right)\nonumber \\
&=& \frac{1}{k^2},\end{aligned}$$ which, as required, is normalized as $\int_1^\infty T(k,N)=1$. The expected proportion of regulators $P_t(k)$ possessing $k$ links is then obtained by integrating the continuous distribution of Eq. \[eq\_k\_minus\_two\] over appropriate ranges $[1,3/2]$ or $[k-1/2,k+1/2]$ to obtain $$P_t(k) = \left\{
\begin{array}{cc}
\frac{1}{3} & k=1 \\
& \\
\frac{4}{4k^2-1} & k>1. \\
\end{array}
\right.$$ These theoretical predictions compare well to simulations of networks of various sizes with linearly increasing numbers of probable links per node and subject to preferential attachment. Fig. \[f\_simulation\_tails\] shows simulated outbound link distributions which are long-tailed and scale free with probabilities scaling roughly as $P_t(k)\propto k^{-2}$ for large $k$. The $P_t(k)$ distribution shows a full one third of regulators have only one link, while 60% have two or fewer links, and 71% have three or fewer links. Fig. \[f\_e\_coli\_long\_tails\] shows the long-tailed distribution $P_t(k)$ expected for a simulated [*E. coli*]{} network of $N=2528$ operons with preferential attachment of links. This figure shows marked similarities to the long-tailed distribution of [*E. coli*]{} shown in Fig. 2(c) of Ref. [@Cherry_03_40]. In particular, the expected number of regulators with $k$ links is $P_t(k) R(N)$ with the number of regulators $R(N)$ obtained from Eq. \[eq\_reg\_density\] (or from observation). For [*E. coli*]{}, this predicts the probable existence of about one regulator possessing link numbers in each of the respective ranges between $[40,49]$ links, between $[50,64]$ links, between $[65,94]$ links, between $[95,169]$ links, and between $[170,700]$ links for instance. (This approximates the connectivity of the global food sensor CRP which regulates up to 197 genes directly [@MandanBabu_03_1234].) The average of the $P(k)$ distribution (as well as the $t_r(j,N)$ distribution) is formally undefined as long as the integration limits are taken to infinity. However, in a network of $N$ nodes, a regulator can practically only regulate a total of $N$ nodes, and this cutoff allows us estimate the average connectivity per regulator (complementing previous estimates following Eq. \[eq\_l\_assignment\]). Using the cutoff and approximating the summation via an integral, the average connectivity per regulator in a network of $N$ nodes is $$\begin{aligned}
\label{eq_pk_average}
\langle k \rangle &=& \sum_{k=1}^{N} k P_t(K) \nonumber \\
&=& \frac{1}{3} +
\frac{1}{2} \ln \left( \frac{4N^2-1}{15} \right),\end{aligned}$$ (or simply $\ln N$ using the continuous distribution of Eq. \[eq\_k\_minus\_two\].) The average number of links per regulator for [*E. coli*]{} from Eq. \[eq\_pk\_average\] is $\langle k
\rangle=7.51$ (or 7.83 using the simpler derivation), which again compares well to the observed value of 5 in [*E. coli*]{} [@Shen_Orr_02_64].
![*A simulation of the proportion of outbound links per regulator $P_t(k)$ in networks of various sizes with linear growth in the probable number of links per node preferentially attached to regulatory nodes. The log-log plot shows slopes of roughly $-2$ in agreement with theoretical predictions (heavy solid line) of a long-tailed scale free distribution $P_t(k)\propto k^{-2}$. \[f\_simulation\_tails\]*](gagen_fig5.eps){width="\columnwidth"}
![*The predicted proportion of regulatory operons $P_t(k)$ regulating $k$ different operons for a simulated E. coli genome with $N=2528$ operons. As expected, most regulators regulate only one other operon, though a small number of regulators can regulate more than 40 operons. This distribution closely approximates the observed proportions for E. coli in Fig. 2(c) of Ref. [@Cherry_03_40] and Fig. 4 of Ref. [@MandanBabu_03_1234], and predicts the probable existence of about one E. coli regulator possessing link numbers in each of the respective ranges between $[40,49]$ links, between $[50,64]$ links, between $[65,94]$ links, between $[95,169]$ links, and between $[170,700]$ links, and so on. \[f\_e\_coli\_long\_tails\]*](gagen_fig6.eps){width="\columnwidth"}
Inherent prokaryote size limits {#sect_size_constraints}
===============================
The accelerating nature of regulatory gene networks necessarily means that these networks must exhibit a transition at some critical network size either to a nonaccelerating architecture permitting continued growth or must cease growth entirely, and we now seek to predict the location of this transition point and compare it to the evolutionary record. We begin by examining an overview of the accelerating genome model. Fig. \[f\_prokaryote\_model\] shows that linear growth in link numbers per node ($\alpha=1$) allows a quadratic growth in the total number of links (Eq. \[eq\_link\_No\]) despite each of the number of regulators (Eq. \[eq\_reg\_density\]) and the number of regulated nodes (Eq. \[eq\_reg\_operons\]) asymptoting to some fraction of $N$ after an initial period of quadratic growth. For large genomes, almost all new nodes will be regulators and densely connected into the existing network which will then multiply regulate almost every node.
![*The quadratic growth in the number of regulatory links $L$, and the asymptoting quadratic growth of regulatory operons $R$ and of regulated operons $O_r$ in relation to the total number of operons $N$. Actual numbers of regulators for 89 prokaryote genomes are shown (solid points), while the non-asymptoting fitted quadratic curve $R_q$ is shown for comparison. The observed maximum size of prokaryote genomes (of order 10,000 genes or about 6,000 operons) lies near the transition point between sparse and dense connectivity as an increasing proportion of operons become linked into the regulatory network. \[f\_prokaryote\_model\]*](gagen_fig7.eps){width="\columnwidth"}
The transition from sparse to dense connectivity occurs as an increasing proportion of operons become linked into the regulatory network leading to the emergence of a single giant component of fully connected nodes. One way to highlight this transition is by determining the proportion of transcription factor which control downstream regulators as such linkages create the single giant component. The proportion of regulators controlling regulators is $$\begin{aligned}
\label{eq_regulation_regulators}
P_{rr}(N) &=& \frac{1}{R(N)} \sum_{k=1}^N
\left[1-(1-l)^k \right] \frac{N}{k}\frac{R(N)}{N} \nonumber \\
&\approx & lN.\end{aligned}$$ Here, the first fraction on the RHS normalizes the proportion in terms of the number of regulators $R(N)$ (Eq. \[eq\_reg\_density\]), the first term in the summation is the probability that node $n_k$ is a regulator, the second term is the average number of regulatory outbound links for this regulatory node $t_r(k,N)$ at network size $N$ (Eq. \[eq\_t\_ii\_dist\] with $\alpha=1$), and the third term approximates the probability that these nodes link to one of the existing regulators under random attachment. (If the very first and very last terms are dropped, the remaining summation over all nodes of the probability that $n_k$ is regulatory with the stated number of links equates to the total number of links in the network $L\approx lN^2$. This is the more accurate version of the calculation leading to Eq. \[eq\_t\_ii\_dist\].) Hence, the proportion of regulators which control transcription factors scales linearly with network size and equals 15% for an $N=2000$ network, 29% for $N=4000$, 44% for $N=6000$, 59% for $N=8000$, 73% for $N=10000$, and 88% for $N=12000$ operons (after which the approximations made break down). Naturally, when most regulators themselves control other regulators, then the entire regulatory network will consist of a single giant component. These ratios compare reasonably well with those observed in [*E. coli*]{} where Ref. [@MandanBabu_03_1234] noted that of 121 transcription factors for which one or more regulatory genes are known, 38 factors or 31.4% regulate other transcription factors. The approximate second line of Eq. \[eq\_regulation\_regulators\] with $N=2528$ for [*E. coli*]{} determines this proportion as $P_{rr}=18.5\%$ while the more accurate top line gives the proportion of regulators which control transcription factors as $P_{rr}=17.7\%$, giving a reasonable match between prediction and observation.
As the proportion of regulators of transcription factors rises, the probable length of regulatory cascades will increase. In fact, the proportion of regulators taking part in a regulatory cascade of length $n\geq 1$ is $$p_n = (1-P_{rr}) P_{rr}^{n-1}.$$ This equation can be obtained from a tree of all binary pathways which at each branching point either terminate with probability $(1-P_{rr})$ or cascade with probability $P_{rr}$. As such, the probable cascade length is negligible when the proportion of regulators controlling regulators is small $P_{rr}\ll 1$ but can become large as $P_{rr}$ itself increases. As $P_{rr}$ is indeed large for networks of size $N>6000$, this again suggests that long cascades of regulatory interactions will lead to the coalescing of a single giant component in this regime. Again, the calculated lengths of regulatory cascades can be compared to those in [*E. coli*]{} where the number of cascades of regulated transcription factors observed in a particular set of regulatory interactions was 23 two-level cascades or 37.7%, 32 three-level cascades or 52.5%, and 6 four-level cascades or 9.8% [@MandanBabu_03_1234]. As one-level or autoregulatory interactions are not included in this observation, the predicted proportions for [*E. coli*]{} are $\bar{p}_n=p_n/(1-p_1)$ with $P_{rr}=17.7\%$ giving 82% two-level cascades, 15% three-level cascades, 3% four-level cascades, 1% five-level cascades, and so on. It is seen that the theoretical predictions overestimate the proportion of two-level cascades and underestimate the number of three-level and higher cascades probably because of selection pressures not included in the model. Lastly, we note that the number of cycles involving closed regulatory loops of size greater than one (i.e. involving more than autoregulation) in the examined portion of the [*E. coli*]{} regulatory network is zero reflecting that feedback loops in these organisms are carried out at the post-transcriptional level involving metabolites such as appear in the [*lac*]{} operon [@Thieffry_98_43; @Shen_Orr_02_64; @MandanBabu_03_1234].
We note that our model is entirely unable to explain the high proportion of autoregulation observed in [*E. coli*]{} with various estimates that 28.1% [@Rosenfeld_02_785], 50% [@Shen_Orr_02_64] and 46.9% [@Thieffry_98_43] of regulators are autoregulatory. The predicted proportion of autoregulators is approximated by replacing the very last fraction ($R/N$) in Eq. \[eq\_regulation\_regulators\] by the term $1/N$ giving the probability that a self-directed link is formed, leading to the expected autoregulatory proportion $\approx
2/N\approx 0.08\%$ for [*E. coli*]{}. This failure likely reflects the action of selection processes promoting spatial rearrangements of entire regulons on the genome and the internal shuffling of genes and promotor units. Such reorganizations of duplicated gene regions (presumably shuffling genes and promotor regions) have been common in [*E. coli*]{} allowing for instance, spatial regulatory motifs whereby the promotors of colocated (overlapping) and often co-functional operons transcribed in opposing directions can interfere [@Warren_03_0310029].
The transition point from sparse to dense connectivity can be roughly located using the continuum approximation [@Barabasi_99_17; @Barabasi_99_50; @Dorogovtsev_01_056125]. These methods have not previously been used for this purpose (to our knowledge) and we first validate their use by deriving the known result that non-growing random graphs of $N$ nodes connected by an increasing number of $L$ undirected links undergo a phase transition from sparse to dense connectivity when $L=N/2$ [@Erdos_60_17]. As the number of links $L$ grows, the $N$ nodes are interlinked into firstly separate islands of size $s_i$ nodes for $i=1,2,\dots$ which eventually link up to form a giant component designated $s_1$ containing essentially all nodes $s_1\approx N$. The largest component grows whenever a newly added link has either its head or tail in island $s_1$ (with probability $s_1/N$) and the other outside it (with probability $(N-s_1)/N$) leading to a size increment equal to the average size of the external islands ($\langle s_{j\neq 1} \rangle$), giving $$\frac{ds_1}{dL} = \left[2 \frac{s_1}{N} \frac{(N-s_1)}{N}\right]
\langle s_{j\neq 1} \rangle.$$ Numerical or analytic integration of this equation with initial conditions $s_1=2$ when $L=1$ and assuming the average size of external smaller islands is $s_1/2$ shows the largest island saturating the entire network when $L=N/2$ as expected. (This simple approach is indicative only and is quite sensitive to for instance, the assumed average size of external islands.)
This result suggests the following transition point in directed regulatory gene networks. Each undirected (i.e. bidirectional) link in random graph theory is equivalent to two directed links allowing bidirectional traffic between any two nodes, suggesting a transition point in directed graphs at roughly $L=N$. This analysis suggests that the largest component is expected to saturate the entire network when link number $L\approx N$ or $N=1/l=13,677$ (see Fig. \[f\_prokaryote\_model\]). In turn, this suggests that for $N<13,677$ a typical network likely consists of isolated trees, while if $N>13,677$ the network likely consists of a single giant cluster where almost every node is connected to all others via intermediate links. When the link number is very large, $N\gg 13,677$, then the network becomes regularly connected [@Albert_02_47]. As prokaryote regulatory networks likely consist of functionally distinct regulated modules [@Thieffry_98_43; @Hartwell_99_c4], it is unlikely that prokaryotic gene networks can successfully operate in the fully connected regime suggesting that prokaryote genome sizes are size constrained $N\leq 13,677$. In fact, the previously noted absence of regulatory cycles in [*E. coli*]{} [@Thieffry_98_43; @Shen_Orr_02_64; @MandanBabu_03_1234] likely reflects the evolutionary importance of maintaining disjoint and non-interfering regulatory units.
These results of random graph theory are suggestive only, and we now turn to consider the size of the largest connected island in prokaryote gene networks featuring directed links whose tails are preferentially attached to regulators and whose heads are randomly distributed over all existing nodes. A further difference is that prokaryote regulatory networks are themselves growing with each added node accompanied by a probabilistic number of links. In addition, we define an island to consist of all nodes which are linked regardless of the orientation of all links and so effectively treat links as being undirected. This is because a regulator can potentially perturb every node downstream to it including those nodes downstream of other regulators and so can modify the regulatory effects of other regulators—essentially, if the downstream effects of different regulators eventually intersect, we count these regulators in the same island. (Other definitions of islands could be used.)
![*The total number of discrete disconnected islands $i_{\rm all}$, the number of islands with respectively two ($i_2$), three ($i_3$) and four ($i_4$) members (left hand axis), and the simulated ($\langle s_1\rangle$) and predicted ($s_1$) size of the largest island measured as a proportion of nodes for various genome sizes (right hand axis). The simulations show the largest island contains $\langle s_1\rangle=50\%$ of all nodes at a critical network size of $N_c=9,029$ nodes. The input parameters of the predicted curve $s_1$ are set so $s_1=\langle s_1\rangle$ at this point.*[]{data-label="f_largest_island"}](gagen_fig8.eps){width="\columnwidth"}
The dominant (but not sole) mechanism by which island $s_1$ can grow is for the newly added node $n_k$ to either (a) be a regulator (with probability $[1-(1-l)^k]$) and establish an outbound regulatory link to some existing node in $s_1$ (with probability $s_1/k$) while at the same time accepting a regulatory link (with probability $[1-(1-l)^k]$) from a node in a different island $s_{j\neq 1}$ (with probability $(k-s_1)/k$), or (b) accept an inbound regulatory link (with probability $[1-(1-l)^k]$) from a regulator in island $s_1$ (with probability $s_1/k$) while establishing a regulatory link (with probability $[1-(1-l)^k]$) to some node in a different island $s_{j\neq 1}$ (with probability $(k-s_1)/k$). (Here, we assume that regulators are uniformly distributed over islands and the number of links within an island scales with the size of the island to crudely model preferential attachment.) The result is that island $s_1$ grows by the size of the second island assumed to be $s_{j\neq 1}$. Altogether, the rate of growth in the size of island $s_1$ is then $$\frac{ds_1}{dk} =
2 \left[ 1 - (1-l)^k \right]^2
\frac{s_1 [k-s_1]}{k^2} \langle s_{j\neq 1} \rangle.$$ For initial conditions, we assume that a first link appears when the genome has $(pg_0^2)^{-1/2}=177$ nodes ($s_1(177)=2$). Simulations show that sufficient small islands are created to ensure $\langle s_{j\neq 1} \rangle$ remains roughly constant and equal to $\langle s_{j\neq 1}\rangle=2.72$, though matching the simulated and predicted curves at the 50% point requires setting $\langle s_{j\neq 1}\rangle=30$. This is reasonable given the approximations made. Fig. \[f\_largest\_island\] shows the size of the largest island $s_1$ as a proportion of all nodes. A single giant component is expected to form at a critical genome size of $N_c=9,029$ operons defined as the point where the simulated proportion of nodes in the giant component is 50%. (Choosing a parameter setting of 40% would also be justifiable and would lead to an exact match between predicted and observed maxima.) Unlike random graph theory, this critical point applies to all growing genomes as it is determined by the value of the link formation probability $l$. Genomes of smaller size than this critical value $N<N_c$ are expected to be sparsely connected so the network consists of multiple discrete connected islands (as in [*E. coli*]{} [@Thieffry_98_43]), while genomes of larger size $N>N_c$ are expected to be densely connected into a single giant component where every regulator eventually perturbs the downstream effects of every other regulator.
Simulations of example genomes of various sizes spanning this critical network size confirm the adequacy of the continuum treatment. Fig. \[f\_largest\_island\] shows the number of all discrete islands as well as the number of islands containing two, three and four components. In the vicinity of the critical genome size $N_c=9,029$, the number of discrete interconnected islands begins to decline as the growing number of links connects more and more islands into the single giant component. The size of the simulated giant component as a proportion of genome size is also shown. This figure suggests that the [*E. coli*]{} genome of $N=2528$ operons should possess a giant component containing about 5% of all nodes (about 100 nodes) which can be compared with the observation that about 70% or 300 operons of the examined regulatory and regulated operons (but not including unregulated and nonregulatory operons) could be loosely grouped into 3-6 “dense overlapping regulons" or DORS while the remaining operons appeared as disjoint systems with most containing 1-3 operons but some containing up to 25 operons [@Shen_Orr_02_64].
The critical network size of $N_c=9,029$ operons or about $N_g=15,349$ genes corresponds to the point where growing regulatory networks exploiting accelerating links can no longer maintain discrete functional units, islands, of interconnected nodes. Larger genomes are densely connected into a single giant component where, eventually, any regulator can perturb the downstream effects of every other node so for instance, it is unlikely that the discrete network motifs found in the [*E. coli*]{} regulatory network [@Shen_Orr_02_64] can survive in this regime. This massive increase in perturbative effects immeasurably increases the difficulty of the evolutionary search process, leading to an expectation that the rate of evolutionary change will drastically slow when growing genome sizes reach criticality $N\approx N_c$. From a biological point of view, it is relatively easy to understand why the critical network size $N_c$ acts as an upper size limit. The accelerating nature of the prokaryote regulation network means that larger networks can add new nodes only be integrating an increasing number of links to gain evolutionary benefits. Of course, the probability of finding $lN$ beneficial links is a rapidly decreasing function of $N$. It is relatively easy to find a beneficial regulator making only of order one link to existing genes (only billions of trials are needed say), but much harder when the regulator is making an average of five links with existing genes (many trillions of trails are needed). Essentially, the more links that must be beneficially integrated, the longer the evolutionary search task and the slower the rate of evolutionary change.
![*The predicted proportion $P_o(k,N)$ of operons with $0, 1, 2, \dots$ regulatory inputs as a function of network size. Small networks mainly possess unregulated operons, while networks of large size have a significantly reduced number of unregulated operons with many operons taking large numbers of regulatory inputs. \[f\_prob\_reg\_operon\_history\]*](gagen_fig9.eps){width="\columnwidth"}
Many other statistical measures suggest that the regulatory mechanisms optimized to perform in a sparsely connected network will not necessarily operate in a densely connected network—evolution cannot foresee later needs. In particular, the proportion of operons $n_j$ which are regulated by $k$ inputs is, using Eq. \[eq\_h\_kNo\_final\], given by $$P_o(k,N)= \frac{1}{N}\sum_{j=1}^N H(k,N) \;=\;
{N \choose k} l^k (1-l)^{N-k}.$$ This distribution increases with increasing network size and is shown in Fig. \[f\_prob\_reg\_operon\_history\] making it clear that small networks mainly possess operons which are either entirely unregulated or regulated by only one or a few regulators. In contrast, large networks ($N>N_c$) have only a small proportion of operons which are unregulated while the majority of operons take between one or more regulatory inputs. It is a more difficult evolutionary task to integrate many inputs to achieve a beneficially regulated output again suggesting that prokaryote regulatory networks featuring accelerating growth in link number are size limited due to their regulatory architecture.
Another way to suggest the strict size limits imposed by the accelerating growth of regulatory links is to consider the probability that the most recently added node $n_{N}$ in a network of size $N$ immediately becomes regulatory. Using Eq. \[eq\_P\_j\_k\_dist\], node $n_{N}$ is a regulator with probability $$P_r(N)=\sum_{i=1}^N P(i,N) \;=\; 1 - (1-l)^N.$$ This probability tends to unity as network size increases, and in particular, surpasses about 50% when networks consist of $N_c$ operons—see Fig. \[f\_prob\_regulator\_No\]. At about this stage, large networks cannot add a new node without it having a significant probability of modifying the dynamics of existing nodes. This immeasurably increases the difficulty of the evolutionary task and again suggests a maximum size limit to prokaryote gene regulatory networks.
![*The rapidly increasing probability $P_r(N)$ that the most recently added node $n_{N}$ in a network of size $N$ nodes is immediately regulatory on its appearance in the genome. For network sizes greater than about $N_c=9,029$ operons, the probability that all new nodes are immediately regulatory exceeds about 50%. \[f\_prob\_regulator\_No\]*](gagen_fig10.eps){width="\columnwidth"}
If the accelerating regulatory networks of prokaryotes were able to operate in the densely connected regime, the evolutionary record might be expected to show prokaryotes of arbitrarily large genome size with a transition in connectivity statistics at some critical genome size of about $N_c\approx 9,029$. Conversely, should these regulatory networks be unable to operate in the densely connected regime, then the evolutionary record should show a maximum size limit to prokaryote genome sizes of about $N_c\approx 9,029$ operons or about $N_g=15,349$ genes, close to the observed upper limit.
Conclusion
==========
In this paper, we generalize models of accelerating networks by including probabilistic links to allow arbitrarily rapid acceleration rates leading to structural transitions in growing networks sometimes severe enough to strictly constrain network size. These structural transitions from sparse to dense connectivity are made more difficult by any additional steric or logical limitations on combinatoric control at any given promotor. Such transitions are in sharp contrast to the stationary statistics and unbounded growth potential of non-accelerating scale free and exponential networks. These probabilistic accelerating networks were applied to model prokaryote regulatory networks which exploit a quadratic growth in the number of regulators and regulatory links with genome size as established via comparative genomics programs. Our models predict a maximum genome size of $N_c\approx 9,029$ operons or about $N_g=15,349$ genes for prokaryotes, closely approximating the observed maximum. We further validated our model by making a detailed comparison of predicted and observed results for [*E. coli*]{}, and achieved satisfactory matches for respectively, the number of observed regulators, an average promotor binding site length of about 7, the long tailed distribution of outgoing regulatory links with an average of between 2.12 and 7.51 (compared to 5), the exponential distribution of incoming regulatory links with an average of around 1.10 (compared to 1.5), the proportion of regulators controlling regulators of around 17.7% (compared to 31.4%), and the probable length of regulatory cascades and the absence of regulatory loops. Our approach is unable to explain the high proportion of autoregulation observed in [*E. coli*]{} [@Shen_Orr_02_64] and this failure likely points to selection for genome reorganizations leading to spatial arrangements of operons allowing joint regulation [@Warren_03_0310029] which is not included in this model. Further, this approach does not include selection pressures ensuring that similarly regulated islands or modules share common functionality [@Shen_Orr_02_64], or other regulatory mechanisms influencing both the transcription and translation of transcription factors including micro-RNAs and other chemical mechanisms and mediators (see for instance [@Vogel_03_6435]).
However, the many successes of the accelerating network model of prokaryote regulatory networks are meaningless if similar results can be achieved via non-accelerating network models. In later work, we will show that the two simplest non-accelerating network models fail to explain either the observed quadratic growth of regulator number with genome size or the detailed statistics pertaining to the [*E. coli*]{} genome [@Gagen_0312022]. In addition, the simplifying assumption adopted here that gene duplications ensure that operons become regulatory only on entry to the genome will be dropped in later work. This will develop a more realistic model including separate physical processes for transcription factor binding to DNA and for establishing regulatory links with regulated operons where links can form at any time.
This work has wider significance due to the still common presumption in molecular biology that “What was true for [*E. coli*]{} would also be true for the elephant" capturing the notion that the mechanisms operating in prokaryotes are exactly identical to those operating in complex multicellular eukaryotes. In this picture, eukaryotes are merely enlarged prokaryotes. The results of this paper indicate that this is not possible—the accelerating nature of regulatory networks necessarily implies that eukaryotes cannot be scaled up prokaryotes and that the (likely) accelerating regulatory networks of eukaryotes must be exploiting novel regulatory mechanisms. The successful modelling of these mechanisms will likely require incorporating computationally complex technologies [@Mattick_01_1611; @Mattick_01_986; @Mattick_03_930] into an accelerating network model, and this also will be addressed in later work.
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| {
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---
abstract: 'We study dynamical systems induced by birational automorphisms on smooth cubic surfaces defined over a number field $K$. In particular we are interested in the product of non-commuting birational Geiser involutions of the cubic surface. We present results describing the sets of $K$ and $\bar{K}$-periodic points of the system, and give a necessary and sufficient condition for a dynamical local-global property called strong residual periodicity. Finally, we give a dynamical result relating to the Mordell–Weil problem on cubic surfaces.'
address: 'Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653 Beer-Sheva 8410501 Israel'
author:
- Solomon Vishkautsan
date: October 2013
title: Arithmetic dynamics on smooth cubic surfaces
---
Introduction
============
In this article we study arithmetic dynamics on smooth cubic surfaces over a number field $K$. The setup is quite simple: Take $X/K$ a smooth cubic surface over a number field $K$, and $f$ a birational automorphism of $X$, also defined over $K$. A dynamical system is induced by applying iterations of $f$ to points in $X(K)$ or $X(\bar{K})$ (where $\bar{K}$ is the algebraic closure of $K$). Such a dynamical system is an example of an *arithmetic-geometric dynamical system* (described formally in Section \[section:dynamical-systems\]). We are interested in two questions: What can be said about the $K$- and $\bar{K}$-periodic points of $f$? What can be said about the interplay between global dynamics over $K$ and local dynamics when the system is reduced modulo $p$, for all but finitely many primes $p$ in $K$’s ring of integers?
In particular, we focus on dynamical systems induced by a simple type of birational automorphism on smooth cubic surfaces, defined by taking the composition of two Geiser involutions of the cubic surface (see Section \[section:cubic-surfaces\]). Such automorphisms are examples of *Halphen twists* (cf. Brown and Ryder[@article:brown-ryder2010], Blanc and Cantat [@article:blanc-cantat2013]). By a theorem of Manin ([@book:manin1986 Example 39.8.4]), the composition of two Geiser involutions is of infinite order in the group of birational automorphisms of the cubic surface, and has the nice property of preserving an elliptic fibration of the cubic surface (by *preserve* we mean that every fiber is mapped to itself under the birational automorphism). Even for this simple type of birational automorphisms, the dynamical properties are not entirely trivial and deserve to be studied carefully.
A complication in studying the dynamics of a birational map $\varphi$ of an algebraic variety $X$ is the *locus of indeterminacy* $\mathcal{Z}(\varphi)$, the set of points where the map $\varphi$ is not defined. Even worse is the fact that the set of points on the variety whose iterations under $\varphi$ land in $\mathcal{Z}(\varphi)$, which we denote by $\mathcal{Z}_\infty(\varphi)$, can *a priori* be the set of all rational points of $X$ defined over $\bar{K}$. A recent result by Amerik states that this in fact cannot happen, but does not guarantee any periodic points lying outside of this set. We also need to define what we mean by periodic points in this setup, since for example for a birational involution $\varphi$ of the projective plane, the image of a point can be undefined for the first iteration of $\varphi$, but fixed under the second iteration. We do not wish to consider points of this type to be periodic, so in this article we will only consider a point to be periodic if it lies outside of $\mathcal{Z}_\infty(\varphi)$.
An arithmetic-geometric dynamical system $D$ (such as the one induced by a birational automorphism of a smooth cubic surface) can be reduced modulo $p$ for all but finitely many primes $p$, inducing *residual dynamical systems* $D_p$ (see Section \[section:dynamical-systems\]). In some systems, an interesting local-global behavior occurs when the system $D$ has no periodic points defined over $K$, but there exist periodic points of bounded period modulo all but finitely many primes $p$, where the bound on the periods is independent of $p$. We consider the more general case, when there exist periodic points of bounded period modulo all but finitely many primes $p$ that are not reductions modulo $p$ of any periodic points over $K$, with the bound on the periods independent of $p$. This property was first described by Bandman, Grunewald and Kunyavski[ĭ]{} [@article:bandman-grunewald-kunyavskii2010 Section 6], and is called *strong residual periodicity*. In the above-mentioned article one can find motivating examples.
Our results in this article are as follows: Let $f$ denote a birational automorphism defined by taking the composition of two Geiser involutions on a smooth cubic surface $X$. We show that the $\bar{K}$-periodic points of $f$ (lying outside of $\mathcal{Z}_\infty(f)$) are Zariski-dense in $X(\bar{K})$ (Corollary \[cor:main-corollary-A\]). The set of $K$-periodic points is contained in the union of finitely many fibers of the elliptic fibration preserved by $f$ (Corollary \[cor:cubic-surface-finite-union-of-fibers-contains-periodic\]), and the number of these fibers is bounded by a number depending only on the degree of the extension $K/\mathbb{Q}$. We further show that if $K=\mathbb{Q}$ then the period of $\mathbb{Q}$-periodic points is bounded by $12$, and cannot equal $11$ (Theorem \[thm:main-theorem-A\]). The fibers containing all periodic points can be found using a sequence of recursively defined polynomials that relate to division polynomials of elliptic curves. We define and use these polynomials to study local-global dynamics of the system and provide a necessary and sufficient condition for strong residual periodicity (Theorem \[thm:main-theorem-B\]). Finally, we provide a result relating to the Mordell–Weil problem on cubic surfaces (see Section \[section:mordell-weil\]): We prove that under mild conditions, the set of periodic points of $f$ is finitely generated by tangents and secants (Theorem \[thm:cubic-finitely-generated\]). We also call the reader’s attention to the very useful Lemma \[lem:group-translation\], which proves that group translations are never strongly residually periodic.
Let us briefly describe the structure of the article: In Sections 2-4 we provide the notations and preliminaries required to prove our results. In Section 5 we describe the dynamics of a product of Geiser involutions. In Section 6 we present a method for counting periodic fibers of such birational automorphisms. In Section 7 we prove a necessary and sufficient condition for strong residual periodicity of a product of Geiser involutions. In Section 8 we discuss the Mordell–Weil problem on cubic surfaces. In Section 9 we provide examples illustrating the various results of the article.
Acknowledgments: This article contains some of the results from the author’s PhD thesis under the joint advisorship of Tatiana Bandman and Boris Kunyavski[ĭ]{} of Bar-Ilan University. Their tremendous help and advice during the writing of the dissertation and this article are greatly appreciated. The author also thanks (in chronological order) Michael Friedman, Igor Dolgachev and Serge Cantat for fruitful correspondence and discussions relating to this article and its results. The author’s research was supported by the Israel Science Foundation, grants 657/09 and 1207/12, and by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev. The author gratefully thanks the referee for his helpful comments and corrections.
Notations
=========
- $K$ is a number field, and $\bar{K}$ its algebraic closure.
- $\mathcal{O}_K$ is the ring of integers of $K$.
- $p$ is a prime ideal in $\mathcal{O}_K$.
- $\mathcal{O}_p$ is the localization of $\mathcal{O}_K$ at the prime $p$.
- $\mathfrak{m}_p$ is the maximal ideal in $\mathcal{O}_p$.
- $\kappa_p$ is the residue field of the prime $p$ (i.e. $\mathcal{O}_p/\mathfrak{m}_p$).
- $\mathcal{L}(x,y)$ is the projective line going through two distinct points $x,y\in\mathbb{P}^3$.
- $\mathcal{P}(x,y,z)$ is the projective plane through noncollinear $x,y,z\in\mathbb{P}^3$.
- $T_x(S)$ is the tangent plane at $x$ for a smooth projective surface $S\subseteq\mathbb{P}^3$.
\[def:indeterminacy\] Given a rational map $\varphi:X\rightarrow{Y}$ between quasiprojective varieties $X,Y$, the *domain* of $\varphi$ is the largest open subset of $X$ for which the restriction of $\varphi$ is a morphism. The complement of the domain is called the *locus of indeterminacy* (or *indeterminacy set*), and is denoted by $\mathcal{Z}(\varphi)$.
\[def:extended-indeterminacy-set\] Let $\varphi$ be a dominant rational self-map $\varphi:X\rightarrow{X}$ of a projective variety $X\subset\mathbb{P}^N_k$. For an integer $n\geq{1}$ we denote $$\mathcal{Z}_n(\varphi) = \bigcup_{i={0}}^{n-1}\varphi^{-i}(\mathcal{Z}(\varphi)).$$ We remark that $\mathcal{Z}_n(\varphi) \neq \mathcal{Z}(\varphi^n)$ (e.g., $\mathcal{Z}_n(\varphi)$ may be an infinite set for a rational map $\varphi$ from a smooth projective surface, but $\mathcal{Z}(\varphi^n)$ is finite, since the locus of indeterminacy of such a map has codimension $\geq{2}$). We also denote $$\mathcal{Z}_\infty(\varphi) = \bigcup_{n=1}^{\infty}\mathcal{Z}_n(\varphi) = \bigcup_{i=0}^{\infty}\varphi^{-i}(\mathcal{Z}(\varphi)).$$ Thus, $\mathcal{Z}_\infty(\varphi)$ is the set of all points whose orbit intersects the locus of indeterminacy $\mathcal{Z}(\varphi)$ (the Zariski-closure of this set is called the *extended indeterminacy set*, cf. Diller [@article:diller1996 Definition 2.1]).
Preliminaries on arithmetic-geometric dynamical systems {#section:dynamical-systems}
=======================================================
In this section, we recall the definitions and properties of arithmetic dynamical systems. We follow Silverman [@article:silverman2008 Section 1], Hutz [@article:hutz2009 Section 2] and Bandman, Grunewald and Kunyavski[ĭ]{} [@article:bandman-grunewald-kunyavskii2010 Section 6] unless otherwise stated.
Let $K$ be a number field. A triple $D=(X, \varphi, F)$ is called an *arithmetic-geometric dynamical system* over $K$ (or *$K$-dynamical system* or *AG dynamical system*) if:
- $X$ is an algebraic $K$-variety;
- $\varphi:X\rightarrow{X}$ is a dominant $K$-endomorphism (or a dominant $K$-rational self-map; note that in this case we allow the function in the dynamical system to be partially defined);
- $F\subset{X(K)}$ is a subset of rational points which we call the *forbidden set* of the dynamical system $D$ (the forbidden set will include points the dynamical behavior of which we wish to ignore, cf. Remark \[rem:forbidden-set\] below).
The dynamics are induced by the components of the system $D$: we study iterations $\varphi^n(x)$ for points $x\in{X(\bar{K})}$. We are particularly interested in *periodic points*, i.e. points $x\in{X(\bar{K})}\setminus{\mathcal{Z}_\infty(\varphi)}$ such that $\varphi^n(x)=x$ for some positive integer $n$. The minimal such $n$ is called the *exact period* of $x$.
Let $\mathcal{O}_K$ be the ring of integers of a number field $K$. A triple $\mathcal{D}=(\mathcal{X}, \Phi,\mathcal{F})$ is called an *$\mathcal{O}_K$-dynamical system* if:
- $\mathcal{X}$ is an $\mathcal{O}_K$-scheme of finite type;
- $\Phi:\mathcal{X}\rightarrow{\mathcal{X}}$ is a dominant $\mathcal{O}_K$-endomorphism (or a dominant $\mathcal{O}_K$-rational self-map);
- $\mathcal{F}\subset{\mathcal{X}(\mathcal{O}_K)}$ is the *forbidden set* of the dynamical system $\mathcal{D}$ (cf. Remark \[rem:forbidden-set\] below).
We say that an $\mathcal{O}_K$-dynamical system $\mathcal{D} = (\mathcal{X}, \Phi, \mathcal{F})$ is an *integral model* of the $K$-dynamical system $D=(X, \varphi, F)$ if:
- $\mathcal{X}\times_{\mathcal{O}_K}K=X$ (this means $X$ is the generic fiber of $\mathcal{X}$);
- the restriction of $\Phi$ to the generic fiber of $\mathcal{X}$ coincides with $\varphi$;
- $\rho(\mathcal{F})=F,$ where $\rho\colon\mathcal{X}(\mathcal{O}_K)\to X(K)$ is the restriction to the generic fiber.
Consider a $K$-AG dynamical system $D=(X, \varphi, F)$ and an integral model $\mathcal{D} = (\mathcal{X}, \Phi, \mathcal{F}),$ as described in the previous section. Let $p$ be a prime of $\mathcal{O}_K$. Then:
- $X_p$, the special fiber of $\mathcal{X}$ at $p$, is called the *reduction of $X$ modulo $p$*. We have $X_p=\mathcal{X}\times_{\mathcal{O}_K}\kappa_p$;
- let $\rho_p\colon\mathcal{X} \to X_p$ be the reduction map (restriction to the special fiber). The image of $a\in\mathcal{X}(\mathcal{O}_K)$ under $\rho_p$ is the *reduction modulo $p$* of $a$;
- $\varphi_p\colon X_p\to X_p$, the restriction of $\Phi$ to the special fiber over $p$, is an endomorphism (or rational self-map) of $\kappa_{p}$-schemes. This is the *reduction of $\varphi$ modulo $p$*;
- $F_p=\rho_p(\mathcal{F})\subset X_p(\kappa_p)$ is the reduction of the forbidden set $\mathcal{F}.$
We call the triple $D_p = (X_p, \varphi_p, F_p)$ the *residual system* of $D$ modulo $p$.
Let $D=(X, \varphi, F)$ be a $K$-AG dynamical system such that $X$ is a smooth and proper $K$-variety and $\varphi$ is a dominant endomorphism. Let $\mathcal{D} = (\mathcal{X}, \Phi, \mathcal{F})$ be an integral model. Then for a place $p$ of $K$ we say $X$ has *good reduction at a prime $p$* if $X_p$ is a smooth and proper scheme; we say that $\varphi$ has *good reduction at a prime $p$* if $\varphi_p$ extends to a dominant $\kappa_p$-morphism. If both $X$ and $\varphi$ have good reductions modulo $p$, we say that $D$ has *good reduction* modulo $p$.
Let us recall some important facts about good reduction: A smooth projective variety $X$ defined over a number field $K$ has good reduction at all but finitely many primes of $\mathcal{O}_K$ (see Hindry and Silverman [@book:hindry-silverman2000 Proposition A.9.1.6]). A similar statement is true for a morphism of a projective variety defined over a number field $K$ (see Hutz [@article:hutz2012 Proposition 1]). For a prime $p$ of good reduction of a dynamical system $D$, reduction modulo $p$ commutes nicely with a morphism, i.e. $\rho_p(\varphi^n(a))=\varphi_p^n(\rho_p(a))$ (see Hutz [@article:hutz2009 Theorem 7]). Therefore we can discuss the reductions of orbits, etc.
\[def:residual-periodicity\] Let $D=(X,\varphi,F)$ be a $K$-AG dynamical system, and let $D_p=(X_p, \varphi_p, F_p)$ be the reduction of $D$ modulo $p$ with respect to some integral model. Let $a\in X_p(\kappa_p)\setminus F_p $ be a periodic point of $\varphi_p.$ Let $\ell_p(\varphi, a)$ be the orbit size of $a.$ Set $\underline{\ell}_p:=\min\{\ell_p(\varphi,a)\}$ where the minimum is taken over all $a$. If there are no periodic points in $X_p(\kappa_p)\setminus F_p, $ we set $\underline{\ell}_p=\infty.$ Let $M$ denote the collection of primes $p$ such that $\underline{\ell}_p=\infty.$ Let $N=\{\underline{\ell}_p\}_{p\not\in M}.$ We say that a $K$-dynamical system $D=(X,\varphi, F)$ or an $\mathcal{O}_K$-dynamical system $\mathcal{D} = (\mathcal{X}, \Phi, \mathcal{F})$ is *residually aperiodic* if the set $M$ is infinite, *residually periodic* if $M$ is finite, and *strongly residually periodic* (SRP) if the sets $M$ and $N$ are both finite. We denote by *SRP(n)* a dynamical system that is strongly residually periodic with minimal periods bounded by an integer $n$ for all but finitely many primes.
\[rem:forbidden-set\] Usually we will take the forbidden set $F\subset{X(K)}$ to be the set of all periodic points in $X(K)$ (or the Zariski-closure of this set), so that SRP describes the situation where we have bounded residual periods that cannot be explained by periodic points in $X(K)$. In case $\varphi$ is rational, we include $\mathcal{Z}_\infty$ in $F$ so that we can exclude points with bad dynamics.
Given a dynamical system $D=(X/K, \varphi, \emptyset)$ (for now we ignore the forbidden set), one can ask about the orbit size of a point $a\in{X(K)}$ when reduced modulo $p$. If $a\in{X(\mathcal{O}_K)}$ is periodic of exact period $n$, then the orbit size of $\rho_p(a)$ divides $n$ (cf. Hutz [@article:hutz2009 Theorem 1]). If $a$ is of an infinite orbit, we would expect that when reducing the point modulo primes $p$, the reduced point $\rho_p(a)$ will have periods that grow together with the cardinality of $\kappa_p$. This is a direct corollary from a theorem of Silverman (see [@article:silverman2008 Theorem 2]):
\[cor:silverman\] Let $D=(X,\varphi, F)$ be an AG dynamical system, and let $a \in X(K)\setminus{\mathcal{Z}_\infty}$ be a point of an infinite orbit. Then the residual periods of $a$ are unbounded over the primes in $\mathcal{O}_K$.
From the corollary we see that strong residual periodicity cannot be explained by one global point of infinite orbit, and in fact we can deduce from Silverman’s theorem that it cannot be explained by a finite set of points of infinite orbit in $X(K)$. The corollary allows us to prove the following simple yet useful lemma:
\[lem:group-translation\] Let $G/K$ be an algebraic group over a number field $K$, and let $D$ be the dynamical system induced by the group translation $\varphi_g(x)=gx,$ for some element $g\in{G(K)}$ of infinite order. Then $D$ is not SRP.
The iterations of $\varphi_g$ are very simple: $\varphi_g^n(x)=g^nx.$ We see that the following are equivalent:
(a) There exists a periodic point $x\in{G(K)}$ of exact period $n$.
(b) The element $g$ is of finite order $n$.
(c) The element $g$ is a periodic point of $\varphi_g$ of exact period $n$: i.e. $\varphi_g^n(g)=g$ (and $n\geq{1}$ is the minimal positive integer satisfying this).
(d) The map $\varphi_g$ is of finite order $n$ in the automorphism group of the underlying variety of $G/K$.
Now, if $g$ is of infinite order in $G/K$, then by the properties above it is of infinite orbit, and we can use the corollary to Silverman’s Theorem (Corollary \[cor:silverman\]) to see that the $\varphi_g$-periods of $g$ modulo primes $p$ are unbounded over the primes. This means that the minimal periods are unbounded (because the equivalent conditions above are also relevant for the reduced system $D_p$ when there is good reduction), so that $\varphi_g$ is not SRP.
Preliminaries on cubic surfaces {#section:cubic-surfaces}
===============================
In order to study arithmetic dynamics on smooth cubic surfaces, we need to recall several classical geometric properties and theorems related to them. We briefly recall the group structure on absolutely irreducible cubic plane curves (cf. Manin [@book:manin1986 Chapter I, Section 1, page 7] and Silverman [@book:silverman2009 Chapter III, Section 2]). Let $C$ be an absolutely irreducible cubic curve in the projective plane $\mathbb{P}^2$, defined over a field $k$. Let $C_{ns}(k)$ denote its set of nonsingular rational points over $k$. Assuming $C_{ns}(k)\neq\emptyset$, we define a binary composition law $\circ:C_{ns}(k)\times{C_{ns}(k)}\rightarrow{C_{ns}(k)},$ by setting $x\circ{y}$ for $x\neq{y}$ to be the third point of intersection of the line $L=\mathcal{L}(x,y)$ with the curve $C$. If $x=y$ then we take $L$ to be the tangent to $C$ at $x$. We can then turn $C_{ns}(k)$ into a group by choosing an element $u\in{C_{ns}(k)}$ and defining $xy:=u\circ(x\circ{y})$ for $x,y\in{C_{ns}(k)}$. With this multiplication, $C_{ns}(k)$ is an abelian group with unit $u$ . If $C$ is smooth, this gives the usual composition law on elliptic curves, and we denote it by $x+y$.
Similarly, we can define a composition law $\circ:S(\bar{k})\times{S(\bar{k})}\rightarrow{S(\bar{k})}$ on a smooth cubic surface $S\subset\mathbb{P}^3_k$ defined over a field $k$ (cf. Manin [@book:manin1986 Chapter I, Section 1, page 7]). This composition is only partially defined. Given two distinct points $x,y\in{S(\bar{k})}$ such that $\mathcal{L}(x,y)$ does not lie on $S$, we define $x\circ{y}$ to be the third point of the intersection of the line $\mathcal{L}(x,y)$ with $S$ (this line has three points of intersection with $S$, by Bezout’s theorem). We note that this operation is commutative but not necessarily associative.
\[def:cubic-good-point\] A point $x\in{S(\bar{k})}$ on a smooth cubic surface $S/k$ is a *good point* if it does not lie on the union of the lines of $S\times_k\bar{k}$ (recall that there are $27$ lines over $\bar{k}$ on a smooth cubic surface, cf. Shafarevich [@book:shafarevich1994-vol1 Chapter IV, Section 2.5, Theorem]). An unordered pair of distinct points $x, y \in S(\bar{k})$ is called a *good pair* if the line containing these points is not tangent to $S\times_k\bar{k}$ and does not intersect the union of the lines on $S\times_k\bar{k}$ in $\mathbb{P}^3$ (cf. Manin [@book:manin1986 Chapter V, Section 33.6]).
It is clear from the definition that a pair of distinct points $x,y\in{S(\bar{k})}$ on a smooth cubic surface is a good pair if and only if $x\circ{y}$ is defined, the three points $x,y$ and $x\circ{y}$ are distinct and all three of them are good.
The *Geiser involution* of a smooth cubic surface $S$ through a point $x\in{S(k)}$, is a map $t_x:S\rightarrow{S}$ sending each $y\in{S(\bar{k})}$ to $x\circ{y}$, when defined (cf. Brown and Ryder [@article:brown-ryder2010 Section 2.2] and Corti, Pukhlikov and Reid [@article:corti-pukhlikov-reid2000 Section 2.6]). We define $t_x$ for absolutely irreducible cubic curves in the same way. It is clear that $t_x$ is a birational involution (it is generally not defined at $x$ itself). A theorem of Manin [@book:manin1986 Theorems 33.7, 33.8] says that the Geiser involutions together with the Bertini involutions and the projective automorphisms generate $Bir(S)$ for a minimal smooth cubic surface $S$ defined over a perfect non-closed field. Some other useful properties of the Geiser involution are that $t_x(y)=t_y(x)$ for a good pair $x,y$, and its locus of indeterminacy is $\mathcal{Z}(t_x)=\{x\}$. Also, $t_x$ is an automorphism when restricted to $S\setminus C_x$, where $C_x = T_x(S)\cap{S}$.
\[thm:txty-on-cubic-curve\] Let $C\subset\mathbb{P}^2$ be an absolutely irreducible plane cubic curve defined over a field $k$. Then:
(a) The product of any two Geiser involutions $t_xt_y$ is a group translation: Given the choice of a group structure on the nonsingular points on the cubic curve, then for any nonsingular point $z\in{C_{ns}(\bar{k})}$ we get $t_xt_y(z) = (y - x) + z$ (or $t_xt_y(z) = x^{-1}yz$ if the group is multiplicative.
(b) For any $x,y,z\in{C}$ we have $t_xt_yt_z=t_w$, where $w=y\circ(x\circ{z})$.
See the proof of Theorem 2.1 in Manin [@book:manin1986].
\[thm:manin-infinite-order\] Let $S$ be a smooth cubic surface over a perfect field $k$, and let $x,y\in{S(k)}$ be a good pair. Then the birational map $t_xt_y$ is of infinite order in $Bir(S)$.
See Manin [@book:manin1986 Example 39.8.4].
We recall some facts about hyperplane sections of a smooth surface $S$ in $\mathbb{P}^3$ defined over a perfect field $k$. If $H$ is a hyperplane in $\mathbb{P}^3$, then a point $x\in{}S\cap{H}$ is singular (on $S\cap{H}$) if and only if $H=T_x$ (here $S\cap{H}$ is viewed scheme-theoretically, since the intersection may not be reduced, cf. Beltrametti et al. [@book:beltrametti-et-al2009 Chapter 3, Section 1.8]). For a smooth cubic surface $S$ in $\mathbb{P}^3$ we know that any hyperplane section will be one of the following (see Reid [@book:reid1988 Chapter 7, Section 1, Proposition]):
\[*a)\]*
an absolutely irreducible smooth plane cubic curve;
a cuspidal plane cubic;
a nodal plane cubic;
an absolutely irreducible conic and a line;
three distinct lines.
By using these properties it is easy to prove the following list of statements about hyperplane sections of cubic surfaces:
\[prop:irreducible-cubic-fiber\]\[prop:plane-section-through-L\_x\_y-points-on-line-are-nonsingular\] Let $S\subset\mathbb{P}^3$ be a smooth cubic surface over a perfect field $k$.
(a) The point $x\in{S(\bar{k})}$ is a good point if and only if the curve $C_x=T_x(S)\cap{S}$ is absolutely irreducible.
(b) Let $x,y$ be distinct good points on $S$. Then $C_x$ and $C_y$ do not have any common components.
(c) Let $x,y$ be a good pair on $S$. Then any plane $H\subset\mathbb{P}^3$ passing through $x,y$ intersects $S$ in an absolutely irreducible cubic curve $C$, and the three (distinct) points $x,y,z$ in $\mathcal{L}(x,y)\cap{S}$ are nonsingular on $C$.
Dynamics of a product of Geiser involutions {#section:dynamics-geiser}
===========================================
In this section we study the global dynamics of a product of two Geiser involutions $t_xt_y$, where $x,y$ is a good pair on $S$. We will show that the dynamics of $t_xt_y$ are determined by its restrictions to the fibers of the elliptic fibration preserved by $t_xt_y$. Using Theorem \[thm:txty-on-cubic-curve\] we see that $t_xt_y$ restricts to a group translation on the nonsingular points of each fiber, making the dynamics of $t_xt_y$ particularly easy to study. As a slight disclaimer, we mention that some of the proofs in this section are classical in nature, so no originality is claimed here (other than applying them to the dynamical setting). Our main results in this section are Proposition \[prop:periodic-point-on-fiber-txty\], characterizing the periodic fibers of exact period $n$, and Proposition \[prop:props-Z\_n(t\_xt\_y)\], proving the existence of $\bar{K}$-periodic points lying outside of $\mathcal{Z}_\infty(t_xt_y)$.
\[prop:cubic-plane-section-invariant\] \[prop:fiber-group-translation\] Let $S$ be a smooth cubic surface, and $x,y$ a good pair. Let $H$ be a hyperplane going through $x,y$. Let $C$ be the hyperplane section $H\cap{S}$ (absolutely irreducible by Proposition \[prop:irreducible-cubic-fiber\]). Then $C$ is invariant under $t_xt_y$ and the restriction of $t_xt_y$ to $C$ is a group translation on $C_{ns}$.
\[*(a)\] That $C$ is invariant is clear from the definition of $t_x$ and $t_xt_y$ is a group translation by Theorem \[thm:txty-on-cubic-curve\].*
Let $S \subset \mathbb{P}^3$ be a smooth cubic surface over a perfect field $k$. An *elliptic fibration* on $S$ is a rational map $\varphi: S\rightarrow{B}$ defined over $k$, such that the geometric generic fiber is birational to a curve of genus $1$. For a field $k$ of characteristic $0$ the base of an elliptic fibration must be of genus $0$ and has a rational point, so it is isomorphic to $\mathbb{P}^1$(see Brown and Ryder [@article:brown-ryder2010 Section 1] for the definition, and a proof of the last statement). Given a good pair $x,y$ on ${S}$, we can define an elliptic fibration by taking the pencil of planes through the line $L=\mathcal{L}(x,y)$: we choose two distinct planes $H_1 = \{f=0\}, H_2 = \{g=0\}$ passing through $L$ (where $f,g$ are linear forms); then the rational map $\varphi=(f,g)$ is an elliptic fibration. We call such a fibration the *linear fibration* associated with the good pair $x,y$. The following is immediate from Proposition \[prop:cubic-plane-section-invariant\].
\[prop:geiser-fibration-stabilized\] Let $S \subset \mathbb{P}^3$ be a smooth cubic surface. Given a good pair $x,y$ on ${S}$, the fibers of the linear fibration through $x,y$ are invariant under the birational automorphism $t_xt_y$.
From Proposition \[prop:fiber-group-translation\] and the proof of Lemma \[lem:group-translation\], we see that the only fibers containing periodic points are those for which $t_xt_y$ restricts to a group translation of finite order (aside from the singular points of the singular fibers, which we will show to be fixed points). Denote by $Fixed(\varphi)$ the set of fixed points of a rational map $\varphi$, then:
\[prop:fixed-txty\] Let $x,y$ be a good pair on a smooth cubic surface $S$ over a perfect field $k$, then the set of fixed points of $t_xt_y$ is equal to $Fixed(t_x)\cap{}Fixed(t_y)$.
It is obvious that if $w\in{Fixed(t_x)\cap{}Fixed(t_y)}$, then $w\in{Fixed(t_xt_y)}$. In the other direction, suppose $t_xt_y(w)=w$. Assume $w\neq{x}$; then we can apply $t_x$ to both sides of the equation and get $t_x(w)=t_y(w)$ (since $w\not\in\{x\}=\mathcal{Z}(t_x)$). We know that $t_x(w)=t_w(x)$, so that we get $t_w(x)=t_w(y)$. Now, if $t_w(x)\neq{w}$, we can apply $t_w$ to both sides of the equation and get $x=y$, a contradiction. Therefore, still under the assumption of $w\neq{x}$, we get $t_w(x)=t_x(w)=w$ and also $t_y(w)=w$ as required. If $w=x$, then we have $t_xt_y(x)=x$, which implies $t_y(x)\in{C_x}$ since $t_x^{-1}(x)=C_x$. The points $x,y, t_y(x)$ are collinear, and since $x,t_y(x)\in{C_x}$, we get that $y\in{C_x}$ as well, which implies $t_y(x)=x$; but then $t_xt_y(x)=t_x(x)$ is indeterminate, a contradiction.
\[cor:txty-fixed-iff-singular\]\[cor:12-fixed-points\] Let $S,x,y$ be as in Proposition \[prop:fixed-txty\]. Then:
(a) A point $w\in{S(\bar{k})}$ is a fixed point of $t_xt_y$ if and only if $w$ is a singular point of the curve $C=\mathcal{P}(x,y,w)\cap{S}$.
(b) The map $t_xt_y$ has at most $12$ fixed points over $\bar{k}$.
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(a) We saw in the proof of Proposition \[prop:fixed-txty\] that $w$ is a fixed point of $t_xt_y$ if and only if $x,y\in{C_w}$. Therefore $C=C_w$, and $w$ is singular on $C_w$ (by Proposition \[prop:irreducible-cubic-fiber\]).
(b) The linear fibration $\pi$ associated with $x,y$ can be blown up at the three points $x,y,z$ in $\mathcal{L}(x,y)\cap{S}$ to give a rational elliptic surface, which only has at most $12$ singular fibers (see Miranda [@article:miranda1990 Section 1]) (Note that no singularities on the fibers of $\pi$ are resolved by the blowup, since $x,y,z$ are smooth on all fibers by Proposition \[prop:plane-section-through-L\_x\_y-points-on-line-are-nonsingular\]). Each of these singular fibers is associated with a fixed point of $t_xt_y$ by Corollary \[cor:txty-fixed-iff-singular\].
We will say that a fiber $C$ in the linear fibration associated with a good pair $\{x,y\}$ is a *periodic fiber* of period $n>0$ if $(t_xt_y)^n$ is the identity when restricted to the fiber $C$, and $n$ is the minimal positive integer satisfying this.
\[prop:periodic-point-on-fiber-txty\] Let $x,y$ be a good pair on a smooth cubic surface $S\subset\mathbb{P}^3$ defined over a perfect field $k$. Let $w\in{S(\bar{k})}\setminus\mathcal{Z}_\infty(t_xt_y)$ be noncollinear with $x,y$, and denote $C=\mathcal{P}(x,y,w)\cap{S}$. Then the following are equivalent:
(a) The point $w$ is a periodic point of exact period $n>1$ of $t_xt_y$.
(b) The curve $C$ is $t_xt_y$-periodic of period $n$ (which is the same as saying $t_xt_y$ is of order $n$ in $Aut(C)$), and $w$ is a nonsingular point of $C$.
(c) The point $y$ is of order $n$ in the group $C_{ns}(k)$ with $x$ chosen to be the unit element (the point $y$ is nonsingular on $C$ by Proposition \[prop:plane-section-through-L\_x\_y-points-on-line-are-nonsingular\]).
Let $w\in{S}$ be periodic of exact period $n>1$. By Proposition \[prop:cubic-plane-section-invariant\], the birational automorphism $t_xt_y$ restricts to an automorphism of $C$. We choose $x$ to be the unit element of the group structure on $C_{ns}(k)$ (note that $w\in{C_{ns}(k)}$ since the period of $w$ is greater than $1$, and then by Corollary \[cor:txty-fixed-iff-singular\] it is nonsingular), and get $$\label{equation:group-translation}
w=(t_xt_y)^{n}(w)=ny+w$$ (see Theorem \[thm:txty-on-cubic-curve\]), from which we get $ny=0$, meaning that the order of $y$ on the cubic curve is $n$ and that $(t_xt_y)^n = id$. If the order of $t_xt_y$ was less than $n$, we would get a contradiction to $n$ being the exact period of $w$. So we have proved that *(a)* implies *(b)* and *(c)*. Similarly, one uses equation (\[equation:group-translation\]) to prove the other implications.
We can say more for the period $n=2$:
\[prop:txty-period-2\] Let $S,x,y,\varphi,w$ and $C$ be as in Proposition \[prop:periodic-point-on-fiber-txty\]. We restrict $\circ$ to the curve $C$, where it is fully defined. Then the following are equivalent:
(a) \[prop:txty-period-2:item-a\] The point $w\in{S(\bar{k})}$ is a periodic point of exact period $n=2$ of $t_xt_y$.
(b) \[prop:txty-period-2:item-d\] $x\circ{x} = y\circ{y}$.
(c) \[prop:txty-period-2:item-e\] $x\circ{x}\in{C_x\cap{C_y}}$.
The statements *(\[prop:txty-period-2:item-d\])* and *(\[prop:txty-period-2:item-e\])* are equivalent, because $x\circ{x}\in{C_y}$ means that the line through $y$ and $x\circ{x}$ has a double point at $y$, and as this line is contained in $\mathcal{P}(x,y,w)$, it must mean that this is the tangent line to $C$ at $y$, but then by definition $x\circ{x}=y\circ{y}$. The statements *(\[prop:txty-period-2:item-a\])* and *(\[prop:txty-period-2:item-d\])* are equivalent since $$(t_xt_y)^2(w)=w \iff 2y+w=w \iff 2y=0 \iff x\circ(y\circ{y})=x,$$ and we can apply $x$ to both sides of the last equation.
\[cor:curves-of-period-2\] Let $S,x,y, \varphi$ be as in Proposition \[prop:periodic-point-on-fiber-txty\]; then there are three periodic fibers of period $2$ defined over $\bar{k}$, counted with multiplicity, in the linear fibration of $S$ through the points $x,y$ (and these contain all periodic points of period $2$).
We show that the period $2$ fibers of $\varphi$ are determined by the intersection of the line $L=T_x(S)\cap{T_y(S)}$ with $S$. The line $L$ cannot lie on $S$, since otherwise it lies on $T_x(S)$ and therefore is contained in $C_x = T_x(S)\cap{S}$; but $C_x$ is absolutely irreducible by Proposition \[prop:irreducible-cubic-fiber\]. Therefore the line $L$ intersects $S$ at three points (counted with multiplicity). None of these three points are collinear with $x,y$ (otherwise $\mathcal{L}(x,y)$ has a double point at both $y$ and $x$). Each point $w\in{S\cap{L}}$ then determines a fiber of the linear fibration, and on this fiber we get $w=x\circ{x}=y\circ{y}$, since $\mathcal{L}(w,x)\subset{T_x(S)}$ and $\mathcal{L}(w,y)\subset{T_y(S)}$. The result then follows from Proposition \[prop:txty-period-2\].
\[prop:props-Z\_n(t\_xt\_y)\] Let $S$ be a smooth cubic surface over a perfect field $k$. Let $x,y$ be a good pair on $S$, and let $C$ be a fiber of the linear fibration through $x,y$. Then:
(a) \[prop:props-Z\_n(t\_xt\_y):item-a\] For any positive integer $n$, the set $C(\bar{k})\cap{\mathcal{Z}_n(t_xt_y)}$ is finite (see Notation \[def:extended-indeterminacy-set\]).
(b) \[prop:props-Z\_n(t\_xt\_y):item-b\] If $C$ is periodic of period $n$, then the set $C(\bar{k})\cap{\mathcal{Z}_\infty(t_xt_y)}$ is finite, and this implies the existence of $\bar{k}$-periodic points outside of $\mathcal{Z}_\infty(t_xt_y)$.
To make notations simpler, we identify all algebraic sets with their underlying set of $\bar{k}$-points. Denote $f=t_xt_y$, and let $z$ be the third point in $\mathcal{L}(x,y)\cap{S}$. We prove the proposition by induction on $n$. For $n=1$ we have $\mathcal{Z}_1(f)=\mathcal{Z}(f)=\{y,z\}$, so the assertion is true. Let $n>1$, and assume that the statement is true for any $m<n$. For $n\geq{2}$ we have $\mathcal{Z}_n(f)=\mathcal{Z}(f)\cup f^{-1}\left[\mathcal{Z}_{n-1}(f)\right],$ so that $$C\cap\mathcal{Z}_n(f) = (C\cap\mathcal{Z}(f)) \cup (C \cap f^{-1}\left[\mathcal{Z}_{n-1}(f)\right]).$$ The first set in the union is finite, so it remains to prove that $C \cap f^{-1}\left[\mathcal{Z}_{n-1}(f)\right]$ is finite. It is easy to see that $C \subset f^{-1}\left[C\right]$, so that $$C \cap f^{-1}\left[\mathcal{Z}_{n-1}(f)\right] \subset C\cap f^{-1}\left[C\cap \mathcal{Z}_{n-1}(f)\right].$$ The set $C\cap \mathcal{Z}_{n-1}(f)$ is finite by the induction hypothesis, so $$C\cap \mathcal{Z}_{n-1}(f) \subseteq \{x,y,z,A_1,...,A_K\},$$ where $A_1,...,A_K$ are points in $C(\bar{k})\setminus\{x,y,z\}$. The inverse image of $\{y,A_1,\ldots,A_K\}$ under $f$ is finite (as can be checked readily from the definition of $t_x$ and $t_y$), so it remains to show that $C\cap{f^{-1}[\{x,z\}]}$ is finite. Now $f^{-1}[\{z\}] = C_y$, which is an irreducible hyperplane section singular at $y$ by Proposition \[prop:irreducible-cubic-fiber\], and therefore has no common components with $C$ (the curve $C$ is irreducible and cannot have a singularity at $x,y,z$ by Proposition \[prop:plane-section-through-L\_x\_y-points-on-line-are-nonsingular\]), so that their intersection is finite. Finally, $f^{-1}[\{x\}] = t_y^{-1}(C_x)$. As before $C \subset t_y^{-1}\left[C\right]$, so that $$C\cap t_y^{-1}\left[C_x\right] \subset t_y^{-1}\left[C\cap C_x\right].$$ The set $C\cap{C_x}$ is finite, so $C\cap{C_x} \subseteq \{x, B_1,...,B_M\},$ where $B_1,...,B_M$ are points in $C(\bar{k})\setminus\{x,y,z\}$ (The curve $C_x$ does not contain $y$ and $z$, since otherwise $x$ is a double point of $\mathcal{L}(x,y)\cap{S}$, which is impossible, since $x,y$ and $z$ are distinct). The inverse image of $\{x, B_1,...,B_M\}$ under $t_y$ is finite. This proves *(\[prop:props-Z\_n(t\_xt\_y):item-a\])*.
To prove *(\[prop:props-Z\_n(t\_xt\_y):item-b\])*, we note that any point of $C$ not in $\mathcal{Z}_n(t_xt_y)$ must be periodic of period $n$, and therefore cannot lie in $\mathcal{Z}_\infty(t_xt_y)$; but $C\cap\mathcal{Z}_n(t_xt_y)$ is finite by *(\[prop:props-Z\_n(t\_xt\_y):item-a\])*, so that $C\cap\mathcal{Z}_\infty(t_xt_y)$ is finite as well.
Division polynomials associated with linear fibration
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We have seen in Section \[section:dynamics-geiser\] that finding periodic points of $t_xt_y$ for a good pair $x,y$ on a smooth cubic surface $S$ defined over a number field $K$, is equivalent to studying periodicity on the fibers of the linear fibration defined by the points $x,y$ (Proposition \[prop:geiser-fibration-stabilized\]). We have also seen that for a fiber to have periodic points of exact period $n$, the fiber itself must be periodic of period $n$, and this is equivalent to $y$ being of order $n$ in the group structure induced by choosing $x$ as the unit element (Proposition \[prop:periodic-point-on-fiber-txty\]). We want to find all the fibers of the linear fibration that are periodic of finite period. In order to do so we employ *division polynomials* of elliptic curves. We first recall the definition and basic properties of these polynomials.
\[def:division-polynomials\] Given an elliptic curve $E$ in Weierstrass form $y^2 = x^3 + Ax+B,$ we associate to it the *division polynomials* $\psi_n, n\geq{0}$ in $\mathbb{Z}[x,y,A,B]$: $$\begin{aligned}
\psi_0 &=& 0 , \qquad \psi_1 = 1, \qquad \psi_2 = 2y, \\
\psi_3 &=& 3x^4 +6Ax^2 +12Bx - A^2 ,\\
\psi_4 &=& 4y(x^6+5Ax^4+20Bx^3 - 5A^2x^2 - 4ABx - 8B^2 - A^3) ,\\
\psi_{2m+1} &=& \psi_{m+2}\psi_m^3 - \psi_{m-1}\psi_m^3 + 1, \quad \text{ for } m \geq 2 ,\\
\psi_{2m} &=& (2y)^{-1}\psi_m(\psi_{m+2}\psi_{m-1}^2 - \psi_{m-2}\psi_{m+1}^2), \quad \text{ for } m \geq 3\end{aligned}$$
Let $E$ be an elliptic curve defined over a number field $K$. Then the division polynomials have the following properties ($E[n]$ is the set of points in $E(\bar{K})$ with order dividing $n$):
(a) $\psi_{2n+1}, y^{-1}\psi_{2n}$ are polynomials in $\mathbb{Z}[x,A,B]$;
(b) the roots of $\psi_{2n+1}$ are the $x$-coordinates of the points in $E[2n+1] \setminus \{\mathcal{O}\}$;
(c) the roots of $y^{-1}\psi_{2n}$ are the $x$-coordinates of the points in $E[2n] \setminus E[2]$.
See Washington [@book:washington2008 Chapter 3, Section 2].
To make the notations easier, we replace $\psi_n$ for even $n$ with $y^{-1}\psi_n$.
\[prop:periodic-fibers-using-division-polynomials\] Let $S$ be a smooth cubic surface defined over a number field $K$, let $x,y\in{S(K)}$ be a good pair, and denote by $\pi:S\rightarrow\mathbb{P}^1$ the linear fibration associated with the good pair $x,y$. There exist polynomials $\gamma_n(t)$ in $K[t]$, for $n\geq{3}$, whose roots in $\bar{K}$ lying outside a finite set $\mathcal{B}\subset\mathbb{P}^1$ correspond to fibers of $\pi$ that are $t_xt_y$-periodic of period $\geq{3}$ and dividing $n$.
We can ensure that the fiber at infinity is non-periodic and nonsingular. Outside this fiber $\pi$ induces a cubic pencil with parameter $t$, whose generic fiber is a smooth cubic curve $C$ in the projective plane $\mathbb{P}^2$ over the function field $K(t)$. The points $x,y$ induce two rational points (which we still denote by $x,y$) on the cubic curve $C$. We choose $x$ to be the unit element of the cubic curve $C$, which induces an elliptic curve group structure on $C$. We note that it is impossible for $y$ on $C$ to be of finite order when $x$ is chosen as unit element, since otherwise $t_xt_y$ is of finite order in $Bir(X)$, contradicting Theorem \[thm:manin-infinite-order\].
We now use a Weierstrass transformation (see Shioda [@article:shioda1995 Section 2]) to map our elliptic curve $(C, x)$ to an isomorphic elliptic curve in Weierstrass form over $K(t)$ $$E: v^2=u^3+A(t)u+B(t).$$ We denote the Weierstrass transformation by $\omega: (C, x) \rightarrow (E,\mathcal{O}).$ The point $y$ is mapped to a rational point on $E$, and we still denote this point by $y$. We can ensure that the Weierstrass form $E$ has coefficients in $\mathcal{O}_K[t]$: This is done by transforming $E$ to a *minimal Weierstrass form* (cf. Silverman [@book:silverman2009 Chapter VII, Section 1]), i.e. an isomorphic copy of $E$ such that the valuation of $A$ and $B$ is nonnegative and minimal (in its isomorphism class) at all places of $K(t)$; then it is possible to get rid of the denominators so that all the coefficients of $A(t)$ and $B(t)$ are in $\mathcal{O}_K$. There exists an open subset $U\subseteq\mathbb{P}^1$ such that for each $t\in{U}$, the specialization of the Weierstrass transformation $\omega$ to the fiber over a specific $t$ will remain an isomorphism of irreducible cubic curves. We include the complement of $U$ in $\mathcal{B}$.
We now have an infinite sequence of division polynomials $\psi_n\in{\mathcal{O}_K[t][u]}, n\geq{3},$ of the elliptic curve $E/K(t)$. We evaluate these polynomials at the $u$-coordinate of the point $y$, and get an element $\tilde{\gamma}_n(t)$ in $K(t)$. The denominator of $\tilde{\gamma}_n$ depends only on $y\in{E}$; thus there are finitely many values of $t$ at which $\tilde{\gamma}_n(t)$ might have poles, which we include in $\mathcal{B}$. We now define $\gamma_n$ to be the numerator of $\tilde{\gamma}_n$, for $n\geq{3}$.
It is clear that the fibers of $\pi$ lying over the roots of $\gamma_n(t)$ not in $\mathcal{B}$, are fibers where $y$ is of finite order $\neq{2}$ and dividing $n$. Thus by Proposition \[prop:periodic-point-on-fiber-txty\] these fibers are periodic of period $\neq{2}$ and dividing $n$.
\[cor:periodic-fibers-using-division-polynomials\] Let $S/K,x,y$ and $\pi$ be as in Proposition \[prop:periodic-fibers-using-division-polynomials\]. There exist polynomials $\Psi_n(t)$ in $K[t]$, for $n\geq{3}$, whose roots in $\bar{K}$ correspond to all the fibers of $\pi$ that are periodic under the birational map $t_xt_y$, of period $\neq{2}$ and dividing $n$.
We only need to check the order of $y$ on the finite number of fibers over the points in $\mathcal{B}$. If a fiber over a point $a\in\mathcal{B}$ is non-periodic, then we factor out $a$ as a root from the polynomials $\gamma_n(t)$. If it is periodic, then we need to make sure it appears as a root of the polynomials $\gamma_n(t)$. It remains to prove that the resulting polynomials have coefficients in $K$. This is true because if we have a point $a$ on whose corresponding fiber we have $y$ of finite order $n$, then the order of $y$ on the fibers over the conjugates of $a$ will be the same, and therefore they will be roots of the same polynomials.
\[cor:main-corollary-A\] Under the assumptions of the last corollary, the $\bar{K}$-periodic points of $t_xt_y$ are Zariski-dense in $S(\bar{K})$.
By Corollary \[cor:periodic-fibers-using-division-polynomials\], there are infinitely many periodic fibers of $t_xt_y$ over $\bar{K}$ (the polynomials $\Psi_p, \Psi_q$ have no common roots for distinct primes $p$ and $q$). By Proposition \[prop:props-Z\_n(t\_xt\_y)\], these fibers must contain $\bar{K}$-rational points lying outside of $\mathcal{Z}_\infty(t_xt_y)$.
We call the polynomials $\Psi_n, n\geq{3}$ in the last corollary the *division polynomials* of the smooth cubic surface $S$ with respect to the good pair $x,y$ (this definition is nonstandard). We now define the 1st and 2nd division polynomials:
1. $\Psi_1(t)$ is defined to be the discriminant of Weierstrass form $E/K(t)$ from the proof of Proposition \[prop:periodic-fibers-using-division-polynomials\].
2. $\Psi_2(t)$ is defined to be the numerator of the $v$ coordinate of the point $y$ on $E/K(t)$.
\[prop:psi1-psi2\] Let $S$ be a smooth cubic surface defined over a number field $K$, and let $x,y\in{S(K)}$ be a good pair.
(a) The roots of $\Psi_1$ in $\bar{K}$ correspond to the fibers of $\pi$ that contain fixed points of $t_xt_y$. This polynomial is of degree at most $12$.
(b) The roots of $\Psi_2$ in $\bar{K}$ correspond to the fibers of $\pi$ that are periodic of period $2$. This polynomial is of degree at most $3$.
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(a) The discriminant of a cubic curve in Weierstrass form is $0$ if and only if the curve is singular. A point $w$ on the surface $S$ is fixed under $t_xt_y$ if and only if the fiber $C=\mathcal{P}(x,y,w) \cap S$ is singular, by Corollary \[cor:txty-fixed-iff-singular\]. There are at most $12$ fixed points of $t_xt_y$ on the surface $S$, by Corollary \[cor:12-fixed-points\], which explains the degree.
(b) A point on an elliptic curve in Weierstrass form is of order $2$ if and only if its $v$ coordinate is $0$. By Proposition \[cor:curves-of-period-2\] there are at most $3$ curves of period $2$.
\[notation:dynatomic-polynomials\] Let $S$ be a smooth cubic surface defined over a number field $K$, let $x,y\in{S(K)}$ be a good pair and let $\Psi_n$ be the division polynomials associated with $x,y$. For $n\geq{1}$ define the polynomials $\Phi_n(t)=\Psi_n(t)\prod\limits_{{d|n, d>2}}\Psi_d(t)^{-1}$. The roots of $\Phi_n(t), n\geq{2},$ in $\bar{K}$ correspond to periodic fibers of $\pi$ of *exact* period $n$.
\[prop:bounded-number-of-periodic-fibers\] Let $S$ be a smooth cubic surface defined over a number field $K$ and let $x,y\in{S(K)}$ be a good pair.
(a) For the case $K=\mathbb{Q}$, the polynomials $\Phi_n(t)$ do not have roots in $\mathbb{Q}$ for $n>12$ or $n=11$.
(b) For a general number field $K$ there exists a positive integer $N$ such that $\Phi_n(t)$ does not have roots in $K$ for $n>N$. The bound $N$ depends only on the degree of the extension $K/\mathbb{Q}$.
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(a) Mazur’s torsion theorem [@article:mazur1978 Theorem 8] lists the possible $\mathbb{Q}$-rational torsion subgroups of elliptic curves over $\mathbb{Q}$. The maximal order of an element of finite order is bounded by $12$. If $\Phi_n(t)$ has a root in $\mathbb{Q}$ for $n>12$, then the fiber over this root is an elliptic curve over $\mathbb{Q}$ with a rational point of order $n$, a contradiction.
(b) Merel’s torsion theorem (see Merel [@article:merel1996 Corollary]), guarantees a bound on the order of the $K$-rational torsion subgroup that depends only on the degree of the field extension. An explicit bound on the order of the torsion subgroup can be found in Parent [@article:parent1999 Corollary 1.8] (this bound is exponential in the degree of the extension of $K/\mathbb{Q}$).
\[cor:cubic-surface-finite-union-of-fibers-contains-periodic\] Let $S$ be a smooth cubic surface defined over a number field $K$ and let $x,y\in{S(K)}$ be a good pair. The set of all periodic points of $t_xt_y$ in $S(K)$ is contained in a finite number of fibers of $\pi$.
All periodic points must be contained in the fibers of the linear fibration lying over the $K$-roots of the polynomials $\Psi_n$. Therefore, by Proposition \[prop:bounded-number-of-periodic-fibers\], there are only finitely many fibers that can contain $K$-periodic points.
\[thm:main-theorem-A\] Under the assumptions of Proposition \[prop:bounded-number-of-periodic-fibers\], for $K=\mathbb{Q}$ the birational automorphism $t_xt_y$ can only have $\mathbb{Q}$-periodic points of exact period $1,...,10,12$. For a general number field $K$, the birational automorphism $t_xt_y$ can only have periodic points of exact period bounded by a number depending only on the degree of the extension $K/\mathbb{Q}$.
The division polynomials of the linear fibration give us an effective method for “finding" all $K$-periodic points of a birational automorphism $t_xt_y$:
(1) First we find the $K$-roots of all division polynomials up to the bounds described in the proof of Proposition \[prop:bounded-number-of-periodic-fibers\].
(2) For the nonsingular periodic fibers, we can then use algorithms to find the generators of the Mordell-Weil of the elliptic curve (cf. Cremona [@book:cremona1997 Chapter 3, Section 5]).
(3) We can parametrize the periodic points on the singular periodic fibers.
(4) For each periodic fiber, we can find all the points in $\mathcal{Z}_\infty(t_xt_y)$ in a finite number of steps (see Proposition \[prop:props-Z\_n(t\_xt\_y)\]).
SRP on cubic surface {#section:SRP-cubic-surface}
====================
We are now ready to prove some results about strong residual periodicity of the AG dynamical system of type $D=(S,t_xt_y,F)$ on a smooth cubic surface $S$ defined over a number field $K$, whose global dynamics were described in the previous two sections. We take the forbidden set $F$ to be union of the Zariski-closure of the set of all periodic points in $X(K)$ and the set of $K$-rational points in $\mathcal{Z}_\infty(t_xt_y)$. In this section we prove two sufficient conditions for SRP, and then we prove that when put together, they are necessary and sufficient.
\[prop:srp-on-cubic-surface-cond1\] Let $S$ be a smooth cubic surface defined over a number field $K$ and let $x,y\in{S(K)}$ be a good pair. If there exists a fiber $C$, defined over $K$, of the linear fibration through the good pair $x,y\in{S(K)}$ such that the set of $K$-rational points $C(K)$ is finite, then $D=(S,\varphi=t_xt_y,F)$ is strongly residually periodic.
The curve $C$ must be smooth, since absolutely irreducible singular cubic curves with a rational point are rational, and therefore have infinitely many rational points. The point $y$ must be of finite order in the group induced by choosing $x$ as the unit element on $C$. All points in $C(K)$, outside of $\mathcal{Z}_\infty(t_xt_y)$, are $t_xt_y$-periodic with period equal to the order of $y$. Denote by $N$ the order of $y$ in $C(K)$. To prove the proposition we will show that for primes $p$ for whom the cardinality of $\kappa_p$ is large enough, there exists a point $w\in{C_p(\kappa_p)}$ that satisfies:
(1) The point $w$ is not a reduction of any periodic point in the Zariski-closure of the subset of periodic points in $S(K)$.
(2) The point $w$ is not a reduction of any point in $\mathcal{Z}_\infty(\varphi)$.
(3) The point $w$ is not in $\mathcal{Z}_\infty(\varphi_p)$ (in particular $w$ is defined under $\varphi_p^N$, and is therefore a periodic point of period at most $N$).
We restrict ourselves to primes $p$ where the system $D$ has good reduction. In particular the elliptic curve $E=(C, x)$ is reduced to an elliptic curve $E_p=(C_p, x_p)$. We can then use Hasse’s theorem on elliptic curves (see Silverman [@book:silverman2009 Thm V.1.1]) to guarantee that the number of rational points in $C_p(\kappa_p)$ is as large as we desire. We can thus guarantee that for all but finitely many primes $p$ there exists a point $w\in{C_p(\kappa_p)}$ such that $w$ is not a reduction of any point in $C(K)$. The Zariski-closure of the subset of periodic points in $S(K)$ is contained in a finite number of fibers of the linear fibration through $x,y$, so that it is enough to choose $p$ such that $C_p$ intersects the reductions of the other fibers only at the points $\{x,y,z\}=\mathcal{L}(x,y)\cap{S}$ (this is true for any prime $p$ such that $\kappa_p$ is large enough). This proves $(1)$.
We proved in Proposition \[prop:props-Z\_n(t\_xt\_y)\] that the intersection of $C(\bar{K})$ with $\mathcal{Z}_\infty(\varphi)$ is finite. We use this and the same reasoning as in the previous paragraph to prove $(2)$.
The set $\mathcal{Z}_\infty(\varphi_p)$ is the set of points whose orbit intersects $\mathcal{Z}(\varphi_p)$. Now, $\mathcal{Z}(\varphi_p)\subseteq\{y_p = \rho_p(y), z_p = \rho_p(z)\}$, so that any point $v\in{C_p(\kappa_p)}$ not in $\mathcal{Z}(\varphi_p)$ that lies in $\mathcal{Z}_\infty(\varphi_p)$, satisfies $ny_p+v=y_p$ or $ny_p+v=z_p$, for some integer $1\leq{n}\leq{N}$. Rewriting this, we get $v=(1-n)y_p$ or $v=z_p-ny_p$ , for some integer $1\leq{n}\leq{N}$. In other words, there are at most $2N$ points in $C_p(\kappa_p)\cap\mathcal{Z}_\infty(\varphi_p)$. The bound $2N$ is independent of the prime $p$, and therefore for primes $p$ such that $\kappa_p$ is large enough, we can find points in ${C_p(\kappa_p)}$ that are not in $\mathcal{Z}_\infty(\varphi_p)$, so that $(3)$ is proved.
The order of $y$ on $C(K)$ is bounded by Theorem \[thm:main-theorem-A\].
See Example \[ex:srp-cubic\] for a dynamical system satisfying the conditions of Proposition \[prop:srp-on-cubic-surface-cond1\].
\[prop:cubic-theta-modulo\] Let $S$ be a smooth cubic surface defined over a number field $K$ and let $x,y\in{S(K)}$ be a good pair. The minimal periods of the residual systems of $D=(S,\varphi=t_xt_y, \mathcal{F})$ are bounded (as in Definition \[def:residual-periodicity\]) if and only if there exists a positive integer $N$ such that the polynomial $\Theta_N(t) = \Phi_1(t)\cdots\Phi_N(t)$ (see Notation \[notation:dynatomic-polynomials\]) has a root modulo all but finitely many primes $p$.
Let $\pi:S\rightarrow\mathbb{P}^1$ be the linear fibration through the pair $x,y$. The polynomial $\Theta_N$ has roots modulo all but finitely many primes if and only if there exist either periodic fibers of $\pi$ of period at most $N$ or fixed points for all but finitely primes. The set of $\mathcal{Z}_\infty(\varphi_p)$ is bounded on periodic curves of period at most $N$ by a bound depending only on $N$, as we have seen in the proof of Proposition \[prop:srp-on-cubic-surface-cond1\]. Thus for primes $p$ such that $\kappa_p$ is large enough, the fiber must contain points outside $\mathcal{Z}_\infty(\varphi_p)$. Therefore, the condition that there exist either periodic fibers of $\pi$ of period at most $N$ or fixed points for all but finitely many primes, is equivalent to the boundedness of the minimal periods of the residual systems $D_p$.
\[prop:srp-on-cubic-surface-cond2\] Let $S$ be a smooth cubic surface defined over a number field $K$, and let $x,y\in{S(K)}$ be a good pair. If there exists a positive integer $N$ such that the polynomial $\Theta_N(t) = \Phi_1(t)\cdots\Phi_N(t)$ divided by all linear factors defined over $K$ (i.e. all $K$-roots are removed), has no roots over $K$, and for all but finitely many primes $p$ the polynomial $\Theta_N(t)$ has a root modulo $p$, then $D=(S, \varphi=t_xt_y, F)$ is strongly residually periodic.
By Proposition \[prop:cubic-theta-modulo\], we know that the minimal residual periods are bounded by $N$ for all but finitely many primes. We need to check that our points of minimal period modulo $p$ are not all reductions of points from the forbidden set.
First we check reductions of $K$-periodic points. We know that the minimal period for all but finitely many primes is at most $N$, and any periodic point over $K$ must either be of larger period than $N$, or belong to a fiber corresponding to a $K$-root of $\Theta_N(t)$. Let $C$ be a $K$-periodic fiber of period $m>N$, then $y$ is of order $m$ on $C$. For all but finitely many primes, the reduction of the point $y$ modulo $p$ will have the same order $m>N$ on the reduced fiber (cf. Silverman [@book:silverman2009 Proposition VII.3.1]). Therefore the reduction of such a fiber cannot give us points of period at most $N$ (by Proposition \[prop:periodic-point-on-fiber-txty\]). Now suppose we have a fiber $C$ of the linear fibration that corresponds to a root $a\in{K}$ of $\Theta_N(t)$. Denote by $\tilde{\Theta}_N(t)\in{K[t]}$ the polynomial obtained from $\Theta_N(t)$ after dividing by all linear factors defined over $K$. Then $a$ is not a root of $\tilde{\Theta}_N(t)$. Further, $\rho_p(a)$ can only be a root of $\tilde{\Theta}_N(t)$ modulo $p$ for a finite number of primes $p$. Therefore the fiber $C_p$ can only agree with the fibers lying over the roots of $\tilde{\Theta}_N(t)$ modulo $p$ for finitely many primes $p$. Thus, periodic points obtained from roots of $\tilde{\Theta}_N(t)$ will not be reductions modulo $p$ of points from the fiber $C$.
Finally, we check the reduction of points in $\mathcal{Z}_\infty(\varphi)$: Given a $\kappa_p$ periodic fiber of period at most $N$, it can contain only a bounded amount of periodic points whose orbit goes through $\mathcal{Z}(\varphi_p)$. The bound for this type of points does not depend on $p$ but only on $N$. Therefore, for primes $p$ such that $\kappa_p$ is large enough, there will be $\kappa_p$-periodic points on the fiber that are not reductions of points from $\mathcal{Z}_\infty(\varphi)$ (the orbit of the reduction of such a point must go through $\mathcal{Z}(\varphi_p)$).
Let $S$ be a smooth cubic surface defined over a number field $K$ and let $x,y\in{S(K)}$ be a good pair. If there exists a number $N$ such that the polynomial $\Theta_N(t) = \Phi_1(t)\cdots\Phi_N(t)$ has no roots over $K$, and for all but finitely many primes $p$ the polynomial $\Theta_N(t)$ has a root modulo $p$, then $D=(S, \varphi=t_xt_y, F)$ is strongly residually periodic.
\[thm:main-theorem-B\] Let $S$ be a smooth cubic surface defined over a number field $K$ and let $x,y\in{S(K)}$ be a good pair. The system $D=(X,\varphi=t_xt_y,F)$ is strongly residually periodic if and only if one of the following is true:
(a) There exists a $K$-fiber in the linear fibration through $x,y$ with finitely many rational points.
(b) There exists a positive integer $N$ such that $\Theta_N = \Phi_1 \cdots \Phi_N$ divided by all linear factors over $K$, has roots modulo all but finitely many primes $p$.
The if part is true by Propositions \[prop:srp-on-cubic-surface-cond1\] and \[prop:srp-on-cubic-surface-cond2\]. If $D$ is strongly residually periodic, then the periods are bounded, and this means there exists a positive integer $N$ such that $\Theta_N$ has roots modulo all but finitely many primes (by Proposition \[prop:cubic-theta-modulo\]). If we divide $\Theta_N$ by all the linear factors over $K$, and we have roots modulo all but finitely many primes, then the second condition is satisfied. Otherwise strong residual periodicity must be explained by a fiber that is defined over $K$, but if such a fiber has infinitely many periodic points, then the entire fiber is in the forbidden set – therefore if the second condition is not met the first condition must be true.
Finite generation of a subset of rational points on a smooth cubic surface {#section:mordell-weil}
==========================================================================
In this section we recall the Mordell–Weil problem for cubic surfaces, and prove a dynamical theorem in a Mordell–Weil flavor, using results we obtained in the previous sections.
Let $S$ be a smooth cubic surface defined over a field $k$. Given a subset of points $\mathcal{S}_0\subseteq{S(k)}$ we can define recursively an increasing sequence of sets $$\label{eqn:tangent-secant-sequence}
\mathcal{S}_0 \subseteq \mathcal{S}_1 \subseteq \mathcal{S}_2 \subseteq \ldots,$$ where $\mathcal{S}_{i+1}$ is defined by adding to $\mathcal{S}_i$ all the points of the form $w\circ{z}\in S(k)$ obtained by taking any two distinct points $w,z\in\mathcal{S}_i$. This is called *drawing secants* through the points in $\mathcal{S}_i$. We can modify this sequence by also including in $\mathcal{S}_{i+1}$ all the points $w\in{S(k)}$ such that $w\in{C_x(k)}$ (recall that $C_x=T_x(S)\cap{S}$) for some $x\in\mathcal{S}_{i}$. This is called *drawing tangents* through the points in $\mathcal{S}_i$. The *span* of $\mathcal{S}_0$ (denoted by $Span(\mathcal{S}_0$)) by tangents and secants (or only secants) is defined as the union of all the sets in the sequence in *(\[eqn:tangent-secant-sequence\])*.
Given a smooth cubic surface $S$ over a field $k$ such that $S(k)\neq\emptyset$, does there exist a finite subset $\mathcal{S}\subset\mathcal{S}(k)$ such that $Span(\mathcal{S}) = S(k)$? If so, then we say $S(k)$ is *finitely generated*.
The Mordell–Weil problem for cubic surfaces is still open, and partial results can be found in Manin [@article:manin1997] and Siksek [@article:siksek2012]. Of course, the answer to the question may very well depend on whether we allow tangents or not, and in fact Manin considers an alternative composition rule in the above-mentioned article.
We proceed to prove a Mordell–Weil like theorem for the set of $K$-periodic points of $t_xt_y$ where $K$ is a number field. In order to obtain “true" finite generation, we restrict ourselves to taking tangents only inside the nonsingular fibers of the linear fibration associated to $x$ and $y$.
\[thm:cubic-finitely-generated\] Let $K$ be a number field, $S/K$ a smooth cubic surface and $x,y\in{S(K)}$ a good pair. If there do not exist singular fibers in the linear fibration through $x,y$ that are periodic of finite period, then the set of $K$-periodic points of the birational automorphism $t_xt_y$ is finitely generated by secants and tangents.
By Corollary \[cor:cubic-surface-finite-union-of-fibers-contains-periodic\], there are only finitely many fibers of the linear fibration containing all periodic points. There are two types of such fibers: the first is a singular fiber containing only one singular point, which is fixed under $t_xt_y$ (the rest of the points on the fiber are non-periodic due to the assumption in the theorem). The second is a smooth fiber that is periodic. The first type is surely finitely generated, and the second is finitely generated by the Mordell–Weil theorem for elliptic curves (see Silverman [@book:silverman2009 Chapter VIII]). Thus we can choose a finite number of generators on each periodic fiber, take the union with all the singular points on the singular fibers, and we are done.
If $K=\mathbb{Q}$ in Theorem \[thm:cubic-finitely-generated\], and the resultants $Res(\Psi_1, \Psi_n)\neq{0}$ for $n=2,3,4,6$, then the set of $\mathbb{Q}$-periodic points of $t_xt_y$ is finitely generated.
The condition $Res(\Psi_1, \Psi_n)\neq{0}$ means there are no fibers that are both singular and of period $n$ (by Proposition \[prop:psi1-psi2\]). If a singular fiber $C$ of the linear fibration is periodic, then $y$ is of finite order in the group $(C_{ns}, x)$. Singular cubics over $\mathbb{Q}$ have group structure on $C_{ns}$ isomorphic to either $\mathbb{Q}^+$ or a subgroup of $L^*$, where $L$ is a quadratic extension of $\mathbb{Q}$ (see Silverman [@book:silverman2009 Exercise III.3.5]). In either case there are no non-trivial torsion elements of an order not in $\{2,3,4,6\}$.
We show in Example \[ex:cubic-singular-periodic-fiber\] that there exist dynamical systems on smooth cubic surfaces where $Res(\Psi_1, \Psi_2)=0$. Of course, this does not mean that the consequence of the corollary is untrue for such examples (i.e. the set of periodic points might still be finitely generated).
Examples
========
In this section we present some examples, illustrating the ideas and theorems presented in the paper. Finding the division polynomials associated with a good pair $x,y$ is unreasonable without the use of a computer. We have implemented the necessary [[Magma]{}]{} [@article:magma] functions to calculate the division polynomials associated with the birational map $t_xt_y$. The code package also contains functions that check the minimal residual periods of $t_xt_y$ for primes up to a given bound (this can provide an indication as to whether a given example is SRP or not). The code can be downloaded at the author’s website <http://www.wishcow.com>. The results of our package can be verified using the [[Magma]{}]{} code package provided by Brown and Ryder [@article:brown-ryder2010], which can generate Geiser involutions for a smooth cubic surface. We start with an example of a strongly residually periodic dynamical system on a smooth cubic surface:
\[ex:srp-cubic\] Let $S/\mathbb{Q}$ be the smooth cubic surface defined by the equation $$\label{equation:srp-cubic-equation}
S: YW^2 + Y^2Z-X^3-4Z^3 = 0.$$ The points $x=[0:1:0:0], y=[0:-2:1:0]$ are a good pair on $S$. We denote by $\varphi$ the birational automorphism $t_xt_y$ on $S$. The forbidden set is defined as in Section \[section:SRP-cubic-surface\]. We will show that the dynamical system $D=(S/\mathbb{Q}, \varphi, F)$ is $SRP(3)$, i.e. there exists a number $M$ such that for any prime $p>M$, the dynamical system has periodic points of period $3$ modulo $p$.
Intersecting the hyperplane $\{W=0\}$ with $S$ gives the cubic curve $C: Y^2Z-X^3-4Z^3,$ which is a smooth cubic. This hyperplane goes through the points $x$ and $y$, and therefore the curve $C$ is invariant under the map $t_xt_y$ (see Proposition \[prop:cubic-plane-section-invariant\]). Choosing the identity of the elliptic group structure to be $x = [0:1:0]$ (or more accurately, the point induced by $x$), we get an elliptic curve with Weierstrass form $v^2=u^3+4.$ Consulting the Cremona database of elliptic curves [@article:cremona2006], we see that $E=(C, x)$ is an elliptic curve with Mordell–Weil rank $0$, and the torsion subgroup is isomorphic to $\mathbb{Z}/3\mathbb{Z}$. The three rational points are $x=[0:1:0], y=[0:-2:1]$ and $z=[0:2:1]$. Since the group is $\mathbb{Z}/3\mathbb{Z}$, the point $y$ is of order $3$ on the elliptic curve. We have already seen (Proposition \[prop:periodic-point-on-fiber-txty\]) that the order of $y$ is the same as the order of $t_xt_y$ in the group $Bir(E)$. Thus we get that even though $t_xt_y$ is of infinite order in $Bir(S)$, it is of finite order $3$ when restricted to the hyperplane section. By Proposition \[prop:srp-on-cubic-surface-cond1\] we get that the dynamical system $D$ is SRP(3).
Let us describe a method for constructing examples with desired dynamical properties (e.g. SRP). We choose an elliptic curve $E: v^2=u^3+Au+B$ with particular properties, for instance with a finite number of rational points as was done in Example \[ex:srp-cubic\]. We then choose two points $x,y\in{E(K)}$ to play the roles of the points $x,y$ on the surface $S$, and search for a smooth surface $S$ containing this curve. We can do this by running over surfaces of the form $$S: W\cdot{H(X,Y,Z,W)}-Y^2Z-X^3-AXZ^2-BZ^3=0$$ where $H$ is a quadratic form, and searching for ones that are smooth and in which $x,y$ is a good pair. Then this surface will have $C=\{W=0\}\cap{S}$ as a fiber in the linear fibration induced by $x,y$, and this fiber $C$ is exactly our curve $E$.
\[ex:srp-cubic-contd\] We continue Example \[ex:srp-cubic\], by proving that the dynamical system $D$ has no periodic points over $\mathbb{Q}$. We use this example to illustrate the method of using the division polynomials of the linear fibration induced by the good pair $x,y$. To find the cubic pencil of the linear fibration of $S$ induced by $x,y$, we set $W=tX$ in equation (\[equation:srp-cubic-equation\]). This is because the line passing through $x,y$ is $\mathcal{L}(x,y)=\{W=0,X=0\}$. We get the cubic pencil $C: t^2X^2Y+Y^2Z-X^3-4Z^3=0.$
The fiber at infinity that we have removed is $X=0$, which is the curve $C_\infty :YW^2 + Y^2Z-4Z^3=0.$ We bring this curve to Weierstrass form and get $E_\infty: v^2 = u^3 - 4u.$ This curve can be checked in the Cremona database [@article:cremona2006] to have Mordell–Weil rank $0$, and torsion subgroup $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. All four rational points on $C_\infty$ can be checked to be in $\mathcal{Z}_\infty(t_xt_y)$ (and therefore are not periodic and are in the forbidden set). The point $y$ has order $2$ in the group $E_\infty(\mathbb{Q})$, so that by Proposition \[prop:srp-on-cubic-surface-cond1\] the dynamical system $D$ is in fact $SRP(2)$.
We can check that the Weierstrass form of the cubic pencil $C$ is $E: v^2=u^3-4t^4u+4.$ The discriminant of $E$ is $\Delta(t) = 4096t^{12}-6912.$ The zeros of the discriminant, as a polynomial in $t$, give us $12$ distinct singular fibers of the linear fibration, all of which are nodal cubics (the coefficient of $u$ is nonzero at all roots of $\Delta(t)$, see Silverman [@book:silverman2009 Proposition III.1.4]). None of the roots of $\Delta(t)$ are rational, so there are no fixed points of $t_xt_y$ defined over $\mathbb{Q}$ (See Corollary \[cor:txty-fixed-iff-singular\]).
The dynamical system is $SRP(1)$ only if the polynomial $\Delta$ has roots modulo all but finitely many primes $p$. We show this is not true. We can check using [[Magma]{}]{} [@article:magma] that the Galois group of the polynomial $\Delta$ has an element of order $12$. Then by the Frobenius density theorem (see Lenstra and Stevenhagen [@article:lenstra-stevenhagen1996 page 32]), there are infinitely many primes $p$ for which the polynomial $\Delta$ remains irreducible when reduced modulo $p$, so that residual periodicity cannot be explained by the fixed points.
To get a fiber of the linear fibration that is periodic of period $2$, we need the $v$ coordinate of the point $y$ to be $0$ when evaluated at $t$. However, the image of $y$ in $E$ is $[0:2:1]$, so we get that outside of the fiber at $\infty$ there are no fibers of period $2$.
We can calculate the division polynomials $\Psi_n, n\geq{3}$ for $t_xt_y$. To find them we take the division polynomials $\psi_n$ for $E$, and evaluate them at $v=0$, since $y=[0:2:1]$. We only need the polynomials $\Psi_n(t)$ for $n\leq{12}, n\neq{11}$ (See Proposition \[prop:bounded-number-of-periodic-fibers\] for why we can skip $\Psi_{11}$, and stop at $\Psi_{12}$). The polynomials are quite large so we do not list them here. A quick check shows that none of these polynomials have roots in $\mathbb{Q}$ outside of $t=0$, which is the fiber of period $3$. This means that there are no fibers of finite period $n\geq{3}$. As we have checked all possibilities, we have proved that $t_xt_y$ has no periodic points over $\mathbb{Q}$.
\[ex:cubic-infinite-rational-periodic-points\] We show an example of a dynamical system on a smooth cubic surface that has infinitely many periodic points. Let $S/\mathbb{Q}$ be the smooth cubic surface defined by the equation $$S: X^3 - 3024XZ^2 - Y^2Z - YW^2 + 81216Z^3 = 0.$$ The points $x=[0,1,0,0], y=[12,216,1,0]$ form a good pair on $S$. The hyperplane section $C = \{W=0\}\cap{S}$ is a fiber of the linear fibration induced by $x,y$. This cubic curve has the Weierstrass form $E: v^2=u^3-3024u+81216.$ This curve can be checked in the Cremona database [@article:cremona2006] to have Mordell–Weil rank $1$, and $y=(12,216)$ has order $3$ in the elliptic curve $E$. This means that the fiber $C$ is periodic of period $3$ under $t_xt_y$, which proves there are infinitely many $\mathbb{Q}$-periodic points for $t_xt_y$ on $S$.
\[ex:cubic-singular-periodic-fiber\] We show an example of a dynamical system on a smooth cubic surface with a singular fiber in the linear fibration containing infinitely many periodic points. This example demonstrates that the condition in Theorem \[thm:cubic-finitely-generated\] is not redundant (in the sense that such systems exist, not that the condition is necessary). Let $S/\mathbb{Q}$ be the smooth cubic surface defined by the equation $$S: X^3 + X^2Z - XYW - Y^2Z - Z^2W - W^3 = 0.$$ The points $x=[0,1,0,0], y=[-1,0,1,0]$ form a good pair on $S$. The hyperplane section $C = \{W=0\}\cap{S}$ is a fiber of the linear fibration induced by $x,y$. This cubic curve has the Weierstrass form $E: v^2=u^3+u^2,$ which is the classic nodal cubic. The point $y=(-1,0)$ on $E$ is the unique point of order $2$ of this curve. This means that $S$ has a singular curve of period $2$ under $t_xt_y$.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
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} |
---
abstract: 'We present first results from pilot observations using a phased array feed (PAF) mounted on the Parkes 64-m radio telescope. The observations presented here cover a frequency range from 1150 to 1480MHz and are used to show the ability of PAFs to suppress standing wave problems by a factor of $\sim10$ which afflict normal feeds. We also compare our results with previous HIPASS observations and with previous H<span style="font-variant:small-caps;">i</span> images of the Large Magellanic Cloud. Drift scan observations of the GAMA G23 field resulted in direct H<span style="font-variant:small-caps;">i</span> detections at $z=0.0043$ and $z=0.0055$ of HIPASS galaxies J2242-30 and J2309-30. Our new measurements generally agree with archival data in spectral shape and flux density, with small differences being due to differing beam patterns. We also detect signal in the stacked H<span style="font-variant:small-caps;">i</span> data of 1094 individually undetected galaxies in the GAMA G23 field in the redshift range $0.05 \leq z \leq 0.075$. Finally, we use the low standing wave ripple and wide bandwidth of the PAF to set a $3\sigma$ upper limit to any positronium recombination line emission from the Galactic Centre of $<0.09$K, corresponding to a recombination rate of $<3.0\times10^{45}\,\mathrm{s}^{-1}$.'
author:
- 'T.N. Reynolds$^{1,2}$[^1], L. Staveley-Smith$^{1,2}$, J. Rhee$^{1,2}$, T. Westmeier$^{1}$, A.P. Chippendale$^{3}$, X. Deng$^{3,4}$, R.D. Ekers$^{3}$, M. Kramer$^4$\'
bibliography:
- 'master.bib'
title: 'Spectral-line Observations Using a Phased Array Feed on the Parkes Telescope'
---
instrumentation: radio telescopes, single dish, instrumentation, extragalactic
INTRODUCTION {#sec:intro}
=============
At centimetre wavelengths, the prime requirements for a sensitive radio telescope include: a large collecting area; low receiver noise; wide bandwidth; good polarisation characteristics; and immunity to radio-frequency interference (RFI). In addition, the ability to quickly survey large areas of sky requires a wide field of view, and the ability to resolve fine details requires a large diameter, or long baselines. The diversity of recent radio telescope design demonstrates that there is no unique solution to the optimum radio telescope design. For example, the Five-hundred meter Aperture Spherical radio-Telescope (FAST) [@Nan2011] combines a multi-feed array with a large monolithic aperture to achieve its science goals, whereas the South African MeerKAT array [@Jonas2009] use arrays of small dishes to achieve a good compromise between sensitivity and field of view.
However, recent developments in Phased Array Feed (PAF) technology mean that the traditional radio telescope feed, usually a large horn-like structure, is no longer the only choice of receptor. Traditional feeds can be large structures, which have low efficiencies, often have low bandwidths, and fundamentally cannot fully sample the sky at any instant. Several groups have therefore experimented with PAFs and the closely-related aperture array technologies [@vanArdenne2010; @Roshi2015; @Warnick2016]. PAFs typically consist of simple receptors closely packed on the focal plane. The voltages from these receptors are then combined in a manner that uniformly illuminates the aperture with higher efficiency than can usually be achieved by conventional means. This approach has been adopted in the CSIRO Australian SKA Pathfinder (ASKAP) telescope [@DeBoer2009; @Hotan2014; @Schinckel2016] and the ASTRON WSRT/APERTIF upgrade [@Oosterloo2009].
Recently, the Max-Planck Institute for Radio Astronomy (MPIfR) procured a CSIRO-built PAF for use on the Effelsberg 100-m telescope. Prior to its installation at Effelsberg, the PAF (henceforth MPIPAF) was installed on the Parkes 64-m telescope for a 6 month commissioning workout. The MPIPAF is the Mk II version from the ASKAP Design Enhancements [ADE, @Hampson2012] project as currently deployed on ASKAP, with some minor physical and electronic modifications designed to make it suitable for installation at Parkes and Effelsberg [@Chippendale2016]. In particular, the radio frequency interference (RFI) environment at these two sites is inferior to that of the ASKAP site [@Chippendale2013; @Indermuehle2016], which lies in the radio-quiet zone of the Murchison Radio-astronomy Observatory [@Wilson2013].
The MPIPAF consists of connected dipoles in a chequerboard pattern [@Hay2008] and can form up to 36 dual-polarisation beams on the sky, simultaneously covering a significantly larger area in a single pointing than traditional receivers. The increased sky coverage permits larger areas of sky to be surveyed to similar sensitivity as traditional receivers, but with less observing time for cryogenically-cooled PAFs. However, one limitation to using a PAF is the increased computing requirements for forming beams and storing and processing the additional data. Only 16 beams were formed at full spectral resolution in our commissioning observations using a firmware module primarily designed for engineering verification of the ASKAP beamformer.
Another limitation of the MPIPAF is that the measured system temperature-efficiency ratio $T_{sys}/\eta \approx 65$K [@Chippendale2016] is a factor of 1.6 times higher than the existing cryogenically-cooled 13-beam receiver at Parkes [@Staveley-Smith1996]. However, this is offset by the larger bandwidth and greater number of beams available. Moreover, the main purpose of the current observations was to test the viability of PAFs in large single dish reflectors and to assess their performance prior to permanently installing a cooled, high-performance version on the Parkes telescope in the future. In this work, we therefore focus on the assessment of the performance of the PAF for spectral line (mainly H<span style="font-variant:small-caps;">i</span>) observations. The MPIPAF has also been assessed for studying pulsars by [@Deng2017].
The first part of our tests focussed on neutral hydrogen (H<span style="font-variant:small-caps;">i</span>) observations of the Large Magellanic Cloud (LMC). Being the closest massive gas-rich galaxy to the Milky Way Galaxy at $\sim50$kpc, it has been the subject of much observational study, including in H<span style="font-variant:small-caps;">i</span> with the Parkes 64-m telescope [@Mcgee1966; @Bruens2005; @Staveley-Smith2003], the Australia Telescope Compact Array (ATCA) [@Kim2003], and other telescopes. We use the Parkes H<span style="font-variant:small-caps;">i</span> survey of the LMC of [@Staveley-Smith2003] as a reference dataset for comparison with our MPIPAF observations.
The second part of our tests focused on observations of extragalactic H<span style="font-variant:small-caps;">i</span>, which is one of the keys to understanding galaxy evolution over cosmic time. At low redshifts ($z\lesssim0.1$), we can detect and measure H<span style="font-variant:small-caps;">i</span> in large numbers of individual galaxies through H<span style="font-variant:small-caps;">i</span> spectroscopy, which involves detection of the redshifted $\lambda$21cm line. An example is the H<span style="font-variant:small-caps;">i</span> Parkes All-Sky Survey [HIPASS, @Barnes2001] which provided a census of southern gas-rich galaxies at $z<0.04$. At higher redshifts, only the very brightest, most massive galaxies will be detected, while the average population of less massive galaxies is too faint to be detected above the telescope noise [@Catinella2008]. However, the H<span style="font-variant:small-caps;">i</span> spectra from optically identified galaxies without individual H<span style="font-variant:small-caps;">i</span> detections can be stacked to detect the average H<span style="font-variant:small-caps;">i</span> emission, as Gaussian noise fluctuations will decrease, leaving the averaged stacked H<span style="font-variant:small-caps;">i</span> signal [e.g.: @Lah2007; @Fabello2011; @Delhaize2013; @Gereb2014; @Gereb2015; @Brown2015; @Rhee2013; @Rhee2016; @Kleiner2016]. For example, [@Delhaize2013] found stacked detections in both the South Galactic Pole and HIPASS data sets, observed on the Parkes 64-m telescope over redshift ranges $0.0405<z<0.1319$ and $z<0.0025$, respectively, using spectral stacking. Our extragalactic observations were designed to examine how well the MPIPAF was able to reproduce existing data, and whether there is any systematic noise floor that prevents detection of weak or distant spectral features.
Finally, the broad bandwidth of the MPIPAF (0.7 to 1.8 GHz) opens up two new science areas. One is ‘intensity mapping’, which is a technique to detect the summed spectral emission from distant galaxies through their power or cross-power spectra [@Pen2009]. We report on these observations in a separate paper. The other is detection of recombination-line emission from positronium in the Galactic Centre. Positronium is an exotic atom composed of an electron and positron first detected in the laboratory by [@Canter1975]. [@Leventhal1978] made the first $\gamma$-ray detection of positronium annihilation from the Galactic Centre [for a review of astronomical positronium studies, see @Ellis2009]. Radio recombination lines (RRLs) of positronium have not yet been detected with radio telescopes [@Anantharamaiah1989]. RRLs can be used to derive properties of diffuse gas within galaxies (e.g.: temperature and density) and are regularly observed for elements such as hydrogen, helium and carbon. The RRL frequencies of positronium can be calculated using the usual Rydberg formula.
This paper is structured as follows. We describe the MPIPAF observations on the Parkes radio telescope, and the data reduction pipeline in Section \[sec:data\]. In section \[sec:results\] we present our results including an examination of standing waves, the system temperature, and RFI in the data. We present a comparison of the data with previous observations of the LMC and individual HIPASS galaxies. We also stack H<span style="font-variant:small-caps;">i</span> spectra for galaxies in the GAMA G23 field and stack hydrogen and positronium recombination-line spectra in the region of the Galactic Centre. In Section \[sec:conclusions\] we present our conclusions. Throughout, we use J2000 coordinates, dates in UTC and adopt a flat $\Lambda$CDM cosmology using ($h$, $\Omega_{\mathrm m}$, $\Omega_{\mathrm b}$, $\Omega_\Lambda$, $\sigma_8$, $n_{\mathrm s}$) = (0.702, 0.275, 0.0446, 0.725, 0.816, 0.968), concordant with the latest *WMAP* and *Planck* results [@Bennett2013; @Planck2015].
THE DATA {#sec:data}
========
Observations {#s-sec:observations}
------------
The observations were taken using ‘Band 2’ of the MPIPAF mounted on the Parkes 64-m radio telescope, which covers a useful band from 1200 to 1500MHz in two orthogonal linear polarisations. Observations were made of the LMC ($63.75^{\circ}\leq \alpha \leq 93.75^{\circ}$, $-70.5^{\circ}\leq \delta \leq -67^{\circ}$; J2000), the footprint of the GAMA survey field G23 ($339^{\circ}\leq \alpha \leq 351^{\circ}$, $-35^{\circ}\leq \delta \leq -30^{\circ}$; J2000), the Circinus galaxy, NGC6744 and the Galactic Centre ($\alpha,\,\delta= 17$:45:40.4,$-$29:00:28.1; J2000). The LMC was observed on 2016 October 6 and 24-25, the G23 field on 2016 September 2-3 and October 4-6 and 24-25, the Circinus galaxy on 2016 August 3, NGC6744 on 2016 September 1 and the Galactic Centre on 2016 September 1. The MPIPAF beamformer output 17 beams, 16 of which were able to be used for these observations (see beam footprint in Figure \[fig:footprint\]). The beam offsets from the central beam in the footprint were set in the beam weights using a pitch of 0.25$^{\circ}$ or 0.35$^{\circ}$. The MPIPAF specifications and observations are summarized in Table \[table:paf\_params\] and Table \[table:obs\_date\], respectively. The LMC and G23 observations were taken using drift scan mode (fixed azimuth and elevation), while the Circinus galaxy, NGC6744 and the Galactic Centre observations were taken using on-off source pointings. Prior to each observation, the calibrator PKS1934-638 was also observed[^2]. We calibrated the flux density using the PKS1934-638 flux model from [@Reynolds1994].
![Footprint of the MPIPAF beams with a pitch of 0.25$^{\circ}$. Only 16 of the beams were used for the observations, the beam labelled 0, dashed circle, was not used.[]{data-label="fig:footprint"}](reynolds_fig1.pdf){width="\columnwidth"}
[@cc@]{} Parameter & Band 2 Values\
\
Bandwidth & 384MHz\
Central Frequency & 1340MHz\
Spectral Resolution & 18.5kHz\
Cycle Time & 4.5s\
Polarisations & 2\
Beams & 16\
[@cccc@]{} Target & Date (2016) & Scan & Integration\
& & Type & Time\
\
LMC & Oct 6, 24-25 & Drift & 7200s\
G23 Field & Sept 2-3 & Drift & 2880s\
& Oct 4-6, 24-25 & &\
Circinus & Aug 3 & ON-OFF & 90s\
NGC6744 & Sept 1 & ON-OFF & 90s\
Galactic & Sept 1 & ON-OFF & 90s\
Centre & & &\
Additionally, we used archival data cubes from the first HIPASS data release [@Meyer2004] and archival Parkes multibeam and ATCA data of the LMC [@Staveley-Smith2003] for comparison with our HIPASS source and LMC observations, respectively.
We also require optical position and redshift information for potential H<span style="font-variant:small-caps;">i</span> sources to attempt blind H<span style="font-variant:small-caps;">i</span> stacking. We queried the NASA/IPAC Extragalactic Database (NED)[^3] to obtain position and redshift information for optically detected sources within the G23 field, which returned source redshift and positions from the 2DFGRS, GALEXASC, GALEXMSC and 2MASX surveys.
Data Reduction {#s-sec:reduction}
--------------
We reduced the data using the data reduction and gridding packages <span style="font-variant:small-caps;">livedata</span> and <span style="font-variant:small-caps;">gridzilla</span>[^4] [for a description of <span style="font-variant:small-caps;">livedata</span> and <span style="font-variant:small-caps;">gridzilla</span>, see @Barnes2001]. Both <span style="font-variant:small-caps;">livedata</span> and <span style="font-variant:small-caps;">gridzilla</span> are designed for reducing Parkes multibeam data, which are in single dish <span style="font-variant:small-caps;">fits</span> format, <span style="font-variant:small-caps;">sdfits</span>. However, the raw MPIPAF data files are in <span style="font-variant:small-caps;">hdf5</span> format, so we first converted the <span style="font-variant:small-caps;">hdf5</span> files to <span style="font-variant:small-caps;">sdfits</span> format using the <span style="font-variant:small-caps;">python</span> package <span style="font-variant:small-caps;">fits2hdf</span> [@Price2015]. We also separated each of the 16 MPIPAF beams in the raw <span style="font-variant:small-caps;">hdf5</span> data files into separate <span style="font-variant:small-caps;">sdfits</span> files, as <span style="font-variant:small-caps;">livedata</span> can only conveniently handle up to 13 beams simultaneously (the number of beams in the Parkes multibeam receiver), and reduced each beam separately.
We performed bandpass correction on both the LMC and G23 field data with <span style="font-variant:small-caps;">livedata</span>. Prior to performing bandpass correction, we used PKS1934-638 to calibrate the flux density scale of the data. We smoothed both data sets using Hanning smoothing and performed bandpass calibration using a second-order robust polynomial and the <span style="font-variant:small-caps;">extended</span> and <span style="font-variant:small-caps;">compact</span> calibration methods for the LMC and G23 data, respectively.
After correcting the bandpass, we gridded the reduced data using <span style="font-variant:small-caps;">gridzilla</span>. We gridded the data using a weighted median, smoothed the data using a Gaussian kernel with full width half maximum (FWHM) of 6 arcmin and a cutoff radius of 13 arcmin and combined the two polarisations. Our final data cubes have a pixel scale of 4 arcmin by 4 arcmin and a spectral resolution of 18.5kHz.
The Galactic Centre and targeted HIPASS source on-off observations were not reduced using <span style="font-variant:small-caps;">livedata</span>. We reduced these on-off data separately using the on-source and off-source pointings, $$\begin{array}{l}
\displaystyle S_{\nu}=\left(\frac{P_{\mathrm{on},\nu}}{P_{\mathrm{off},\nu}}-1\right)T_{\mathrm{sys}}(\nu),
\end{array}
\label{eq:on_off}$$ where $T_{\mathrm{sys}}(\nu)$ is the system temperature determined from observing the calibrator PKS1934-638 prior to observing the science target, $P_{\mathrm{on},\nu}$ is the on-source pointing and $P_{\mathrm{off},\nu}$ is the off-source pointing. We accounted for the frequency dependence of $T_{\mathrm{sys}}$.
The one exception to this is the Circinus galaxy, which did not have a calibrator observation prior to or after the science observation on August 5. For Circinus, we used a PKS1934-638 observation from August 8 as the nearest calibrator observation. However, we believe that the calibration is reliable as only the central beam (beam 8) was used for this observation which was fairly stable over the August and September observations (see Section \[s-sec:tsys\] and Figure \[fig:tsys\]).
We then used <span style="font-variant:small-caps;">gridzilla</span> to grid the reduced Galactic Centre observation similarly to the LMC and G23 field data. However, we gridded each beam separately, using <span style="font-variant:small-caps;">gridzilla</span>’s weighted median statistic, <span style="font-variant:small-caps;">WGTMED</span>, for RFI suppression and used a top-hat smoothing kernel with a FWHM and cutoff radius of 12 and 6 arcmin, respectively. We did not use <span style="font-variant:small-caps;">gridzilla</span> for the targeted HIPASS sources as they lie in RFI free regions of the spectrum and simply combined the two polarisations from each beam with a <span style="font-variant:small-caps;">python</span> script.
RESULTS {#sec:results}
=======
Standing Waves {#s-sec:waves}
--------------
Standing waves are introduced as a result of broadband signals entering the telescope along multiple paths and creating an interference pattern. The principal (‘on-source’) standing wave at Parkes is created by radiation reflecting from the feed towards the apex of the telescope. The frequency interval of this standing wave is $c/2F$, where $F$ is the focal length (Parkes $F/D=0.41$), and corresponds to 5.6MHz at Parkes. Reflections off other parts of the dish and feed support legs result in standing waves at other frequencies. Standing waves from off-source interference can be even more complex. As with other baseline artefacts, the effect of standing waves can be mitigated by careful calibration. However, this becomes more difficult if the phase or amplitude of the wave shifts over time [@Briggs1997]. Both [@Delhaize2013] and [@Kleiner2016] noted the presence of standing waves in Parkes multibeam data and further corrected by fitting and subtracting high-order polynomials from their spectra. Standing waves can even be problematic for radio interferometers [@Popping2008].
These results can be understood by noting that the amplitude, $a$, of the standing wave relative to the power in the direct signal, $A$, is $a/A=2\gamma$, where $\gamma$ is the voltage ratio of the delayed and undelayed signals. So a scattered power of only 0.01% will give rise to a standing wave amplitude ratio of 2% for the multibeam, and a scattered power of only 0.0001% will result in an amplitude ratio of 0.2% for the MPIPAF. The large apparent difference in the reflection coefficient of the two receivers ($\sim100$) is partly a result of the increased efficiency of the MPIPAF. The higher efficiency of the MPIPAF leaves less energy available for multipath reflections from the feed, but this can explain at most a 1.4 times reduction in reflected power compared to the multibeam, as suggested from a comparison of the measured multibeam [@Staveley-Smith1996] and MPIPAF [@Chippendale2016] feed efficiencies of 50-64% and 64-75%.
However, this cannot be the only factor. We must also consider that the MPIPAF fully samples the focal plane, such that neighbouring beams, which are separated by 15 arcmin on the sky, also have the same high efficiency. On the other hand, the multibeam feed array undersamples the aperture plane, and neighbouring beams are separated by 28 arcmin, or $\sim 2$ beamwidths [@Staveley-Smith1996]. Therefore, the efficiency of this receiver for hypothetical beams separated by 15 arcmin is effectively zero – i.e. most incident radiation is reflected. The low MPIPAF standing wave amplitude must therefore be a result of the low amounts of power reflected from the focal area around a given beam, and not just the power reflected from the beam itself. A more accurate analysis of the excellent standing wave performance of the MPIPAF relative to the traditional multibeam array requires a full electromagnetic simulation and diffraction analysis, and is outside the scope of this paper.
Modulation of the primary beam pattern by standing waves on interferometers is a related problem [@Popping2008]. Using APERTIF on the Westerbork Synthesis Radio Telescope, [@Oosterloo2010] have shown that this can also been suppressed using PAFs. Aside from the lower standing wave, additional suppression can be obtained by utilising the frequency flexibility inherent in beamforming. For the MPIPAF, the beamformer weights can be adjusted in 1MHz sub-bands. As the primary reflection standing wave period of 5.6MHz is well sampled by the 1MHz resolution of the beamformer and the 90mm spatial sampling period of the MPIPAF chequerboard at the focal plane (spacing between PAF element feeds) it may be possible to suppress the standing wave even further via more advanced beamforming techniques in the future.
![Sample PKS1934-638 spectra prior to bandpass correction from the MPIPAF (blue) and multibeam (brown) central beam, panel (a), and an off-axis beam, panel (b). The angular offsets of the off-axis beams, MPIPAF beam 1 and multibeam beam 10, are $\sim0.5^{\circ}$ and $\sim1.5^{\circ}$, respectively. The MPIPAF spectra have been offset by 7 and 27Jy (the median difference between the MPIPAF and multibeam spectra for the central and off-axis beams, respectively) for ease of comparison with the multibeam spectra.[]{data-label="fig:standing_wave_spec"}](reynolds_fig2.pdf){width="\columnwidth"}
![Amplitude of the power in the standing wave spectral feature in the PKS1934-638 spectra in Figure \[fig:standing\_wave\_spec\] from Fourier analysis. The amplitude of the standing wave is shown for the MPIPAF (blue) and multibeam (brown) spectra in the central beam, panel (a), and an outer beam, panel (b). The vertical dashed lines indicate 5.5MHz and 5.7MHz (black and grey, respectively).[]{data-label="fig:standing_wave_amp"}](reynolds_fig3.pdf){width="\columnwidth"}
System Temperature {#s-sec:tsys}
------------------
We determined the system temperature, $T_{\mathrm{sys}}$ using the calibrator observation of PKS1934-638 directly preceding the target observations. [@Chippendale2016] determined $T_{\mathrm{sys}}/\eta$ for the MPIPAF to be in the range $\sim45-60$K and to be relatively stable between the two measured polarisations from observations with the dish pointing near zenith. In the three months of observing (August-October), however, we find $T_{\mathrm{sys}}/\eta$ to be significantly higher (increasing with decreasing frequency) and less stable, ranging from $\sim70-140$K (mostly between$\sim70-110$K), with significant variation between the two polarisations and the observation dates and beams (see Figures \[fig:tsys\_comparison\] and \[fig:tsys\]). We also note that adjacent beams are separated by $0.25^{\circ}$, or $\sim 1$ beamwidth, so there is some overlap. We measure a mean correlation in the spectral noise between adjacent beams, in the same linear polarisation, of $\sim 10$%. As expected, this is slightly less than the level of correlation of $13-20\%$ previously measured for adjacent ASKAP BETA beams, which are separated by $0.78^{\circ}$, or $\sim 0.7$ beamwidths [@Serra2015].
Some variation in $T_{\mathrm{sys}}/\eta$ is to be expected as new beam weights were not made for each spectral line observation and the state of the hardware was not carefully controlled between observations. The probable cause of the $T_{\mathrm{sys}}/\eta$ variation are delay slips (mostly single-sample) between different ports of the MPIPAF digitiser when the digital receiver is power cycled [@Bannister2015]. In a production system such as ASKAP or Bonn, this can be calibrated using an on-dish noise source.
The central beam (beam 8) appears to be one of the most stable beams, remaining roughly constant during the August and September observations and at a different value for the October observations (dashed and solid lines Figure \[fig:tsys\], respectively). The $T_{\mathrm{sys}}$ values for polarisation A were slightly more constant than those for polarisation B. If a PAF is permanently installed on the Parkes telescope, the $T_{\mathrm{sys}}/\eta$ noise can be lowered below the [@Chippendale2016] values by cooling the receiver systems. Simulations suggest that a next-generation ‘rocket’ design can achieve $T_{\mathrm{sys}}/\eta=20-25$K [@Dunning2016], which is a 40K improvement on the MPIPAF and a $15-20$K improvement on the Parkes multibeam receiver.
![Comparison of polarisation A and B $T_{\mathrm{sys}}/\eta$ variation with frequency for the central beam (beam 8) from October 4 observations with the test values upon installation on Parkes [@Chippendale2016]. The higher values measured by us at low frequencies are a result of the beamformer delay slips noted in the main text.[]{data-label="fig:tsys_comparison"}](reynolds_fig4.pdf){width="\columnwidth"}
![Polarisation A and B $T_{\mathrm{sys}}/\eta$ average values over the frequency range $\nu=1400-1420$MHz in each of the 16 beams over observations from August, September and October (panels (a) and (b), respectively). August and September are dashed lines and October are solid lines.[]{data-label="fig:tsys"}](reynolds_fig5.pdf){width="\columnwidth"}
Radio Frequency Interference {#s-sec:rfi}
----------------------------
There is significant RFI present in the observations, as the Parkes telescope is located in New South Wales and suffers interference from radio, television, mobile phones as well as satellites. In the frequency range of our observations, the main contributors are the navigational satellite signals (e.g. GPS) at $\nu<1290$MHz (see Figure \[fig:rfimit\_spec\]). This is in contrast to the excellent RFI situation at the Murchison Radio-astronomy Observatory (MRO) where, with the exception of satellite RFI and tropospheric ducting events [@Indermuehle2016], low-frequency observations are largely free of terrestrial contamination.
Since PAFs can fully sample the focal plane and have flexible beamforming capability, they are perfect for the application of advanced RFI mitigation and suppression techniques [see @Fridman2001 for an overview]. Indeed, the ASKAP Boolardy Engineering Test Array (BETA) was used to test one of these RFI mitigation methodologies, a spatial filtering technique based on projecting out the interferer signature [@Hellbourg2012]. This test demonstrated the effectiveness of the projection algorithm in suppressing RFI contamination in ASKAP PAF data [@Hellbourg2016]. This success encouraged us to attempt a similar technique for the MPIPAF data taken at the Parkes site. Full mitigation typically requires much higher temporal and spectral resolution than we have in the MPIPAF data ($\sim$microseconds and $\sim$kHz vs. 4.5seconds and 18.5kHz, respectively). However, we were able to demonstrate again the power of the projection method in some of our observations [Figure \[fig:rfimit\_spec\] shows spectra taken before (red) and after (blue) applying RFI mitigation. See @Chippendale2017 for further details].
For most of our observations, RFI contamination was removed from the spectra using more conventional threshold flagging techniques, by excluding individual spectral channels with flux density, $S_{\nu_{\mathrm{obs}}}>5\sigma_i$, where $\sigma_i$ is the channel RMS noise. We excluded all optical sources with redshifts placing them within the frequency range of GNSS satellites ($1240-1252$MHz). We also excluded, by manual inspection, all spectra containing occasional GPS L3 emissions at $1376-1384$MHz. We did not exclude sources falling within the receiver breakthrough at $\sim1350$MHz as this appeared relatively stable and, unlike other RFI regions had a reasonably low channel RMS ($\sim0.06-0.08$Jy, c.f. clean spectral channels RMS $\sim0.04$Jy).
The Large Magellanic Cloud {#s-sec:lmc}
--------------------------
The LMC was observed as a check of the accuracy of the flux density and frequency calibration of the MPIPAF, as well as a basic check of the reduction pipeline. We compare our results with accurate archival observations from [@Staveley-Smith2003], which used the Parkes multibeam receiver. As the observed LMC emission is over the range $\nu\sim1418-1419.5$MHz, the LMC observations are in the RFI free section of the band and we can use all channels containing LMC emission.
We gridded the MPIPAF data without any further calibration or adjustment except to apply a Gaussian smooth of $\mathrm{FWHM}=7$arcmin to remove small residual scanning lines which can still be seen faintly in Figure \[fig:lmc\_mom0\] at $-68^{\circ}<\delta<-67^{\circ}$. Unlike the previous multibeam observations, no cross-scans were taken to mitigate against such artefacts.
In Figure \[fig:lmc\_mom0\], we compare the MPIPAF column density map with contours from the archival multibeam observations from [@Staveley-Smith2003]. The multibeam contours match well with the MPIPAF image. One thing of note in our map with the MPIPAF data are the edge effects at the top and bottom of the map which are due to the gridding process under/over-estimating the flux along the edges.
Figure \[fig:lmc\_pixels\] shows a comparison of the brightness temperature values in the individual pixels of the MPIPAF and multibeam image cubes, excluding any boundary regions. There is excellent agreement, with the MPIPAF temperature having a small ($\sim1.25$K) zero-point offset. The zero-point offset is mainly due to the in-scan bandpass calibration procedure adopted. This has the effect of removing uniform background emission.
![Pixel-by-pixel comparison of temperatures in the MPIPAF and multibeam image cubes. The number of pixels compared is 96,000. The line of best fit is shown in solid red. The shading indicates the data point density, with lighter shading indicating increasing density.[]{data-label="fig:lmc_pixels"}](reynolds_fig8.pdf){width="\columnwidth"}
GAMA G23 Field {#s-sec:gama}
--------------
We extracted the spectrum for each optically identified galaxy listed in NED for the G23 field in the redshift range $0.003\leq z\leq0.23$, covering the available bandpass of the MPIPAF band 2. In each channel, we averaged the flux from a 9 pixel box ($3\times3$pixel $-$ $12\times12$arcmin) centred on the galaxy to ensure we did not lose any flux. We then extracted 600 channels ($\sim11$MHz) around the central redshifted frequency for each galaxy.
### HI Stacking {#ss-sec:stacking}
We perform H<span style="font-variant:small-caps;">i</span> stacking using the H<span style="font-variant:small-caps;">i</span> mass spectra rather than the originally extracted flux density spectra. We computed the observed-frame H<span style="font-variant:small-caps;">i</span> mass spectrum following Equation 1 from [@Delhaize2013], $$\begin{array}{l}
\displaystyle \frac{M_{\mathrm{H\,\textsc{i},\nu_{\mathrm{obs}}}}}{M_{\odot}\,\mathrm{MHz}^{-1}}=4.98\times10^7\left(\frac{S_{\nu_{\mathrm{obs}}}}{\mathrm{Jy}}\right)\left(\frac{D_L}{\mathrm{Mpc}}\right)^2,
\end{array}
\label{eq:mass_spec}$$ where $S_{\nu_{\mathrm{obs}}}$ is the observed-frame flux density and $D_L$ is the luminosity distance.
To stack spectra, all spectra must be shifted and aligned at the rest frequency, 1420.406MHz. We do this by shifting the spectral axis from observed to rest frame (i.e. $\nu_{\mathrm{rest}}=\nu_{\mathrm{obs}}(1+z)$) and to conserve total mass, $$\begin{array}{l}
\displaystyle M_{\mathrm{H\,\textsc{i},\nu_{\mathrm{rest}}}}=\frac{M_{\mathrm{H\,\textsc{i}},\nu_{\mathrm{obs}}}}{1+z}.
\end{array}
\label{eq:shift_spec}$$
We compute the stacked H<span style="font-variant:small-caps;">i</span> mass spectrum as done by [@Delhaize2013], $$\begin{array}{l}
\displaystyle M_{\mathrm{stacked,i}}=\frac{\sum_i w_i M_{\mathrm{H\,\textsc{i},\nu_{\mathrm{rest}},i}}}{\sum_i w_i},
\end{array}
\label{eq:stacked_spec}$$ where $M_{\mathrm{H\,\textsc{i},\nu_{\mathrm{rest}},i}}$ is an individual galaxy’s H<span style="font-variant:small-caps;">i</span> mass in channel $i$, $M_{\mathrm{stacked,i}}$ is the final stacked mass in channel $i$ and $w_i$ is the weight given by $$\begin{array}{l}
\displaystyle w_i=\frac{1}{\sigma_i^2 D_L^4},
\end{array}
\label{eq:weight}$$ where $\sigma_i$ is the RMS noise in channel $i$, which we calculated using the <span style="font-variant:small-caps;">miriad</span> task <span style="font-variant:small-caps;">imstat</span>. We removed RFI contamination from the extracted spectra as described in Section \[s-sec:rfi\] (excluded channels with $S_{\nu_{\mathrm{obs}}}>5\sigma_i$ and excluded spectra containing GPS satellite RFI at $1376-1384$MHz). We then fit and subtracted a 4$^{\mathrm{th}}$-order polynomial to the stacked spectra to leave a flat baseline in the final spectrum.
We stacked the $M_{\mathrm{H\,\textsc{i}}}$ spectra in redshift bins of 0.05 up to $z=0.20$, with the final bin, $0.20 \leq z \leq 0.23$. We determined the RMS noise in the stacked spectra by randomising and reassigning each redshift to a different pair of coordinates from the input NED catalogue, ensuring that no redshift was assigned to its original coordinates. We then extracted and stacked these mock spectra identically to the galaxy spectra. The stacked mock spectra should not result in a positive detection, as our mock spectra are not centred on galaxies, and should give an approximation of the RMS noise. We performed the randomised stacking 10 times in each redshift bin and inspected each mock stacked spectrum for a possible signal mimicking a detection.
There was no detection in the $0.00 \leq z \leq 0.05$ bin after excluding two direct detections at $z=0.0043$ and $z=0.0055$ (see Section \[ss-sec:detection\]). Due to the increasing RFI levels at $z>0.10$, there were no direct or indirect H<span style="font-variant:small-caps;">i</span> detections in these redshift bins.
We found a detection for $\sim1100$ stacked galaxies at $0.050 \leq z \leq 0.075$. This signal was not found to be mimicked in the mock spectra from random lines of sight. In Figure \[fig:stacked\_spec\], we plot the stacked galaxy spectrum (blue) and the random mock spectrum (brown). Both spectra exhibit similar residual baseline curvature.
Our stacked H<span style="font-variant:small-caps;">i</span> detection spans the range from $\sim1419.8-1420.8$MHz (shown in Figure \[fig:stacked\_spec\] by the dashed green lines), which is significantly narrower than the [@Delhaize2013] South Galactic Pole stacked spectra (i.e.: $\sim1$MHz vs. $\sim3.6$MHz for $z=0.05-0.075$ and $z=0.04-0.13$, respectively). Our stacked detection is most likely narrower than the results of [@Delhaize2013] because of our smaller sample size [i.e.: $\sim1/3$ that of the South Galactic Pole region from @Delhaize2013] and lower redshift range, hence less confused sources entering our sample.
We integrated over the emission region to calculate the average H<span style="font-variant:small-caps;">i</span> mass of our stacked galaxies using $$\begin{array}{l}
\displaystyle \langle M_{\mathrm{H\,\textsc{i}}}\rangle=\int^{\nu_2}_{\nu_1} \langle M_{\mathrm{H\,\textsc{i}},\nu}\rangle d\nu,
\end{array}
\label{eq:mass_avg}$$ where $\nu_1$ and $\nu_2$ are the edges of the emission region. We find an average integrated H<span style="font-variant:small-caps;">i</span> mass of $\langle M_{\mathrm{H\,\textsc{i}}} \rangle = 1.24 \pm 0.18 \times 10^9 h^{-2} M_{\odot}$. Similar to [@Delhaize2013], we computed the error in $\langle M_{\mathrm{H\,\textsc{i}}} \rangle$ by integrating the $\langle M_{\mathrm{H\,\textsc{i}}} \rangle$ random mock stack. We find our $\langle M_{\mathrm{H\,\textsc{i}}} \rangle$ value to be lower than the South Galactic Pole value of $\langle M_{\mathrm{H\,\textsc{i}}} \rangle = 6.93 \pm 0.17 \times 10^9 h^{-2} M_{\odot}$ from [@Delhaize2013], indicating we have detected lower mass galaxies.
We investigated the noise behaviour of the stacked MPIPAF data by stacking the mock random line of sight flux density spectra, as described previously. We calculated the RMS of a stack of $N$ randomly chosen spectra to determine the noise behaviour with increasing number of stacked spectra. To estimate the noise behaviour we fix the weighting of each spectra using the mean RMS calculated from 1000 individual mock spectra ($\sigma=0.142$Jy) over the frequency range $1321-1352$MHz (corresponding to the redshift range of the stacked H<span style="font-variant:small-caps;">i</span> detection). Figure \[fig:rms\_spec\_num\] shows the change in RMS noise in the stacked spectrum with number of spectra included in the stack with the error bars calculated as the $1\sigma$ standard deviation of the RMS for 100 random stacks of $N$ spectra. We find the noise decreases as expected for Gaussian noise with a gradient of $-0.49 \pm 0.01$ for the MPIPAF data. This is similar to the noise behaviour for the Parkes multibeam data (gradient $\sim-0.5$) from [@Delhaize2013].
![Stacked $M_{\mathrm{H\,\textsc{i}}}$ spectrum (blue) for 1094 galaxies at $0.05 \leq z \leq 0.075$. The average mock spectrum from randomising the redshifts of the NED catalogue and stacking the spectra (shown in brown). The noise level in the mock spectrum is lower than that of the data as it is the mean of 10 simulations. The dashed green and grey lines indicate the left and right edges of the stacked H<span style="font-variant:small-caps;">i</span> emission determined by visual inspection and the rest frame H<span style="font-variant:small-caps;">i</span> line, respectively.[]{data-label="fig:stacked_spec"}](reynolds_fig9.pdf){width="\columnwidth"}
![The RMS noise in the stacked flux density signal vs. the number of stacked spectra. The error bars denote $1\sigma$ errors on the RMS. The dashed line shows the expected trend, assuming Gaussian noise, in decrease in noise with number of spectra with a gradient of $-0.5$.[]{data-label="fig:rms_spec_num"}](reynolds_fig10.pdf){width="\columnwidth"}
Comparison with HIPASS Detections {#s-sec:hipass}
---------------------------------
We present results of targeted observations of two HIPASS sources, J1413-65 (Circinus) and J1909-63A (NGC6744) from August 3 and September 4, respectively, observed with the MPIPAF, in addition to two HIPASS sources we detected in the GAMA G23 field, J2242-30 (NGC7361) and J2309-30 (ESO469-G015). We obtain HIPASS spectra from the archival HIPASS data cubes from the first HIPASS data release [@Meyer2004].
### Targeted HIPASS Observations {#ss-sec:target}
The Circinus galaxy, was only observed with the central MPIPAF beam providing a single spectrum. We therefore compare the MPIPAF observation with the pencil beam spectrum along the same line of sight through the galaxy from the archival HIPASS data cube, as this galaxy is resolved and not contained within a single beam. The MPIPAF and HIPASS spectra agree well in shape and flux density with the only difference in that we detect some additional flux on the high frequency end of the spectrum ($\nu \gsim 1418.5$MHz, top panel of Figure \[fig:hipass\_specs\]). This is most likely due to a slight position offset between the nearest HIPASS data cube pixel to the MPIPAF spectrum position (HIPASS pixel position: $\alpha,\,\delta=213.41^{\circ},\,-65.35^{\circ}$, MPIPAF line of sight position: $\alpha,\,\delta=213.36^{\circ},\,-65.31^{\circ}$).
Although the NGC6744 observation utilized all 16 MPIPAF beams, not all 16 beams lie upon the galaxy. NGC6744 has an angular radial size in H<span style="font-variant:small-caps;">i</span> of $\sim15$arcmin [$\sim30$arcmin in diameter, @Ryder1999], while 13 of the MPIPAF beams have angular separations $>30$arcmin from the centre of NGC6744. We integrated the flux from the three beams with angular separations $<24$arcmin. We also calculated the total integrated flux density from the archival HIPASS cube within a $13\times13$ pixel box centred on NGC6744 (maximum angular separation 24arcmin, matching the separation of the MPIPAF beams). The spectral shape and flux density of the integrated MPIPAF spectrum agrees with the HIPASS spectrum (lower panel of Figure \[fig:hipass\_specs\]).
![Targeted HIPASS galaxy line of sight spectra of Circinus and NGC6744, panels (a) and (b), respectively. The Circinus spectrum is from a single line of sight, while the NGC 6744 spectrum is the integrated line of sight spectrum from the 16 MPIPAF beams. The MPIPAF and HIPASS spectra are shown in blue and brown, respectively. The dashed green lines indicate the edges of the galaxy emission determine by visual inspection.[]{data-label="fig:hipass_specs"}](reynolds_fig11.pdf){width="\columnwidth"}
### G23 HI Detections {#ss-sec:detection}
We have two direct H<span style="font-variant:small-caps;">i</span> detections of HIPASS detected galaxies, NGC7361 and ESO469-G015, at $z=0.0043$ and $z=0.0055$ (Figure \[fig:g23\_specs\] panels (a) and (b), respectively). We found these direct detections through visual inspection of the H<span style="font-variant:small-caps;">i</span> spectra extracted from the G23 field based on optical identifications from NED. Unlike the targeted HIPASS galaxy observations, we can compare the total integrated flux for these two galaxies as they are completed covered by the drift scan. We compare our MPIPAF spectra with the HIPASS spectra integrated over the same area from the archival data cubes ($20\times20$arcmin) for these two galaxies in Figure \[fig:g23\_specs\] (blue and brown lines, respectively). The integrated MPIPAF spectrum of NGC 7361 shows very good agreement with the HIPASS data both in spectral shape and flux density. While the integrated MPIPAF spectrum of ESO 469-G015 has a similar spectral shape, it has a slightly lower flux density. Nevertheless, both spectra agree within the combined uncertainties (Table \[table:g23\_fluxes\]).
![Direct H<span style="font-variant:small-caps;">i</span> detection of HIPASS galaxies NGC7361 and ESO469-G015 at $z=0.0043$ and $z=0.0055$, panels (a) and (b), respectively. The MPIPAF and HIPASS integrated spectra are shown in blue and brown, respectively. The dashed green lines indicate the edges of the galaxy emission determine by visual inspection.[]{data-label="fig:g23_specs"}](reynolds_fig12.pdf){width="\columnwidth"}
[@ccc@]{} Galaxy & MPIPAF & HIPASS\
& Flux \[mJy\] & Flux \[mJy\]\
\
NGC7361 & $0.098\pm0.014$ &$0.097\pm0.009$\
ESO469-G015 & $0.041\pm0.012$ & $0.049\pm0.011$\
Galactic Centre Hydrogen and Positronium Recombination Lines {#s-sec:galcent}
------------------------------------------------------------
[@cccc@]{} H$n\alpha$ & $\nu_{\mathrm{H}}$ \[MHz\] & Ps$n\alpha$ & $\nu_{\mathrm{Ps}}$ \[MHz\]\
\
165 & 1450.58 & 131 & 1446.81\
166 & 1424.60 & 132 & 1414.29\
167 & 1399.24 & 133 & 1382.75\
168 & 1374.48 & 134 & 1352.14\
169 & 1350.29 & 135 & 1322.42\
170 & 1326.67 & 136 & 1293.57\
171 & 1303.60 & 137 & 1265.55\
172 & 1281.06 & 138 & 1238.33\
173 & 1259.03 & 139 & 1211.89\
174 & 1237.51 & 140 & 1186.20\
175 & 1216.48 & 141 & 1161.23\
176 & 1195.92 & &\
177 & 1175.82 & &\
178 & 1156.17 & &\
We inspected the Galactic Centre spectra from each beam at the 14 Hydrogen and 11 positronium radio recombination line frequencies predicted from the Rydberg equation within the band (Table \[table:rydberg\]). We were unable to use large sections of the spectra due to RFI contamination. Also present in the spectra are 1MHz beamformer ‘jumps’, some of which were not removed during bandpass calibration. These 1MHz ‘jumps’ have also been seen in early ASKAP data and are not caused by RFI or the Parkes dish, but are due to the discretization of the beamformer weights. We were able to model and remove the spectral shape of the ‘jumps’ with a first order polynomial fit to each 1MHz spectral interval as the ‘jumps’ occur at exactly 1MHz intervals (e.g.: 1380.5MHz, 1381.5MHz, 1382.5MHz, etc.). We find detections for the H$165\alpha$, H$166\alpha$, H$167\alpha$, H$168\alpha$ and H$170\alpha$ hydrogen recombination lines in individual MPIPAF beams (Figure \[fig:gc\_stack\](a) shows the stacked signal for these five lines in all 16 MPIPAF beams). In Table \[table:h\_recomb\] we list calculated line parameters from Gaussian fits to the spectra in Figure \[fig:gc\_stack\](a). Our calculated peak intensity line temperatures agree with previous single dish Galactic Centre hydrogen RRL studies (Table \[table:h\_recomb\]). All other hydrogen recombination lines lie within regions with high RFI contamination and are undetected.
For positronium, however, we did not have any clear direct detections above the noise. In an attempt to improve the signal to noise, we stacked the Galactic Centre spectra from each beam centred on the predicted frequencies of positronium recombination lines to look for a stacked detection. We were unable to stack spectra at all predicted frequencies due to the presence of RFI, as mentioned above. We excluded the RFI contaminated sections of the spectra. This reduced the number of stacked positronium spectra to 64, centred on the predicted Ps$131\alpha$, Ps$132\alpha$, Ps$133\alpha$ and Ps$135\alpha$ lines. For each spectrum, we extracted 450 channels centred on the predicted line frequency and fit a $2^{\mathrm{nd}}$-order polynomial to remove the baseline from the stacked spectrum. We find no detection in the stacked positronium spectra in either emission or absorption as shown in Figure \[fig:gc\_stack\](b) and set a $3\sigma$ upper limit on the stacked recombination line signal of $<0.09$K. Using this upper limit, we calculate the recombination rate to be $<3.0\times10^{45}\,\mathrm{s}^{-1}$, assuming the positronium line would have a FWHM of 4.2MHz, as the positronium line width is thermally broadened to 30 times the hydrogen line width at 1400MHz. The positronium upper limit improves upon the results of [@Anantharamaiah1989], who placed a $3\sigma$ upper limit of $<29.3$K (recombination rate $<1.1\times10^{44}\,\mathrm{s}^{-1}$, assuming a line width FWMH of 4.2MHz) on the detection of the positronium Ps$133\alpha$ line from the Galactic Centre from VLA observations. It should be noted that the recombination rate upper limit from [@Anantharamaiah1989] is lower than our value due to the differing flux limits ($<3.4$mJy vs. $<84$mJy from the VLA and Parkes, respectively) which is a result of the arcsecond vs. arcminute resolution of the VLA ($\sim12\times6$arcsec) and Parkes ($\sim15\times15$arcmin), respectively.
![Stacked Galactic Centre hydrogen and positronium spectra, Panels (a) and (b), respectively. For hydrogen we stacked the spectra from all 16 MPIPAF beams, for individual recombination lines and recovered a detection for the $165\alpha$, H$166\alpha$, H$167\alpha$, H$168\alpha$ and H$170\alpha$ lines. The positronium spectrum is the combined stack of the Ps$131\alpha$, Ps$132\alpha$, Ps$133\alpha$ and Ps$135\alpha$ lines in all 16 MPIPAF beams and does not show a detection.[]{data-label="fig:gc_stack"}](reynolds_fig13.pdf){width="\columnwidth"}
[@ccccccc@]{} RRL & FWHM \[MHz\] & $T_{\mathrm{L}}$ \[K\]$^a$ & $T_{\mathrm{L}}$ \[K\]$^b$ & $T_{\mathrm{L}}$ \[K\]$^c$ & $T_{\mathrm{L}}$ \[K\]$^d$ & $T_{\mathrm{L}}$ \[K\]$^d$\
\
$H165\alpha$ & $0.11\pm0.01$ & $0.30\pm0.02$ & ... & ... & ... & ...\
$H166\alpha$ & $0.11\pm0.01$ & $0.37\pm0.02$ & 0.65 & 0.14 & $0.09 - 1.03$ & $0.29-0.76$\
$H167\alpha$ & $0.13\pm0.01$ & $0.42\pm0.02$ & 0.64 & ... & ... & ...\
$H168\alpha$ & $0.17\pm0.02$ & $0.54\pm0.04$ & 0.61 & ... & ... & ...\
$H170\alpha$ & $0.12\pm0.01$ & $0.54\pm0.02$ & ... & ... & ... & ...\
Dish & & 64m & 43m & 43m & 76.2m & 25.6m\
Diameter & & & & & &\
CONCLUSIONS {#sec:conclusions}
===========
We have presented results of using a modified ASKAP phased array feed (PAF) mounted on the Parkes 64m radio telescope for performing H<span style="font-variant:small-caps;">i</span> mapping and stacking.
- The standing wave amplitude, resulting from interference with reflected waves, is substantially reduced in the PAF data compared with conventional receivers. We estimate an amplitude reduction by a factor of $\sim 10$ compared with the multibeam receiver. This reduction represents the higher efficiency and full focal-plane sampling of the PAF.
- The system temperatures during our observations are higher and less stable than those during initial tests by [@Chippendale2016]. This is most likely due to delay slips in the digital receiver which were not monitored or corrected for during the observations, but can be corrected using an on-dish noise source to estimate the delays and applying a compensating phase slope to existing beamformer weights.
- The lower frequency ($\nu<1290$MHz) data contain significant but well-known satellite RFI contamination. We demonstrate that, even with the low temporal and spectral resolution of our observations, significant mitigation is possible.
- We have compared observations of the Large Magellanic Cloud with archival Parkes multibeam data, and found excellent agreement.
- We have demonstrated that noise continues to decrease with time for long observations with a PAF. In particular, we find a stacked detection of extragalactic H<span style="font-variant:small-caps;">i</span> in the GAMA G23 field in the redshift range $0.05 \leq z \leq 0.075$.
- Two direct H<span style="font-variant:small-caps;">i</span> detections in the GAMA G23 field at $z=0.0043$ and $z=0.0055$ of NGC7361 and ESO469-G015 are also noted. Both integrated spectra show good agreement in spectral shape with archival HIPASS data and the measured fluxes agree within the statistical uncertainties.
- From targeted observations of HIPASS sources Circinus and NGC6744, we found reasonable agreement with the archival HIPASS line of sight spectrum of Circinus. Some of the difference may be due to the different (and not optimal) MPIPAF beam shapes. The integrated line of sight spectrum of NGC6744 agrees well with the integrated HIPASS spectrum.
- We find clear direct detections of five hydrogen recombination lines: $H165\alpha$, $H166\alpha$, $H167\alpha$, $H168\alpha$ and $H170\alpha$. We do not find a detection of positronium recombination lines in the Galactic Centre observations, but set a $3\sigma$ upper limit of $<0.09$K, corresponding to a recombination rate of $<3.0\times10^{45}\,\mathrm{s}^{-1}$.
The above demonstration, whilst limited in scope, demonstrates the viability of PAFs on large single dish telescopes. The main areas that need improvement for a permanent installation are the system temperature, which requires cryogenic cooling, and a mechanism for ensuring a stable and reproducible beamforming methodology in the presence of RFI [@Chippendale2017]. The flexible beamforming capability of PAFs is nevertheless enormously powerful and can in itself, as already demonstrated, reduce the impact of RFI.
This research was conducted by the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020. The Parkes radio telescope is part of the Australia Telescope National Facility which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSRIO. The MPIPAF is a collaboration between CSIRO Astronomy and Space Science (CASS) and MPIfR of the Max Planck Society. We wish to thank A. Brown for improving the duty cycle of the 18.5 kHz resolution spectrum integrator, M. Marquarding and E. Troup for contributions to software systems development and integration, particularly for observation and telescope control, and Dr. K. Bannister and C. Haskins for support of software implementation of RFI mitigation. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
[^1]: [email protected]
[^2]: Except for Circinus which used a calibrator observation from five days later (August 8), see Section \[s-sec:reduction\]
[^3]: <https://ned.ipac.caltech.edu/>
[^4]: <span style="font-variant:small-caps;">livedata</span> and <span style="font-variant:small-caps;">gridzilla</span> are supported by the Australia Telescope National Facility and are available at <http://www.atnf.csiro.au/computing/software/livedata/>.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report an ultra-bright lensed submillimeter galaxy (SMG) at $z=2.0439$, [*WISE*]{} J132934.18+224327.3, identified as a result of a full-sky cross-correlation of the [*AllWISE*]{} and [*Planck*]{} compact source catalogs aimed to search for bright analogs of the submillimeter galaxy SMMJ2135, the Cosmic Eyelash. Inspection of archival SCUBA-2 observations of the candidates revealed a source with fluxes (S$_{850 \mu m}$= 130 mJy) consistent with the [*Planck*]{} measurements. The centroid of the SCUBA-2 source coincides within 1 arcsec with the position of the [*AllWISE*]{} mid-IR source, and, remarkably, with an arc shaped lensed galaxy in [*HST*]{} images at visible wavelengths. Low-resolution rest-frame UV-optical spectroscopy of this lensed galaxy obtained with 10.4 m GTC reveals the typical absorption lines of a starburst galaxy. Gemini-N near-IR spectroscopy provided a clear detection of H$_{\alpha}$ emission. The lensed source appears to be gravitationally magnified by a massive foreground galaxy cluster lens at $z = 0.44$, modeling with Lenstool indicates a lensing amplification factor of $11\pm 2$. We determine an intrinsic rest-frame 8-1000-$\mu$m luminosity, $L_{\rm IR}$, of $(1.3 \pm 0.1) \times 10^{13}$ $L_\sun$, and a likely star-formation rate (SFR) of $\sim 500-2000$ $M_\sun yr^{-1}$. The SED shows a remarkable similarity with the Cosmic Eyelash from optical-mid/IR to sub-millimeter/radio, albeit at higher fluxes.'
author:
- 'A. Díaz–Sánchez'
- 'S. Iglesias-Groth'
- 'R. Rebolo'
- 'H. Dannerbauer'
title: 'Discovery of a lensed ultrabright submillimeter galaxy at $z=2.0439$'
---
Introduction {#sec:intro}
============
Submillimeter galaxies (SMGs) provide important clues on the formation and evolution of massive galaxies in the distant universe [@Ca14]. These high-z dusty starbursts, peaking at redshift z=2.3-2.5 [@Ch05; @Si14; @St16] are massive and most probably the progenitors of present-day ellipticals (e.g. [@Iv13]). Given the typical sizes of starbursts (less than a few kpc, e.g. [@Ho16]) even with ALMA and [*HST*]{} it is difficult to achieve the desirable spatial resolution to carry out detailed studies of star-forming regions in these dusty galaxies.
Strong gravitational lensing by massive galaxy clusters can enhance the apparent brightness of SMGs [@Sm97; @Sm02]. Wide [*Herschel*]{} surveys as H-ATLAS, HerMES or the South Pole Telescope and the [*Planck*]{} space mission provided an increasing number of bright, lensed SMGs [@Ne10; @Vi13; @We13; @Ca15; @Ha16; @St16] that may allow their study at a resolution of 100 pc, close to the size of GMCs, and thus understand better their ISM properties (see e.g. [@Vl15; @Sw15]) and compare then with galaxies in the local universe. Among SMGs it is outstanding the serendipitously discovered lensed galaxy SMM J2135-0102 at z=2.3259 (the Cosmic Eyelash [@Sw10; @Iv10; @Da11]).
In this Letter we present the first result of a search for the brightest examples in the sky of dusty star-forming high redshift galaxies using data of the [*Planck*]{} and [*WISE*]{} space missions. We report the discovery of [*WISE*]{} J132934.18+224327.3 as a magnified SMG at $z=2.044$ with spectral energy distribution (SED) remarkably similar tothat of the Cosmic Eyelash but higher apparent brightness at all wavelengths from visible to radio. We adopt a flat $\Lambda$CDM cosmology from [@Pl14] with $H_0=68$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_m=0.31$ and $\Omega_\Lambda = 1 - \Omega_m$
The search {#sec:search}
==========
In order to find bright lensed SMGs in the full-sky, we carried out a cross-matching between the [*AllWISE*]{}[^1] and [*Planck*]{}[^2] full-sky compact source catalogs. As a reference we adopted the SED of the Cosmic Eyelash [@Sw10], from the MIR to the submillimeter. Our aim was to identify candidates in these catalogs with MIR colors similar to the Cosmic Eyelash and strong submillimeter fluxes. First we built a full-sky selection of galaxies verifying the color criteria $W1-W2>0.8$, $W2-W3<2.4$, $W3-W4>3.5$ and detection at $S/N>3$ in the four bands of the [*AllWISE*]{} catalog, we limited the sample to galactic latitude b $\geq$20$^{o}$. These color criteria were employed by [@Ig17] in their NIR/MIR search for lensed submillimeter galaxies. These criteria also select SMGs with SED similar to the average SED of 73 spectroscopically identified SMGs from [@Mi10; @Ha11] and to the ALESS SMG composite SED [@Sw14]. Only SMGs at $z=2-2.8$ are expected to fulfill these color conditions.
Then, we requested the selected objects to have a [*Planck*]{} source detected within 5 arcmin with submillimeter flux ratios consistent with those expected for SMGs at $z=2-3$. We found 8 [*WISE*]{} candidates with SEDs consistent with the adopted references, however the [*Planck*]{} sub-millimeter/millimeter maps for most of these sources clearly show contamination by galactic dust emission. Inspection of the [*Planck*]{} maps and available images from different optical and near-IR catalogs identified [*WISE*]{} J132934.18+224327.3, as the most likely counterpart of the submillimeter [*Planck*]{} source PCCS2 857 G007.94+80.29 and very promising candidate to an ultra-bright SMG analog to the Cosmic Eyelash.
A subsequent search in CADC[^3] provided detections by SCUBA-2 at James Clerk Maxwell Telescope of a sub-millimeter source with coordinates consistent within 1 arcsec with those of [*WISE*]{} J132934.18+224327.3 which could be responsible of the observed [*Planck*]{} submillimeter fluxes. In fact, the position of the source coincides with a strong lensing cluster SDSS 1329+2243 at $z=0.44$ [@Bo14] which has been observed by [@Jo15] with JCMT/SCUBA-2 and reported a source in snapshot observations at 850 and 450 $\mu$m with fluxes S$_{450}\approx$605 mJy and S$_{850}\approx$130 mJy, consistent with being the main counterpart of the [*Planck*]{} source. [@Jo15] reported arc structures in Keck images suggesting it could be a lensed submillimeter source. Additional detections in the radio band are found in FIRST[^4].
We used Vizier at CDS[^5] to find the most likely counterpart of the [*WISE/Planck/SCUBA-2*]{} source in the near-IR and visible. In [*HST-ACS*]{}[^6] images a lensed galaxy is also detected at less than 1 arcsec of [*WISE*]{} J132934.18+224327.3. We postulate this lensed galaxy is the optical counterpart of the strong submillimeter and mid-IR source (see Fig. 1 and Fig. 2a). Hereafter we designate this lensed galaxy as ‘Cosmic Eyebrow‘.
Observations and data reduction {#sec:obser}
===============================
GTC Spectroscopy {#sec:gtc}
----------------
We have obtained spectroscopy of the lensed galaxy with the optical imager and spectrograph OSIRIS at the 10.4 m Gran Telescopio de Canarias (GTC), on the night of 2017 April 18 in clear conditions and dark moon with $0.8^{\prime\prime}$ seeing. Observations were made using the low-resolution, long-slit mode with a $0.8^{\prime\prime}$ slit and the R500B grism. In this configuration the resolution is $R\sim 540$ and we have 3.54 $\AA$/pix from $\sim $ 3700 to 7200 $\AA$. The slit was oriented along the line of the arc structure of the lensed source (P.A. of 64.22$^{\rm o}$). We grouped observations in two observing blocks of two exposures of 1385 seconds each with a 60 second acquisition image per block in $g$-band (see Fig. 2). The seeing limited image shows an arc-like galaxy in the expected position for which we measured $g_{\rm AB}=22.85 \pm 0.03$ for region 1 and $g_{\rm AB}=23.43 \pm 0.04$ for region 2. The spectroscopic observations, with a total integration time of 5540 s, were reduced using the noao/twodspec and noao/onedspec packages of IRAF to yield fluxed, wavelength-calibrated spectra. We extracted spectra for the full arc region along the slit and separately for encircled regions 1 and 2. Our GTC spectroscopy reveals that the two regions in the arc are the same source. In Fig. 3 we show the individual spectra for each arc region and the total arc spectrum. The spectra of the two arc regions show strong absorption features. We derive the redshift from the fit of well measured lines of SiII $\lambda 1260.4$, OI $\lambda 1302$, CII $\lambda 1334.5$, SiII $\lambda 1526.7$ and AlII $\lambda 1670.7$, for the full arc we find $z= 2.0448 \pm 0.0004$.
Gemini Spectroscopy {#sec:gemini}
-------------------
Archival GEMINI NIR-spectroscopic observations were found in CADC for region 1 of the arc. The spectrum was obtained using the cross-dispersed (XD) mode of the Gemini Near-IR Spectrograph on the 8.1 m Gemini North telescope on the night of 2014 May 10 (under PI: Jane Rigby program ID: XGN-2014A-C-3) with the “short blue” camera, 32 l/mm grating and $0.68^{\prime\prime}$ slit. The slit used in this XD mode is $7^{\prime\prime}$ in length and the orientation was with P.A. of 89.7$^{\rm o}$ in region 1. The telescope was nodded (typically $3.5^{\prime\prime}$ distant) in an ABBA-type pattern. We took the raw and calibration data from the CADC and reduced them using the Gemini IRAF package version v1.13.1 and following [@Ms15] and the instruction on the Reducing XD spectra webpage from GEMINI observatory[^7]. The total integration time for the spectrum was 3600 seconds and two telluric stars of spectral type A0V were observed immediately before and after of the galaxy. Only in the K band we have found useful data with a clear detection of H$_{\alpha}$ in emission. In the inset of Fig. 3, we show the spectrum near the $H_\alpha$ emission line from which we obtain a redshift of $z=2.0439 \pm 0.0006$ in agreement with the rest-frame UV-spectrum. This redshift is consistent with the value published in [@Og12] ($z\approx2.04$) for a source coincident in position with the lensed galaxy.
Analysis and discussion {#sec:dis}
=======================
Lens modeling {#sec:lens}
-------------
We have used Lenstool[^8] [@Kn93; @Ju07] to perform a mass reconstruction of the foreground cluster, assuming a parametric model for the distribution of dark matter. This model was constrained using the location of the multiple images identified in the cluster. We used a simple model with a single cluster-scale mass component, as well as individual galaxy-scale mass components centered on each cluster member. For each component, we used a dual pseudo-isothermal elliptical mass distribution (dPIE, also known as a truncated PIEMD, [@Li05]). Sextractor and visual inspection of the [*HST/WFPC3*]{} images provided an identification of six families of multiply lensed background galaxies, arclets with 2-6 components each, based on proximity, colors and shape. The redshift of the family of Cosmic Eyebrow (A in Fig. 1) was fixed at $z=2.044$, and for the other families was set as free parameter. When modeling the lensing, the redshift of 3 families (B, C and E in Fig. 1) was found very close to $z=2.044$, so we fixed it to $z=2.044$ and only for 2 families the redshifts were considered a free parameter. We checked the model with the reconstruction of the arclets, and found good agreement between the positions for image arcs and the reconstruction of each component of the families. The data are best fitted with a cluster-scale potential of ellipticity $e = 0.226$, position angle PA =47.00$^{\rm o}$, and velocity dispersion $\sigma_{\rm PIEMD} =830$ km/s. The enclosed mass within an aperture of 250 kpc is $M=1.8 \pm 0.5 \times 10^{14}M_\sun$ with an Einstein radius of $\theta_{\rm e}= 11. \pm 0.4 ^{\prime\prime}$ at $z=2.044$. The amplification is obtained from the comparison between the main characteristic of the galaxies in the source plane and in the image plane, we find an amplification factor of $11\pm 2$ for the two main members of the Cosmic Eyebrow family, which are our spectroscopy regions on the arc indicated in Fig. 2. The mean amplification factors for each family with $z=2.044$ are A $\sim 11\pm 2$, B $\sim 7\pm 2$, C $\sim 14\pm 2$, E $\sim 7\pm 2$, the largest amplification is for the brightest member of family C $\sim 20\pm 2$, for the other two families the mean amplification factors are D $\sim 4\pm 2$ ($z\sim 2.9$) and F $\sim 5\pm 2$ ($z\sim 1.0$).
SED {#sec:sed}
---
We plot the SED (Fig. 4) assuming the lensed source in the [*HST*]{} and GTC images is the counterpart of the SCUBA-2 and [*WISE*]{} detections (all positions coincident within 1 arcsec) and list the photometry data in table 1. We calculate the upper limit for $K_s$ and $H$ bands from UKIDSS catalog[^9] and for 60 and 100 $\mu$m from [*IRAS*]{} Sky Survey Atlas[^10]. This SED is consistent with a source at redshift $z=2-2.5$ and, as expected, it is very well fitted by the SED of the Cosmic Eyelash in that redshift range. At all frequencies from optical to radio, Cosmic Eyebrow is brighter than the Cosmic Eyelash.
We calculate the rest-frame 8-1000-$\mu$m luminosity, $L_{\rm IR}$, from direct integration of the data fit, and obtain an intrinsic luminosity for our galaxy $L({\rm Eyebrow})= (1.3 \pm 0.1) \times 10^{13}$ $L_\sun$, indicating a SFR of $\sim 2000$ $M_\sun yr^{-1}$ [@Ke98], which assumes a Salpeter IMF, assuming a Chabrier IMF would possibly be a factor of $\sim$ 1.8 lower [@Ca14].
We also use the MAGPHYS code that allows us to fit simultaneously the ultraviolet-to-radio SED, so we can constrain physical parameters [@Cu15]. The fit is shown in (Fig. 4), we would benefit from measurements in photometric bands between 22 and 350 $\mu$m in order to establish which SED describes better the Cosmic Eyebrow. From this fit we give the median-likelihood estimates (and confidence ranges) of several physical parameters corrected for lensing amplification, stellar mass $log(M_\ast/M_\sun) = 11.49^{+0.06}_{-0.05}$, star formation rate $log(SFR/M_\sun yr^{-1}) = 2.73^{+0.02}_{-0.01}$, mass-weighted age $log(age_M/yr) = 9.28^{+0.01}_{-0.01}$, average V-band dust attenuation $A_V = 3.30^{+0.02}_{-0.02}$, H-band mass-to-light ratio $log(M_\ast/L_H) = 0.14^{+0.01}_{-0.01}$, total dust luminosity $log(L_{\rm dust}/L_\sun) = 13.27^{+0.01}_{-0.02}$, luminosity-averaged dust temperature $T_{\rm dust}/K = 40^{+7}_{-2}$ and total dust mass $log(M_{\rm dust}/M_\sun) = 9.19^{+0.15}_{-0.08}$. This physical parameters compare well with the average properties of the ALESS SMGs in [@Cu15]. The SFR obtained with MAGPHYS is lower than estimated with the [@Ke98] relation. This is partially due to the use of the Chabrier IMF and to the additional dust heating caused by the relatively old stellar populations of our galaxy which increases the dust luminosity at fixed SFRs [@Cu15].
[lcc]{}\[b!\] 0.3921 & 0.0022 $\pm 0.0001$ & [*HST/WFC3*]{}\
0.45 & 0.004 $\pm 0.002$ & GTC/OSIRIS\
0.5887 & 0.0043 $\pm 0.0001$ & [*HST/WFC3*]{}\
1.0552 & 0.0252 $\pm 0.0004$ & [*HST/WFC3*]{}\
1.5369 & 0.0614 $\pm 0.0006$ & [*HST/WFC3*]{}\
1.644 & $<$ 0.112 & UKIDSS\
2.199 & $<$ 0.110 & UKIDSS\
3.4 & 0.37 $\pm 0.01$ & [*WISE*]{}\
4.6 & 0.45 $\pm 0.02$ & [*WISE*]{}\
12 & 0.7 $\pm 0.1$ & [*WISE*]{}\
22 & 10.6 $\pm 0.8$ & [*WISE*]{}\
60 & $<$ 100 & [*IRAS*]{}\
100 & $<$ 300 & [*IRAS*]{}\
450 & 604 $\pm 86$ & SCUBA-2\
850 & 127 $\pm 11$ & SCUBA-2\
350 & 1298 $\pm 200$ & [*Planck*]{}\
550 & 692 $\pm 100$ & [*Planck*]{}\
850 & 271 $\pm 90$ & [*Planck*]{}\
21.4 & $ 3.56 \pm 0.14$ & FIRST\
Spectroscopy {#sec:spectdis}
------------
The UV rest-frame spectrum of the Cosmic Eyebrow resembles that of Lyman break galaxies (LBG) [@Sh03] and display the strong interstellar absorption features typical of the spectra of starburst galaxies [@Ca13]. We identify in the observed spectrum low-ionization resonance interstellar metal lines such as SiII $\lambda 1260.4$, OI 1302+Si II $\lambda1304$, CII $\lambda 1334$, SiII $\lambda 1526$ and AlII $\lambda 1670$ which are associated with the neutral interstellar medium. We also detect high-ionization metal lines of Si IV $\lambda\lambda 1393,1402$ and CIV $\lambda\lambda 1548, 1550$ which are associated with ionized interstellar gas and PCygni stellar wind features. The CIV feature exhibits a strong interstellar absorption component plus a weaker blueshifted broad absorption and marginal evidence of redshifted emission associated with stellar winds which likely originate in main-sequence, giant and supergiant O stars [@Wa84]. Our data do not show evidence for the He II $\lambda 1640$ which could be an indication of a low ratio of Wolf-Rayet to O stars [@Scha98].
In the spectrum of the region 1 of the arc we find $H_\alpha$ at $19982 \pm 4$ $\AA$ and the \[NII\] $\lambda 6583$ at $ 20041 \pm 5$ $\AA$. Full width at half maximum (FWHM) line widths are $386 \pm 20$ km s$^{-1}$ and $331 \pm 20$ km s$^{-1}$, after correction for instrumental profile we estimate the intrinsic width of these lines is $150 \pm 25$ km s$^{-1}$. The low \[NII\]$\lambda 6583$/$H_\alpha$ ratio of $0.33 \pm 0.04$ suggests a star-forming region [@Ke13]. The EW$_{rest}$ ($H_\alpha$)= $111 \pm 10$ $\AA$ compares well with values reported in other well known SMGs of similar luminosity [@Ol16].
Conclusions {#sec:concl}
===========
A cross-match of the [*AllWISE*]{} and [*Planck*]{} compact source catalogs aimed to identify the brightest examples of submillimeter galaxies in the full sky uncovered [*WISE*]{} J132934.18+224327.3, the Cosmic Eyebrow, a $z=2.0439 \pm 0.0006$ galaxy with spectral energy distribution from mid-IR to submillimeter similar to that of the Cosmic Eyelash and higher observed fluxes. Archival data of SCUBA-2 and [*HST*]{} reveals a multiple lensed galaxy at the position of the strong submillimeter source. Follow-up observations with GTC/OSIRIS provided a precise redshift determination of the lensed galaxy and a rest-frame UV spectrum with clearly identified low-ionization resonance interstellar metal lines which is consistent with that of starburst galaxies. Near-IR spectroscopy with Gemini-North showed H$_{\alpha}$ in emission at a strength consistent with values reported for other SMGs. This galaxy is gravitationally magnified by a massive cluster at $z = 0.44$, modeling with Lenstool indicates a lensing amplification factor of $11\pm 2$. The intrinsic rest-frame 8-1000-$\mu$m luminosity of the lensed galaxy is $(1.3 \pm 0.1) \times 10^{13}$ $L_\sun$ indicating a likely SFR of $\sim 500-2000$ $M_\sun yr^{-1}$. The SED of this new SMG resembles the Cosmic Eyelash from optical-mid/IR to sub-millimeter/radio, albeit at higher intrinsic luminosity. It is one of the brightest examples of SMGs so far reported which may enable detailed high spatial resolution studies of star-forming regions in such dusty galaxies.
Based on observations made with the GTC telescope, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. This work has been partially funded by projects ESP2015-69020-C2-1-R, ESP2014-56869-C2-2-P, and AYA2015-69350-C3-3-P (MINECO). H.D. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (MINECO) under the 2014 Ramón y Cajal program MINECO RYC-2014-15686. We thank David Valls-Gabaud for his suggestions and comments on various aspects of this work. We are grateful to the anonymous referee for his useful comments on the paper.
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[^1]: <http://wise2.ipac.caltech.edu/docs/release/allwise/>
[^2]: <http://pla.esac.esa.int/pla/#home>
[^3]: <http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/>
[^4]: <http://sundog.stsci.edu/first/catalogs.html>
[^5]: <http://vizier.u-strasbg.fr/viz-bin/VizieR>
[^6]: <http://archive.stsci.edu/>
[^7]: <http://www.gemini.edu/sciops/instruments/gnirs>
[^8]: <https://projets.lam.fr/projects/lenstool/wiki>
[^9]: <http://wsa.roe.ac.uk/>
[^10]: <http://irsa.ipac.caltech.edu/applications/IRAS/ISSA/>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we introduce a three-dimensional mathematical model of collagen contraction with microbuckling based on the two-dimensional model in [@Dallon:2014:MMC]. The model both qualitatively and quantitatively replicates experimental data including lattice contraction over a time course of 40 hours for lattices with various cell densities, cell density profiles within contracted lattices, radial cut angles in lattices, and cell force propagation within a lattice. The importance of the model lattice formation and the crucial nature of its connectivity are discussed including differences with models which do not include microbuckling. The model suggests that most cells within contracting lattices are engaged in directed motion.'
author:
- 'E. J. Evans'
- 'J. C. Dallon'
bibliography:
- 'lattice.bib'
title: 'A Three-Dimensional Mathematical Model of Collagen Contraction'
---
Introduction
============
Fibroblast populated collagen lattices have been widely studied since first introduced by Bell et. al. [@Bell:1979:PTS] with the aim of better understanding cell extracellular matrix interactions and wound contraction. The contraction of the lattice is an irreversible cell mediated process. In the first 48 hours, remodeling of the matrix is mainly mechanical and there is little cell proliferation [@Ehrlich:2006:EMW; @Greco:1992:DCD], extracellular degredation (although matrix metalloproteinases appear to play a role in contraction via cell dynamics) [@Martin:2011:EMI], or extracellular matrix production [@Redden:2003:CCC]. There are three proposed mechanisms responsible for the contraction of the collagen lattice in the first 24-48 hours: cell elongation, tractional forces due to cell locomotion, and cell contraction [@Dallon:2008:RFC].
Previous models related to our work can be divided into two categories. The first category is models of collagen and fibrous structures [@Wyart:2008:EFS; @Sharma:2016:SCG]. These models use discrete formations of the the fibrous structure and are most closely related to the work here. In Wyart et. al. the authors, while not modeling collagen lattices, use two-dimensional random networks of springs to model fibrous networks and numerically derive material properties of the models. One of the model parameters which they investigate, which is pertinent to our study, is the average coordination number. It is defined to be $z=\frac{2N_c}{N}$ where $N_c$ is the number of bonds connecting nodes and $N$ is the number of nodes. A system is isostatic or rigid when $z=2d$ where $d$ is the dimension. They found that systems where $z<2d$ have a stress-strain relationship with a zero plateau and then a strain-stiffening region. This is commonly found in fibrous networks including collagen lattices [@Licup:2015:SCM; @Sharma:2016:SCG]. More recently, Sharma et. al. [@Sharma:2016:SCG] both modeled and measured material properties of collagen gels. In their model, they use a lattice of nodes connected by elastic elements which have a Young’s modulus and a bending modulus. The elastic energy of the system is minimized when the system is deformed. The model discussed here does not have a bending modulus; the fibers are elastic ropes or include microbuckling, and cell interactions are included.
The second category is models of cells interacting with fibrous tissue. These models include models focused on alignment [@Barocas:1997:ABT; @Dallon:1998:CAC; @Olsen:1999:MMA; @Dallon:1999:MME; @Schluter:2012:CMS; @Reinhardt:2014:AMT; @Notbohm:2015:MFP] and those concerned with contraction [@Simon:2012:MRB; @Simon:2014:CMM; @Dallon:2014:MMC]. Of these, some are force based [@Barocas:1997:ABT; @Simon:2012:MRB; @Simon:2014:CMM; @Reinhardt:2014:AMT; @Dallon:2014:MMC; @Notbohm:2015:MFP], some are continuum descriptions [@Barocas:1997:ABT; @Dallon:1998:CAC; @Olsen:1999:MMA; @Dallon:1999:MME; @Schluter:2012:CMS; @Simon:2012:MRB; @Simon:2014:CMM], and some more closely resemble our model by treating the fibrous tissue as a discrete structure and focus on forces [@Reinhardt:2014:AMT; @Notbohm:2015:MFP]. In [@Reinhardt:2014:AMT], the matrix is modeled with springs and torsional forces, whereas in [@Notbohm:2015:MFP] the matrix is modeled with elastic ropes which they call fibers with microbuckling. The model here is more similar to that in [@Notbohm:2015:MFP] but the focus is on lattice contraction and a larger space scale than either of the other two models.
Mathematical model {#sec:notation}
==================
The model is a three-dimensional extension of one of the two-dimensional models presented in [@Dallon:2014:MMC].
Model of cell–motion
--------------------
The equation of motion for the center of mass of the $i$th cell with $K$ I-sites is given by $$C {\bf x}_i^{\prime} = -\sum_{j=1} ^K\alpha(||{\bf x}_i-{\bf y}_{p_{i,j}}||-\ell)\frac{{\bf x}_i-{\bf y}_{p_{i,j}}}{||{\bf x}_i-{\bf y}_{p_{i,j}}||},$$ where ${\bf x}_i^{\prime}$ represents the velocity of the cell, $C$ is the drag coefficient, and ${\bf x}_i$ is the location in $\mathbb{R}^3$ of the cell center for $i=1,\cdots,N$. The spring constant $\alpha$ is the same for all cells and all I-sites and, together with the spring rest length $\ell$, define the forces exerted by the cell. In this paper we will fix the number of I-sites per cell $K$ at 50 and $\ell=0$. The Reynolds number is low and therefore, because of the relative magnitudes of the coefficients, the expected acceleration term on the right hand side of the equation is set to zero. For an illustration of how the cell is modeled mathematically see Figure \[fig:cell\].
The I-sites are constrained to attach to lattice nodes, which are specific locations in $\mathbb{R}^3$, that change with time (see Section \[sec:lat\]). The lattice node locations are denoted by ${\bf y}_k$, and I-sites from the same cell or other cells can be attached to the same node location. The set of indices $p_{i,1},p_{i,2}, \cdots,p_{i,K}$ specify the lattice nodes associated with the I-sites of cell $i$.
![This figure illustrates how the cell is modeled mathematically in this paper. We consider a cell as a center of mass with attached springs. The other end of the springs are attached to I-sites which can interact with the extracellular matrix (membrane bound integrin based adhesion sites). In the simulations for this paper $K=50$; that is, there are 50 I-sites per cell. Although this figure is presented in the plane, the actual simulations occur in a three-dimensional environment.[]{data-label="fig:cell"}](Modelcell2.pdf){width=".8\linewidth"}
If an I-site maintains its connection to a lattice node indefinitely, the cell reaches an equilibrium position and the velocity of the cell is zero. We therefore require the I-sites to detach from the collagen lattice and reattach. The duration of attachment is taken from a Poisson-like distribution with mean attachment of 60 seconds. Upon detaching the I-site immediately reattaches to a node in the lattice; thus, for all times in our simulation, $K$ attachment sites are maintained.
In a change from earlier work [@Dallon:2014:MMC], the determination of the location of the next I-site is dependent on the direction the cell is moving. Specifically the I-site is placed in a cone with vertex angle 4 degrees, in the direction of motion. The exact distance from the center of mass is determined from a uniform distribution between $0$ and $115.726$. The placement of the I-site is discussed in further detail in Section \[sec:isiteloc\]. The I-site then attaches to the closest lattice node. For more information about the I-sites, the reader is referred to a related model discussed in [@Dallon:2013:FBM].
Collagen lattice {#sec:lat}
----------------
The collagen lattice is modeled by nodes which are connected with elastic ropes to form a network of spring-like connections (see Figure \[fig:collagen\]). In [@Notbohm:2015:MFP] they call it microbuckling.
in [0,4,...,12]{}
in [0,3,...,6]{}
in [0,3,6]{}
\(n) at (,,) ; (n) – (n); (n) – (n); (n) – (n);
;
;
;
To create the collagen lattice, first $M$ nodes are placed in a prescribed domain and a minimal connectivity value $\mathcal{M}$ is specified. Then, for each node in the lattice the closest $1.5 \times \mathcal{M}$ nodes are selected. From this selection of nodes exactly $\mathcal{M}$ nodes are sampled without replacement and are connected to the node. Recall the cell I-sites are constrained to be at lattice nodes.
The equation of motion for the lattice node $k$ is $$\gamma {\bf y}^{\prime}_k(t)=\overbrace{\sum_{m=1,m\neq k}^M {\bf
f}_{k,m}(t)}^{\mbox{force due to lattice entanglement}}+\overbrace{\sum_{i=1}^N{\bf
c}_{i,k}(t).}^{\mbox{force due to cells}}
\label{equ:lattice}$$ Observe that there are two types of forces acting on lattice nodes. The first (given by the first summation on the right hand side) is force due to connections with other nodes in the lattice. The second type of force (given by the second summation on the right hand side) is force due to interactions with cells. Note that a cell only exerts force directly on a lattice node if the cell has an I-site that is attached to the node.
Forces exerted by other nodes in the lattice can be classified into two types, those forces resulting from normal links and those resulting from compacted links. Forces exerted from normal links are spring-like in that if the connection is stretched, the force acts in proportion to the stretching. If the connection is compressed however, no force is exerted by a normal link. Compacted links allow for the compaction of collagen and are the result of a non-reversible process. These links differ from normal links in two key ways. First, the spring constant is much stiffer than the spring constant for normal links, making the forces exerted by compacted links much higher than the forces exerted by normal links with equivalent amounts of stretching. Second, compacted links resist compression. The existence of these two types of links are due to the nature of collagen. When the collagen fibrils are pulled, they resist the pulling due to their association with other fibrils. Yet if a cell exerts forces at two points along the same fibril drawing the two points closer, the fibril is not compressed but becomes slack between the two points similar to a rope, i.e., it exhibits microbuckling.
With these two types of links defined, the force due to a lattice connection between node $k$ and node $m$ is defined as: $${\bf f}_{k,m}(t) = \left \{
\begin{array}{ll}
0&\mbox{${\bf y}_k$ and ${\bf y}_m$ are not linked,}\\
0&||{\bf v}_{k,m}||<\ell_{k,m} \,\,\parbox[t][][t]{.2\linewidth}{ and the link\\[-2pt] is normal,}
\\[10pt]
-d_{k,m}{\bf \hat{v}}_{k,m}&
\ell_{k,m}
\leq ||{\bf v}_{k,m}||\,\,\parbox[t][][t]{.2\linewidth}{ and the link\\[-2pt] is normal,}\\[10pt]
-d_{k,m}^*{\bf \hat{v}}_{k,m}&
\mbox{if the
link is compacted.}
\end{array} \right .$$ Here ${\bf y}_m$ is the location of lattice node $m$, ${\bf v}_{k,m}={\bf y}_k-{\bf y}_m$, ${\bf \hat{v}}_{k,m}={\bf v}_{k,m}/||{\bf v}_{k,m}||$, $d_{k,m}=\beta(||{\bf v}_{k,m}||-\ell_{k,m})$ the signed magnitude of the force generated by a normal link between node $k$ and node $m$, $d_{k,m}^*=\beta^*(||{\bf v}_{k,m}||-\ell_{k,m}^*)$ the signed magnitude of the force generated by a compacted link between node $k$ and node $m$, $\ell_{k,m}$ is the rest length of the connection between node $k$ and node $m$ and is set as the initial distance between the nodes at the beginning of the simulation, $\beta$ is the spring constant for normal links, $\beta^*=d_{\beta}\beta$ is the spring constant for compacted links, and $\ell_{k,m}^*=d_{\ell}\ell_{k,m}$ is the rest length for the spring connecting node $k$ with node $m$ when the link is compacted. Initially all links are normal and become compacted if the distance between two linked nodes becomes small enough, that is, if $d_{k,m} <
d_{p}\ell_{k,m}$. When links are compacted the rest length of the spring is shortened ($d_{\ell}<1$), the spring constant is increased ($d_{\beta}>1$), and the link resists compression.
The forces exerted on lattice nodes due to the cell $i$ are defined by: $${\bf c}_{i,k}(t) = \sum_{j=1}^K \alpha(||{\bf x}_i-{\bf y}_{p_{i,j}}||-\ell)\frac{{\bf x}_i-{\bf y}_{p_{i,j}}}{||{\bf x}_i-{\bf y}_{p_{i,j}}||}\delta(p_{i,j}-k),$$ where $\bf{x}_i$ is the cell center location, $\bf{y}_{p_{i,j}}$ is a lattice node location, $\alpha$ is the spring constant and $\ell$ is the rest length of the integrin. Here $\delta(0)=1$ and $\delta(x)=0$ for any non-zero $x$ and indicates whether the $j$th I-site of cell $i$ is interacting with node $k$.
Results
=======
It became clear when extending the model to three dimensions that the lattice formulation and connectivity are crucial to the model results. We kept the collagen parameters essentially the same as the two-dimensional model. Only two parameters differ, one is a property of collagen and the other is a property of the cell. The collagen property that differs is the viscous drag on the collagen nodes, $\frac{\gamma}{\beta}$. The other parameter is the cell force $\frac{\alpha}{\beta}$. The collagen property $\frac{\gamma}{\beta}$ used here is 0.0863 compared to 0.114 in the two-dimensional model and for the cell strength $\frac{\alpha}{\beta}$ the value used here is 0.07 and the value used in the two-dimensional model was 2.239. For the other parameters see the caption for Figure \[fig:comparedata\].
The first objective of our work is to match the experimental observations detailed in [@Dallon:2014:MMC]. The results for lattices with 3,750, 10,000, 30,000, and 100,000 cells per mL, gathered over a period of 40 hours, are shown in Figure \[fig:comparedata\]. For the numerical simulations, we assume that only fibroblasts exist in the collagen lattice. Note, we do not assume stress dependent attachment mechanisms and stress dependent contraction mechanisms detailed in [@Dallon:2014:MMC].
Lattice connectivity results {#sec:CLR}
----------------------------
When creating a three-dimensional collagen lattice, the three-dimensional analog of the Delaunay triangulation did not create suitable connectivity between nodes in the collagen lattice. Specifically, there was no guarantee that the nodes would be connected to nodes that were close neighbors. For this reason we developed our own simple algorithm that guarantees that a node is connected to a minimum of $\mathcal{M}$ neighbors. There are two key values in our algorithm: first the minimal number of neighbors each node is connected to and second exactly to which neighbors the node is connected. We address these individually.
Both the overall contraction rate as well as the final contraction amount is dependent on the interconnectivity of the collagen lattice. Physically, the greater the number of connections that exist between elements in the collagen lattice, the greater the stiffness of the resulting lattice. Several preliminary calculations were performed to determine approximately the optimal minimal connectivity of the collagen lattice. In these calculations we were interested in determining a connectivity that would approximate the shape of the graph for each of the four experimental cell densities. In Figure \[fig:colcomp\], the simulated contraction is plotted against the experimental data. Observe that when the collagen lattice is not stiff (due to low connectivity), those simulations involving high cell densities contract more than they should, and when the collagen lattice is too stiff (that is, the minimal number of connections is too high), the lattice does not contract enough at lower cell densities. In our experiments in silico, we found a minimal connectivity between nodes of 30 connections to be ideal. This is an order of magnitude higher than that found in [@Licup:2015:SCM; @Sharma:2016:SCG] and 5 times higher than that predicted by Maxwell’s criteria for rigidity [@Wyart:2008:EFS]. In [@Notbohm:2015:MFP] a two-dimensional model with microbuckling (the type of spring-like forces used here) was essential to produce correct force propogation with connectivity between 2 and 8. But for three-dimensional models with microbuckling, they found that a connectivity of 14 gave reasonable results. We therefore checked our model without microbuckling (i.e., true springs which resist compression). The results are shown in Figure \[fig:SpringDensity\]. All the parameters are the same except there is no microbuckling, and the connectivity of the lattices are 3, 4, and 6. It is clear that the lattices are more stiff with lower connectivity than the model with microbuckling. Thus a much higher connectivity should be used when microbuckling is assumed.
Force propagation distances
---------------------------
In [@Notbohm:2015:MFP] the authors concluded that microbuckling was essential to match experimental data regarding force propagation and to see tethers between cells. Experimental data indicated that force decreased with a rate proportional to $r^{n}$ where $r$ is the distance from the cell and $n=-.52$. In a three-dimensional model, they found connectivities of 3.5 and 14 gave results $n=-.82$ and $n=-.67$ respectively. In their paper, the three-dimensional lattice was simulated with a regular grid of nodes connected in a cubic type pattern. Our model, in contrast, models a three-dimensional lattice with nodes in an irregular pattern and connections are randomly chosen as previously explained. For our lattice, we calculated $n=-.858$ and $n=-.267$ with minimum connectivities of 6 and 30 respectively. As the connectivity of our lattice increased, the distance the force propagated also increased as was seen in [@Notbohm:2015:MFP] in the presence of microbuckling. Our lattices with a minimum of 30 connections per node propagated force greater distances than in the experimental data. This could be in part due to the sparse number of nodes in our model when compared to the volume of the lattice. In order for the cell attachment sites to attach to lattice nodes in an approximately radially symmetric manner, the attachment sites had to be on the order of 100 microns away from the cell center. The results were obtained by fitting data from 10 different lattices with nonlinear least squares.
Direction of cell motion {#sec:isiteloc}
------------------------
Prior to determining the appropriate value for the connectivity of the lattice, an important change to the placement of the I-sites occurred as compared to previous models by the authors. In the current model, the direction of placement of the I-site is determined by the velocity of the cell. In prior models, the placement of the I-site was dependent on the prior location of the I-site. Specifically in the current model, when an integrin detaches from the collagen lattice, the current velocity of the cell is determined. The I-site is then located with probability 0.8 in a cone with an opening angle of 4 degrees in the direction of the cell velocity, see Figure \[fig:dircell\]. (Initially, each cell is given a random velocity.) This results in directed cell motion. The introduction of the placement of the I-sites in the direction of motion allowed the model to reproduce two important features seen in biological experiments, namely, fibroblast distribution at the end of the simulation and the behavior of the collagen disc in the presence of a radial cut. Moreover adding directed cell motion eliminated the need for the cells to become inactive. In previous work [@Dallon:2014:MMC], in order to match the contraction of lattices in time and with different cell densities it was necessary to have the cell become inactive. If the cells remained active they would continue contracting the lattice well beyond what was seen experimentally. Thus it was postulated that a mechanosensing mechanism inactivated the cells. This is no longer assumed.
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### Cell distribution at the conclusion of the simulation
Experimentalists [@Simon:2012:MRB; @Ehrlich:2000:DMH] have observed that as cells contract a collagen lattice, the cells are more concentrated near the boundary of the lattice and the lattice is also more compacted around the edge. Figure \[fig:firstlast\] shows simulation results for the initial cell distribution on the left and final cell distribution on the right for each cell density. It is easily seen that the cells are more dense near the boundary at the end of the simulations.
![The initial and final configuration of the lattices are shown for four different densities with the cell center marked as solid circles. The top row is the 3750 cells per mL, the second row 10,000 cells per mL, the third row is 30,000 cells per mL, and the fourth row is 100,000 cells per mL. On the left, the initial configuration is shown and on the right, the configuration is shown after 40 hours. []{data-label="fig:firstlast"}](./4kTime0dens.pdf "fig:"){width=".34\linewidth"}![The initial and final configuration of the lattices are shown for four different densities with the cell center marked as solid circles. The top row is the 3750 cells per mL, the second row 10,000 cells per mL, the third row is 30,000 cells per mL, and the fourth row is 100,000 cells per mL. On the left, the initial configuration is shown and on the right, the configuration is shown after 40 hours. []{data-label="fig:firstlast"}](./4kTime40dens.pdf "fig:"){width=".34\linewidth"}
![The initial and final configuration of the lattices are shown for four different densities with the cell center marked as solid circles. The top row is the 3750 cells per mL, the second row 10,000 cells per mL, the third row is 30,000 cells per mL, and the fourth row is 100,000 cells per mL. On the left, the initial configuration is shown and on the right, the configuration is shown after 40 hours. []{data-label="fig:firstlast"}](./10kTime0dens.pdf "fig:"){width=".34\linewidth"}![The initial and final configuration of the lattices are shown for four different densities with the cell center marked as solid circles. The top row is the 3750 cells per mL, the second row 10,000 cells per mL, the third row is 30,000 cells per mL, and the fourth row is 100,000 cells per mL. On the left, the initial configuration is shown and on the right, the configuration is shown after 40 hours. []{data-label="fig:firstlast"}](./10kTime40dens.pdf "fig:"){width=".34\linewidth"}
![The initial and final configuration of the lattices are shown for four different densities with the cell center marked as solid circles. The top row is the 3750 cells per mL, the second row 10,000 cells per mL, the third row is 30,000 cells per mL, and the fourth row is 100,000 cells per mL. On the left, the initial configuration is shown and on the right, the configuration is shown after 40 hours. []{data-label="fig:firstlast"}](./30kTime0dens.pdf "fig:"){width=".34\linewidth"}![The initial and final configuration of the lattices are shown for four different densities with the cell center marked as solid circles. The top row is the 3750 cells per mL, the second row 10,000 cells per mL, the third row is 30,000 cells per mL, and the fourth row is 100,000 cells per mL. On the left, the initial configuration is shown and on the right, the configuration is shown after 40 hours. []{data-label="fig:firstlast"}](./30kTime40dens.pdf "fig:"){width=".34\linewidth"}
![The initial and final configuration of the lattices are shown for four different densities with the cell center marked as solid circles. The top row is the 3750 cells per mL, the second row 10,000 cells per mL, the third row is 30,000 cells per mL, and the fourth row is 100,000 cells per mL. On the left, the initial configuration is shown and on the right, the configuration is shown after 40 hours. []{data-label="fig:firstlast"}](./100kTime0dens.pdf "fig:"){width=".34\linewidth"}![The initial and final configuration of the lattices are shown for four different densities with the cell center marked as solid circles. The top row is the 3750 cells per mL, the second row 10,000 cells per mL, the third row is 30,000 cells per mL, and the fourth row is 100,000 cells per mL. On the left, the initial configuration is shown and on the right, the configuration is shown after 40 hours. []{data-label="fig:firstlast"}](./100kTime40dens.pdf "fig:"){width=".34\linewidth"}
In Figure \[fig:comparefinaldist\] we showcase the effect of undirected integrin placement (on the left) and the mechanosensing rules of the prior paper which inactivate the cells (on the right) in the three-dimensional setting. In both cases, the cells do not appear to be aggregating near the periphery of the lattice unlike the two-dimensional simulations [@Dallon:2014:MMC]. Additionally, as the lattices contract they do not maintain a nice circular shape as is commonly seen experimentally and in the two-dimensional simulations. In Figure \[fig:comprop\] we compare the collagen contraction for the mechanosensing rules to the new rule of directed motion. Again the directed cell motion gives results which more closely match the experimental data.
![The final configuration after 40 hours of the lattices are shown for 100,000 cells per mL with the cell center marked as solid circles. On the left is the final configuration in the case of undirected integrin placement and on the right the final configuration is shown when the cells become inactive due to mechanosensing. []{data-label="fig:comparefinaldist"}](./nomove.pdf "fig:"){width=".34\linewidth"}![The final configuration after 40 hours of the lattices are shown for 100,000 cells per mL with the cell center marked as solid circles. On the left is the final configuration in the case of undirected integrin placement and on the right the final configuration is shown when the cells become inactive due to mechanosensing. []{data-label="fig:comparefinaldist"}](./tension.pdf "fig:"){width=".34\linewidth"}
In order to quantify the aggregation of cells near the periphery, we compare the interior cell density and the boundary cell density at the conclusion of the simulation with 100,000 cells per mL. In particular, we consider the interior of the domain to be the 90% of the domain closest to the center, and the boundary to be the remaining 10% of the domain. We then counted the number of cells in the interior and the number of cells at the boundary for 10 simulation runs with the same initial lattice configuration but different random initial positions for the cells and different instantiations for the random variables for the directed cell motion. The results confirm the same characteristic of the final cell distribution with the cell distribution at the boundary over 14 times more dense than the density in the interior of the domain.
### Radial cuts
Experimentally [@Simon:2012:MRB] radial cuts are used to determine the presence of residual type stress. When a radial cut is introduced at the beginning of an experiment, only a small angle is initially seen (presumably due to the width of the blade). When cuts are introduced one to five days into the experiment, the observed behavior is that a “Pac-Man"-like shape forms where the angle of the opening is on average about $20\pm 6$ degrees [@Simon:2012:MRB]. We replicate this experiment by introducing a radial cut and measuring the opening angle as the simulation progresses over 40 hours. Assuming the center of the disc to be located at the origin, we determine the opening angle by calculating, for each collagen node with positive $x$ value, the angle determined by $\arctan(y/x)$. We then classify these calculated angles into two categories, those with negative angle values (corresponding to negative $y$ values) and those with positive angle values (corresponding to the nodes with positive $y$ values). Our opening angle is then defined to be $\min(\text{positive angles}) - \max(\text{negative angles})$. In Figure \[fig:angleplots\] the opening angle, as a function of time, is given for differing cell densities. For cell density of 50,000 cells per mL, 10 simulation runs with the same initial lattice configuration but different random initial positions for the cells and different instantiations for the random variables for the directed cell motion were performed and the average ending angle was 16 degrees, corresponding nicely with the experimental data.
Directed cell motion was not only necessary to obtain the approximately correct angle opening measurements, but also necessary to preserve the expected topology of the disc with a radial cut. When the cell motion is undirected, the disc does not maintain the expected circular shape, but instead develops a bulge opposite the radial cut. Moreover, the radial opening develops teeth-like protrusions along the boundary, see Figure \[fig:teeth\].
Discussion
==========
We have shown that when changing from a two-dimensional model to a three-dimensional model of a fibrous structure like collagen, the construct of the model lattice is crucial to model results. For the two-dimensional model, a Delaunay triangulation worked well, but in order to match contraction data over a period of time for gels with several different cell densities, the model cells needed to become inactive or they would continue contracting the gels beyond what is seen experimentally [@Dallon:2014:MMC]. When moving to a three-dimensional model, the analog of the two-dimensional Delaunay triangulation gave strange topology. We changed the topology of the lattice and found that in order to match the previously mentioned data, with almost the same parameters for the collagen properties as the two-dimensional model, the connectivity was much higher than that predicted by theory and other discrete models of collagen [@Wyart:2008:EFS; @Sharma:2016:SCG; @Licup:2015:SCM] which did not include microbuckling. When modeling with spring-like connections which do not resist compression, the connectivity must be much higher in order to match experimental data. To the best of our knowledge, all the connectivity values for collagen are determined by modeling and there is no experimental data for connectivity of collagen lattices.
We also found that with the new formulation of the lattice in three dimensions, the cells do not need to become inactive as in the two-dimensional gels if they exhibit directed motion. In other words, the cells need to be engaged in directed cell motion (the direction does not matter) by placing new attachment sites in the direction of motion. With this assumption, the model results matched the final cell distribution data, the angle in the slit gel experiments, the contraction data, force-distance propagation data, and the overall morphology of the gels.
In conclusion, we have developed a three-dimensional model which, with reasonable assumptions, replicates the results of several different biological experiments. The model assumes microbuckling, directed cell motion, and the same collagen properties as a similar two-dimensional model. When modeling a three-dimensional lattice with microbuckling, the connectivity results of models with true spring interactions do not apply. Assuming that cells are engaged in directed cell motion gives better results than undirected cell or force-sensing cells which inactivate. Finally, connectivity and lattice generation greatly affect model results and need more systematic investigation.
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